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Folks,

After a long wait, I have received correspondence from Owen Jorgensen about my idea of converting the so-called "Marpurg" Quasi Equal Temperament to Equal Temperament. With only slight disappointment, the idea apparently works.

The slight disappointment is in the fact that the results, if followed literally, result in a very nearly perfect ET with all notes either perfectly where they would be in ET or within one cent to as close as 0.02 cents.

This means that if the directions are followed, there are only 2 M3s to tune and all the rest are 4ths and 5ths, all tuned as pure first, then adjusted from that point to beat exactly the same as another 4th or 5th until all notes from F3-F4 are within a tolerance that would score a perfect 100 on the PTG Tuning Exam.

Furthermore, my website is up and running again with my latest articles posted in PDF format, down-loadable for anyone who can make use of the information provided. Thanks goes to Ricardo Carlos of Chihuahua, Mexico for his generous help.

The new site may be accessed at: www.billbremmer.com

In order to avoid making this post too long, I will put a summary of the ET via Marpurg sequence in the following post. It will include the deviations from true ET which were calculated by Professor Owen Jorgensen RPT. Also available on the website are my often requested articles on Midrange Piano Tuning, Octave Types and the EBVT. There are graphs by Jason Kanter which provide valuable insights to the properties of each kind of temperament idea, including ET.

Here is the summary page from my newest article. The full, detailed instructions and comments may be viewed at: www.billbremmer.com

Summary Sequence for ET via Marpurg

1. Tune A4 to A-440 pitch.
2. Tune the contiguous M3s F3-A3, A3-C#4, C#4-F4 and F4-A4 exactly the same as in Standard Equal Temperament.
3. Temporarily Tune A#3 from F3, a beatless 4th.
4. Temporarily Tune C4 from F3, a beatless 5th
5. Temporarily Tune D4 from A3, a beatless 4th.
6. Temporarily Tune E4 from A3, a beatless 5th.
7. Temporarily Tune F#3 from C#4, a beatless 5th
8. Temporarily Tune G#3 for C#4, a beatless 4th.
9. Temporarily Tune G3 from C4, a beatless 4th, then compare it to the G3-D4 5th, notice the strong beat and flatten G3 until both the G3-C4 4th and G3-D4 5th beat exactly the same.
10. Temporarily Tune B3 from F#3, a beatless 4th, then compare it to the B3-E4 4th, notice the strong beat and sharpen B3 until both the F#3-B3 4th and the B3-E4 4th beat exactly the same.
11. Temporarily Tune D#4 from A#3, a beatless 4th, then compare it to the G#3-D#4 5th, notice the strong beat and sharpen D#4 until both the G#3-D#4 5th and A#3-D#4 4th beat exactly the same.
12. You now have the so-called “Marpurg” Quasi Equal Temperament.
13. To effect a nearly perfect Equal Temperament, continue as follows:
14. Notice the beatless F3-A#3 4th and compare it to the strongly beating A#3-D#4 4th. Sharpen A#3 until both the F3-A#3 4th and the A#3-D#4 4th beat exactly the same.
15. Notice the F3-C4 beatless 5th and compare it to the strongly beating G3-C4 4th. Flatten C4 until both the F3-C4 5th and the G3-C4 4th beat exactly the same.
16. Notice the A3-D4 beatless 4th and compare it to the strongly beating G3-D4 5th. Sharpen D4 until both the A3-D4 4th and the G3-D4 5th beat exactly the same.
17. Notice the A3-A4 beatless 5th and compare it to the strongly beating B3-E4 4th. Flatten E4 until both the A3-E4 5th and the B3-E4 4th beat exactly the same.
18. Notice the F#3-C#4 beatless 5th and compare it to the strongly beating F#3-B3 4th. Sharpen F#3 until both the F3-C#4 5th and the F#3-B3 4th beat exactly the same.
19. Notice the G#3-C#4 beatless 4th and compare it to the strongly beating G#3-D#4 5th. Flatten G#3 until both the G#3-C#4 4th and the G#3-D#4 5th beat exactly the same.
20. The results will be a very nearly perfect Equal Temperament!

Below are the theoretical results of following the above directions literally:

Note Cents
C 0.05
C# 0.00
D 0.16
D# 0.78
E 0.05
F 0.00
F# 0.60
G 0.56
G# 0.05
A 0.00
A# 0.16
B 0.56

Numerical calculations provided by Professor Owen Jorgensen RPT

Bill, looks like I get to have the first comment!

You said Professor Jorgensen thought it was clever, and I agree.

If an experienced tuner makes the fourths a little faster than the fifths in the first round of tempered 4ths and 5ths, that wouldn’t cause greater errors later, or would it? Also, the first step is critical. If the contiguous thirds aren’t dead on, those errors continue. But it seems that any M3 errors would be averaged out somewhat as the sequence progresses. Is this true?

Thanks for posting, I had been looking forward to it.

Thanks for your interest, Upright. Actually I had you in mind the whole time. One of my main interests is helping people learn to tune better whether they are PTG members or not.

I know you have stated many times that you use the Braide-White sequence because it is familiar to you. You are not alone. I started with that sequence too but I found that no matter what I did, the result was uneven M3s and M6s and unresolvable 4ths and 5ths.

The problem is always how much to temper. In order to have it work out correctly, you have to temper each 4th and 5th exactly right and by exactly the same amount each time. Unfortunately when using that sequence, you don't have immediate feedback at each step which tells you how much error there may be in each note tuned.

While there are some very skilled and experienced tuners who have always used that sequence and seem to know just how much to temper each time, all the rest must use very complex ways of checking and cross checking to correct the results. In the end, whether you have tuned contiguous M3s initially or not, for the temperament to be correct, they must have that 4:5 relationship.

Therefore, it makes sense to do them first. They serve to exactly and precisely divide the octave into three equal parts. When you tune a 4th and 5th from each of these notes, you avoid making a compound or cummulative error because you do not tune a 4th or 5th from a previously note tuned as a 4th or 5th which may have been tuned erroneously.

Other popular sequences such as the Baldissin-Sanderson idea also begin with contiguous M3s. However, that sequence has you tune two full octaves from A2-A4 first. For anyone trying to pass Part 1 of the PTG Tuning Exam, this forces the tuner to cross unnecessarily into the Bass which is outside of the Midrange. For smaller pianos, it also forces you to cross outside of the range you are concentrating on, into an area with different inharmonicity and which can be left for last rather than first. Also, that literature does not tell you how to self correct using the F3-F4 octave the way mine does.

Upright, I suggest that you and anyone else who finds tuning the contiguous M3s correctly a difficult process to follow the step by step instructions which are included in the detailed instructions on my website. Don't make the mistake of skipping tuning A4 first. There are ONLY two M3s to tune: F3-A3 and A3-C#4. The other two intervals are octaves which are easy. Once you have the skill devoloped, it will seem as easy to do as the rest.

I suggest keeping it simple at first. Make your octaves sound "pure" (beatless). If this is already your preference, fine. I also suggest that you try the sequence literally and make every interval beat as equally as possible. If you start right off trying to make 4ths beat faster than 5ths, you risk introducing error which cannot be resolved easily.

It would be a better idea to use the equal beating method and listen afterwards to hear how smooth the temperament is. At that point, use your skills remedy any errors you can actually hear. You may use the chart provided on the website for a clue to which notes may be slightly sharp or flat. The note D#4 will probably be the greatest error at close to -0.8 cents. Try sharpening that note very slightly first. Only after having followed the directions literally the first few to several times would I recommend trying to finesse the idea by tuning proportionately beating 4ths and 5ths.

Bill, Thanks for thinking of me. We all should work to improve our tuning. I am interested in different temperments sequences because I like puzzles.

I am tempted to start a companion Topic to “Tuning – an Art?” I would call it “Tuning – a Philosophy?” There probably would not be much interest. My approach, or philosophy to tuning is that any method should work well for challenging pianos, and then it will work even better on quality pianos. Yours seems to be that any method should work for a novice trying to pass the PTG exam, the present objective standard of tuning. I am not trying to be judgmental here, just pointing out differences in our thinking.

I do not believe that contiguous M3, nor any other intervals, can be tuned accurately enough, on the first pass of a temperment sequence, to have the greater than 1/8 cent accuracy needed to have progressively faster beating 4ths and 5ths. This is especially true when the temperment crosses into wound strings, and when the scale should have wound strings in the temperment but doesn’t. In these cases, compromises must be made that result in the M3rds having a ratio different than 4:5. (Actually, I don't think it is possible to have all intervals beat progressively faster on a challenging piano.) Like you claim is a fault of the Braide White sequence, it can’t be known how much to temper the M3s at the beginning of the sequence, either.

I believe the 4th and 5th intervals have the advantage over M3rds of having a check without tuning any other notes. I’ve learned more about this since we last posted to each other. The check can even be used to determine if there is much of a difference in inharomicity between the strings being tuned. Consider the E3, A3 4th. If you tune it to a just 4:3 4th and then compare the beating of the E3, C4 interval to the A3, C4 you can get an idea of the difference in inharmonicity of the strings, and how much compromise will be needed. If the beat rates are the same, or nearly so, little compromising of SBI and RBI will be necessary. The more practical way is to tune the 4th so it sounds good and the check gives close to a 7:8 ratio in beat speed.

I believe the Braide White temperment is great, but not for everybody. Likewise the contiguous 3rds temperments are not for everybody. While the Braide White sequence is fine, the explanation and checks could be improved.

Now back to your ET via Marpurg sequence. Something I find interesting is that after the contiguous M3rds are tuned, the next permanently tuned note creates a diminished 5th in the temperment. This is a required interval for the M3-M6 check. I believe the M3-M6 test is essential to prevent an “egg-shaped” temperment: one where the intervals beat progressively faster, but not evenly. The sooner this check can be made, the fewer fine adjustments will be required in additional passes through the temperment. Of course, the disadvantage to this sequence is the number of temporarily tuned notes.

Thanks again for your work and consideration. I sure hope other members will post on this too.

[QUOTE]Originally posted by UprightTooner:
Bill, Thanks for thinking of me. We all should work to improve our tuning. I am interested in different temperments sequences because I like puzzles.

Upright, the many sequences there are indeed are like puzzles and although you say you are interested in the various "puzzles", the only one you seem to embrace is the very most difficult one to solve. The fact is that you can create an ET starting on any note using any conceivable sequence if you know what you are doing.

"I am tempted to start a companion Topic to “Tuning – an Art?” I would call it “Tuning – a Philosophy?”..." There probably would not be much interest. My approach, or philosophy..."

Actually, I believe that ET is a misguided goal. It doesn't make any piano sound the best that it can. That is why I never use it and haven't for nearly 20 years. But as a PTG member interested in helping Associates upgrade to RPT, I am interested in helping them achieve that goal. I also am interested in helping those who believe in ET to actually be able to tune it correctly.

Correct or true ET means: all 4ths and 5ths bearing the same amount of tempering (not some pure or nearly so and others beating noticeably) and all contiguous M3s having a 4:5 ratio (slightly slower/slightly faster). All ascending and descending M3s and M6s result in a very smooth progression or digression. M3-M6 and "inside M3 and outside M6" tests are quasi equal beating.

"I do not believe that contiguous M3, nor any other intervals, can be tuned accurately enough, on the first pass of a temperment sequence, to have the greater than 1/8 cent accuracy needed..."..."I believe the 4th and 5th intervals have the advantage over M3rds of having a check without tuning any other notes..."... "...close to a 7:8 ratio in beat speed."

A change of 1 beat per second can be heard in a M3 with less than 0.2 cents change in pitch of either one of the notes which comprise that interval. It takes at least 10 times that much, 2.0 cents to make a difference of 1 beat per second in a 4th or 5th. Having a correct ratio of 4:5 M3s from F3-A4 is easily done and is the very most accurate way to create a framework for an ET.

That is why I teach it as do many other methods, including the well-known, Baldassin-Sanderson method. I don't know of any method at all which uses a 7:8 ratio of anything but even if there are, this would be far more difficult to perceive and control than the 4:5 ratio of contiguous M3s.

"I believe the Braide White temperment is great, but not for everybody. Likewise the contiguous 3rds temperments are not for everybody. While the Braide White sequence is fine, the explanation and checks could be improved."

Once again, only the most highly skilled tuners can tune it accurately and when it is tuned accurately, the contiguous M3's result in a 4:5 ratio, even on "poorly scaled" instruments. I can do it on such instruments and I show my students how it can be done, so your reasoning that it is not possible is not true.

"Now back to your ET via Marpurg sequence. Something I find interesting is that after the contiguous M3rds are tuned, the next permanently tuned note creates a diminished 5th in the temperment. This is a required interval for the M3-M6 check..."

What???

"I believe the M3-M6 test is essential..."

I never use it because it only yields quasi equal beating intervals. It is not essential.

"The sooner this check can be made, the fewer fine adjustments will be required in additional passes through the temperment."

Using the Braide-White sequence, you must tune 5 notes, all unverifiable before you have a single M3 and a single M3-M6 test. If either seems incorrect, you have no way of discovering where the error is since it can be in any one or all of the previously tuned notes.

"Of course, the disadvantage to [ET via Marpurg] sequence is the number of temporarily tuned notes.

Not so. Each step takes you closer without the need for any Rapidly Beating Interval checks. Those may be used when the sequence is complete. If you don't like the idea of "temporarily tuning" anything, you can use the "Marpurg Shortcut" which is included in my article called "Midrange Piano Tuning".

If you prefer to guess at how much a 4th and 5th should be tempered, that is the one to use. The possibility of compounded and cumulative errors is completely eliminated. In the Braide-White sequence, that kind of error is a near certainty.

"Thanks again for your work and consideration. I sure hope other members will post on this too."

You're welcome and I hope so too.

Bill: Thanks for the reply. I really enjoy these discussions with you.

"Correct or true ET means: all 4ths and 5ths bearing the same amount of tempering (not some pure or nearly so and others beating noticeably) and all contiguous M3s having a 4:5 ratio (slightly slower/slightly faster). All ascending and descending M3s and M6s result in a very smooth progression or digression. M3-M6 and "inside M3 and outside M6" tests are quasi equal beating."

"A change of 1 beat per second can be heard in a M3 with less than 0.2 cents change in pitch of either one of the notes which comprise that interval. It takes at least 10 times that much, 2.0 cents to make a difference of 1 beat per second in a 4th or 5th. Having a correct ratio of 4:5 M3s from F3-A4 is easily done and is the very most accurate way to create a framework for an ET."

These two statments are like comparing apples and oranges. Which M3rd you are talking about that changes 1bps with a change of pitch of 0.2 cents? The important thing is whether the M3rd still fits between the adjacent ones. It takes a change of 7/8 cent to one note to match the beat speed of the adjacent M3rd. Likewise with a 4th or 5th. Which 4th or 5th are you talking about? It only takes 1/8 of a cent to one note to match the beat speed of an adjacent 4th or 5th. A change of 2.0 cents would match the beat speed of the interval an octave away.

"That is why I teach it as do many other methods, including the well-known, Baldassin-Sanderson method. I don't know of any method at all which uses a 7:8 ratio of anything but even if there are, this would be far more difficult to perceive and control than the 4:5 ratio of contiguous M3s."

The 7:8 ratio is the change of beat speed between two intervals a whole step apart. The Braid White (sorry about the previous misspelling) sequence uses the 7:8 ratio as the check between the first two RBIs that are tuned. I know you have heard of that method. Now you have heard of another. I find it easier to percieve and control the 7:8 ratio than the 4:5 ratio. It’s a personal preference.

“Once again, only the most highly skilled tuners can tune it accurately and when it is tuned accurately, the contiguous M3's result in a 4:5 ratio, even on "poorly scaled" instruments. I can do it on such instruments and I show my students how it can be done, so your reasoning that it is not possible is not true.”

This is anecdotal. My anecdotes are different. I didn’t say it wasn’t possible. I just believe that it results in compromising more important intervals on challenging pianos.

" "Now back to your ET via Marpurg sequence. Something I find interesting is that after the contiguous M3rds are tuned, the next permanently tuned note creates a diminished 5th in the temperment. This is a required interval for the M3-M6 check..."

What??? "

The M3-M6 is formed from a Dominant 7th chord, which contains a diminished 5th. In your sequence, F A and C# are tuned first. The next permanently tuned note is G. G-C# is a diminished 5th. The Braid White sequence does not produce a diminished 5th until the 6th note. I can’t think of a sequence that produces a tuned diminished 5th quicker. I think it’s interesting.

“I never use it because it only yields quasi equal beating intervals. It is not essential.”

I agree that the M3-M6 check is not between equal beating intervals. If they are equal beating, I expect an error of 1/3 cent. That does not mean that it is not useful. If the tuner can hear that small amount of difference, it is a great cross check. If they cannot, it is an error of less than ½ cent.

“Using the Braide-White sequence, you must tune 5 notes, all unverifiable before you have a single M3 and a single M3-M6 test. If either seems incorrect, you have no way of discovering where the error is since it can be in any one or all of the previously tuned notes.”

Yes, 5 notes for a M3, but only 4 notes for another RBI, the M6th. Also the M6th is composed of notes 1 step away from the starting point and 2 steps away. Actually, you don’t get to use the M3-M6 check until the 7th note. Or maybe you are referring to the test with the 7:8 ratio?

As described in Braid White’s book it may seem difficult to discover errors. By using the check I mentioned, it is easy find an error if you can hear the 7:8 ratio. There is also the 2:3 ratio and 3:4 ratio for checking 4ths and 5ths.

“ "Of course, the disadvantage to [ET via Marpurg] sequence is the number of temporarily tuned notes.

Not so. Each step takes you closer without the need for any Rapidly Beating Interval checks. Those may be used when the sequence is complete. If you don't like the idea of "temporarily tuning" anything, you can use the "Marpurg Shortcut" which is included in my article called "Midrange Piano Tuning". “

It is still temporarily tuned notes.


Thanks again Bill. I hope you are enjoying this as much as I am.

Bill,

Thanks for taking the time to post your latest project. I don't have time right now to really delve in and comment as much as I'd like, but you did mention one thing I'd like to ask you about:

Do you have reasons for personally not favoring a full two-octave system like the Baldassin-Sanderson? You mentioned bass inharmonicity being different, and I suppose that's why I haven't gone to a full two-octave system. Although, in addition to using a system that spans from F3 to A4, I've started to experiment a bit with including C#3 in the initial series of contiguous thirds. It just seems to me that flexibility when going into the bass is a good thing, and I wonder if two-octave systems somewhat sacrifice that. Maybe if I tuned more concert grands than verticals I'd think otherwise. I can see how more plain wire strings might make one think differently.

As far as the Braid White sequence, I still occasionally trot out the similar one I used for years. It can be fun for a change, like for a quick store tuning or something similar. I've learned though, that for consistency and accuracy from one piano to the next, it just can't be trusted as much as my newer system, a sequence very similar to the A3-A4 contiguous thirds-based system Jim Coleman came up with. The main difference is that my system also includes notes down to F3.

Upright, I think there are some things you still don't appreciate about a contiguous thirds-based system. Your arguments sometimes seem skewed. For example, you say the initial series of thirds lacks checks, but in fact the series itself is largely self-correcting, provided one starts with a good initial A3-A4 octave. In addition it provides, in the beginning of the temperament sequence, an accurate reading of the piano's inharmonicity over the whole temperament area, something progressing through a Braid White-style sequence can't do. However, it's not my intention to argue or convince you to switch. I do suspect, though, that you haven't given a contiguous thirds-based system sufficient trial time. It's not as easy as starting up with the Braid White system. All contiguous thirds-based systems aren't the same, either. Bill's previously-publicized system for example (not the one above, which I haven't really considered yet), like any of the better ones, uses fourths and fifths right along with thirds and sixths, after the initial series of thirds. So it includes information from both slow and fast-beating intervals. There's also the objective standard of how many test intervals each tuned note has, and how quickly they become available.

Hopefully I'll have more time this week to look over this thread.

That's all for now,

Jeff

Actually, I should make a disclosure:

After tuning A3 to A4 I tune D4 before tuning a series of thirds. Then I tune F3 and F4, then C#4. That gives me my series of thirds. (Unless I want to also throw in C#3 at this point). Next I tune A#3, then F#3 and F#4. By that point I typically have enough information from both slow-beating and rapid-beating intervals to know how things are going to go, and also to have validated my series of thirds. It's rare, but sometimes within these first notes I'll find something that causes me to change one or more of the series of contiguous thirds.

I'm not saying my way is necessary in order to validate the series of thirds early, or that it's the only way; but it does seem to help me.

Jeff

Jeff: Thanks for joining in. Having a third poster might help Bill and I from butting heads too much.

It’s obvious that you have studied temperment sequences. I understand how contiguous M3rd are self-correcting. I just don't believe that they (or any other intervals) can be tuned accurately for a really good temperment on the first pass. I believe that any really good temperment will require further adjustment after a first pass. So, the argument that the tempering of 4ths and 5ths can't be verified until later in the sequence is also true for contiguous M3rds.

Could you explain what you mean by:

"In addition it provides, in the beginning of the temperament sequence, an accurate reading of the piano's inharmonicity over the whole temperament area... "

I have been able to use contiguous M3rd to set temperments, and I see the advantages. There is one disadvantage that I cannot deal with. It's based on M3rds! I lean toward 4ths and 5ths being right when I make compromises. If I start with M3rds, I end up having to change them anyway on challenging pianos. It seems I make compromises differently than others, so a different temperment sequence would be expected. I'd rather start with 4ths and 5ths and then compromise the M3rds. If I tuned mostly well-scaled pianos, I probably would switch to a contiguous M3rd sequence.

Not knowing if the 4ths and 5ths are tempered correctly at the beginning, is a problem with the Braid White sequence. But by using the additional checks I mentioned, it is a much smaller problem.

I'm really glad you joined in, Jeff.

Jeff, tuning the D4 early on in the sequence is not a bad idea at all. If you have the A3-A4 octave set, you can find the point where the D4 makes a pure 4th and 5th between A3-A4 and by sharpening it slightly, you will temper both intervals. It can still be a question of how much tempering will be correct but it is also easy to hear when the 4th and 5th beat proportionately and correctly. Therefore, it is easy to place the D4 with a fairly high amount of reliability. The same would be true of the note E4.

If you then proceed to construct the contiguous M3s from F3-A4, you can then fit the A#3 and C4 between F3-F4 in the same manner. That would, at that point, provide a very nice framework of 8 notes, all very conveniently spaced: F3-A3-A#3-C4-C#4,D4,E4 and F4. At that point, you will have two adjacent sets of M3's A3-C#4/A#3-D4 and C4-E4/C#4-F4.

The M3s created by the initial contiguous M3s are generally more reliable but Upright is correct in saying that nothing is absolute. PTG Tuning Exam Master Tuning Committees can spend a half hour or more on the contiguous M3s alone. I have always said that one can never be absolutely sure when they are correct but it is very obvious when they are to any appreciable degree incorrect. In other words, when they seem correct to you, you will probably be within an acceptable tolerance, when they seem incorrect, you will probably be outside that tolerance.

Therefore, the contiguous M3s at this point can serve to correct any error there may have been in tempering the A#3, C4, D4 and E4. Of course, the whole arrangement can point out a slight flaw in the contiguous M3s as well. Bear in mind that any note once tuned can drift slightly from where it was initially tuned at any time. Master Tuning Committees often make corrections of less than 0.5 cents in this sensitive area. They are quite difficult to make and take hold but they can be and are aurally detectable. Any and all types of aural checks are used between one committee member or another.

So, after determining that this fine framework is correct, the rest of the notes can hardly be placed with significant error. None of the notes tuned in this particular sequence could have been subject to any kind of compound or cumulative error. So, continuing, you can tune F#3 from A#3 (tuning a 4th is considered more reliable than tuning a 5th because there is only one set of audible coincident partials) ((thanks, Dr. Coleman for this observation)) and compare it to the C#4 as a 5th, then compare adjacent M3s F3-A3/F#3-A#3 and contiguous M3s F#3-A#3/A#3-D4. That is 3 checks for one note!

Then, you can tune B3 as a 4th from F#3 and compare it again as a 4th from E4. In this case, the two 4ths should be apparently equal beating. (Although 4ths theoretically increase very slightly in speed, for all practical purposes, these two 4ths beat very similarly). The G3-B3 M3 can then be compared with the two adjacent M3s below it, F3-A3/F#3-A#3. There are also two adjacent m3s, F#3-A3/G3-A#3.

G#3 can then be tuned as a 4th from C#4. This will yield adjacent M3s and m3s to all the way to D#4 which would be the last note to be tuned at which point all intervals can be tested.

This would be another very reliable way to construct a temperament since it strictly avoids the compounding of errors.

To answer the question about the Baldassin-Sanderson idea: I am not well versed in it and have actually avoided memorizing it, wanting to pursue my own thought process, not wishing to copy someone else's idea and call it my own. Jim Coleman, Sr. RPT has been a long time mentor of mine. I learned much from him starting back in 1979. He continually rethinks the whole temperament process and has produced a number of ideas, all good ones. I don't know if he has done anything the same or similar to what I wrote about above but what is there is just another possibility in my mind.

I spoke to him last Summer and asked him about what advantage a two octave temperament from A2-A4 may have. His answer involved the transition of differing inharmonicity between the low tenor and the high bass. Typically on any piano but especially on smaller pianos, the inharmonicity rises the lower one descends in the plain wire, then abruptly becomes very low in the higher wound strings, including those which may be on the tenor bridge.

For the tutoring sessions we both gave at the PTG Convention in Kansas City in 2006, we were both provided a small, Yamaha Grand with two unisons of plain wire in the low tenor. These would not have met the specifications for a PTG Tuning Exam piano but we both agreed that they would indeed be even more useful because we could show the students how to deal with this kind of transition.

At that same event, I was involved with a Tuning Exam Master Tuning and I did project down to A2 across the break and constructed contiguous M3s from A2 to F3 after those from F3 to A4 had been determined. The D3 and E3 could be placed between A2 and A3 as above and the C#3 could be tested against both the F#3 and G#3 above it.

This is all perfectly acceptable and works just fine but I still don't believe it is necessary. The primary purpose of my teaching in the last five years has been to help those PTG Associates who normally use Electronic Tuning Devices to learn to tune the Midrange well enough to pass Part 1 of the PTG Tuning Exam. This part must be done aurally but an ETD may be used in Part 2.

Only the notes C3-B4 are involved in Part 1, the Temperament and Midrange, according to the PTG Tuning Exam. Therefore, tuning anything lower than C3 is unnecessary and even counter productive for Part 1 whether the Examinee tunes Part 2 aurally or not.

What both Jim and I showed our students was that to have 4ths, 5ths M3s, m3s and M6s and also contiguous M3s and m3s (contiguous m3s beat at a 5:6 ratio), the octaves must be stretched wider as one descends from the F3-F4 octave through all of the plain wire strings. This does NOT mean creating an obvious beat in the octaves. It means just a wider type but still acceptable octave. For example, if the octave between F3-F4/A3-A4 is a 2:1 or 4:2 type, the octaves in the low tenor among the plain wire strings simply needs to get a little wider, perhaps as much as a 6:3 type.

If that is not done, the 5ths and m3s will beat faster and the 4ths and M3s will beat slower. If a wider octave in the temperament octave (F3-F4/A3-A4) is used, such as a compromise between a 4:2 and 6:3 or as much as a 6:3 (or beyond that for the so-called ET with "pure" 5ths) the compromise necessary in the lower plain wire is simply less.

When the wound strings are reached, the amount of stretch necessary in the octave becomes abruptly less in order to keep all intervals in line. It could even require a very slightly narrow octave if the initial octaves chosen in the temperament region were of the very conservative, 2:1 type.

The ratios of contiguous M3s and m3s still remain constant although their beat speed becomes markedly lower than in the F3-F4 octave. When descending into the Bass section, this process is simply continued. In my opinion, there is no particular advantage to having tuned A2 prematurely. I can even see the possibility of creating a dilemma if the A2-A3 octave were have been made too wide. Contiguous M3s could still fit properly but 4ths would beat too rapidly and 5ths too slowly and there would be a conflict in the transition from plain to wound strings.

One final comment about the contiguous M3s being self correcting. This is inherently true. But the key to it being self correcting is that BOTH sets of octaves, the A3-A4 AND the F3-F4 must be exactly as possible the same size or type. An ET (and contiguous M3s)can be constructed within any size octave from slightly narrow to wide enough to create the ET with pure 5ths idea and anywhere in between.

But one cannot hope to have a correct succession of contiguous M3s when the F3-F4 and A3-A4 octaves are of substantially different sizes. That is where I think most people make a mistake if one is made. I have even seen a PTG Tuning Exam Master Tuning with that kind of error.

If you are sure that both octaves are of the same size (type) and the contiguous M3s do not progress properly, it means that BOTH F3 and F4 must be changed to correct the arrangement. After that, if C#4 can be placed so the arrangement is correct, the correction has been valid. If it still doesn't work out, one will find that upon each attempt at making it work will bring one closer to the goal.

It will usually take the tuner only one or two corrections to get an acceptable arrangement. Each correction will involve a smaller and smaller change. Of course, you might imagine that the process could be infinite and indeed it can seem so, particularly for a Master Tuning Committee. But it is at that point when the corrections being made are less than 0.5 cents. When one is at the point of less than one cent error, the arrangement is already at a superior level.

Upright: "I'd rather start with 4ths and 5ths and then compromise the M3rds."

Let me first say that I have no doubt that you produce acceptable, if not superior quality tunings. We are really splitting hairs in a lot of what we have discussed. But one of the non-equal temperaments I use fits your above description! It is the 1/9 Comma Meantone. It can only be done with an ETD which is set up to tune each 5th 2.4 cents narrow, that is, a mere 0.4 cents narrower than in ET. It is such a slight amount that the 5ths do not seem any narrower than one might produce by simply tuning ET with a very conservative octave. My SAT III however, sets the temperament octave at a compromise between a 4:2 and a 6:3 by default.

The result is really the same as if one had erroneously tuned each 5th slightly narrow. Each "error" would compound the previous. Since all of the 5ths are slightly too narrow to be ET, there comes a point where one of the 5ths cannot be reconciled. In a meantone temperament, that place is generally left between G# and D#. That is commonly called the "wolf" 5th because in any of the stronger meantones, it beats harshly as a wide 5th and sounds out of tune. In the classic 1/4 Comma Meantone, it is 45 cents wide!

But the 1/9 Comma Meantone is very mild and it just so happens that the wide 5th is the same amount wide as all of the other 5ths are narrow. Therefore, if you play all of the 4ths and 5ths, they will sound perfect, virtually the same as if the temperament were in ET. However, if you play the M3s and M6s, they will be mildly but quite perceptively uneven, at least to a piano technician. However, they do correspond to the Cycle of 5ths and it gives the temperament a very mild and pleasant key coloration.

The point of this parable is that the effect of compound and cumulative errors which often happens when one Slowly Beating Interval is tuned from another without a good way to verify and correct either one, the result is uneven and irresolvable Rapidly Beating Intervals.

I have seen in nearly every one of your postings that you have probably encountered this dilemma. A true ET *must* have both SBIs and RBIs reconcile and that includes contiguous M3s. No kind of interval can be favored over another.

You are not the only person who does not like to start with contiguous M3s. One person I tutored told me, "I don't like to pull that F3-A3 M3 out of thin air". To me, it seems that one is pulling it out of far thinner air to only arrive at it after having tuned five consecutive notes, none of which can be checked against anything else before creating that one, lonely F3-A3 M3 which also cannot be checked against anything else.

You would have the F3-D4 M6, yes but it is formed from an unverified 4th made from A3-D4. The F3 is obtained by then tuning G3 from D4 (unverifiable and a difficult interval to tune correctly at that since a 5th in this range has two sets of audible coincident partials), then C4 from G3 and finally another difficult to place 5th, F3 from C4. That gives you a chain of 3 intervals, all of which may contain a slight error.

You have absolutely no way of knowing at this point whether the F3-A3 M3 is correct or not. If it *seems* too fast or slow, you can guess at correcting it but all it will be is a guess. You can then back up through the C4 and G4 but if your D4 was too wide or narrow, you will resolve a dilemma between those notes but will encounter more later on.

Perhaps then you tune F4 from F3. Do you make any attempt to insure that it the same size and type of octave as the A3-A4? If you then proceed to tune A#3 from F3, you have again, one lonely M3, A#3-D4. You still have no adjacent or contiguous M3s to listen to, compare or make fine adjustments to. You probably tuned the A3-E4 5th first (there's that unreliable 5th again with its two sets of audible coincident partials), so you would have a C4-E4 M3. So, you have two M3s a whole step apart. If either one sounds wrong, to which note can you refer for a reliable check? You have the G3-E4 M6 but do you move G3, C4 or E4? What will be the consequence of that?

So, onward you press, tune D#4 from A#3. luckily this time it is a 4th! Now you finally have 2 adjacent M3s: B3-D#4/C4-E4 but still not one set of contiguous M3s! Now you tune G#3 from D#4 (another troublesome 5th)but you finally have one lonely set of contiguous M3s: G#3-C4/C4-E4.

If the upper M3 isn't slightly faster than the lower M3, the relationship isn't correct. If the two are exactly the same, the relationship is incorrect but only slightly so. If the bottom M3 is faster than the top M3, the relationship is very incorrect and if the top M3 beats obviously much faster than a 4:5 ratio could ever be rationalized to be, the relationship is also very incorrect. If the relationship is found to be incorrect, which note or notes do you change and what will be the consequences of that?

You do have at this point, two more adjacent M3s, G3-B3 and G#3-C4. But once again, if they don't progress smoothly, it would be very difficult to locate the error. Even if they do, they are isolated. You have the F3-A3 but then there is a whole step gap to the G3-B3 and then another whole step gap to the A#3-D4. It would be very difficult at this point to verify any continuity. The two adjacent M3s could sound good together but they still could be faster or slower than they need to be when the entire temperament octave is completed. You are still a step away from ever being able to hear the critical contiguous M3s, F3-A3, A3-C#4, C#4-F4 and F4-A4.

I'll grant you this, you haven't tuned any M3s yet and so none have been pulled out of thin air. They have just appeared, either very luckily correct or unfortunately too wide or too narrow as they may be with no good way to correct any of them yet.

At last, the next to last note you tune, you get to tune C#4 from G#4 and luckily, it is a 4th. You now can listen to the contiguous M3s which all of the other modern temperament ideas suggest is best to start with. Did all of the previous 12 notes result in an even 4:5 progression from F3 to A4? I understand how these may be difficult to discern. But surely if the progression is very incorrect, it will be obvious. Surely as well, the small difference between a 4:5 ratio of four evenly spaced M3s would be far easier to correct and verify than some 7:8 ratio somewhere else which has no other contiguous or adjacent intervals with which to compare it.

So, knowing that as an indisputable fact, the contiguous M3s from F3 to A4 MUST have the proper relationship in order for the temperament to be considered *equal* and not quasi equal, Victorian, a mild meantone, some kind of hodge-podge or at the very worst (but oh, so very often the case), Reverse Well, you must now correct the contiguous M3s and go back and fix all the other 12 notes so that they also fit before you can tune the final note, F#3.

Doesn't it make far better sense to do this first rather than last? Even if you don't believe you can get them perfect the first time, doesn't dividing the F3-F4 octave into 3 (at least reasonably) equal parts make sense? From those notes, 4ths and 5ths can then be tuned without compounding and accumulating errors. Further results can serve to point out very small errors in the initial contiguous M3s.

The PTG Tuning Exam Master Tuning Committee attempts the ultimate in perfection. An Examinee need only come to within one cent for a perfect score and this is, in fact, seldom seen.

There have been a number of recent reports about using non-equal temperaments in various situations. Most of those used which the casual listener and/or the pianist never become aware of consciously are different enough from true ET that they fall below the tolerances for the PTG Tuning Exam. It takes quite a substantial amount of temperament error for most people to be aware of it.

This is the reason why many aural tuners whose work would not pass the PTG Tuning Exams are still able to make a living, some are even concert tuners. Having said that, people still do seem to recognize when one technician's work makes the piano sound better and last longer than another's. The reasons can be very complex.

That is why it is important to be able to accomplish what you *intend* to accomplish. And that is, in fact the theme of this year's PTG Annual Convention: The Intentional Technician. I hope as many technicians as possible can attend it because there will be many opportunities to observe a myriad of ideas and techniques to accomplish what we all intend: a piano to look, sound, respond and please the piano playing public to the best of our knowledge and ability.

Bill: Thanks for the very thoughtful and detailed post. I very much appreciate the "blow-by-blow" critique of the Braid White sequence. I've skimmed though what you wrote and some questions come to mind, but I want to wait a day or two until I've digested your post. I don't want to disrespect it with any hasty comments.

Regards,

OK, Upright, I hope you will and that you won't simply try to pick apart what I said and come up with defenses. I'm not trying to be mean to or belittle you, even if it may seem like that. I often seem to come across as a little too blunt and if I have, I apologize for that.

The fact is, that the contiguous M3s MUST prove out whether you deliberately set them or arrive at them by tuning 4ths and 5ths. If they don't as a result of tuning 4ths and 5ths, it means that there has been error in the tempering of those intervals.

I really don't expect that there would be a great deal of compound and cumulative error in your tempering and I give you the benefit of the doubt that there isn't. But I do think it would be better if you could hear the contiguous M3s far sooner in your temperament construction than at the very end of it.

Here is one thing you might try: From A4, tune the A3 as an octave. Then, tune the D4 and E4 as 4ths and 5ths as I mentioned in my previous post. From both the D4 and E4, you can get A3 by balancing the tempering between those two notes.

I am sure you have at least some idea of what the F3-A4 M3 should sound like. If you now try to place F3 from A3 as a M3, you will also have a M6 between F3 and D4. Listening to both those intervals, the F3-A3 M3 and the F3-D4 M6 may give you a better clue to whether you have the F3 placed anywhere reasonably well

Now, go back to the Braide-White sequence and fill in the G3 and C4 as you have always done. Now listen to the resultant 5th from F3-C4. How well does it fit? At this point, I think you should be able to adjust F3 sharper or flatter to make it fit if it doesn't and you will also still have the F3-A3 M3 and the F3-D4 M6. You can balance all of these until everything works out without any 4th or 5th being either too pure or beating noticeably.

Now tune F4 from F3, an octave and check that both the F3-C4 5th and the C4-F4 4th are properly balanced. Fit in A#3 between F3 and F4 as a 4th and 5th. Tune the B3 as a 4th down from E4. That will create a G3-B3 M3. Does it progress with F3-A3 at that very slightly fast 7:8 ratio that you are able to recognize? If not, check back with the B3-E4 4th and if necessary, the G4-C4 4th (but the latter should be more reliable).

Now tune F#3 as a 4th from B3. This will create 3 adjacent M3s: F3-A3, F#3-A#3 and G3-B3. They should all progress very smoothly.

Now, you can tune C#4 from F#4 and you will have the 4 contiguous M3s from F3-A4. You won't have to tune A3-C#4 as a M3 but you can see if it fits properly by listening to the contiguous M3s but also to all adjacent M3s but especially G#3-C4, A3-C#4 and A#3-D4. They should confirm each other but if they don't, I think it will be easy to spot what you can move so they do.

Now, you would only have one note left to tune: D#4 which you can balance as a 4th and 5th from G#3 and A#3 below it.

In this idea, I only had you estimate one M3, the F3-A3 but you quickly had a way to verify it by both previous and subsequent notes you tuned. Compound and cumulative errors were avoided yet there was at least some of the the pattern you are used to available.

Let me know if this helps you because if it does, there was another person who asked if just one M3 could be used and is any of the M3s are estimable, it is the F3-A3 M3.

Another way of getting the F3 without estimating any M3s at all would be to use two tuning forks, the A for and the C fork. Theoretically, there would be a small error in doing so but practically, it will work. I saw Bill Garlick RPT demonstrate this over 20 years ago.

If you tune the A4 to the fork, tune an octave to A3 and then tune C4 to a C fork, you can immediately tune F3 and F4 from C4, balance the 4ths and 5ths and then fit in A#3 between F3 and F4. Also fit in D4 and E4 between A3 and A4. That will give you: F3-A3-A#3-C4-D4 and E4. Half of the notes will be tuned and also be very reliable. From each of these, you can tune more 4ths and 5ths and discover RBI checks as you go and you will also avoid compound and cumulative errors. B3 can be tuned from E4 and G3 can be balanced between C4 and D4. F#3 can then be tuned from B3 and C#4 from F#3. That will create the contiguous M3s from F3-A4.

Bill : I have had time to read your post more carefully. There are a few changes to my tuning method I’m going to try. Again, Thank You.

We’ve each had our turn at picking at each other’s posts. Despite any entertainment value, it wasn’t very productive. Let me try to go about what I am saying a different way.

The “blow-by-blow” of the Braid White sequence seems familiar. I think this is your standard explanation. That is a good thing because it should be more accurate than something done on the spot. All through your critique I found myself saying “Yes, but…” It made me think about how everything has advantages and disadvantages.

There is an advantage for a sequence that produces accumulative errors. It will show errors that cannot be perceived in single steps. Like sawing 12 pieces of wood, each 1 inch long. They each seem to measure 1 inch separately, but when stacked together they might measure only 11-7/8 inches. Also when tuning one note from the last one tuned and so on, there is only one error chain to trace back. The disadvantage is how far you might have to go back to find the mistake, and how much you might have to fix when it is found.

The way I actually use the Braid White sequence is when I find my latest note is in error, I put it about where it should be, and work backwards though the sequence to the original error and then forward again to touch things up. When I am done with this first pass, I have a good idea of how fast the various intervals should beat, and how I want to make my compromises. Then when I go through the sequence again, the knowledge gained form the first, rough pass is used on a second finer pass.

I won’t repeat what I posted to Jeff about my experience tuning with M3rds and why I prefer to use 4ths and 5ths. I do want to add a few things, though. Thinking about the accumulative errors of the Braid White system made it clear in my mind something about the contiguous 3rds. They are like a tree with three limbs. One limb you know is correct, and you know the other two are close. As whatever sequence you choose progresses, you refine the position of the other two limbs by checking how far off the branches (additional tuned notes) are from the other limbs. The problem is: which limb is wrong, or are both wrong? I can’t help but prefer a sequence that is consecutive to one that branches out.

I know your definition of ET. Mine is based more on concept than on tests. To me, ET means that all keys have the same color. Unfortunately, that makes it subjective. An untrained ear might hear a horrible temperament as being equal. An exceptional ear may hear different colors in a tuning that meets your definition of ET. On a piano with a challenging scale, trained ears might want different compromises to be made to the temperament in order to have all keys have as close to the same color as possible. My ears need to hear the 4ths and 5ths beating like a heartbeat within a chord. The 3rds don’t give me as much sense of color. This is the reason that when I make compromises on challenging pianos, I favor the SBI. I also like to keep my octaves wide. Favoring the RBI tends to narrow them across the break. Enough said on this. This is where the “train jumped the tracks” on the last topic we posted to each other on temperaments.

You mentioned how 5ths have two audible beats. Yes, I remember noticing and mentioning this to my tuning teacher in one of my first lessons. He replied very quickly to always listen to the slowest beat. It wasn’t something new to him. I believe this is an advantage of 5ths, not a disadvantage. They have a color all by themselves like no other interval. Recognizing and controlling this color can result in more precise tuning that just listening for a beat rate. I especially like to use this property when setting octave stretch.

I want to make some comments on the M3-M6 test for the benefit of the other readers. I’m sure you already know all this, Bill. The M3-M6 test is based on the inversions (but not the root position) of the Dominant 7th chord. The beat rate of the interval of the root and third of the chord is compared to the beat rate of the fifth and seventh. They beat at almost the same rate, theoretically. I really like using this test to do a fine touch-up of a temperament. No, I don’t try to make them beat equally. I do listen for any that beat much differently. This indicates errors that are not always obvious by just listening for intervals beating progressively faster.

Now Bill, I have some questions that I have no way of finding the answer to by myself.

When using an ETD, on a typical piano, how close does the average tuner get to the target pitch?

When setting a set of contiguous M3rds, on a typical piano, how close does the average tuner get to the ideal pitch?

And last, why is it so important to you that I tune with contiguous M3rds?

Regards,

I don't know when I might try this.

I respectfully disagree with your choice on the F3-A3 M3. Depending on the piano, F3 could be a wound string with much less inharmonicty than A3, or unwound with much more. I'd suggest the A3-C#3 M3. It would have a more consistant beat rate from one piano to the next.

Bill,

Thanks for your reply to my question, and also for your additional thoughts.

Maybe sometime I'll have the chance to let you review the whole temperament sequence I've been using and get your reaction. This isn't the time, since this thread is supposed to be somewhat linked to your ET via Marpurg idea.

There's nothing terribly new or different about my sequence; but it did come after I had a very clear personal idea to pursue. Not long after, I discovered Jim Coleman had come up with a system long before that was very similar. I discussed this with him via e-mail, wondering if the same basic idea was behind his system. His system did provide more clarity, and I ended up switching a step or two around after studying his way.

It's been a busy week, so I must apologize for not having the time yet to delve into your new system and make comments directly related to the topic.

Jeff

Jeff,

I'd be interested in knowing your method. How about a new topic. When you have time, of course.

Regards,

I can only repeat as I have done so many times before, Upright, the contiguous M3s from F3-A4 is self correcting, accounts for and incorporates both inharmonicity and the size of octave chosen. No matter how erroneous the initial estimate for the F3-A3 M3 may be, if you follow the instructions, not skip or disregard any of the steps, the notes F3-A3-C#4-F4 and A4 will be more accurate and reliable than with any other method. This includes pianos with either very high or very low inharmonicity on the F3 string.

From what you have said each and every time, it seems to me that you have not read, followed and understood the process. You have only told yourself that you can't do it. You still believe that tuning a series of 4ths and 5ths, none of which are anywhere close to being reliable and none of which account for and adjust for either octave size or inharmonicity will lead you to a more accurate arrangement on a poorly scaled piano.

It simply isn't true. It is evidenced by what you keep saying. You tune ET *but* you compromise the M3's. Yeah, I always tune ET too *but* I change all of the intervals so they sound well-tempered. That is what people in the 19th Century said. The Braide-White temperament sequence is obsolete and a hold-over from the kinds of temperaments that were considered to be ET in their time but not by today's standards.

Each and every person whom I have ever examined at a PTG Tuning Exam who used the Braide-White sequence either failed out-rightly or had only marginally passable results. On the other hand, those who know how to set the contiguous M3s always had superior results.

How can you be right and Bill Garlick, all other instructors at the North Benett Street School of Piano Tuning, all of their students who all seem to turn out with superior tuning skills, Dr. Sanderson, Jim Coleman, Rick Baldassin, most any other PTG tuning instructor, virtually all PTG Tuning Exam Master Tuning Committees, the majority of university technicians, concert tuners, etc., etc., and me, who has used such a system for 25 years and taught it to countless numbers of aspiring technicians, many of whom subsequently achieved perfect or near perfect Temperament and Midrange score on their tuning exams be wrong?

Read and study the material on my website and follow the instructions for setting the contiguous M3s word by word and line by line. Don't skip and disregard steps. When you change F3, you must also change F4. You must ascertain that both the A3-A4 and F3-F4 are of exactly the same size and type. If you haven't understood what that means and how to do it, the information is in my material. If you can't do that, you cannot have accurate results. Once again, the method accounts for *both* inharmonicity, whatever it is on whichever string *and* the octave size which you have chosen, whether it is what you call "pure", slightly narrowed, slightly expanded or greatly expanded enough to create the ET with pure 5ths, on *any* kind of piano from Bosendorfer, Fazioli, Steinway and Shigeru Kawai to Whitmore, Wurlitzer, Poole, Acrosonic and Jesse French. If you do that, you will find out how simple and easy it actually is.

I am not going to argue with you anymore about this. I happen to know what I am talking about.

Bill, I have gotten contiguous M3rds to work on well scaled pianos. I don't like it on the challenging pianos I usually tune.

Hope you don't think you wasted your time. I have read everything you wrote carefully and have learned from it.

Regards,

Oh, really?