Hello Forum Members,

I have written a new article on tuning the F3-F4 temperaments sequence in ET. I would like to publish here and excerpt of it involving the initial contiguous M3s, F3-A4. Many of the modern ways of constructing ET begin with these steps, so they can be applied to any number of sequences that technicians today are using. What I have tried to do here is explain just what to listen for in the clearest, most easy to understand manner. So, it is nothing new, it is only presented in the manner which I have developed in my own teaching techniques.

I will start the article excerpt in my next post with its introduction, the post the actual sequence in a post to follow that. I hope this helps those who have been struggling with this concept.

Tips on Tuning the Contiguous M3s from F3 to A4

You have probably heard of the 4:5 Ratio of Contiguous M3s. The interval, Major third is abbreviated as, M3. This is a scientific principle that came to light when Dr. Al Sanderson was developing the first practical ETD for tuning pianos. You will not find it in any of the older and classic books on tuning. Not only is it an extremely valuable tool for identifying errors within the temperament as well as anywhere in the piano where M3s can easily be heard, it serves as the ultimate tool for precisely and unequivocally dividing the temperament octave into three equal parts. Once the octave is divided in that way, the rest of the temperament can hardly go wrong, no matter which sequence you may choose to tune the rest of the notes.

Contiguous M3s are defined as any two M3s which share a common note between them. Example: F3-A3 and A3-C#4.

Of course, the 4:5 ratio does not mean four beats per second versus five beats per second; instead, it means that for every four beats of the lower M3, there are five beats in the upper M3 of any two contiguous M3s, anywhere on the piano where they can be heard. Naturally, this applies to ET (but also the Marpurg Temperament) and not any other Historical Temperaments but it applies no matter how much or how little stretch there is in the tuning. Therefore, it applies to other versions of ET such as the ET with “pure” (beatless) fifths and to Bernhard Stopper’s popular idea which has beatless octave and fifths (P12).

Do not be concerned if you are not sure what the 4:5 ratio of contiguous M3s should sound like. You need only know that it is what would be deemed a small difference. It is not the very smallest discernible difference but neither is it an obviously large discrepancy. Indeed, contiguous minor thirds (m3) have a smaller ratio, 5:6. The difference between any two consecutive M3s (those right next to each other) is even smaller and the difference between any two consecutive m3s is smaller yet.

So, the small difference found in the 4:5 ratio of contiguous M3s is one that is easily heard and identifiable. If any two intervals beat exactly the same, they are known as Equal Beating. If any two contiguous M3s beat equally or very nearly so, the relationship is not correct and certainly, if the lower or bottom M3 of any two contiguous M3s beats faster than the upper or top M3, the relationship is very incorrect.

Normally, the F4-A4 M3 beats too fast to be useful when comparing M3s. In that range of the piano, M6s and M10s are much easier to hear. However, when constructing these four initial contiguous M3s, the top F4-A4 M3 is an essential tool in determining whether the bottom three contiguous M3s, F3-A3, A3-C#4 and C#4-F4 are correct. The bottom three may sound acceptable by themselves but the top F4-A4 M3 can serve to identify the very smallest error.

The top F4-A4 M3 sounds very fast, at perhaps the limit of your perception or discernibility. Therefore, if it beats gently and is easily heard, it is too slow and that means the bottom F3-A3 is also too slow and the C#4-F4 M3 will likely beat the same or faster than the F4-A4 M3.

If the top F4-A4 M3 is beating too fast, its beats may be so fast as to not be heard; they will be an indiscernible blur and the interval would likely have a “sour” sound to the ear. In that case, there would be an obviously large difference in beating between the contiguous M3s C#4-F4 and F4-A4. The A3-C#4 and C#4-F4 M3s will also either beat too nearly the same or be inverted.

Very important clues to how large or small the error is between any two contiguous M3s are equal or very similar beating and inverted beating. If any two contiguous M3s beat exactly the same or very nearly the same, there is a small error, perhaps about one cent. If the bottom of any two contiguous M3s beats faster than the upper, the error is larger, perhaps 2 cents or more. The more inverted any two contiguous M3s are in their beating, the larger the error.

If you are a technician who normally uses an ETD and are struggling with aural tuning concepts in your quest to pass the PTG tuning exam or are simply wishing to improve your aural tuning skills and you spot a small error, visualize what making a 1 cent or so correction would be like using your ETD. The pattern would be moving only slowly and slightly. It would take only the slightest application of tuning hammer technique and test blows to make that correction. Once you’ve applied the techniques, listen again to the relationship of the two contiguous M3s to determine if you accomplished that very small correction or made a correction that was too large.

As an exercise in listening, use your ETD programmed as you typically do to tune the three notes F3, A3 and C#4. Listen to that small difference between the rates of beating in the two contiguous M3s. Now, set your ETD on A3 and tune A3 just one cent flatter. You’ll surely hear the beat rates and ratio change! Try also sharpening A3 from its normal position and try sharpening and flattening F3 and C#3 similarly so that you become accustomed to hearing the significant difference that only a one cent change makes in the beating of a M3 and the ratio of beating of two contiguous M3s.

In the following sequence, Major thirds are abbreviated as M3, fourths are abbreviated as P4 and fifths as P5. The “P” stands for “perfect” even though fourths and fifths in ET are always tempered. The word, perfect, comes from music theory, not tuning theory. You will note that there are no other intervals used such as sixths (M6) or minor thirds (m3). Those intervals may be used later when the entire temperament sequence is complete to correct any small errors either in the initial sequence of contiguous M3s or elsewhere. The correctness of the initial contiguous M3s depends of course on your perception but any pitch at any time may drift slightly from where you initially put it without you being aware of it. That is especially true when you are making substantial changes to the pitch of each string.

The Sequence for Tuning the Contiguous M3s F3-A4

Tune A4 to the Tuning Fork.

Tune A3 to A4, a beatless sounding octave, approaching from the wide side of beatless, just to insure the octave is not any amount narrow.


Estimate the F3-A3 M3. Any reasonable estimate will work. If you have no idea what the usual 7 beats per second specification sounds like, simply sharpen and flatten F3 until you hear a moderate amount of beating (the F3-A3 M3 must be a widened interval, not narrowed), neither obviously slow or obviously too fast and “sour” sounding. Anywhere within that range is a close enough estimate at this point.

Tune F4 from F3, a beatless sounding octave, again approaching from the wide side of beatless to avoid a narrow octave. Both F3 and F4 may have to be slightly readjusted but the next note you tune will tell you whether they do or not and whether they need to be flattened or sharpened.

Play the A3-C#4 M3. Sharpen or flatten C#4 until the A3-C#4 M3 is a widened interval and beats at first exactly the same as the F3-A3 M3. At this point, you know that the C#4 must be sharpened slightly because contiguous M3s can never beat exactly the same as one another. Sharpen C#4 so that the A3-C#4 M3 beats a small amount faster than the F3-A3 M3 does presently.

Now, listen to the sequence of contiguous M3s: F3-A3, A#-C#4, C#4-F4 and F4-A4. Do all M3s progress slightly in their rate of beating? If not, the clue to which way to now move F3 will be in any error found in this slight progression of beating from one M3 to the next.

If F3 is too sharp, the C#4-F4 M3 will beat the same or slower than the A3-C#4 M3. The F4-A4 M3 will also be the same or slower than the C#4-F4 M3. If you have the two contiguous M3s A3-C#4 and C#4-F4 beating the same or very nearly so, only a very slight adjustment flatter of F3 is needed. If the rate of beating is inverted, the C#4-F4 M3 beats faster than the F4-A4 M3, a larger adjustment flatter is needed.

If F3 is too flat, the A3-C#4 and C#4-F4 will be inverted and the C#4-F4 M3 will reveal a large difference in beating.

After adjusting the F3 flatter or sharper as is indicated, immediately retune F4 to F3, again approaching a beatless sounding octave from the wide side. Then, listen to the F3-A3 M3 and adjust C#4 if need be so that the A3-C#4 M3 beats slightly faster than the F3-A3 M3.

Listen to the sequence of contiguous M3s again and make any small adjustments necessary to achieve the small and smooth progression of beat rates. When you believe you have it, it will be within the range of tolerance of the PTG exam for each note! A one cent change in any of these notes from when they are correct will be very obviously wrong. So, when the sequence sounds correct and acceptable to you as that small progression from each M3 to the next, you are well within an acceptable range, if not absolute perfection.

Bill, would this not be easier to do for a beginner by shifting everything one octave lower and slowing the beat rates in half; or would the transfer past the break into the wound strings create the need for undesirable refinements that will replicate themselves with ill effect in the octave tuning sequences upwards? Your description of procedure is good and I would only think the hardest hurdle for newcomers to this temperament sequence would be discerning differences in the highest two CM3rds and what is regarded as "very fast beats" as opposed to "sour". I suppose having an ETD to help stay in safe boundaries would help maybe in this case.

When tuning small pianos (whitney spinets) I have dropped the temperament down an octave so that it takes place entirely on wrapped strings so I don't have to deal with the double transition (plain trichords to plain bichords to wrapped bichords) in the normal temperament octave.

Bill:

You know that I am a 4ths and 5ths tuner. But I hope that my comments are taken as they are meant: as constructive criticism.

In Paragraph 4 of the “Tips” post you wrote:

“Do not be concerned if you are not sure what the 4:5 ratio of contiguous M3s should sound like. You need only know that it is what would be deemed a small difference. It is not the very smallest discernible difference but neither is it an obviously large discrepancy. Indeed, contiguous minor thirds (m3) have a smaller ratio, 5:6. The difference between any two consecutive M3s (those right next to each other) is even smaller and the difference between any two consecutive m3s is smaller yet.”

I had to read this a couple of times to understand it because the word “contiguous” is so similar to “consecutive”. A different word than consecutive, although accurate, may be more suitable. Chromatic comes to mind.

Also, it seems that you are saying that the beat rate ratio between consecutive M3s is larger than m3s. Like any chromatic beat rates the ratio is based on the 12th root of 2. Since the beat rate of both M3s and m3s approximately double for each octave, the chromatic beat rate ratios are the same, about 15/16.

In paragraph 9 of the “Tips” post you wrote:

”Very important clues to how large or small the error is between any two contiguous M3s are equal or very similar beating and inverted beating. If any two contiguous M3s beat exactly the same or very nearly the same, there is a small error, perhaps about one cent. If the bottom of any two contiguous M3s beats faster than the upper, the error is larger, perhaps 2 cents or more. The more inverted any two contiguous M3s are in their beating, the larger the error.”

It takes 2 cents of error (rather than 1 cent) to cause two M3s in a set of CM3s have the same beat rate. Consider the F3-A3-C#4 set of CM3s as an example with the beat rates of the M3s being approximately 7 bps and 9 bps. Now I know that we would not adjust A3, but this is just an example. The partials that cause the beat rates include the 4th and 5th partials of A3, which are A5 and C#6. A5 (A880) changes 1 hz for each 2 cents. So if A3 is raised by 2 cents, then the M3s in the F3-A3-C#4 set of CM3s will both beat approximately 8 bps.

Something that you probably know, but other readers may not know is the use of the m6 that is formed by a set of CM3s. It is understandable if this is not taught to avoid confusion, but let me mention it anyway. Because the test for an 8:4 octave is the m6-M3 test, in the temperament octave (where there is less difference in octave types) the beat rate of an m6 (F3-C#4) can be substituted as an approximation for the beat rate of the M3 (C#4-F4). Perhaps this would only be useful when first tuning F3 and C#4. But it could save a few steps by not having to tune F4 a couple of extra times until F3 and C#4 are more certain. The same thing can be done with the beat rate of A3-F4 substituting for F4-A4.

Best wishes with your new article. I do have one other comment, though. It seems a bit wordy. But then I don’t like Mozart (too many notes).

Ryan:

Uh, the tenor break on Whitney spinets is E3-F3 with no wound strings in the tenor although F3 and F#3 are unwound bichords. Kimball spinets are another story.

Bill...its an interesting read...thanks for taking the time to place it before us for consideration...

I've only had time to read it in cursory manner...absorbed about the first 1/2 so far...I'll read it again more carefully as time permits...but it looks like a great way to understand contiguous M3's.

...another take on (or new set of) useful tests, at very minimum...

RPD

Bill,
Great timing. I just happen to be trying to learn to aurally tune a temperament. The temperament setting technique I'm using starts off with contiguous M3s, and I've been trying to learn a good system to get them right.

I will play around with your system, and perhaps come back with some more suggestions. But for now, I've got only one. I would add the target beat rates in beats per second for all the M3s you mention, instead of just some of them.
Thank you,

Thanks for all the comments, especially yours, Jeff. I should perhaps change what I said to 2 cents instead of just one but I need to be sure about that before I do. I know from my experience working with master tunings that the committee often makes corrections of less than one cent. In my own experience, I know that making a change of 0.5 cents is quite small. Still, I have read other comments from other places that seem to discount "errors of 1 or 2 cents" as being significant.

Therefore, I would really like to pin down what I am saying for it to be accurate and thus credible. If, for example the F3-A3 and A3-C#4 M3s beat equally, how much would the A3 need to be flattened for these two intervals to beat at a 4:5 ratio? My guess was about 1 cent. You seem to say 2 and if that is true, I will change what I have written. The article is not due to be published until next Fall, so there is time.

The other point you mentioned about the word to use, "consecutive" vs. "chromatic" I will think about. I have to say I have pondered about which word to use as well. I'll ask for opinions on that.

Also, Owen Jorgensen told me once that the ratio of contiguous m3s is 5:6 and I have taken that as a fact. I had only assumed that the difference between consecutive/chromatic M3s and M3s would be different but by a very small amount. You seem to indicate that the difference is the same, 15:16. I can consult with Owen Jorgensen on these matters for a ruling. I admit that I don't really look into the math on these things and I did not provide a ratio for either. I just know that the difference between any two M3s or m3s which are next to each other is a very small gradation.

When a master tuning committee plays a series of either M3s or m3s and spots a slight error in the evenness of a progression, the correction made can be very difficult because of how small it really is. I don't believe I have ever witnessed an aural correction that was smaller than 0.3 cents and those have been the very slightest aurally detectable adjustments.

Somehow, it seems to me with just a gut feeling that a ratio of 15:16 between any two rapidly beating intervals is simply too small to be discernible. I will inquire about this. My gut feeling is that it is about half of what you claim, something like 7:8 and I still doubt that both M3s and m3s which are consecutive or chromatic, whichever you prefer as a description could be exactly the same ratio. I just seem to know that from having listened to so many master tunings.

Lastly, I am very sorry to say this but I have never known anyone at any time to have used the m6 interval for any diagnostic tool. For the most part, I use 4ths and 5ths just like you do and most other people do. I then use M3s both contiguous and consecutive/chromatic and I use m3s in the lower part of Octave 3 and perhaps upper part of Octave 2 but only in extremely fine tuning of those areas. I do use M6s in the upper part of Octave 3 and lower part of Octave 4 but have never found a use for m6s anywhere.

There was a slight omission in my post: "Now, listen to the sequence of contiguous M3s: F3-A3, A#-C#4...". The very last interval should read: A#3-C#4. I would like to correct this but there is no "edit" tab for me to do so. I also had a few typos in a post I made a few days ago that I would like to correct. Does anyone know how I can edit either of my posts?

Regards,

Perfect timing...I just started studing tuning.

Bill, I beleive there is a time frame for editing...but I'm not sure how long it is...sorry

Thanks for posting....

Les

Bill:

Of course you should check what I stated for accuracy. You know I thought I was wrong once, but it turns out I was mistaken.

Because of my navigational background I can't help but look into the accuracy of anything numeric, and also to find some way to cross check anything for blunders. So I’d appreciate hearing what Mr. Jorgensen has to say on the subject.

Sometimes this place works really well....to wit, I get to sit here and read and absorb this stuff while you good folks do the work!

Life is good...thanks Bill and Jeff for this dialogue...RPD

And thank you for the many fine post on things that I have little experience with but much interest!

I know there is resistance to the idea of counting beats for many tuners but don't you think that learning how to accurately count 7 would help get F3 placed much more accurately and minimize any guess work?

What! And do anything suggested by Saint William Braid White? Blasphemy!

Thanks, I'm muddling my way through it.

BTW, there's a small typo in the post titled "The Sequence..."
In the 6th paragraph, (that begins "Now, listen to the sequence... "), there's an A# that should be A natural.

Thanks again!

I'd whole heartedly agree with you Gene. We used to have a huge set of tuning forks at college that you whacked with a little wooden mallet. You would then scurry off to the nearest tuning room hoping not to be interrupted and lose that beat rate and have to walk back. We were introduced to it the day we started learning to tune A so I'm sure my instructors thought it was important and I always have since.

Excellent article Bill. Since I have never cared much for math, (some people like and quickly comprehend math, others dont), the discussion of the math is a little maddening to me.

But I agree with Gene and Emmery, bps for the thirds would be helpful. Personally, I have never found a good method to "count beats," but in my opinion, one can readily learn to recognize by hearing what the various bps of thirds should sound like. And I think it is important to remember when tuning M3's it is important to perform checks of the 4ths and 5ths, etc.

But great article as usual from you.

Richard Brekne has posted some wave files of beat rates to his great web site.
http://home.broadpark.no/~rbrekne/beats.html
Also there is the Sanderson Beat Rater - a bit on the expensive side but it is effective.
Also I like to set the metronome at 120 and get my foot to tappin with it - I have found that about as fast as I can say one two one two one two three out loud without fouling up the words I get 7 and it fits within the 120 (two ticks) of the metro and this tempo goes nicely with a Sousa March and is easily memorized with some practice. Got to get the foot tappin to get it into the memory.
Having tried all I prefer the wave files as they give the wa wa sound which is better than the beat rater as it is a tick sound -just too difficult to take it with me.

Sorry Bill - I did not intend to hijack your most excellant thread with this beat stuff.

No problem, Gene, it is just that my entire point is that it is not necessary. I have never counted 7 beats per second in my life! With this approach, you can put 2 or 3 beats per second or 15-20 beats per second and it won't really matter, the system will correct itself.

The problem I see with trying to learn what exactly 7 beats per second sounds like is that the rate itself is an approximate estimate. The true, exact rate is determined both by actual inharmonicity and the chosen amount of stretch whether it is conservative or really wild. So, as I see it, there is no point in learning or teaching what exactly 7 beats per second sounds like. Anyone gets a close enough idea of what it is with experience anyway.

Jeff, I tried the F3-A3 and A3-C#4 intervals as equal beating first, then I tried flattening A3 by 1 cent and got a very good representation of a 4:5 ratio, then I tried flattening A3 1 more cent and I got an obviously larger ratio. I did the same by sharpening F3 and then C#4 after retuning the two contiguous M3s to equal beating. I got the same results again. Therefore, I consider what I wrote about an ETD user visualizing making a correction of 1 cent when two contiguous M3s are equal beating as correct. This article is about trying to teach ETD users *practical* techniques. It is not about theoretical tuning theory (which never quite works in reality) nor is it about mathematical theory which the average person and myself do not grasp and had no real application when doing the mechanical and artistic job of tuning a piano.

I'll find out later if consecutive m3s have the same ratio or not as consecutive M3s (which I do not believe they do) and also what those ratios are (which I can hardly believe could be anywhere near as small as 15:16.

Bill, I think you make a valid point. These thirds will correct themselves no matter where they are initially set. When you align each of the thirds within the temperament, if each progressively beats a little faster, it will all work out. The method you are suggesting simply sets up the initial structure. I think I am stating your intent correctly.

I think the practice of setting a temperament based upon progressively faster thirds, and the use of M3's as the primary tuning intervals during temperament tuning, may be a radical concept for tuners used to tuning aurally using 4ths and 5th with a variety of checks. These same checks would apply to tuning thirds. Not sure my foregoing comments makes a lot of sense. I am one of those people who know what I hear, but have a hard time putting it into words.

This is not something new. It follows from the ratio of the major third interval. If your fundamental has frequency f0, then your major third f1 is 5/4 * f0, and the major third f2 above that is 5/4 * f1 = 5/4 * 5/4 * f0. This means that the frequencies of f1 and f2 are 5/4 times f0 and f1 respectively, so the beat rate or difference between them 5/4 as fast.

Except that if you are tuning equal tempered major thirds, the ratio is not 5/4 or 1.2, but 2^(1/3) = 1.25992104989. However, if you are not counting too carefully, 5/4 is close enough. Close enough to ignore the effect of inharmonicity, which is present, but negligible compared to the effect of tempering. After all, if you go up 5/4 from A440, you get to C#550, and 5/4 from that will be F660. But 5/4 from that is A792, so some adjustment needs to be made.

I am a great believer in "reasonably close" so the discrepancy in the math does not bother me. Everything will come out once you reconcile the thirds with the octaves.

Thanks for stepping in, BDB, I know you can do the math. Although I cannot decipher what you wrote, I can't even begin to, I am aware that *theoretically* the ratio of contiguous M3s is not 4:5 (or 1:1.25) but 1:1.26. Big deal.

What is important in learning aural tuning is to perceive and control beats. I've worked as a tuning examiner now for 18 years. Whenever there are errors in an exam, particularly when an examinee has failed, I have consistently observed that the individual can perceive the errors, the problem was in having a suitable plan to control and distribute beats in the first place.

Through reading and observation, I can conclude that most any experienced piano technician can perceive when two rapidly beating intervals(RBI)beat exactly the same to within less than 0.5 cents. It is unfortunate therefore, that ET does not permit any two RBIs to beat exactly the same, we always have to look for that small difference. It becomes a question of judgment about whether the *difference* is correct or not.

Therefore, I like to downplay even whether one can perceive a 4:5 ratio correctly or not, hence my phrase, "Do not be concerned if...[you can't or don't really know]..." because virtually any error of 1 cent or more will be quite evident when comparing any two contiguous M3s in any range of the piano where M3s are discernible. In other words, as I often say, "You may never be entirely sure when the relationship is absolutely correct but you will always know when it is not correct". That goes all the way up to a master tuning committee that spends an entire hour hemming and hawing over just F3-A3-C#4-F4 and A4. I know, I have been there and done it many times over. As far as I was concerned, it was fine when we started and it took an entire hour to make it a cat's whisker better.

I must say that I consider what one of my local colleagues calls "beating your head against the wall", the absolute perfection of ET a useless and futile effort except in one and one case only: the production of a master tuning for an exam. Otherwise, perfecting ET to the degree that some people like to theorize about never will make any piano sound any better. There are surely other aspects of piano preparation that would benefit far more for the time spent.

Since I never do tune any pianos in ET and haven't now for 20 years, I don't waste my time even thinking about the 4:5 ratio of contiguous M3s because the temperaments I use never has any two intervals in that configuration. I only spend time and energy on it in an effort to help people pass the tuning exams. Otherwise, I wouldn't even think about it.

This does not mean that I disagree with the vast majority of technicians who strive to tune a good and proper ET. If that is what they believe in, that is OK with me. I only am interested in technicians learning how to do what they want to do and believe in better.

I read a discussion that took place on the CAUT list in a recent PTG Journal. My eyes rolled when I saw a major contradiction in thinking. The gist of it was, that as long as 4ths and 5ths are even, a little unevenness in 3rds and 6ths wouldn't matter. Well, strictly speaking if any 3rds and 6ths are uneven, then so are the 4ths and 5ths and vice-versa, as far as *EQUAL* temperament is concerned.

The fact is that the person who wrote that was describing exactly what I normally do: my 4ths and 5ths are "kinda, sorta pretty even" (as I once heard a now retired symphony conductor describe it) and so are my 3rds and 6ths. The difference I make in my tuning is that I channel all of my best efforts at perfection into a different model from ET, my "one or two cent differences" are not randomly placed, they are specifically and willfully targeted at an entirely different goal from that of ET.

So, it does not even matter to me what the ratio of consecutive (or chromatic, however one may think of it) Major or minor 3rds is in ET because in aural tuning, there are no ETDs, there are no calculators, there are no ancient books, there is only the technician, a piano and a tuning hammer. I know what smooth and even progressions sound like and I know all of the techniques that are practical and useful to produce them. I also know of many other ways to tune where those techniques are completely invalid but where different techniques apply. None or what I do as a skilled piano technician invokes any mathematical theory, it only involves skilled use of a tuning hammer and careful listening.

I appreciate and thank you for your contribution to this discussion.

All:

I am a bit of a “Fool On the Hill.” The numbers do mean something to me while I tune. And I can’t imagine needing to count beats after learning to set a temperament, by whatever aural method. Wouldn’t you just remember and know what they sound like and just know when a departure from the usual beat rates exists and sometimes for good reason?

Tuning a proper M3 is pretty simple. A fourth up, a fifth down, a fourth up, and a fifth down (or visa versa). When the fourths beat about 1 bps wide and the fifths about 1/2 bps narrow it is going to be a pretty good M3. It probably won’t be perfect, but until all the other notes are tuned, you can’t be certain anyway.

A very accurate ET temperament is important to me for tuning the rest of the piano, otherwise the fifths, octaves, twelfths, and double octaves will not all sound even which makes a piano sound truly in tune with itself. The beat ratio of fourths to fifths, and the M3 inside M6 outside test (and its inversions) will show errors that chromatic beat rates do not. There is math that shows this to be true, but I’ll keep them “On the Hill” with me this time, unless someone asks.

As I said, the math is not terribly important, but if you are not going to be accurate with it, you should just say that it is approximate, rather than saying that this, that or the other is true due to some mathematics. It will make for a better article. Especially when it is not true.

Complete accuracy is impossible. For me, it is sufficient that whatever I am tuning has the characteristics of equal or whatever other temperament, because at some point, you have to say it is close enough, or you need to do some error analysis. That would be tougher to write about.

I have learned to accurately count 7, 8 and 9 bps.
As 7 is a good place to tune the first f3-a3 on any piano, it gets me very close the first attempt and I am comfortable with that - many times it is right on and needs no more changing. If f3 is a wound bichord it will be slower, if f3 is the lowest plain steel trichord on a spinet it will be faster but not much, if f3 is a plain steel trichord on the typical test piano or larger it will be almost exactly 7 - most of the time.
I tune with fast beating intervals and use the 4ths and 5ths as checks and tests.
I also tune using a variation of the Braid White Temperament that Tremaine Parsons developed and use fast beating intervals to check, it works out very well.
In my opinion, tuning with 4ths and 5ths or fast beating intervals is kind of like the cart before the horse scenario.
As long as you use both it will come out ok as they all must fit.
I agree that fast beating intervals should progress evenly and 4ths and 5ths should all have the same character. I say this because the character of the 4ths and 5ths will vary on a piano if the focus is on even fast beating intervals as scaling issues need to get flushed out somewhere.

Gene:

Interesting what you posted about the effect of F3 being a wound string on the beat rate of F3-A3. I find the opposite to be true.

I like your way of talking about the "character" of 4ths and 5ths. There is more there than just the beats.

The variation of the BW sequence I am using right now tunes A#3 after F3 and before G3. What is the Tremaine Parsons variation? I would really like to know.

Yes, F3-A3 does beat about 7 bps on just about any piano. The effect of iH on beat rates, especially in the temperament octave, is largely self-correcting. I often read that the beat rates are different than theoretical because of iH and octave stretching. But have never read why this is so nor an example of what the difference really is for a typical or a specific piano. And they certainly don’t seem very different from piano to piano. Of course when there is a scaling problem something has to be different. The difference then is mostly due to the chosen compromises, I believe.

Jeff,
The BW variation is c4, f3, g3, d4, a3, e4 then test for 7, 8 9 and 10 bps(or smooth progression) with f3/a3, f3/d4, g3/e4 and c4/e4. Then tune b3 and test inside outside f3/d4-g3/b3. Next tune f#3 and c#4 and test inside outside g3/e4-a3/c#4. Next tune g#3, d#4 and test inside outside f#3/d#4-g#3/c4. Next tune a#3 and f4 and test inside outside g#3/f4-a#3/d4 and the f3/f4 octave should be slightly wide with M3/m10 test. There are certainly more tests that can be done like contiguous M3 and others.
I find that I can usually get smooth M3's and good sounding 4ths$5ths that are almost consistant. (still talking small spinet type pianos here)
Scaling issues usually only show up in the lower tenor and through the bass. Closer to the end of the tenor bridge the ih can get very high with plain steel and lower with wound bichords.
Hi ih would force any interval wider - would it not?
So you can have 7 at f3/a3 but the f3/f4 octave could not be correct. Testing fast beating chromatic intervals across the break which would you trust, M3, M6 or 10ths? The farther the upper test note is from the lo tenor the more reliable it is because it is out of range of the high ih strings - so 6th or 10th would be a good choice for this.
So what do you want at the break - smooth fast beating intervals, clean octaves, consistent 4ths and 5ths? Cannot have it all.
This is why I think it is good to learn aural tuning on small poorly scaled pianos. When you finally get to tune a well scaled piano it is amazing how nicely it comes together.

Thanks for that, Gene, that is the way I first learned to tune too. I managed to qualify as an RPT and to train to be an examiner using that system in 1983. So, I am not saying it can't be done or that countless numbers of technicians haven't performed successfully that very same way. Tuning is in fact still being taught that way.

However, around the same time as I had used that system successfully, the idea of tuning a chain of contiguous M3s first came into realization by a relatively few technicians. Dr. Sanderson, Bill Garlick and Rick Baldassin come to mind. When I heard of it and tried it, I struggled with the concept too, not really knowing a good way to do it. When I talked with Owen Jorgensen about it sometime ago, he told me that he had used the system you describe for his entire career at Michigan State University, tuning all kinds of pianos but notably the problem scale of Baldwin Hamilton verticals. He said he had to use elaborate techniques to cope with the irregular scaling.

What he said next made an impression on me. He said that by the time he had heard of the contiguous M3s idea, he was near the end of his career of performing routine tunings (in ET), so he had never taken it upon himself to employ that method in his routine work but he did immediately see the value of it: It serves to divide the octave into three unequivocally equal parts at the outset of the temperament construction. Thus, it solves the most difficult part of that construction at the outset; first, rather than last.

I've written about this many times yet I still don't seem to get my point across. The system you describe by definition requires you to make an estimate, another estimate, an estimate upon a previous estimate and another estimate upon two previous estimates before you ever have a M3 to consider and it is formed by an estimate and another estimate made upon two previous estimates. At that point you only have a M6 to compare it with. That interval is formed by two estimates. The resulting comparison can sound reasonable and perhaps luckily be correct but the previously tuned pitches can still all be slightly erroneous and lead to further complications later on. Sooner or later, the F3-A3-C#4-F4 will be in place but if they are determined to be incorrect and then remedied, it requires backing up through all of the previous pitches which lead to those errors. The problem with backing up through a sequence to correct errors is that it only leads to more estimates and estimates made upon a previous estimate.

While the sequence is simple to learn, that is if the technician really knows the 4ths and 5ths and understands the cycle of 5ths but today's technician who has started out using an ETD easily becomes frustrated trying to learn to discern all of that and particularly the idea of "counting" beats.

I often wonder how a Spanish speaking person would "count" uno, dos, tres, quatro, cinco, seis, siete in just one second? For that matter, how do you do it in English and how do you "count" 9 and 12 beats in just one second? The truth is, no one really "counts" them but those who use that concept learn to perceive those quantities and time intervals after a lot of practice.

The contemporary ETD user rightly asks, "What for?" The usual answer given is that aural tuning is important to know. Even if the ETD user tries, they find the experience frustrating to learn and often give up. So, ETD users while claiming to have an "ear", often do not really perceive beats very well and have no sound or organized basis for aural correction of errors or whether any custom programming of the ETD results in any real improvement or not.

So really, my premise is that teaching such a person to learn to construct contiguous M3s at the outset of a temperament sequence is actually far easier and more immediately rewarding for that one group of people to learn in spite of it being more difficult and confusing for people who have already learned and are generally successful with the classic methods of temperament construction.

It has been my experience that the group of technicians about whom I have been writing, the technician who first learned to tune with an ETD, can in fact readily and easily perceive when two Rapidly Beating Intervals (RBI) beat exactly the same, regardless of their rate, as long as that rate is within a reasonably discernible range. They can also perceive very slight differences with very little effort. Therefore, I de-emphasize the notion of "counting" but emphasize the perception of equality, then small and very small differences. Ratios, whatever they are exactly according to mathematical theory also are not really important. What is important is the perception of the amount or difference of any one interval compared to another and another.

For those who have already learned to use older methods successfully, there is perhaps no need to change one's ways. Only if that kind of system has been proven to be consistently faulty and frustrating because of that would there be any need to try to adopt a more modern method of temperament construction.

Bill,
You do not need to convince me of the value of contiguous M3's.
I was taught to use them by Peter Clark about 10 years ago.
In 1968 I took the Niles Bryant corrispondence course and learned the BW sequence and was given a pendulum device that swung at 1/second on one side and 3/5sec on the other. Guess how long it took to get reasonable results from that.
Peter's sequence starts with a3/a4 a hair wide and f3/a3 at 7 (here is where I count). Then f3/d4 at 8 (count here too) and checked with p4 a3/d4. (here is where the character of the p4 p5 begins to take shape) (you can also check the a3/f4 p5 to get back to the original octave.
Next tune a#3 to d4 at 9 (count here too) but also check with f3/a#3 p4 and an even progression of 7, 8 and 9 from f3/a3, f3/d4 and a#3/d4. Next I lock in the f3/f4 octave same width as a3/a4 and test with f3/a#3 p4 and a#3/f4 p5 and the standard octave tests. Next it is easy enough to place c#4 a little slower than a#3/d4 and a little faster than f3/d4. Or/and use the sequence of 7, 8, 9, and then slower than 9 from the intervals tuned. If this sounds good then the first set of contiguous M3's are locked in. I must admit that I have a difficult time with the contiguous M3's all by themselves without the additional references created by the above sequence.
However, you have professed this approach for beginners so I know that it certainly is a valuable technique. I just need to practice it more. I can get stuck on one approach to my detrement.
But once the contiguous M3's are correct you have told the temperament that it will favour a smooth progression of thirds and the 4ths and 5ths will fall where they may, is this a correct assumption?
I like the BW variation and will use it at times especially on small poorly sclaed pianos where it really does not matter how even the M3's are and the focus is to get it to sound as good as possible.
It is good to have more than one technique.
Frank French tutored me through a historic temperamant sequence and it was quite easy - forgot it however for lack of practice.

Gene, thanks for your post. I totally agree with what I think it was you who said that for the temperament to be truly equal, both slowly and rapidly beating intervals must be addressed. An example of when M3s can sound perfect but 4ths and 5ths erratic is the "Marpurg" temperament. It is not an equal temperament and it would not pass the tuning exam.

In a recent journal article, there was a comment about as long as 4ths & 5ths are "even" a little unevenness in 3rds & 6ths doesn't matter. Well maybe and maybe not depending on what you want from the sound. If I were to tune the EBVT and told anyone to check my tuning and they assumed I had intended to tune ET, they would say my 4ths and 5ths sounded pretty good and so did my 3rds & 6ths except here and there, there is a bit of a problem with both.

The truth is, that to tune ET accurately, no one can solely rely on or even emphasize either slowly or rapidly beating intervals. They must work together and compliment each other or the results will be something other than ET, be those results good, bad or indifferent.

Now, to another question I received from private mail. How can this system of contiguous M3s be adapted for use with a C fork?

Here's how:

1. Tune C5 to the fork (or skip that, it won't really matter much).

2. Tune C4 to C5 (or directly from the fork if you choose).

3. Tune C3 from C4.

4. Estimate the C3-E3 M3.

5. Tune E4 from the estimated pitch of E3.

6. Listen to the E3-G# M3 and adjust it first so that the C3-E3 M3 and the E3-G# M3 beat exactly the same, then sharpen G# slightly (about 1 cent)so that the E3-G#3 M3 beats a small amount faster than the C3-G#3 M3.

7. Listen to the entire sequence. Do all M3s beat progressively faster? Are there any two which beat exactly or very nearly the same? Is there any one pair of contiguous M3s where the upper beats faster than the lower?

The results will clue and cue you which way to move your initial estimate of E3. Remember to always re-tune E4 to E3 after you have moved E3. Listen again to the E3-G# M3 and adjust it if necessary so that it beats a small amount faster than the C3-E3 M3 as it sounds now after the correction you made.

It usually only takes one or two corrections to arrive at a suitable, evenly progressing sequence. The key here is the 4th set of contiguous M3s. The top pair reveals whether the bottom three are imbalanced or exaggerated.

It is of utmost importance to work with a sequence of 4 contiguous M3s instead of just 3. The reason is that you have 2 pairs of octaves instead of just one. Those two pairs serve to greatly diminish any possible error. The 4th pair of contiguous M3s includes one note which is outside of the temperament octave itself but it plays a key role in confirming that the 4 notes within the temperament octave (in this case, C3-E3-G#3 and C4)are correctly distributed. It is far easier to make a substantial error if only the 3 contiguous M3s within the temperament octave are considered.

May I offer some tests at this point? C4-E4 should beat twice as fast as C3-E3, and C3-E3 should beat as fast as C3-E4.

Thanks BDB but in this outline sequence, I left out many details I have written before. Certainly, however, anyone can use octave verifying checks at any time if they choose to do so. Having said that, the article I have written in its totality tells how to get a piano which has been detuned for a tuning exam or a piano which is off pitch quickly to a state which then can accept fine corrections.

In either case, an exam detuning or a pitch raise (or lowering, we are now approaching the humid season across most of North America), every note is presumed to be quite off from its target pitch. If one were to employ fine checks at this point, the test notes which are needed would also need to be moved to a convenient position. Therefore, I suggest as a roughing in technique, to simply tune the octave beatless but approach it from the wide side so as to insure it is not narrow.

In writing I have done about fine tuning techniques, I emphasize that the F3-F4 and A3-A4 (or C3-C4 and E3-E4 octaves when tuning a C based temperament) need to be verified as being the same type or quality of octave. That means they should both be 2:1, 4:2 or 6:3 octaves or somewhere in between any of those, just as long as they are confirmed using the same checks. When I analyze a temperament numerically, I look for no more than 0.5 cents difference between those two octaves before I proceed any further.

The article I will publish when it is ready is only about arriving at a rough tuning of the temperament and midrange. It will clearly state that the fine checks you speak of can then be used more effectively to refine the temperament as time permits during an exam or on the second pass during a pitch correction tuning.

By the way, the octave checks you suggest only apply to the most conservative kind of stretch, which of course is valid, if that is what you intend to do. The first check you cite, "...should beat twice as fast..." would be difficult to verify. Equal beating is easily perceived by nearly any piano technician, even those who customarily only tune using an ETD. "Twice as fast", presumed to mean, exactly twice as fast would involve a judgment call that I don't believe any human being has the ability to determine with precise accuracy. (Yeah, it sounds twice as fast to me, how about you?)

The second check you mention is, in fact an example of equal beating. If a M3 and M10 beat exactly the same, the octave will sound beatless to the ear and can most likely be no amount more stretched than a 4:2 octave. Some tuning exam master tunings are indeed constructed that way but most, from my experience have central octaves slightly wider than that. I believe most ETD default programs provide for a slightly wider octave than 4:2.

(I recall reading once that someone who had worked on a master tuning as part of a committee was told by the conductor of that master tuning, "We want NO stretch!" Now, this was his recollection but I take it to mean that the conductor wanted what you describe, equal beating M3-M10 tests). (Ah! the *perfect* tuning: "pure" octaves!) (Do we not care what problems emphasizing one interval over another will create?) (Could it be that crafting a compromise among all intervals, including octaves may result in a better overall compromise?) (Sorry for the digression but I felt it quite important to this discussion).

When I construct a master tuning, I aim for that very small difference, an exact compromise between a 4:2 and 6:3 octave. In that case, neither the M3-M10 nor the m3-M6 test are equal beating. On lower inharmonicity pianos, the two may really be indistinguishable and apparently equal beating but on most pianos, the two tests offer an extremely small difference, about as small as is discernible to most technicians. (Don't ask me for a ratio because that would depend on the exact inharmonicity profile of the piano in question).

I vividly recall Dr. Sanderson telling me and also reading it in published material that Dr. Sanderson worked with the idea of a "4:2 octave + 1 cent". Generally, that proved to be a suitable compromise between a 4:2 and a 6:3 octave.

These are extremely small distinctions. Therefore, they "put the cart before the horse" with regards to taking a substantially out of tune piano to a state where these kind of refinements can reasonably be executed.

Thanks to all who have contributed thus far. I have enjoyed the discussion.

Gene:

I do not see a difference between the variation that you posted and the BW sequence in his book.

”Hi ih would force any interval wider - would it not?”

Well yes, but does that mean that the beat rate will change as you would expect when widening an interval? Oddly enough it depends on the iH slope and the interval in question. As I mentioned, the effects of iH on beat rates are largely self-correcting. It was just during the past few days that I realized that there was an easier way of looking at this by using cents rather than Hz.

I am going to use the piano model from Young’s paper on inharmonicity. The paper on the internet doesn’t specifically say what piano this is, because the graphs are missing, but it seems to be an S&S D. C3 has an iH of 0.1 cents which doubles each 8 semi-tones. [Octave Error Edit begins] So if we use F4-A4 as an example, the iH for F4 = 0.4362, A4 = 0.6169 and F5 = 1.2338. When applying iH, it is multiplied times the square of the partial in question. So when determining the stretch for a F4-F5 4:2 octave, we find that the octave is stretched 2.004 cents, or 0.1703 cents per semi-tone. So since F4-A4 is 4 semi-tones wide it is stretched by 0.6812 cents. But how does iH affect the partials that cause the beating? The fifth partial of F4 is sharp by 10.905 and the fourth partial of A4 is sharp by 9.8704 cents. [Octave Error Edit Ends] So iH has widened the interval with octave stretch by 0.6812, but narrowed the nearly coincident partials by 1.0346. The net effect on beat rate is a narrowing of 0.3534 cents. So even though the interval is wider, it beats as if it is about a third of a cent narrower!

As far as compromising across a challenging break, as you say, it depends on what you want. On the Verituner site there is this comparison of compromises when tuning across the break of a Baldwin console: Click Here. Unfortunately, it does not show the beat rates of the fourths and fifths. I deliberately tune challenging breaks similar to the SAT. I prefer to keep an even “character” to the fourth and fifths and let the M3s jump in beat rate if necessary. I cannot do this when tuning with CM3s.

Edit: Jump in beat rate across the break, but nowhere else.

Jeff,
I did type up the wrong sequence - sorry about that.
here is the BW variation:
c4/c3 octave a hair wide. g3 between c3/c4, d3to g3, a3to d3, e3to a3, b3to e3 and test c3/e3>c3/a3>d3/b3>g3/b3 gradual increas in bps.
next tune f#3to b3 and test d3/f#3=c3/a3, next c#3to f#3 and g#3to c#3 and test e3/g#3=d3/b3, next d#3 to g#3 and a#3 to d#3 and f3 to a#3 and test a3/f3=d#3/c4 now f3 should fit nicely as a p4/p5 between c3/c4. Hope I got it right this time - having a senior moment.

I have a stock ssd scale and it differs a bit from your ih numbers:
f3=.18, a3=.245 and f4=.753 also the ih does not quite double at the 8th semi tone but close enough. Maybe Youngs paper is of a different piano? Or do ssd's vary that much - na - couldn't be.
I had a sneeking suspicion that a smooth ih curve in a well scaled piano would follow the same scaling ratio that is used to select wire sizes and speaking lengths: 39to37 and increase 1/2 size every 6 notes and it does - that is at the top of the piano where you can get an ideal scale. Farther down past note 60 or so all bets are off. Most piano makers do not take the time to carefully carve the bridge to get the ideal scale anyway - in my opinion.
Not easy for me to imagine how ih and beat rates work together but what does come to mind is: what is happening to the fundamental friquency and the other partials? You can make all of the 5th partials line up but the others will not cooperate and ih will dictate how much or how bad it will sound.

I'll add to the above ih curve/scale design with this example:
the speaking length of this ssd is 54mm at note 88 and the ih is 14.7 wire size 13-1/2. Multiply 54*39/37 = 56.91(the ideal spl for note 87. Do the same to the ih 14.7*37/39=13.9 and if you continue this down the scale and go up a half size every 6 semitones the scale will be quite smooth and ih will follow along very closely. Problems happen when the spl cannot be increased at the correct ideal ratio and the compromises begin and for almost all pianos-small or large this happens at around note 60 or so.
Also, I am not certain how the ih values reflect what is really going on - do you know what the units associated with ih are? I think the values are of only one partial?? Or is it a more generalized value that tries to describe all partials?? I need to dig back into the Calculating Technician book.

Gene:

Thanks for the corrected BW variation. Won’t be much use to me, though. I am happy when F3 is unwound. An unwound C3 is a treat.

Whooops! I worked out the iH for F4, A4 and F5 by mistake. Well it will give a sort of comparison.

Lets see what your numbers do:

The effect of a 4:2 octave stretch will widen the interval 0.04 cents, and the effect of iH on the partials will change the beating as if the interval was 0.58 cents narrower for a net change of beat speed as if the interval was about 1/2 cent narrower.

Now to go do some editing - Thank You!

Well 39/37 is the 13.17th root of two, so if a chicken and a half lays and egg and a half in a day and a half...

All I know about piano scaling is the Henway.

The iH number is the difference in cents between what a string should vibrate at and what it actually does. The number of cents that each partial vibrates more than it should is calculated by multiplying the iH by the square of the partial number. By measuring the frequencies of two partials, the iH can be calculated, because they will not be powers of the same number.

excuse my feble math brain but - following your explaination:
ih for 4th partial of f3=.4362*(4*4)=6.97 and ih for second partial of f4=1.2338*(2*2)=4.93
Am I following you correct?
Difference is 2.04? maybe not the correct way to look at or calc it?
I am lost.
I do have a master tuning for an ssd that has been refined several times and I have the 4th partial of f3 at +1.5 cents and the second partial of f4 and +1.5 cents.
This master tuning is not from the recorded stock ssd scale that I gave ih numbers for but it is probably very close.
When I tune the ssd the f3/f4 octave is ever so slightly wide of equal beating - probably less than a hair but wide.
Opinion, analysis?

Gene:

Yes you are following me correct, including the original octave error… the calculated iH of F4 is .4362 and F5 is 1.2338. This makes the F4-F5 4:2 octave 1202 cents wide. How many cents sharp or flat, from theoretical, the pitches are does not come into play. Maybe this is where you are confused?

In your master tuning, since the fourth partial of F3 and the second partial of F4 are both +1.5 cents from theoretical, then they are at the same pitch and it is a 4:2 octave. And according to the figures I worked out from the iH you gave me, this would be an octave that is about 0.5 cents wide.

But my purpose of putting out these figures was to show that even if an octave is stretched to satisfy iH, that does not mean that the wide intervals will beat faster than theoretical, but can beat slower, depending on the iH slope, iH values, the octave type, and the interval in question.

Bill:

You wrote –

” I've written about this many times yet I still don't seem to get my point across. The system you describe by definition requires you to make an estimate, another estimate, an estimate upon a previous estimate and another estimate upon two previous estimates before you ever have a M3 to consider and it is formed by an estimate and another estimate made upon two previous estimates. At that point you only have a M6 to compare it with. That interval is formed by two estimates. The resulting comparison can sound reasonable and perhaps luckily be correct but the previously tuned pitches can still all be slightly erroneous and lead to further complications later on. Sooner or later, the F3-A3-C#4-F4 will be in place but if they are determined to be incorrect and then remedied, it requires backing up through all of the previous pitches which lead to those errors. The problem with backing up through a sequence to correct errors is that it only leads to more estimates and estimates made upon a previous estimate.”

I’ve also read this many times in your posts, and I do get your point, but am able to get a more accurate ET by tuning with 4ths and 5ths and checking with 3rds and 6ths. You see I consider a set of CM3s to be estimates also. You seem to support this when remarking on how long it takes a tuning committee to tune the first set. And then when a 4th or 5th is tuned to one of these notes, it also becomes an estimate upon an estimate. Just because tuning by 4ths and 5ths is an estimate upon an estimate does not mean tuning with CM3s is not. The 1 bps of a temperament 4th represents 2 cents. I think they can be more accurately tuned than a set of CM3s as a starting point.

To me the advantage of tuning 4ths and 5ths is you know where the errors are coming from. With a close, but not quite perfect set of CM3s, it could be one or both of the notes tuned to the starting note that is slightly in error. Yes, when tuning thru a 4ths and 5ths sequence there are accumulative errors, and that is what I like about it. Each note that is tuned gives more and more checks that lead sequentially back to the starting note. With each note that is tuned the accuracy of the temperament can be determined with greater precision. Very small errors may not be noticed individually, but when compounded they can be identified and dealt with.

I would like to contribute my "2 cents" worth on this topic.

I have been a C fork man and an F-F staunch temperament tuner for my entire life. 40 years total now, 35 of these full time.

I was unconvinced that there was anything better, or any other method or any tool that could be of better use than my ear. After all, I could do an excellent tuning this way. I could tune two concert grands or more together as perfectly as you could get them so, why try something new right? Well, that was my mentality. However, I have come to learn over the years that there is always something better, someone with a better way, someone with more experience in certain areas, a better tool or a better aide. Especially with today's computer technology.

I was also unconvinced that EDT's had any value to them whatsoever. Not after hearing some tunings that were produced from many tuners that had no clue how to properly use them or that had ears that couldn't decipher those notes that were wrong.

I had this mentality until I was convinced that they perfected them and even more so, when I finally bought one for myself. I worked with it, with my own ears along with the advice of many here (thank you for that advice) and by working with Dean Reyburn at his house for free, I might add. While none of them are "perfect," none of us are either, they are a great addition to the human ear and it does make life considerably easier to tune more than one piano together but, only, in conjunction with the human ear.

That said, I think too that without actually trying Bills method, which I just started myself 2 or 3 days ago, there is no valid point for argument against it.

While I at first, was the first one here to disagree with him I have since changed my mind after trying his method. I am even more convinced than ever that he was and still is correct and I was not. While there are many ways to achieve a good result, some ways are better than others.

Today for example, I started tuning an Everett spinet. F-3 and F#-4 bass, ran into the tenor section. RCT would not set the middle section of the piano correctly no matter what I tried in this case, due to a very, very, poor scaling design on this piano and a TON of false beats. It was listening to the wrong stuff "this time."

I proceeded to try Bill's method instead and it turned out great.

Previous to tuning this piano, I tuned a Yamaha G1 that I have been tuning yearly for 30 + years. I started with A-4 as Bill recommends. I then tuned A-3 to A-4. Then, I tuned F-3 to A-3. Then I tuned F-4 to F-3. Then I tuned C#4 to A-3. (I think that's right) I checked it all with the CM3's up to A-4 for consistency and correctness. After re-doing a couple of them I proceeded to verify just exactly how far off I was with my RCT. I used NO 4ths or 5ths for verification. Only Bill's method.

Now, bear in mind, that this is ONLY the 3rd TIME that I have tried it his way.

The worst note of the 5 notes was 0.58 cents off. The best was dead on with RCT and the other 3 were 0.48 cents off or less. Regardless, 1 was perfect, 3 were less than 1/2 cent off and 1 was slightly over 1/2 cent off from matching RCT.

If I am writing the numbers correctly, I think I am.. (?) None of these notes were even close to being 1 cent off. And that is only on my 3rd very inexperienced try.

From what I see so far, in my opinion,it is indeed easier than a regular temperament. The reason being is that each piano sets a temperament differently. We have to give and take A LOT on Baldwin Hamilton's and other Baldwin studios as well as Story & Clark's and other brands within that F-F temperament section because each one sets differently. On Baldwins, I've tuned those using the C-4 and C-5 AND F-F temperament in order to get it to come out correctly.

With Bill's method, it just seems to fall into place.

That's what I've encountered thus far. So, as I said, unless one has actually tried it, an argument against it in my opinion, actually becomes worthless and baseless.

Thank you, Jerry,

That is accepting the RTC is perfect too. Remember that even a master tuning committee often uses an ETD to construct the tuning which will be improved by the committee. Usually, the corrections made are less than 1 cent. I have rarely found a CM3 sequence generated by an ETD to be completely satisfactory. It too is only an approximation based upon estimates.

As I have said many times, the method accounts for whatever inharmonicity there is because it incorporates it as the sequence develops. I am sorry to say that no matter how well one may be able to "count" 7 beats per second, that exact figure is only an approximation and would only be actually correct in a rare circumstance.

There is only one estimate which is ever made in the sequence as I have described it. After that, there are two more steps which, in fact are not estimates. Those two steps either confirm or direct in which direction either sharp or flat and how much, to re-tune the initial estimate. After that, there are no more estimates, only sure and unequivocal placement of each note, the first time, every time.

Master tuning committees always take what seems to be an unusual amount of time because they are trying to improve what already would be considered a perfected ET. The closer to perfect the temperament already is, the more time it takes to improve very small errors. But when few if any errors are found, the process goes quicker. With older methods, it used to take a committee as long as 8 hours.

Now, most master tuning committees use a CM3s foundation and complete the process in half to one quarter the time which used to be spent sorting out errors created by making an estimate and then other estimates upon previous estimates upon estimates. It could never be clear where any error may have been made because there was likely to be an error in each and every estimate. Therefore, a system which requires only one estimate to be made and that estimate is either verified or corrected before proceeding any further, is clearly more reliable and efficient.

For some 25 years now, I have observed that 9 in 10 aural tuners who use a 4ths and 5ths method do not really tune ET. Most often, in spite of what they may know theoretically or intend, they tune a backwards version of a Well Temperament which is often called "Reverse Well". Reverse Well is virtually impossible as a result when the temperament sequence begins with tuning a series of 4 CM3s.

A 4ths and 5ths sequence is far better suited to tuning a Historical Temperament than it is ET. That is where such a sequence had its origin. Throughout tuning history from the 17th Century on, there were people who tried to find a method to accurately tune ET and all of them failed. Braide-White claimed to have the answer but those who have used his method mostly failed as well. Only the most highly skilled technicians who used knowledge learned elsewhere which Braide-White did not provide in his book to find ways to correct the inevitable errors his system would generate ever succeeded. The rest only believed strongly in what they were doing and never looked beyond what they thought they already knew.

Jerry:

I have tried CM3s. Not sure if your post meant that the only reason someone wouldn't prefer them is because they hadn't tried them.

Tuned my first Kemble yesterday. There’s not many around here. I guess they are a Yamaha knock off. About a 45 inch console with a D3-D#3 tenor break and two wound bi-chords in the tenor. F3-A3 needed to be much slower than 7 bps even with F3-C4 being almost pure. But I knew that from forming this M3 by tuning fourths and fifths, which are less effected by iH because they use lower partials. In fact when I tuned the first fifth (F3-C4) I had a good idea what was going on because of how much faster the 6:4 beat compared to the 3:2. Likewise, F3-D4 had to beat a bit slower than G3-B3. All the RBIs were progressive on the unwound notes when I finished the temperament, but I deliberately left a jump to faster RBIs when crossing into the wound strings. This was necessary for the fourths and fifths to sound their best. I could have tuned this using CM3s, but I would not have liked the result as much. I think F3-C4 would have ended up wide and there would be problems in the adjacent octaves.

It is really just a personal preference.

Bill:

William Braid White mentions that there are many other tests that a tuner can discover and use for himself, and gives a few examples. You always paint an inaccurate picture of his method, and by inference imply that other methods are better. But just because there is a worm in one apple doesn’t mean there aren’t worms in another.

And another thing!

Proponents of CM3 tuning often use the figure of about 1/2 cent as the accuracy of the notes that are tuned. But what accuracy is needed to tune an Equal Temperament where all RBIs beat progressively faster? I know the answer, but it is really up to the proponents to show that the accuracy that a set of CM3s are tuned to, is sufficient to set an Equal Temperament. Not to mention how the remaining three sets of CM3s are placed correctly.

(No I am not in a good mood today, but what's new?)

Tooner, it seems to me that you never did read and follow the directions. You may estimate the F3-A3 M3 at 7 beats per second but if the piano tells you to change it, then you do. Jerry read and followed the directions on an oddly scaled piano and got results that worked within what are considered reasonable tolerances. With your method, you were scrambling all over the place.

I quote Owen Jorgensen, "The value of using the CM3s is that it serves to divide the temperament octave into three precisely equal parts from the outset. Therefore it solves the most difficult part of temperament tuning before proceeding any further".

The Braide White book does not mention or explain how to do this. The first attempts by technicians in the 1980's also did not explain how to use *both* the F3-F4 and A3-A4 octaves to control the process.

If anyone uses the method as I have described it in the initial posts in this thread, they can use any other sequence that is familiar after that. The F3, A3, C#4 and F4 can be considered to be the most reliable notes tuned and serve to correct any estimate made thereafter tuning either 4ths & 5ths or 3rds & 6ths. Any known, familiar and useful checks, whether Braide White mentioned them or not may be used at any time.

Now, that is the last I am going to say about that to you, Tooner. Past history has shown me that you are likely to respond, "I still do not believe the CM3s are accurate". You will claim that you can tune to an accuracy of 1/8 cent using 4ths and 5ths and have the math to prove it. You will cite numerous examples of when you tried the CM3s and they didn't work as well as your method.

No one is forcing you to use CM3s. But if by tuning your method, the CM3s don't bear a 4:5 relationship and some of your intervals are favored over others, then your temperament is by definition, a non-equal temperament. When I tune the EBVT, the CM3s don't bear a 4:5 relationship but they are not supposed to. That is intended and by design. It is a non-equal temperament.

So, from now on, I will not respond to any of your arguments against the use of CM3s, no matter what they are. None of your math or anecdotal evidence will persuade me. I know what I am doing from many years experience in research, consultation and practice. I will teach this method to my students as I have for years. The results I produce are the reason why I am asked time and again to be a tuning tutor. You are free to gather your own students and produce results with them.

You are also free to prove that you actually can tune an equal temperament using your methods. You know how that can be done and it is not by writing about it. You have succeeded in preventing me from mentioning just how it can be done but you already know by now how you can do it and I suggest you do it and prove it to yourself.

Bill:

You have side-stepped the question, attempted to produce a villain in order to glorify a saint, and dramatically ended the discussion at the place of your own choosing.

Oh, and did Mr. Jorgensen actually use the term "CM3s"? After all, you supposedly quoted him.

” You are also free to prove that you actually can tune an equal temperament using your methods. You know how that can be done and it is not by writing about it. You have succeeded in preventing me from mentioning just how it can be done but you already know by now how you can do it and I suggest you do it and prove it to yourself.”

This reminds me of a short story: The Lumber Room

I have better things to think about. CM3s are pretty simple and crude.

The title of this is "A New Look at Tuning Contiguous M3s". Its validity of purpose is served if it even helps one single person who is having difficulty using other methods or simply finds it more practical. I don't doubt this because we are all different in our capacities and routes we take in learning. For me, I am tinkering with it because I think it could prove useful in some situations. Never hurts to add to your arsenal. It might get shelved away because I have grown so accustomed to my own sequence. It is however a different way of describing the process and could help those who are challenged in the deciphering stage of the game or providing at the least, an alternative approach to learning in the present stale condition of tuning pedagogy.

Excellent post Emmery. I completely agree with you.

Tooner, as you say, maybe it's just because you are having a "bad day." But you again show yourself to be what so many of us think you are. I had determined some time ago to just ignore you. Maybe I should do that now, but you again show yourself to be somewhat of an idiot.

It became quite evident to me sometime ago, that you have no interest in contributing to anything other than to those threads which justify your vain imagination when it comes to piano tuning. Perhaps you do have some valid questions regarding the tuning of M3's. But why is it necessary to turn this into a personal attack on Mr. Bremmer or those, who would post with the intent of helping you better yourself.

Actually, I dont think you want to better yourself. You simply try to justify your pitiful existence, and add to the "inharmonicity" of this forum. For that reason, I personally have slowed down on my contributions to this forum, and due to the vindictivness of this post, it most likely will be one of my last.

It will surely be my last directed toward you, regardless of the type of attack you mount as this thread progresses. This is my absolute last word to you, my friend

Thank you Emmery. I do not recall whether you are an ETD user or not. I have also heard from may technicians who do use a traditional sequence successfully and others who tune the CM3s first and then use the sequence that is most familiar to them. The F3,A3,C#4 and F4 or the C3,E3,G#3 and C4 from a C fork will serve as strongly reliable pitches on which the rest of any conceivable sequence can rely.

This thread isn't about the rest of the sequence, only about how to tune the first series of CM3s. Here are some suggestions about how to show to your self that it really works:

If you are an ETD user, program your ETD normally. Choose a well scaled piano for this exercise and tune F3, A3, C#4, F4 and A4 as carefully as you can to the program. I suggest listening carefully to the F3-A3 M3 it produces and get that sound in your ear. There is no need to "count" the beats but if you believe that helps you, that is certainly OK if you do. Then flatten F3 by 1 cent and follow the procedure carefully. Be sure to listen again to the F3-A3 M3 and hear how a change of just 1 cent did obviously change the beat rate of the F3-A3 M3. You will then see and hear that you cannot make a series of 4 CM3s which progress properly. Then try the opposite, sharpen F3 by 1 cent and follow the procedure carefully. Again, you will see and hear how an error of just 1 cent becomes an disfunctionally large error.

Then, by ear, after the second exercise, imagine how a 1 cent error would appear on your ETD display (a fairly slow pattern)and how carefully you would need to flatten F3 to change it by just 1 cent. Listen to the resultant F3-A3 M3 and if it sounds like a reasonable estimate, then follow the procedure carefully again.

If it seems to work, so much the better but if it appears that you made too much of a change, make again the very small change you need to correct F3 and try again. When you believe that you have a correct series of 4 CM3s, compare your results to what your ETD provides and you should find either little or no significant discrepancy. In other words, all pitches well under 1 cent from where they should be ideally.

That will prove to you the precision with which the CM3s can be set aurally.

If you are a strictly aural tuner, simply make your first F3-A3 M3 a little faster than you instinctively know it should be (such as 8 beats per second). Then do the same making the F3-A3 M3 beat just a little slower (such as 6 beats per second).

This is the exercise I take almost all of my students through. It shows them just how the piano itself will tell them how much and in which direction (sharper or flatter) the initial estimate needs correction.

Now, where I believe this procedure may be of the most benefit to you Emmery is not so much the well scaled piano but the poorest, most irregularly scaled piano. Once you believe you know and understand the procedure thoroughly, try it on the kind of piano you have had the most difficulty with in the past.

The initial F3-A3 estimate you make may seem right at first as it was with well scaled pianos but the rest of the sequence will tell you how much and in which direction you must tune differently for this particular piano. When the CM3s are correct, the octave is indeed equally and precisely divided. All other intervals will fit into place, all equally tempered, even if the beat rates vary slightly from what they are on well scaled pianos. You will have no need to favor one kind of interval over another and the temperament will be truly equal.

Please let us all know how this has worked for you.

Ron:

I have been pointing out problems in Bill's arguments for CM3s over 4ths and 5ths. That is not the same thing as a personal attack.

Unright said:



Jeff, I see the question quite differently. Your above contention is a neat argument, which I know you are very capable of. However, I don't think it adequately addresses what I believe may be the single greatest advantage of a CM3-based system, the general truth which most or all more specific arguments can be subsumed beneath:

CM3-based systems begin with the premise that accuracy in temperament is more easily achieved by dealing first with the big picture, then refining as one proceeds into the smaller. A more-or-less equal subdivision of the octave into CM3s is the start. A Braid White-style sequence, on the other hand, attempts to arrive at an accurate larger picture by starting with the smaller. I'll start with "the bird's eye view" of a situation, when it's available, anytime.

The main advantage of the CM3 system is that it always proceeds with more understanding of the big picture, more quickly giving feedback on how each step fits -- or will eventually fit -- with this big picture. (It isn't even really necessary, for this to be true, that the first CM3 chain be perfect at the outset.)

To say much more would probably just muddy the water. I still marvel at the above-stated truth, and what it’s made possible for me.

I did want to say one other thing: Somewhere above, you brought up the truth that slow-beating intervals, like fourths and fifths, have less inharmonicity than rapid-beating intervals. You gave this as a reason for preferring fourths and fifths to tune with.

It so happens I recently had an interesting exchange with Jim Coleman Sr. about this. Something he contributed to the Accu-tuner manual seemed to conflict with the preference you mentioned, of leaning toward fourths and fifths because of their lower inharmonicity. After wondering off-and-on about this, I finally e-mailed Mr. Coleman after viewing a video on tuning -- maybe twenty or so years old -- by another well-known figure in the PTG. This other person said that, because of higher partials having more inharmonicity, he favored octaves first, then fifths and fourths, followed by thirds and sixths.

I asked Mr. Coleman if perhaps iH in the higher partials -- even if greater -- was more consistent somehow, and this was why he himself preferred slightly imperfect SBIs in the temperament -- if they allowed more evenly-progressing RBI's. He agreed with this. He said it isn't absolutely clear to him why this is true, but he suspects it may have to do with bridge terminations.

Good luck on your own path, Jeff.

Jerry G., good for you! I admire your willingness to learn and try new things, then publicly share your experiences. I used a C-fork Braid White-style system myself, for over twenty-five years. Like you, my first attempts with CM3's came after reading Bill's articles. I ended up altering his sequence, but the basic ideas are still mostly the same. I hope you continue to find useful things in your experimentation.

Jeff S.

Thanks Jeff S.

Well, I have to thank Bill B for taking the time to explain to me privately so that I in turn, can TRY to teach my son "somewhat" before his session with Bill this summer, I probably never would have gotten it. I have my son Jerry Jr., set up to be tutored by Bill at the convention in July.

I never thought I would hear myself say that there was a better way than ET. But, there is.

After literally FORCING myself to read this stuff about Bill's method (I never have enjoyed reading about tuning stuff) using the EBVT and other sequences regarding it I thought I would be a fool not to try it.

Technology changes, theories change, ideas change and, so must we to keep ourselves updated and our minds YOUNG. But, then again, from what you read on some of my posts, you know that I'm just little kid at heart anyway!

Jeff S.:

Thanks for wishing me luck and understanding that I need to take my own path. If I dealt with only well scaled pianos I would use a variation of Bill's ET via Marpurg. The variation that I worked out can produce an ET that is very accurate, but does not work well with changes in iH.

Since actually crunching the iH numbers recently, I am understanding more what others have said about the beat rates of 4ths and 5ths (and many other things, also), but I don’t want to get too far off-topic.

Higher partials verses lower partials seemed to be another way of saying RBIs versus SBIs, but I am not so sure now. Only the 2nd partial is used by the SBIs alone. And the 3:2 fifth is the only interval having a beat rate that acts oddly. But if we consider the fifth to be a 6:4 interval, then it is only the beat speed (and therefore width) and not the partial numbers that differentiate the SBIs from the RBIs.

To me the biggest question is what to do about the compromises across a break. Again, just recently, it occurred to me that the fourths should take preference. This is so that the 12ths and double octaves, even if they do not beat equally, will not beat obviously. The fourth is the third “leg” of the 12th – double octave triangle. If they take a jump in beat rate, so will the 12ths and/or double octaves.

Thank you, Jeff S. It was thirty years ago now that I learned that 3rds & 6ths were to be considered the "fine tuning" intervals and 4ths & 5ths the "coarse tuning" intervals. In other words, each needed to be balanced with and controlled by the other.

M3s have 5:4 as their coincident partials. Since they beat rapidly, any change of just one cent (and as little as 0.2 or 0.3 cents) can easily be heard when comparing either chromatic M3s or CM3s. Therefore, CM3S can and do divide the octave into 3 equal parts with great precision. It only depends on the skill and perception of the technician to make that equal division.

However, to continue constructing the rest of the temperament sequence by only using 3rds & 6ths could easily result in a non-ET. For one thing, after tuning the series of CMTs, you can't tune another M3 until you have tuned at least one other kind of interval. You could tune a M6 from F3 and then continue with M3s and M6s and not listen to 4ths & 5ths but that has never been what I have suggested.

In each of the three sequences I have presented on my website, the "up a 3rd, up a 3rd, down a 5th", the Marpurg Shortcut and the ET via Marpurg, all of the rest of the notes tuned are 4ths & 5ths and are then controlled and finely adjusted by 3rds & 6ths. If there does happen to be a slight error in the initial CM3s either because of an error in judgment or instability, it will reveal itself and can then be corrected. I have found this to be true a few times during master tunings.

When an error was detected, it was usually an amount of 0.5 cents or less. Of course, that kind of error can result from slight instability and not from initial tuning error. Either way, any error in CM3s does eventually reveal itself. Certainly what is never considered is that any particular piano cannot support correctly tuned CM3s. It is ALWAYS possible to divide an octave (of any size or type) into 3 exactly equal parts on any piano from a Fazioli to a Whitmore, from a Steinway D to a Starck Spinet.

Constructing a sequence by using a series of several 4ths and 5ths first and then trying to detect and resolve errors when 3rds and 6tfs become formed is far more perilous. One can only make estimates of very slow beats, 1 or 2 beats per second or "3 beats in 5 seconds" or whatever. You can NEVER know what the exact rate should be because that rate is determined by two independent factors, the actual inharmonicity of the piano and the width of the temperament octave. An error of 1 cent is not nearly as evident among 4ths and 5ths as it is among 3rds & 6ths. Indeed, as John Travis wrote in his book, "Let's Tune Up", there is a natural tendency to "err towards the just 5th" as he put it.

There is more than one reason for that. 5ths, especially in the 4th Octave quite often do have 2 sets of audible beats. If the inharmonicity is high, it can be confusing which set of beats is being heard, the 3:2 or the 6:4. Just as with octaves, if one set is chosen over the other, one will be right and the other wrong.

Although I have usually seen it suggested that the lower set of coincident partials of 5ths be favored, should not each 5th where both sets of partials can be heard be a compromise between both? That would mean using extremely fine listening techniques to strike the balance between the two the way a compromise between a 4:2 and 6:3 octave is formed. That is usually an extremely small distinction. How many technicians, especially those ETD users struggling to learn aural tuning could do that? How many of those could even grasp and remember the techniques?

For that reason, Jim Coleman, Sr. suggested to me that a 4th whose coincident partials are 4:3 (with the second set 8:6 being virtually inaudible)is a far more reliable slow beating interval to tune than a 5th.

Both of my "Marpurg" sequences are meant to be rough tuning sequences to get the temperament close enough to where truly fine checks are of any use at all. The title "Up a 3rd..." is somewhat misleading because M3s are not tuned, only 4ths and 5ths (more 4ths than 5ths) are tuned (after the initial set of CM3s, of course) but when they are, there is always either a m3 or M3 to check each 4th or 5th and as the sequence progresses, more and more checks become available. The entire sequence checks and corrects it self as it is constructed. No errors are ever corrected by "backing up" through 4ths and 5ths to "even them out". Doing that only means making estimates and estimates upon estimates again.

If the latter is done, checking to see if the 3rds are smooth does not point to where the errors are specifically, it only reveals that there are errors. The conclusion can easily be made that the particular piano cannot support smoothly ascending M3s and M6s but this is never true. Any piano can.

By contrast to my "Up a 3rd..." sequence, a typical 4ths & 5ths sequence requires estimates, estimates upon estimates and estimates upon estimates upon estimates before any check is available. (Sorry that I must keep emphasizing that point). One estimate may be a little narrow, another a little wide, another a little narrow and another a little wide. The resulting M3 and M6 may sound OK but each note which lead to it is incorrect and can be by 1 cent or more. The tests used for 4ths & 5ths only reveal whether the interval is wide or narrow but not whether it is tempered enough or too much.

In a Pianotech discussion that was reprinted in a recent issue of the PTG Journal, one writer (whose opinion I had seen written virtually the same way some 10 years earlier) wrote that 1 or 2 cent errors don't matter, as long as 4ths & 5ths are fairly even, a little unevenness in 3rds and 6ths is unimportant. Apparently that technician's opinion hadn't changed in a decade or more even though that same person has consistently argued that ET and ET only is the one acceptable way to tune the piano. I'm fairly sure that person became an RPT before the standardized tuning exam we have had in place since 1980 or so. I'd bet that same person would be astonished at the temperament score if the exam were attempted today. Is the definition of ET, "kinda, sorta pretty even"? (and then only on "good" pianos?)

It is not as far fetched as it may seem, particularly considering what John Travis pointed out. Tuners instinctively don't like the tempered sound of a narrowed 5th and quite often make them "cleaner" sounding than they should be. They will disfavor other intervals to compensate. They may well blame the inability to get "smooth" M3s and/or find that the CM3s don't work out and blame it on irregular scaling (or that CM3s aren't even useful) but the truth is that a perfect ET can be constructed on the most irregularly scaled pianos. The key to doing that is to begin the temperament construction with the CM3s from F3-A4 or C3-E4.

I found this to be true on that little Everett piano that I attempted to tune with RCT that I mentioned in a earlier post. A regular F-F temperament would not work on this piano either doing it in only as a regular ET. But, doing it Bill's way which I am STILL trying to figure out and understand, did make it come out as it should both ways. I won't be working again until Thursday (long story) but, I'll fool around with it some more at that time, time permitting.

Bill:

You wrote: ”You can NEVER know what the exact rate should be because that rate is determined by two independent factors, the actual inharmonicity of the piano and the width of the temperament octave.” So, how much error is produced if the theoretical beat rates of fourths and fifths are used? Can you give an example?

Anyone who could use theoretical beat rates to tune any piano would get the same results as a Strobe Tuner would provide. However, since those beat rates are all irrational numbers that no human being could reliably and consistently produce, the results would not even be as good as what a Strobe Tuner would produce. We all know that a Strobe Tuner does a poor job of constructing a temperament even on the best piano. The more irregularly scaled the piano is, the worse the results.

So, the more that anyone firmly believes in and adheres to the absolute folly of tuning 4ths and 5ths according to theoretical beat rates and expects the outcome to be perfect, the more frustration, imperfect and most probably poor results will be the story of that person's life as a piano technician.

I hope you folks don't mind my butting in here, but I have to say that this is one of the best threads I've seen on PW.

When I was still an "aural only" piano tuning student just over 30 years ago, I too was taught 4ths & 5ths first, then later 3rds & 6ths (and then 10ths and so on). When I went on my own, I'd use an A fork and set the D3 - D4 octave rather than the F3 - F4. (I'd use a C 523.3 fork if I wanted F3-F4.)In the early years, I tuned oh-so-many Whitney spinets. I wanted to deal with all the problems right in the temperament octave as soon as possible. So I'd have trichords, unwound bichords, and wound bichords in the D-D temperament and try to work it out early. I'd use CM3s in the D3-D4 octave and build from there.

It never made sense to me to begin with F3 - F4 from the A440 fork even on the better pianos... too much back tracking. I hope you don't mind my saying so, but I think you have come up with a brilliant solution... using a "fourth CM3" to verify F3 via the F4-F3 octave, after making sure F4-A4 increases consistently. It's no extra work; F4-A4 will be tuned anyhow when the F3-F4 CM3s are set.

I always thought the Holy Grail of tuning was good beat speed progression throughout all conceivable intervals (though in practice "all" may be unrealistic). All octaves and multi-octaves should be stretched so that they are as clean as possible with each other ( octave, double-octave, triple-octave, etc.), but without sacrificing (too much) the integrity of the single-octave. The only absolute is A440; Everything else is subject to negotiation in pursuit of an artistic result. As the temperament is expanded downwards and upwards, errors in the original temperament will inevitably show themselves. So, the temperament octave can continue to be refined. Ideally, the entire keyboard becomes the temperament... becomes the compromise. All intervals are of equal importance in the ideal tuning except that the octaves may not beat obviously.

Or, so I thought!

I must say that I never really paid attention to the arguments that consideration of 3rds should be more heavily weighted somehow than 4ths or 5ths. And, it never occurred to me that 4ths & 5ths should be more heavily weighted somehow than 3rds. You folks are making some very interesting and novel (to me) arguments on both sides.

I'll be following this thread closely to see how this develops; I may need to start doing things a little differently.

Bill:

Thanks for the comments and additional thoughts.

I thought I'd go back to a couple of things you said toward the beginning of this thread.

In your first post you said this:



Then, in the posted introduction to your article, you said this:



The impression one gets is that the idea of basing the temperament on a series of CM3s originated fairly recently, with the above-mentioned people. That may not be what you're trying to convey, but that's the way it comes across, if you don't mind me saying so.

As you may know or suspect, while the people you mentioned may be regarded as modern popularizers of this system, there were technicians doing it before them. For example, Rick Baldassin himself related, in a June 1988 PTG Journal article (reprinted in the PTG tuning exam sourcebook), that he'd recently encountered a man named Donn Foli who, about thirty years earlier, published (perhaps sometime in the 1950's?) the same temperament system Baldassin was using at the time (in the 1980's), including the CM3 chain.

But apparently it goes back even further. I believe, Bill, you yourself have somehwere mentioned the name Oliver Faust in connection with the CM3 idea. To elaborate on this, let me quote from a March, 1981, PTG Journal article by Carl Root (also reprinted in the PTG tuning exam sourcebook).

Referring to contiguous thirds, Root says:

"This sequence is simply the pivoting, or hinging, of M3s above or below a common note. Frank W. Hale is credited with inventing it and taught it at Oliver C. Faust's tuning school in Boston. It must have seemed a radical departure from the P4-P5 temperaments of the time, but its many adherents today attest to its validity."

As I said, Bill, you may already be aware of all this. But, in the interest of your article's clarity, I thought I'd bring it up. I notice on my copy of the Carl Root article I scribbled his e-mail address in pencil, although I no longer remember why. I suppose that means he's still available, if you should wish to consult him sometime about his sources.

As always, thanks for your continuing contributions.

Jeff

Bill:

I was ready to concede that we do not have enough common ground to continue this discussion. But your mention of Strob-o-tuning gives us another way to approach the subject.

Let’s say we take a piano and tune the first partials to the theoretical ET frequencies. Will the resulting beat rates also be the theoretical beat rates? NO! So the question remains, if the piano is tuned to theoretical beat rates, what will be the result? You say it will be in error. I again ask, how much error and can you give an example?

Maybe I'm missing something, but wouldn't it be as impossible to compute the exact error, without knowing the iH of the particular piano, as it'd be to compute the beat rates?

Jeff S.

Joe:

Thanks for showing interest in this discussion. I will try to keep going, but it seems futile. The idea of tuning with a preference of one interval and having all intervals be correct, but tuning with a preference of another interval and having all other intervals be incorrect defies logic.

You seem to have an open mind on this. Consider a set of CM3s. If we take the beat speed of an upper M3 and divide it by the beat speed of the M3 an octave lower the result will be about 2. If we find the cube root of this number (about 5/4), we can multiply it by the beat speed of the lower M3 to find the beat speed of the next higher M3. We can continue this and have the beat speed of all the M3s in the set. If we want to, we can now say that the octave is divided into three equal parts. But is it?

All we can really say is that the beat speed of the intervals is logarithmic. The beat speeds of the intervals are caused by the difference in frequency of the nearly coincident partials. The frequencies of the partials are determined by the frequency of the fundamental, the partial number and the iH times the square of the partial number. The higher partials are affected by iH more than the lower partials. Now if we consider the beat rates of other intervals, what can we expect? Will the beat rates be progressive? Yes, but only if the iH is progressive. If the iH is not progressive (if there is a poorly scaled break) the other beat rates will not be progressive.

But you know that without the theory, otherwise you would not have come up with a temperament sequence that spans the break on a Whitney spinet.

Now for extra credit, consider a long string of CM3s, where the beat speed ratio of M3s an octave apart are not all equal due to tuning different octave types in different parts of the piano…..

Yes, that is why I asked for an example. The fascinating thing that I noticed, when really crunching the numbers by including iH, is that the effect of iH is largely self-correcting when considering beat rates. But it is up to Bill to defend his statement about beat rates being incorrect.

Joe, thank you for joining the discussion. Many tuning instructions for an F3-F4 temperament from an A fork say to tune A4 first, then tune A3 from A4. As do many technicians, I used to skip the first step and just start from A3 until I realized that in itself was a mistake. It could cause a 1 or more cent error.

Jeff Smith: thank you for that information and you are right, the error that would be caused by using theoretical values would depend on the actual inharmonicity of the piano itself, so I view Jeff D's question as merely bait. There is no answer for that because it would be a different answer in every case and the example would be each and every piano.

Regarding first use of CM3s: I was not aware of these earlier uses and knowledge of CM3s but I am glad you told me because I will need to change the wording in my article to reflect it. I had suspected that Bill Garlick did not come up with the idea on his own but probably gained it through research. Oliver Faust was the first to use the "Up a 3rd..." sequence and he was a contemporary of William Braide White. However, it was Braide White's book that most people studied, not Faust's. The two never communicated according to Owen Jorgensen. At the time, there were two piano technician's organizations, BR belonged to one, Faust to the other. There was very little if any interaction between the two organizations in the early 20th Century, again, according to Jorgensen.

This reminds me, however of some information that was published a few years ago in the PTG Journal about how a sequence for ET had been published in Chopin's time. The editor at that time wrote a comment that concluded that Chopin then would have known about ET and used it. Today, any break through information spreads like wildfire through the internet but in Chopin's time, just because somebody wrote a book or a paper does not mean that many people ever read it or used the idea. In any case, according to Owen Jorgensen, that sequence would have more than likely resulted in a quasi ET, not the real thing. It would have been difficult to really tune a true ET using those instructions, just as it is using William Braide White's.

So, the possibility of effectively using the CM3s has always been there and any mathematician could have discovered them at any time. Marpurg, (18th Century) in fact, came close but instead of a 4:5 ratio, he thought they should be equal beating. Close, but not quite. (It is possible to put 3 equal beating M3s within an octave but this does NOT divide the octave into 3 equal parts, the 4:5 ratio does). So, what I said is essentially correct, it took Dr. Sanderson to reveal this concept to PTG and through that organization, the knowledge became widespread and commonly known.

You mean you used to tune pianos to A 440.25 instead of A 440? What a disaster that must have been!


So before Dr Sanderson came along, PTG tuners trying to tune equal temperament would tune A - C# something like equal-beating to C# - F, and F -A somewhat like equal-beating to C# - F, where presumably they would not notice that the next A - C# would either beat a lot, lot faster, or that the octave C#s would be awfully narrow?

Bill:

Bait or not, consider how "Modern Tuning Theory" says that fourths beat at the same speed throughout the piano, about 1 bps, just as non-iH theory says they should n the temperament octave.

Jeff --

I'm not at home right now, so I don't have my copy of W. Braid White or any other "old" source handy. However, I do remember being taught thirty years ago that fourths and fifths should both slowly increase in speed, including in the temperament octave. I remember the F3-C4 fifth was supposed to be 3 beats in 5 seconds, and the rate increased (I think) by 1/2 a beat every other fifth, as one progressed upward chromatically. I'm afraid I don't remember for sure now, but I think fourths progressed about the same, with the F3-A#3 fourth being 4 beats in 5 seconds. I think by the A3-D4 fourth was 1 beat per second. It's been awhile since I've thought like that ...

Also, I'm not sure to what degree the "1 bps fourths throughout the piano" idea is held with unanimity by "modern tuning" theorists. For example, some teachers and writers I'm aware of are more concerned with how fourths relate in speed to fifths, rather than what exact beat speed they're supposed to have. Also, I think the 1 bps idea is often used just as a general rule of thumb, rather than something to adhere to exactly (particularly if more weight is given to how RBIs progress).

David Anderson is an interesting case. In some ways you'd probably like the way he tunes. He uses a fourths and fifths temperament, checking with RBIs. But, he apparently doesn't bother with checking progressive RBIs anywhere else in the piano. However, he's damn picky about the unanimity in beat speeds of fourths and fifths. All fourths beat the same, but he actually prefers they beat a bit faster than 1 bps. Fifths are very nearly clean. (Not sure how rigidly he adheres to those guidelines in the bass.) In the class I took with him, he only had time to tune the temperament and the bass. I would've been very interested to hear how he tuned the treble.

Virgil Smith also likes the idea of fourths and fifths beating roughly the same throughout the piano, but I'm not sure how rigidly he adheres to the idea. He also uses RBI checks.

I think there are still prominent teachers and writers who want fourths to slowly increase in speed as one tunes out of the temperament (I think Bill B. is one).

Jeff S.

Beat rates increase geometrically as you go up the scale, no matter what the interval. They double every octave.

Contiguous fifths increase by 3:2, roughly. Contiguous fourths increase by 4:3.

Jeff S. and BDB:

Not too long ago I was involved in a thread on pianotech about whether the beat rate of fourths increases or not. David Anderson and other notables contributed. I crunched some numbers (including iH) and posted them that showed that with 4:2 octaves the beat rate does increase, doubling each octave at first, but then not doubling but still increasing. And doing the same thing with 2:1 octaves and 8:4 octaves, the beat rate of the fourths would change in how often they would double in an expected way. Someone else set a temperament on a piano, determined the octave ratio, used the twelfth root of the ratio, applied it to the partials, tuned the piano, and had fourths that stayed at the same beat speed. His octave type narrowed as he tuned upward.

But there was something peculiar about my numbers. I noticed that when calculating the 4:2 octave for two contiguous octaves, taking the 12th root of the 1st partial ratios of each octave, and raising the root to the 7th power to determine the 1st partial of the 5th in each octave; that the resulting 5ths did not produce a 4:2 octave!

Since then I have studied more. Earlier in this Topic I showed how the effects of iH on beat rates are largely self-correcting, especially in the temperament. Also, the 3:2 fifth is an odd case in that it can become wide in the high treble, as Mr. Scott demonstrated.

So, a piano can be tuned so that the fourths all beat the same and the fifths will increase in beat speed, but it would be best to start with a wide temperment octave because the octaves will have to be a narrower and narrower type. And a piano can be tuned so that the fourths increase in beat speed and the fifths will stay about the same beat speed, but it would be best to start with a narrow temperament octave because the octaves will have to be a wider and wider type. But, I believe, the decision to make is how busy do you want the octaves to sound for how high the treble is tuned.

I become very annoyed with Bill’s promotion of things that he prefers by falsely degrading things that he does not prefer. It reminds me of the New and Improved versus Old Brand X comparisons on TV. All I can do is challenge him (and hopefully challenge other people’s minds) into proving what he is saying is actually true for Old Brand X and not true for the New and Improved.

So the question remains: If the beat rates for non-iH theoretical tones are used for tuning, what is the error, especially in the first fourth and fifth that are tuned? Of course the next question will be: Is this error more than tuning the first set of CM3s?

You cannot show anything about inharmonicity that is independent of the piano being considered. Inharmonicity depends on the physics of the piano.

By the way, why is the difference between the 2:1 or 4:2 octave and the 8:4 octave supposed to be significant, while everyone ignores the difference between the 3:2 fifth and the 12:8 fifth?

Thank you for your point here, BDB, which is a point I made earlier. I would remind everyone that the subject of this post is about a clearly defined way to tune the CM3s which minimize any possible error when doing so. A discussion of other intervals belongs on a new thread if that is where you want to take it.

Quite right. You can even be more precise by saying that inharmonicity isn't something "a piano" has, but is something a string has, although Young did find that some generalizations are possible:

http://www.afn.org/~afn49304/youngnew.htm

edit: dang...how do I get the link to 'work'?

Fourth time's the charm...LOL

BDB:

When I did the math I used a mathematical model of a piano from Young's paper. iH at C3 is 0.1 which doubles every 8 semi-tones. Other scales produce different results above the temperament octave, but about the same beat speeds in the temperament.

Actually, I do consider the 6:4 fifth when tuning. The m3-M3 test is great for determining how to compromise across a break. And even when listening directly to a fifth, when iH is low the 6:4 beats pretty much twice the speed of the 3:2 and is not very noticeable. But when iH is higher, like the lower unwound notes on short and poorly scaled pianos, the difference is more obvious. Then it seems best to tune the 6:4 so it beats about 1 bps so that the m3s and M3s progress more in step with each other and not worry so much about the 3:2 fifth. This will also show up in better inside M3 outside M6 tests, and its m3 - M3 inversion.

Bill:

If you had stuck to the talking about CM3s, others might have also. You are the one that brought in other intervals as a comparison.

I, for one, would appreciate some specific examples of the math, like if you start from A 440, what is E if it is a 3:2 fifth or a 6:4 fifth in equal temperament.

I do not believe that you can use any assumption about inharmonicity across the break between the wound and plain strings. They are completely different animals.

BDB:

I should post the math tomorrow, but the A-E will not be either a 3:2 fifth or a 6:4 fifth unless we are tuning one of these types of beatless fifths. And with a more conservative stretch the nearly coincident partial matches will have different beat rates, of course.

I think there are some valid assumptions about iH across a break. On a poor break the iH slope is flatter or even negative above the break. Below the break the iH will have more slope and probably less iH. So, a poor break is characterized by a larger difference in octave types (and other interval types) in the lower unwound strings, and some intervals will be less progressive than others when crossing the break.

You should be able to figure out what the beat rate is as the difference between the tempered fifth and the perfect fifth using any or all of these ratios. If you start at A 440, you should be able to give frequency numbers for E figured as a perfect fifth, a perfect fifth plus inharmonicity assuming a 3:2 fifth or assuming a 6:4 fifth, and as a tempered fifth in those three ways.

Yes, I suppose I could do all that... and maybe you could too.

I am not sure what independent use the frequencies (of the first partials I assume) for these the notes of these different intervals would be.

I was planning on showing the math that calculates the 3:2 beat rate and the 6:4 beat rate when a 4:2 octave is used, with a constant first partial ratio. This would include many equal tempered frequencies, but no Pythagorean frequencies. My thought is of doing this for A3-E4, so I would include A3. An octave needs to be created, anyway. And a new Topic would be in order as this would be completely separate from CM3s.

But there would not be much sense in doing this if you are looking for something else.

Bill,

I am glad you are writing an article about CM3. If you remember me I used to be an ETD tuner, and thanks to your articles in your site and your posts here in PW I am now an aural tuner. The only use I make of my ETD now is to tune A 440 accurately and for pitch raises, but I fine tune the pianos aurally. I prefer what I tune now by ear than what I get with my ETD.

The problem with my ETD is that it has to be configured to match each different piano. And I must deal with stretching different types of intervalls along the scale and it is not a simple task to design those "styles" as they are called in Verituner terminology.

I was never able to tune Braid White type sequences, based on 5ths and 4ths because even if I could detect errors I always get lost when trying to correct them.

It was the CM3 approach which enabled me to tune by ear. I tune the temperament following a Sanderson-Baldassin sequence, using the CM3 approach, then I tune the rest of the piano by tuning "mindless octaves" to the top and to the bottom.

I seldom tune well scaled pianos and I find the CM3 approach ideal to tune those spinets and baby grands I am faced with all the time. Things just fall into place each time.

I studied, in the Randy Potter's course, several temperaments using the CM3, but it was until I read your explanations on how to tune F3-A3-C#4-F4-A4 that I really understood how to do it and was able to do it. And then I became free of my ETD which was an unflexible tyranus trying to tune each piano the same way, regardless of individual particularities of the piano in turn as temperament octaves with bridge breaks, wound strings, high iH solid strings, etc.

I think this article you are writing is of great importance because if one follows the instructions one really gets the results as expected. That is unique in the tuning litterature. I have read tones of articles, books, manuals, courses, etc. which tell you to tune a "7 BPS wide M3rd" or "3 beats in five seconds" or "7, 8 and 9 BPS" or a "slightly wide octave" or a "one nudge narrow fifth" and a "1 BPS wide 4th"

All of that stuff is meaningless for the begginer. It may have sense to an experienced tuner with a good number of pianos tuned. But for someone learning to tune you cannot simply ask him to "temper a fifth 2 cents narrow".

You are the only one who explains in detail how to set the temperament octave or tenth in a way a begginer can understand. You are the only one who explains how to tune and refine the tuning of F3 or the lower note in the CM3.

You are right, no need to count beats, the piano will tell if the estimated rate is correct or if it needs to be changed and it is the relationship between rates which is important not the rates themselves. And that is why the CM3 is better not only for begginers but for all of those seasoned tuners who resist to quit using old fashioned systems that have only a historical value.

One must evolve as technology developpes new ways of doing things!

Congratulations Bill, keep going!

Thank you Rafael,

Your command of the English language is to be congratulated as well. I am currently working with three students. One lives near enough to come to my home for sessions, the other two do it by telephone. It is amazing that a common speaker phone is adequate. I can hear the beats and coach the student. I will also have a dozen or so students at the PTG Convention next month. Many of my students from past years went on to pass the tuning exam successfully and commented that the concepts I taught them were the key to doing it.

The CM3 material I posted here is part of a longer article called, "ET via Marpurg". There is an article on my website which is the very same subject but I am rewriting the entire idea for better clarity for the PTG Journal. It is due to be published next Fall. One of my students has very good editing skills and is helping me to get the most clarity possible out of the writing. Just yesterday, he performed the basic sequence over the phone and without any further correction (which he is well able to do), the initial sequence scored a very respectable 90.

Time and again, the simple sequence which requires one estimate and one estimate only, proves itself to work. The initial estimate is either verified or corrected after the next two steps (which are not estimates). Thereafter, the rest of the sequence is completed using only 4ths & 5ths, none of which are estimates because they are all tuned beatless initially, then compared to a corresponding interval and then are made to beat equally. None of these steps are considered to be estimates because any person with only the most basic skills at using a tuning hammer can perform these steps to a tolerance well under the limit of what is needed to construct a temperament with accuracy.

It has been demonstrated over and over that a person with virtually no knowledge of how to construct an ET can do so very well using a method where only beatless and equal beating intervals are used (with the only exception being the initial estimate of the F3-A3 M3). A novice can hear when an interval is beatless or equal beating to a tolerance of under 0.5 cents. Therefore, the accuracy required is built in to the procedure instead of be required to solve a complicated maze of problems for which there may be no ultimate solution.

One of my students who is now an RPT first learned to tune using a 4ths & 5ths sequence from a popular correspondence course. When he joined PTG, his skills were not adequate to pass the tuning exam. However, upon learning to use the CM3s and the mindless octaves idea, he passed the exam easily. Today, he tunes so well that during my illness, I referred him to three very important jobs which I could not do. Two were in concert halls with Steinway concert grands and one at a fine restaurant with a Kawai grand. His work was praised and accepted as being as fine as any veteran technician could have provided even though he only has a few years experience and only tunes about 1/4 as many pianos as I usually do per week. He was also hired to tune multiple pianos at solo and ensemble events late last winter. He executed each tuning with remarkable precision, all completely by ear.

I could go on and on citing other similar success stories from people of have learned these simple and practical techniques. By contrast, I virtually never hear of anyone these days who is new to the field and has learned to tune aurally using a 4ths & 5ths method and gone on to be a skilled aural tuner and RPT. The lure and allure of the ETD becomes too strong. Learning what it takes, "counting beats", complicated checks where comparisons are still a judgment call and may not reveal or solve an error are all too much for many technicians today.

The survival of aural tuning skills depends upon material which is accessible, easy to learn and which yields superior results after a few trials rather than years of practice. I am pleased that you have also been a beneficiary of this effort to improve aural tuning education.

Just for fun I've made some numbers, with MS OFFICE EXCEL in my computer, to evaluate the accuracy of the CM3 in terms of what a not so much trained ear can hear in the beats when tuning the CM3rds.

When I say just for fun it's because I am convinced of the goodness of that system. I don't need these numbers. One only have to try to see how accurate and easy this system can be. But I guess these numbers will be interesting to some tuners in this forum.


The numbers are (without iH):
F3 = 174.61 Hz
A3 = 220.00 Hz
C#4 = 277.18 Hz
F4 = 349.23 Hz
A4 = 440.00 Hz

The beats are:
F3-A3 = 6.93 BPS
A3-C#4 =8.73 BPS
C#4-F4 = 11.00 BPS
F4-A4 = 13.86 BPS

The ratio between contiguous thirds are:
F3-A3/A3-C#4 = 4/5
A3-C#4/C#4-F4 = 4/5
C#4-F4/F4-A4 = 4/5


Now what happens if we sharppen C#4 by 1 cent?

The frequency becomes C#4 = 277.34 Hz

The beats become:
F3-A3 = 6.93 BPS
A3-C#4 = 9.37 BPS
C#4-F4 = 10.20 BPS
F4-A4 = 13.86 BPS

And the ratios become:
F3-A3/A3-C#4 = 3/4
A3-C#4/C#4-F4 = 1/1
C#4-F4/F4-A4 = 3/4

It is obviously wrong: we have equal beating between A3-C#4 and C#4-F4 and the other contiguous thirds are in a 3/4 ratio which is easyly identified as wrong.

We can also see that a detuning of more than 1 cent will invert the ratio for A3-C#4/C#4-F4, that is the lower third will beat faster than the upper 3rd.



What if we lower C#4 by 1 cent?

The frequency becomes C#4 277.02 Hz

The beats become:
F3-A3 = 6.93 BPS
A3-C#4 = 8.09 BPS
C#4-F4 = 11.80 BPS
F4-A4 = 13.86 BPS

And the ratios become:
F3-A3/A3-C#4 = 6/7
A3-C#4/C#4-F4 = 2/3
C#4-F4/F4-A4 = 6/7

The difference in speed between A3-C#4 and C#4-F4 is enormous, while the other contiguous thirds beat almost at the same rate. It is also easyly recognized as wrong.



If we detune C#4 by only 1/2 cent sharp we have:

C#4 = 277.26 Hz

The beats become:
F3-A3 = 6.93 BPS
A3-C#4 = 9.05 BPS
C#4-F4 = 10.60 BPS
F4-A4 = 13.86 BPS

And the ratios become:
F3-A3/A3-C#4 = 3/4
A3-C#4/C#4-F4 = 6/7
C#4-F4/F4-A4 = 3/4

The error is also easyly heard, the inner thirds are beating about the same rate (6/7), while the low and up extremes have a huge ratio of 3/4.



And last for C#4 detuned 1/2 cent flat we have

C#4 = 277.10 Hz

The beats become:
F3-A3 = 6.93 BPS
A3-C#4 = 8.41 BPS
C#4-F4 = 11.40 BPS
F4-A4 = 13.86 BPS

And the ratios become:
F3-A3/A3-C#4 = 5/6
A3-C#4/C#4-F4 = 3/4
C#4-F4/F4-A4 = 9/11

Here also the error is obvious. The difference between 5/6 and 3/4 is quite audible.



Let's resume:

C4 in tune => ratios of 4/5, 4/5, 4/5
C#4 sharp by 0.5 cents => ratios of 3/4, 6/7, 3/4
C#4 flat by 0.5 cents => ratios of 5/6, 3/4, 9/11

The smallest of these differences is found between 5/6 and 4/5. The difference between 3/4 and 6/7 is greater and easier to hear. So the tuner only has to be able to discern the difference between a 3/4 ratio compared to a 5/6 ratio. That is easy to achieve. Even more, there is no need to quantisize these ratios, one doesn´t have to know if it is a 3/4 or a 4/5 or a 5/6 ratio. One should only be able to identify an ordered smooth progression of the beats when going up or down the CM3rds.

In the same way if we detune F3 (and F4 accordingly) we'll come to similar results: the errors are obvious for a detuning of half a cent sharp or flat.



So, we can conclude that a tuner with little experience is able to identify errors of less than half a cent in the tuning of the CM3rds F3-A3-C#4-F4-A4.

And most important of all: he is able to correct those errors because he always knows:

1.- which note is in fault
2.- in which way: sharp or flat
3.- by how much: More than 1 cent if the ratios are inverted. Less than one cent if the ratios are not inverted but in a wrong progression.

I think this system is the Holly Grial for piano tuners! There is no other system so accurate and so easy done.

I'm currently learning this system and those five notes are referred to as the pivotal notes in a temperament from F3 to A4. The rest of your temperament is based off of those pivotal notes.

I can’t help but smile. The pivotal notes are considered to be Written in Stone, and then I read your signature line: “Change is the only thing that’s constant anymore...”

Thank you very much Rafael. I could not have done what you did, so I very much appreciate your effort. It reveals very clearly what I always knew by instinct. Any novice *can* hear equal beating and also *can* hear small differences and certainly can hear when differences are too large. They can also clearly hear when the bottom of two CM3s is faster than the top.

Jeff, STOP saying that they are "set in stone"! I never said that, nobody but you ever did. The CM3s are always subject to review and correction. After all, any pitch can drift a small amount after it has been set then need correction. Rafael makes the point very well, however that these pivotal pitches can easily be set to within a very small margin of error. If you continue to claim that you still don't believe they are accurate and therefore the idea has little or no merit, I will once again ignore whatever you say about it.

You can tune however you want and use whatever method you want. But I am teaching CM3s and that is that. I am teaching it to a student this very afternoon at 3 PM. I am NOT teaching Braide White! I am NOT teaching BW for all of the reasons I have previously stated. You may get your own students, write your own theories, articles, etc., do whatever you want but just drop trying to challenge the validity of this idea because you will get nowhere doing it.

Bill:

Sorry that you could not share in what was merely ironic humor.

You are not the only one to teach tuning with CM3s. Others do consider the first set of CM3s to be “written in stone”.

The following is from the “Aural Tuning” section of Appendix E of the SAT-III manual starting on page 44 and available at:

http://www.accu-tuner.com/pdfs/2009_SAT_III.pdf

”Step 6. Check that the three major tenths formed on the seven notes tuned so far also in the ratio of 4 to 5. Also check the C# and F octaves with both the third-tenth and minor-third-sixth tests. Scale problems will show up at this stage, and it may be necessary to compromise slightly the perfectly rising thirds to get satisfactory octaves and tenths.

Step 7. Fill in the six untuned notes between F3 and C#4 to get a nine-note mini-temperament, but be sure not to change already tuned notes. Tune up a fourth from F3 to A#3, down a third from A#3 to F#3, up a fourth from F#3 to B3 and stop. Then tune down a fourth from C#4 to G#3, up a third from G#3 to C4, down a fourth from C4 to G3 and stop. Check the G3-B3 third, which is the test interval for this tuning. If it is too small, you must expand your fourths, and vice versa. With just nine notes to worry about, it is always possible to get five perfectly rising thirds and four matched fourths no matter how poorly scaled the piano may be. The beat rates may not be very close to theoretical, but they will be right for the given piano and its inharmonicity
characteristics. So tune the piano, and let the beat rates fall where they may!”

In step 7 it says, “…but be sure not to change already tuned notes.” Not everything is about you, Bill.

Also I notice in step 6, “…it may be necessary to compromise slightly the perfectly rising thirds to get satisfactory octaves and tenths.” This is the first time I remember reading this in a CM3 sequence and is my main reason for preferring another sequence. But there seems to be a conflict with step 7 were it says: “…it is always possible to get five perfectly rising thirds and four matched fourths no matter how poorly scaled the piano may be.”

You act as if I am trying to get you to change your teaching method. That is not true. Why do you do this? What I do is point out that you do not show, other than anecdotally, that the problems that you point out in other sequences do not apply to yours.

Well that statement can sure cause some problems!

It is not unusual at all to have to go back and tweak a "pivotal tone". For me, the temperament sequence is only the first step in setting the temperament. The second step is the refining phase. This is where I go from my course to my fine sandpaper, so to speak.

This is also why some time ago I switched from using temperament strips to single mutes. I almost always tune unisons as I go. I get better stability and it allows me to use what Virgil Smith called the "Cracking the Unison" technique to finely adjust interval sizes.

Well, I didn't make the statement, just quoted it. But there is a difference between changing the pitch of a note because it drifted and changing it because after other notes were tuned, there is a reason to change it from what it was originally.

I’ve been tempted to try tuning unisons as I go, but usually when I am tuning in octave 5, I choose to make some small changes to octave 4 and even octave 3 for a better compromise between octaves, 12ths and double octaves. I’ve tried the “cracking the unisons” technique, but have not found it necessary or useful. If an interval needs to be changed, it is because there is a compromise necessary between two intervals. I want to hear both intervals (one at a time, of course) to get the compromise.

I really haven't much time for this today but I take the "be sure not to..." phrase to mean not to accidentally change any notes which have already been set, at this point. I agree with Ryan's first paragraph. With the method I am now teaching, all notes of the sequence and the rest of the midrange are moved into position without yet doing any checks at all. Once that is done, the very next step is to give all notes some firm test blows and then review and correct if necessary the initial CM3s.

If, at that point, the rest of the temperament octave seems nearly perfect, finding and correcting any small errors can be done. If the temperament seems generally rough, I recommend going through the "Up a 3rd, up a 3rd, down a 5th" sequence where any small errors become evident with each step.

Unfortunately, I find the material Jeff quoted from to be confusing, difficult to read and follow. That is why I developed my own system. There are no others which really satisfy me in every respect although people do find methods that work for them, so if that material has worked for some and they use it, no argument from me.

I do think the statement about poorly scaled pianos is important. It is the very same thing that I have said. There is no contradiction in it. The octave can be divided into 3 equal parts on any piano, regardless of scaling issues. From that point, all other intervals can be tempered equally. It is as simple as that. The final results will vary from poor to well scaled pianos, of course but a perfected ET is still possible.

From what I have understood Jeff to say, he feels that ET is not possible on poorly scaled pianos, so he favors one kind of interval over the other. That is precisely what the use of a cycle of 5ths sequence leads to. The result is inherently a quasi equal temperament and is really what all of the 19th Century and even some 18th Century theorists (like Marpurg) always ended up with.

While Braide-White's idea looks good in writing and theory, the most common result is less than perfect. That is what I have been working to change. I am not interested in changing anyone's way they tune if they are successful with it and are satisfied. I am interested in teaching people who cannot yet produce satisfactory results. Since I am committed to a certain method, I am working with that method. I've already done the other and have experienced first hand what the problems are, so I have looked for something which works better and which can be learned easily.

Bill:

I sincerely hope we can just “agree to disagree” on the idea of favoring one interval over another on poorly scaled pianos. I understand how you see that favoring SBIs will lead to a quasi-equal temperament (although I see this as really a localized well-temperament) because I see favoring M3s on poorly scaled pianos also leading to a quasi-equal temperament. I think of it as a “Marpurg error” where M3s are progressive but other intervals are not. The quote from the SAT III manual seems to address this problem.

Well, Jeff, I have told you before that if what you do makes the piano sound better in your opinion, then that is a good enough reason to do it. After all, I don't tune in ET at all and I also mostly use 4ths and 5ths. Out side of the temperament octave, I never use anything but 4ths & 5ths or their extensions.

Having said that, the student I had this afternoon scored a perfect 100 using my system: CM3s then only 4ths & 5ths, no checks of any kind. He didn't have to stay long after that because there was nothing more to correct. I recommended that he schedule a tuning exam later this summer. That is one more ETD user who has seen the light.

When he left, just for fun, I detuned the temperament octave again and ran through the BW sequence, estimating each tempered 4th or 5th as best I could but did not use any checks. I just went through the sequence the same as I would through the ET via Marpurg. My result: a score of -122.5!!! All of my 4ths & 5ths sounded just fine except when I got to the end of the sequence where the last one couldn't be reconciled.

Now, on a true PTG exam, the score wouldn't be negative, just a big Zero. I can tell you this, however, I have seen several exam results that looked just like mine did today in my 19 years of doing this work. I've seen examinees sweating bullets trying to reconcile the irreconcilable results they got from trying to use the BW system. Struggling to try to fix something here or there as the examiners walked in the door saying, "Your time is up", they all said, "If I only had a few more minutes!" They all gave up after that and went back to using an ETD, resigned to the fact that they could not learn to tune aurally.

It's all about the success vs. failure rate which I have personally witnessed. I have to go with what I believe in and what works for me. Everyone does.

Sir!

If you did not use any checks, YOU DID NOT FOLLOW THE BW SEQUENCE!!! You have once again misrepresented his method.

As you said, and I agree, stick with what works. I remember clearly the first time I tried the BW sequence. It was at my second tuning lesson and the sequence worked well for me, but not quite perfectly. That took two more tries.

I have much more difficulty with CM3 sequences. The best one I know is your ET via Marpurg, but I have to use additional checks, and also make the SBIs beat correctly unequal. (My standard may be higher than 100% on an entry level exam.) I still would not use it on a poorly scaled piano, but that’s just me.

Jeff, I admit that I did not do what BW says to do, I did what those trying to use what they have gathered from the method and haven't practiced enough do. I could surely use the BW sequence and end up with a perfect score, I did when I passed my qualifying exam in 1983. What I did last night was tune several intervals, estimating each and when there was a chack available, I clearly heard how wrong the interval was but I pretended I was someone who didn't know the difference and moved on.

Even if I had "backed up through it" (and was a person who didn't really understand it), I may have made it better but would have probably left 1 and 2 cent errors throughout. Just going ahead, I had 3, 4, 5 and 7 point errors. As Rafael showed, if you can perceive the small difference that a 4:5 ratio is, you can't even make a full 1 cent error with Contiguous M3s.

The ET via Marpurg sequence is theoretically not perfect if followed exactly as described. That is fully disclosed in my article. It is only intended to bring each note approximately to pitch but still, all will be within the 1 cent error tolerance of the exam.

My student's results after completing the sequence showed no points off, a score of 100. Yet, we could still hear slight irregularities and that is what is expected. We did correct them and he saw how small the changes were. I understand what you say about wanting to be so perfect all of the time but I can assure you that many concert technician's temperaments are routinely not as close to perfection as were my student's on just one run through of the sequence.

So, how do you measure success? I would say that you do it by learning something and accomplishing goals one at a time. He met the challenge and nailed it. I would call the session a success by any standard.

For the Marpurg sequence to work correctly, you do have to get the CM3's right. When you get them right, there could be no more than 0.25 to 0.5 error in any of those notes. So, they are not "set in stone" as such, they are merely very reliable which is always what I have said. It then depends upon tuning each 4th & 5th that is to be tuned initially pure as being really pure and not a bit narrow or wide. To do that, you would really need to use test intervals but if the piano is detuned or off pitch, all of the notes needed for that are too far off to be useful.

So, you can only depend upon the beatlessness of the interval and nothing more. With that, there is still very little possibility for error, most likely less than 0.5 cents. After the pure intervals, the Marpurg depends on getting the next set of 4ths and 5ths exactly equal beating. There is also a very small margin of error in that. Then, the intervals that were originally tuned as pure are again adjusted to equal beating, each having a very small possibility for error.

If you try to second guess the sequence and make 4ths & 5ths proportionately beat, that is where you would do yourself more harm than good and most likely end up with results that are more flawed than if you follow the sequence as written.

It seems to me that from what you have written so many times about the CM3s is that instead of following the steps as I have written them, you have immediately thought of a better way and done that only to prove to yourself that they "cannot be accurate" as you have so many times stated. You must not be getting the to pairs of octaves to be the same size and instead of correcting your initial estimate, you are fudging something else, otherwise, with as much practice and skill as you have, you would get them right each and every time.

If you did tune the CM3s first and got them right and then followed your preferred sequence after that, each time you tune an interval that could relate to F3, A3, C#4 or F4, you would have a far firmer basis upon which to correct any intervals formed by estimates. You could consider those notes to be more reliable than any others but not "set in stone".

The ET via Marpurg sequence, as stated many times before, requires one estimate and one estimate only. No other notes besides F3 are ever estimates. You seem to want to consider the whole thing as an estimate and that is up to you. But if you want to take it that far, yes, the whole world and everything we do is an estimate, nothing is ever perfect nor will anything ever be perfect. Multiply it all out to 100 places and prove it. Every frequency of ET is irrational, so none of them are ever right, they are all estimates.

Someone that did not know you better might think you are saying that the problem with the BW sequence is in not using it properly rather than a problem with the sequence itself. They should know better. That is my contention, not yours.

“Every frequency of ET is irrational, so none of them are ever right, they are all estimates.” I could not agree more!

But with the following, I have to respectively disagree:

”….. It then depends upon tuning each 4th & 5th that is to be tuned initially pure as being really pure and not a bit narrow or wide. To do that, you would really need to use test intervals but if the piano is detuned or off pitch, all of the notes needed for that are too far off to be useful.”

The M3-M6 test for the 4:3 fourth and the M6-M10 test for the 3:2 fifth, like the well known octave tests, do not require that the test notes be in tune, just reasonable. But for the fifth, I prefer the m3-M3 test for the 6:4 partial match. If there is much difference between the 3:2 and 6:4 partial matches, it is due to poor scaling. I believe it is then better to use the higher partials of the 6:4 fifth, anyway, for better compromises with the RBIs.

And I will admit that I have a hard time keeping the octaves even with CM3s, but it is a stability problem with how I tune. I need to hear that slight bit of change as the pin is flexed. I can hear this with SBIs, but not RBIs.

I agree. What I find on most small pianos is that as you approach the last plain wire strings in the tenor section, the thirds will get slower than normal (think Baldwin Hamilton! :o) . This is because the inharmonicity of the lower note is higher due to the lower tension, thick, short, plain wire strings. If you try to keep the thirds at their theoretical rates the octaves will often be unacceptable.

Once the lower note of the third switches from a plain string to a wrapped string, the inharmonicity goes down substantially, causing the 3rds to suddenly increase in speed. Once I understood this it made tuning consoles and spinets SO much easier.

This is the reason that the 5ths often sound somewhat out of tune, (even when they are not), in this same area of short scaled pianos. Even if the 3/2 coincidental partial is wide, the 6/4 may be annoyingly narrow.

For this reason I let the octaves be as wide as I can possibly stand it in this section to help the 5ths and 3rds sound more "normal".

Ryan:

YES!

I did not understand this until a couple of years ago when I read about it on this Forum. I have no fear of poorly scaled pianos anymore. For me the m3-M3 test solves the problem. The other thing that happens is the M6s slow down even more because of how many notes they span, unless they span a break in which case they beat much faster. Knowing this helps to make allowances when using the outside M6 – inside M3 test and its inversions.

What surprises me is that this is not taught more! What you say about the 6ths makes total sense. Once you can wrap your brain around this concept you can predict what may happen with beat speeds in these pianos. It makes it much more fun! I use to go nuts going around in circles trying to make those 3rds beat the "right" speed. Since beginners are mostly tuning these types of pianos, this concept will hopefully become more common knowledge.

Jeff, if the piano is detuned for an exam or a substantial pitch raise, the test notes are NOT in a convenient place, (as I have said over and over), therefore, if you want to use them, you have to adjust them too and it is simply not necessary and furthermore is counter productive. No audible beat in a 4th or 5th puts it within a narrow enough range to be workable. The system clearly works just as written. You always want to re-invent it to something "better" before you even try it and then fail and go on to say it doesn't work.

Ryan, a perfect ET can still be constructed across the kind of break you describe and it doesn't compromise any interval all that much. If F3 is a wound string and all strings are above it are plain wire, you can still construct the series of CM3s within that F3-F4 octave, construct the rest of the temperament within that octave and proceed below F3 without any compromise that would cause the temperament to be something other than ET. The M3-M6 tests will work too. All one has to do is follow the steps and not second guess them or re-invent the whole thing on the fly. But if you try to impose theoretical beat rates on everything (which is in any case, humanly impossible), it will fail every time.

I started learning how to tune on an acrosonic spinet. Lowest plain wires are on G3. I always tried to get close to 7bps on the F3-A3 third and set my CM3's. I took the Randy Potter course and CM3's were one technique that was advocated and it said that short scales may require a slower F3-A3 third due to "inharmonicity"- but didn't go into a lot of detail beyond that.

Anyway, the E3-G#3 M3 slowed way down no matter what! Every time! As if TIGHT tuning pins and a student hammer didn't make it hard enough!
It wasn't until I slowed the F3-A3 M3 to near 6bps that everything started to fall in line.

It was reading Bill's articles on his website that helped it click a little better for me, even though I was familiar with CM3's. It was still tricky at first but now has become virtually indespensible for me.

I haven't tried the Marpurg variations yet, so I can't speak to that but CM3's are a good tool. When the chain is "off" you can really hear it and it usually is fairly easy to hear the culprit.

I usually try to get 6 CM3's from A2-A4 as per the Baldessin-Sanderson method Gadzar mentioned. This goes part of the way towards dealing with scaling issues over breaks early on. It takes me a little longer but is usually worth it. Then I go onto a Potter F-A temperament which is basically up a 6th, down a 3rd, down a 3rd three times plus a few more steps to make a 10th(F3-A4) which is kind of backwards from Bill's up a 3rd, up a 3rd, down a 5th. And it can be tuned as a 3rds/6ths sequence or a 4ths/5ths sequence or a little of both and you don't have to change the order of the sequence.

There's another variation I've been experimenting with involving establishing the width of a 12th early on and balancing that with CM3's.

I have to say that learning the CM3's gets you REAL CLOSE. If anything, it's trying to get the first 2 octaves to match that may get tricky when that F3 is a wound string. But, once you get the "feel" or the "ear" as to the roughly 4:5 progression of CM3's the temperament usually falls right into place.

When I have a lot of trouble with a difficult (to tune) piano, because it's not well scaled, or it has dead strings, false beats or any other problem, I take out my Verituner and I tune the temperament with it. Technologie is a good thing!

But the point here is to find a sequence which guarants enough accuracy in a reasonable short amount of time. If I have to make 29 tests to see if an interval is correct that’s fine with my search of perfection but it’s not good for business.

If I can get a good set of CM3rds without tests and then an acceptable temperament with only a few tests: That is proficiency! If after that I have enough time to refine my tuning, then there are tones of tests which can be used to do that.

The important is to be able to produce aurally a good temperament in a short amount of time in a reliable way.

Thank you very much, CJ. Once the F3-F4 octave has been tuned, if one wants to establish another octave below that, tuning F2 to F3, then filling in the A2 and C#3 notes really works the same as it does with the octave above it, even on such pianos as an Acrosonic. Tuning octaves from A2 to A3 and from C#3 to C#4 gets those notes close, then it is a matter of listening to whether there is that small, slower/faster relationship and making slight adjustments if necessary but still maintaining "pure" sounding octaves in all of them.

Then, filling in the notes in between in similar fashion, octaves first, then checking 4ths & 5ths, then as more final and finer checks, the CM3s and minor thirds, the scale can be perfectly equal and still maintain all proportions. The method for doing it really does have an effect on the outcome. The method will work on all pianos.

That method begins with the F3-F4 octave CM3s and using the F3-F4 octave as a unit, then placing C#4. If it is determined that the chain of CM3s does not quite work out, no matter to how small of a degree, the initial F3-A3 M3 is adjusted but because F3 has moved, so must F4 by the same amount. The F3-F4 octave must serve as a unit in itself and therefore does not involve any estimate whatsoever. Then C#4 may or may not need to be moved as well.

I'm afraid that none of the CM3 schemes I have seen written up explain the technique in quite the same way. It is more like, "Play around with these notes until everything seems balanced". That may work but then again, it may not. That is the critical flaw I have seen with all other CM3 schemes.

Both in a roughing in procedure and in the finest review and correction, the series of 4 CM3s from F3-F4 MUST be used (not just 3). In the finest review and correction, it must be determined that both the F3-F4 and A3-A4 octaves are of the same type, size or width (all three of these terms are synonymous).
When you have smoothly ascending and descending CM3s spread over a chain of 4 intervals (again, very importantly, 4 not just 3) AND you have determined that both the F3-F4 and A3-A4 octaves are of the same type, those five notes are as close to ultimate accuracy as is humanly possible.

The same can be replicated from F2 to F3 on any piano that exists, regardless of scaling. If this seems to contradict what any other description or method involving CM3s says, if only slightly, then it is a reason to validate the subject of this thread, a "New Look at Tuning CM3s".

There really is something new about it (sorry BDB). Yes, the basic principal was known in the early 20th Century, I read that again in Owen Jorgensen's "Tuning" book yesterday and noted I had put a paper clip on that chapter but had forgotten about it. It is still true to say however, that it was not brought to light until the early 1980's by Dr. Sanderson. I recall hearing Rick Baldassin talk about it and the "move the notes around" business back then too.

Every so often, I review older material I have known for many years to see if there is anything more I can pull from it. First, I found something with the CM3s I could apply a specific technique to. Then, I took a look at the "Marpurg" temperament and asked myself, "What if I applied the same equal beating technique that is used to tune the G3, B3 and D#4 of that temperament to the notes which were tuned as pure 4ths & 5ths? Would they then be moved to the correct position for a true ET? The preliminary results I got aurally were that it worked.

A theoretical analysis by Owen Jorgensen revealed that it was still slightly imperfect but within a very small margin. He told me that he had never heard of such an idea before but that it was very clever indeed. Slightly imperfect not withstanding, that was still good enough for me to put forth as a valid technique.

Trials with many, many people who otherwise did not have the skill to tune an acceptable temperament revealed that what they could produce on their own with no coaching, yielded results that are acceptable by PTG's standards, often within the superior range and yesterday for the first time with no error at all according to PTG's standards and by using PTG's methods for scoring.

Show me or tell me who, where and when ever did that using the Braide White sequence! Who ever tuned a temperament octave where and when, using no checks whatsoever, that yielded results that close to absolute perfection?

Thanks again, Rafael, you wrote your post as I was writing mine but you seem to very well understand my point and purpose.

Bill:

If a piano is so far off that the test notes for fourths and fifths are unusable (You do not mention how far off this is, so neither will I) then the test notes for octave types will be too far off, also. But, I don’t remember reading in any of your instructions about tuning these test notes so that the correct octaves can be tuned.

And I suppose that anyone that has a tuning sequence that works well for themselves can say to someone that it does not work well for: “The system clearly works just as written. You always want to re-invent it to something "better" before you even try it and then fail and go on to say it doesn't work.” Your statement is outrageous.

Jeff, for the UMPTEENTH time, the ET via Marpurg is a ROUGH tuning sequence that includes tuning the CM3s as the first few steps. This thread is about tuning those CM3s and any test notes for the two pairs of octaves will be too far off to be useful.

If the piano is detuned for a PTG tuning exam, the test notes you would want to use when FINE tuning are ROUGHLY 4 cents off, either sharp or flat at random. If you are using the sequence for a pitch raise, the test notes you would be using during FINE tuning are however far off the whole piano is, like maybe a 1/2 step. You wouldn't use such test notes when you change the pitch of a piano even 10 cents. I wouldn't use them if I were changing the pitch even 2 cents.

An octave is an octave until you get to the point where you decide to define the octave as a certain type or a compromise of certain types. For those who are struggling to learn aural tuning, the difference between those types may be far too much to try to manage. Therefore, I VERY CLEARLY SPELL OUT (if you ever bothered to actually read what I wrote, you would not have written what you did) that the octave is to be tuned beatless as it is approached from the wide side. You make the octave obviously wide, then you narrow it just to the point where it stops beating.

Most novices can understand that and do it. Throwing all kinds of fancy checks and extremely small distinctions between 2:1, 4:2 and 6:3 octaves at an aural tuning novice loses them and is also completely USELESS when performing a pitch raise or when moving notes into a workable position on a PTG exam detuned piano. If you tune the octave the WAY I DESCRIBE, the most likely result will be a nicely workable 4:2 octave. The difference there may be between the F3-F4 and A3-A4 octave is also likely to be insignificant when using the technique the WAY I DESCRIBED IT, for the purposes I described it, as a positive and very nearly perfect step in the right direction, not the way you want to do it in your ultimate quest for the holy grail of perfect ET.

Now to address your second paragraph. I very clearly remember you writing, more than once but I am not going to bother to look up either post, that you said you had figured out a way, nearly instantly, of taking the ET via Marpurg sequence, AS I DESCRIBED IT, (the results being EXPECTED to be slightly imperfect as they most often are)and created your OWN version of it that was perfect.

But since the ET via Marpurg sequence begins with and DEPENDS upon the CM3s which you CONSISTENTLY claim can NOT EVER be accurate (as much as 2 cents off per note by your own mathematical wizardry), then that must mean that your "perfected" version of what I wrote and which I teach and will continue to teach, CAN NOT EVER work either.

You never posted your "perfected" version of the ET via Marpurg. You only claimed that it would yield perfect results where as what I labored long and hard at does not.

Therefore, what I want to see is YOUR version of the ET via Marpurg which I suppose does not use CM3s since they can never be accurate and which does not use any checks whatsoever of any intervals and no estimates of any kind. The miracle system. Anyone can do it. You thought it up right then and there, so let us all be the beneficiaries of your wisdom. We all want to see the way you would tune a detuned piano or a piano needing a pitch raise by tuning each note exactly and precisely right the first time with no corrections of anything at all being necessary.

I want to see those octave types and 4th & 5th checks all factored in too. Remember, I say they aren't necessary and can't even be used during a pitch raise or for roughing in a detuned piano. So, you must have figured out a way to do everything with no checks and no estimates whatsoever. So, let's have it. What is this marvelous method?

Bill:

I am interested in discussing, not arguing. Your gauntlet will not be picked up.

In http://www.pianoworld.com/forum/ubbthreads.php/topics/623061/1.html I posted:

”But first, I noticed that the ET via Marpurg sequence has the ability to refine the first CM3 other than by using intervals with beat rates halfway between. When the just 4ths and 5ths are created on one note, they also create chromatic m3s or M6s with the other 2 notes. The other two notes can then be adjusted so that the chromatic m3s or M6s beat at the correct 15:16 ratio, the same as any other chromatic intervals. After retuning the just 4ths and 5ths, and checking the octave type with the other notes, the same refining can be made to each M3 in the CM3. Of course, if the CM3 is very accurate, these are just checks and not refinements.”

The basic idea is that listening to chromatic RBIs can create a more accurate temperament than listening to RBIs 4 semitones apart. Your ET via Marpurg sequence creates chromatic additional RBIs when the SBIs are tuned, so why not listen to them to check the original set of CM3s? Even though these chromatic RBIs are not properly tempered, they will all be the same width (if the M3s are all the same width) and the beat speed progression will prove this.

For triumpteenth time, Tooner, you do all the checks after all the notes are filled in. Each of those notes will be very close if not within tolerance and they will not be subject to cumulative error as you suggestion definitely will. An aural tuning novice attempting the tuning exam may have used up all available time at that point. If he/she has been struggling with a myriad of checks the whole way through, the sequence may not even be complete by that time. If it is a pitch raise, is the precision you insist upon with each step really necessary? Won't the results be skewed even as you progress through the sequence?

I asked Owen Jorgensen about your "15:16 ratio". He said it was only approximate, not really all that accurate. So, if someone who is not really accustomed to all of the RBI checks that you insist upon fumbles with them, 15:16 being a significantly smaller difference than 4:5, how much more accurate would those results really be?

It boils down to this: why not do it the easy way, the way which anyone can hear, control and get to within an amazing degree of accuracy first, then use any known way of refinement rather than doing it the hard way and lousing the whole thing up?

We seem to be talking about two different things. I am talking about the accuracy of CM3s. You are talking about passing an exam. But if the 15:16 ratio is only approximate, then the 4:5 ratio is only a guess.



I know you assume that I must not have tried this, or I would be a convert. This shows you have a conceited opinion of your methods.

I have tried many, many things, including your methods, and feel that I have wasted a great deal of time. Here is the problem I encounter by starting with a set of CM3s, roughing a temperament in and then polishing it up. Sooner or later the question comes up whether the F and/or the C# (assuming that A is used as a pitch source) are really where they should be.

For the way my mind works, I need a direct path back to the pitch source, not two possible paths, just one. I have found situations where one could be high, the other could be low, or both, and no way to know for sure.

It is like drawing a twelve pointed star by making a rough triangle and then later, after sketching in the other nine points, trying to make it more accurate but not knowing if the first triangle really is a equilateral triangle or not.

It is too bad that you do not see how highly I value your ET via Marpurg idea. It makes it possible to tune a very accurate set of CM3s. If I dealt with mostly well scaled pianos, I would use a variation of the sequence.

Tooner,

Excuse my irruption in your discusion with Bill, but I think I have something important to say in what you are talking about.

I remember once you have said me to tune fifths using m3-M3 test looking for a 8/7 ratio. The m3 should beat 8 times in the same time the M3 beats 7 times.

My question is: If you are able to accurately adjust such a small difference in the beating of m3s and M3s, why can't you adjust an accurate 5/4 ratio between two contiguous M3s? At first one can conclude that's easier than the more thin ratio 8/7!

Gadzar:

I am glad you jumped in for a number of reasons.

I do not take note of how long it takes an interval to beat a certain number of times, and then compare it to how many beats another interval produces in the same amount of time. What I do is notice how close the beat rates are to each other, and compare that to what I know the “closeness” should be. The smaller the closeness, the more accurate I can hear the difference. As an analogy, I could draw a freehand mark on a piece of paper one inch from the edge more accurately than one 5 inches from the edge.

But even if a set of CM3 were tuned to an exact beat ratio of 4:5 (which I think Bill says not to do), this would cause an error. The beat rates would not quite double in an octave, and the octave would be narrow.

But the main idea is that the beat rates of chromatic intervals will show errors that intervals many semi-tones apart will not. And in the chromatic m3 and M6 tests that are available in Bill’s ET via Marpurg, although the ratio should be 15:16, this does not need to be listened for. If one chromatic set of m3s or M6s beats, say 7:8, another will beat closer to 1:1. When all three sets have a small, audible, and equal progression, the set of CM3s is very accurate.

Another reason I am glad you jumped in is that it gives me an opportunity to correct something I said a while ago about the m3-M3 test for a 6:4 fifth. There hasn’t been much interest in this test, so I did not see any reason to mention it before. I had erroneously thought that regardless of the position of the test note, that the ratio would be 8:7. Actually, like all other tests, the difference in beat rate of the test intervals is the beat rate of the 6:4 fifth. The m3 could be 2 bps and the M3 1 bps, with a ratio of 2:1, but if the correct beat speed of the 6:4 fifth is 1 bps, then the fifth is correct. However, once when the beat rate of the fifth is correct (by whatever method), then the test note can be correctly placed with a beat rate ratio of 8:7. If someone knows what the progression of RBIs that are two semitones apart sounds like, this can be useful. This is all very useful to me, especially when tuning challenging pianos.

Don't waste your time any more, Tooner, it is obvious you have your own opinion which you have had before this thread ever started. That opinion is that the CM3s aren't accurate but the Braide White sequence is. You won't read the material and do what it says, you make up your own version of it which you did again in your last post and then proclaim it doesn't work. You have done this over and over again.

There are other popular sequences which use the CM3s. One has been written up recently in the PTG Journal. Both mine and that one have similarities. Both will be videotaped at the upcoming PTG Convention which you are not attending. The classes for people wishing to take the PTG tuning exam also teach the CM3s. What will not be taught and not be taped and has not been written about in the PTG Journal is your favorite obsolete idea that is 100 years old. CM3s are taught in PTG's study material for passing the tuning exam as they are at the North Bennett Street school, Chicago School, Randy Potter and other schools.

So, now that you have chosen to insult me by calling my work and endeavors "conceited", I will not respond to you on this subject any further. I have already tried to do that but you protested saying I was cutting off the discussion right in the middle of it. There was no middle, there was no beginning. But this is the end. You still don't believe the CM3s can be accurate no matter how many times and ways anyone tries to explain it to you.

So, just keep tuning the way you do. Start your own thread about that. I surely won't argue any point you make. I know you have done it before but it went nowhere, so try again. See if you get anyone to agree with you. Write your own article. Submit it to the PTG Journal (they do take articles from non-members if they consider them to have merit).

You always insist on having the last word, so go ahead and have it but as you do, consider than nothing whatsoever you have said about either the CM3s or the ET via Marpurg sequence has convinced me or persuaded me to change anything about what I teach. If any of my students brings up the Braide White sequence, I will say what I have said here about it and not consider any of your arguments. In my mind, none of them have ever had any merit whatsoever.

I know that, excuse my english, it is difficult to me to express my ideas clearly.



When tuning CM3s you first tune A3-A4 and F3-F4 octaves the width you exactly want them to be, then you divide these two octaves into CM3s, so the ratio is not exactly 5:4 but very close. And the octaves are the width you have decided, they won't be narrow. There are no cumulative errors (as in Braid-White sequence).



Obviously, it is the same with all test notes, the test note can be way off tune and the test still works, in fact you can have 2:1, 3:2, 4:3 which are really easy to hear and check accurately, with a little sense of rythm. And you can have 5:4, 6:5, 7:6, 8:7... ...16:15, which are more difficult to estimate accurately.

But I have one question: Why the difference must be 1 BPS? Can't it be 1.1 BPS, or 0.9 BPS, or whatever value near 1 BPS is demanded by the iH of that specific piano a that specific point in the scale?

As I said before, the nice thing about CM3s sequence is that first you set your octave width, acordingly to the specific iH present in that piano at that point and then you divide the octave into M3s! Well, not exactly that, in fact you divide a M10th into 4 M3s: F3-A4,A4-C#4, C#4-F4, F4-A4. Thus the ratio 4:5 is an approximate value which must be refined accordingly to iH, by adjusting your initial estimate of F3 up or down to achieve a smooth progression of beat rates of M3s up to A4.

This is far more accurate than assuming 1 BPS for a 4th, which is an arbitrary value pulled out from thin air! (Where have I listened that phrase?).



Challenging pianos: it is all about them!

If you are tuning a well scaled piano you seldom hear a difference between a 6:3 octave and a 4:2 octave in the temperament area, so the temperament is easy to tune. And all of the tuning falls into place without much troubles. I am convinced the CM3s sequence is best suited to tune bad scaled pianos with a lot of iH, i.e. spinets and baby grands, where you find jumps in iH from one note to the next in the temperament area.

I will stop posting to this Topic out of respect to the Original Poster.

"Obviously, it is the same with all test notes, the test note can be way off tune and the test still works"

This is only true to a certain point. Indeed, in my article, "Midrange Piano Tuning", I suggest more than once that the tuner adjust the test note until the interval beats very rapidly, at or near the limit of discernibility. The idea is that the more rapidly the two intervals beat, the better one can hear any slight difference. If the intervals beat slowly such as one or two beats per second, one may not be able to detect any difference there may be.

Having said that, some of my students have said that if they tune the test note so that it beats as rapidly as I suggest, they become distracted by the "sour" sound. So, they prefer to have the test note to be very approximately where it should be in a fine tuning. Certainly, any tests which involve minor thirds cannot be very far off.

In my fine tuning sequence, I suggest an octave size that is a compromise between a 4:2 and a 6:3 octave. For the A3-A4 octave, that involves the minor third A3-C4. If C4 is already where it should be, the A3-C4 m3 will beat at or nearly the limit of discernibilty, so I suggest temporarily sharpening C4 very slightly so that the A3-C4 m3 can be better heard.

If one is performing a pitch raise of just 10 cents, all test notes will be unusable unless they are also adjusted with each step. During pitch raises, the pitch is expected to drift as the tuning is accomplished. This makes the use of test notes which in fine tuning, are used mostly to confirm that an interval is correct more than to form it, unnecessary and counterproductive during a pitch raise. The greater the change of pitch, the more useless and counterproductive they become. Indeed, a finely tuned interval during a large pitch raise creates more error than an interval that is purposefully mistuned to anticipate a drop or rise in pitch as the case may be.

Regarding an a tuning exam detuned piano, each pitch is detuned alternately flat and sharp by approximately 4 cents. The examiner makes no effort to precisely detune the piano. So, some pitches may be a little more than 4 cents flat or sharp. Therefore, any test notes are likely to be too far off to be useful as they are. Any test interval will either beat far too rapidly or far too slowly to be useful unless it is also adjusted as the sequence progresses.

A test note must also be tempered in the right direction. A M3, M10 or M17 which is narrow, for example is not valid. Although this is not likely on an exam detuned piano, it could be the case but if the task is to lower the pitch of a piano, it may likely be what is encountered.

This would make tuning a 4ths and 5ths sequence extremely cumbersome, especially for an aural tuning novice. I suspect that those examinees I have observed whose results had errors so large as to fall far below a passing score, even to the point of a negative score (that is marked as a zero) had this problem.

Imagine trying to tune a series of 4ths and 5ths and trying to use a test note for each that is so far off that it too had to be adjusted by a fairly large amount with each step! If the person is barely familiar with the test note process, this would be extremely cumbersome, confusing and defeating. If there were a way of simply tuning each pitch to a very nearly precise point first until all 13 notes of the temperament octave is completed, then with all pitches very nearly correct, the test notes would then be far more useful.

However, as I demonstrated to myself last week, using a 4ths and 5ths sequence alone with no checks can produce disastrous results. An aural tuning novice could end up fighting with the entire arrangement the entire time and lose the battle in a very sorry way. When I would work with such individuals after the fact, it became clear to me that the problem was never once that the tuner could not perceive the errors, they just could not control them. As I say in my article, the essence of aural tuning is the perception and control of beats. Aural tuning novices can always perceive beats but they often do not know what to do to control them.

I have also determined that aural tuning novices cannot manage such fine distinctions as a compromise between a 4:2 and a 6:3 octave. Since a 4:2 octave will work just as well for tuning exam purposes, I teach novices to tune octaves as beatless, approached from the wide side. Using that technique, a test interval will generally confirm that the octave is the 4:2 type.

Therefore, when roughing in the temperament octave on a detuned piano, a test interval is not helpful but if the tuner progresses to the point where a fine tuning of the temperament octave can be done, the test for the 4:2 octave is easy enough to learn and execute. Even an aural tuning novice can confirm that both the F3-F4 and A3-A4 octaves are of the same size and therefore be quite certain the the CM3s from F3-A4 are as precisely tuned as is humanly possible.

At that point, the rest of the temperament octave can be checked and refined. I never once had a student who could not at that point, distinguish the very smallest of errors and also correct them. They really feel a sense of accomplishment when for the first time in their career, they have produced a temperament octave that is perfect to within PTG's tolerances and have done so entirely by ear.

I never use any such tests for 4ths or 5ths. Those tests were originally meant to confirm that a 4th or 5th is beatless when tuning Historical Temperaments which require beatless intervals. When both test intervals beat equally, the 4th or 5th is confirmed as beatless. Their use in tuning ET is most effective when trying to prove an error rather than to confirm an interval is correct. If a 5th is slightly wide or a 4th slightly narrow, for example, it may sound acceptable when played alone but the test note will reveal that it is tempered in the wrong direction. Other interval tests will confirm a very obvious error in such a case.

In testing for equally tempered intervals up and down the temperament octave in the fine tuning process, it is not necessary to use 3:2 or 6:4 tests for every 5th. If any 5th is not tempered enough or too much, unevenness will be quite apparent in other intervals. Such a test can be useful in correcting an isolated error but really not necessary. Causing the 5th in question to conform to all other related intervals is sufficient. The test for the 5th would merely be redundant.

Trying to construct a temperament by using test intervals for 4ths and 5ths each step of the way would not only be extremely cumbersome but would not guarantee accuracy. Since the test must prove two slightly different beat rates rather than equality, it becomes a judgment call in each case. A ratio of 8:7 for example (if that is really what it is) is very difficult to judge as correct. Even if it is judged as correct, there is no other control over that judgment.

The series of 4:5 CM3s on the other hand, has 4 controlling intervals, each being the judge of its partner and all 4 being the final judge. Using a test for 4ths and 5ths in each step only makes each interval sound correct with itself and not necessarily correct with any related intervals or in the perspective of the entire temperament octave. Therefore, in my opinion, tests for 4ths and 5ths in temperament construction have very little value at all. They serve at best to confirm a large error rather than a small one.

I shudder to think of the struggle I would impose on a student and myself if I were to try to persuade them to do this using a classic 4ths and 5ths sequence! Most of them would just get lost in the whole maze and give up trying to learn to tune aurally. Using a method that produces consistently superior results however, encourages them to go ahead and take that test. They are permitted to use their ETD in Part 2 of the exam to tune the outer octaves and to ensure that the midrange is stable enough to pass the stability test.

I do not feel in any of these cases that I have taught merely the techniques required to pass the exam and that really, the technician has not truly learned aural tuning concepts. What I have done instead is to link very likely for the first time for these people the perception (which they already have) and the ability to control beats. They have then found the confidence needed to judge whether their ETD tunings are the best they can be. They often go on to do custom programming of the ETD which they were not confident in doing up to that point. Many of them go on to serve on Master Tuning Committees and attempt a qualifying exam to train as examiners which must be done entirely by ear.

I used to think that dropping the aural tuning requirement for RPT would be acceptable as many PTG Associates have desired and even demanded. I no longer think that would be acceptable because I have seen many times over what benefits learning aural tuning skills has provided for people who previously did not possess them. This can also be true for those who for whatever reason, choose not to join PTG or take the tuning exam. Any ETD user can and should learn the basic aural tuning skills it takes to tune the midrange accurately. Once that is accomplished, outer octave tuning can be mindlessly simple and easy.

They can also make sense out of the value there may be in tuning non-equal temperaments in the future. They can use 4th and 5th test intervals effectively to confirm beatless intervals when they are required. They will be able to understand the difference between ET and the non-ET and the purpose for using a non-ET. Using correction figures with an ETD becomes more than just inserting some irrational looking figures and producing crazy sounding results. Those results will have a meaning and a purpose behind them when the tuner possesses aural tuning judgment.

Aural tuning judgment is the goal of aural tuning instruction and education. With judgment, a technician can understand why some tunings sound better than others and not just wonder why.

Come on UnrightTooner, do you leave the discusion when things become difficult to you?

You are right Bill, the test note can and thus should be adjusted where it produces the optimum beat rate audible to the tuner. What I was trying to explain is that the test note can be moved and the test still works. Sorry about my extrapolations.

And for the sake of that "respect" you have now launched another thread called "Errors" where you intend to highlight the supposed erros in what Bill states?

I think the best I can do is ignore all of your posts from now on. Sorry.

Rafael, that is what I prefer Tooner to do. I will not challenge him on whatever he comes up with. I'll leave that to anyone else who wants to do that or who wants to agree with him. I am firmly resolved and satisfied in my approach to tuning ET as an idea that is practical and which works for the people I am trying to help to learn to tune ET.

I have mucked around a bit in my spare time here experimenting with Bill's suggested sequence (on a challenging piano no doubt) and would have to say it was near impossible to not let my biases interfere. Still, I could see it being useful as an alternative or even complimentary method of getting a decent temperament for those who are new to it or struggling with other methods. I say this not because it worked any better for me than what I have learned and used for many years, but rather, it did not reveal any large hidden surprises or errors that left me in a lurch.
Tuning temperaments are like finding your way out of a desert. Having a good sequence is much like having a map. Fixed theoretical beat rates on the other hand are analogous to a mirage in a desert. Knowing that its just a mirage does not change the fact that it still looks like real water; it will always be the logical place to go to a new traveler if everything else still looks like desert. The problem lies with the assumption that the water/mirage will lead you in a straight line out of the desert...which in most cases does not.
I'm sure that the best people to assess the value of using this method are the very people in need of it, neophytes. Those who have learned to do it with other methods like myself have the added problem of divorcing ourselves of our biases and in a sense "unlearn" and subsequently devalue what we had ultimately conquered. With human pride being what it is, its not an easy path to take. It would be a worthwhile experiment for a tuning school to separate its students into 2 groups with one using this method exclusively, just to see if does better than the other group.

Thank you Emmery for your statements. Your last suggestion is quite good. I have always said that if people are successful with another method which is habitual, then that is a better approach for that individual. So, I am quite pleased that you gave it a try. I am sure that the equal beating approach was unfamiliar to you but I think you can imagine that for an aural tuning novice, it is no more unfamiliar than any other method.

I do get the comment quite often, "These 4ths are equal beating now but they beat faster than I am used to hearing." I cover that in my written material but I am sure that comment can be read and forgotten easily. The fact is that at the point when they are heard that way, those 4ths are still 2 cents wider from where they will end up at the end of the process, so they naturally will sound too fast at that point. It is important therefore to have strictly pure intervals where they are directed and strictly equal beating where that is directed. The more closely those directions are adhered to, the better the end result will be.

This process is specifically designed to tune a tuning exam detuned piano and NOT for fine tuning. It follows that it can be useful for a pitch change. Once all 13 notes of the temperament have been initially tuned and error is noted, it would not be productive to start the ET via Marpurg sequence all over again. That would actually involve detuning notes which are already where they need to be or very nearly so.

It is better at that point to follow some other known sequence step by step and whenever a note is found to be slightly incorrect, to correct it at that point and use whatever checks are known and available at that time. If the result is very nearly perfect and only slightly flawed, diagnostic checks that prove an error can be used to make small corrections.

I've tried both (being a hobby mathematican, theory teacher, pianoplayer and former chess player that was the only solution that could get me to sleep at night :-D

I started with the counting of beats in F3-A3-C#4-F4-A4, just to find out they very seldom lined up as they should, except if I got into "ultrastrech mode" and adjusted the F4 very wide. On the contrary, when in similar situations the F3 is rised to get a beat interval of 6bps, even 5bps, things fell into place. I learned to like bills idea of not counting, just comparing the contigious 3rd sentence! The feel for the slight acceleration is musical in some way, and hey - I come from the land of Räikkönen and Häkkinen, so i know what gears ratio is :))

Regards,
Pati