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.