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I don't have to include iH in my maths to see what happens.

Tooner, I tune pianos in the way I want them to sound. To speak clearly:

In theoretical E.T. F3-G#3 beats slower than F4-A4 and faster than C#4-F4, right?

So, I tune pianos to sound that way. With F3-G#3 beating faster than C#4-F4 and slower than F4-A4. The actual beat rates? I don't care. But the relationship between them? Yes! I do care!

I also tune narrow fifths, beating slower than fourths, wide fourths of course, and progressing (wide) major thirds, tenths and major sixths. With narrow minor thirds, and...etc, etc, etc. That's what I tune in pianos. I like to call it E.T. I repeat for you: If it is calculated as E.T., and sounds as E.T. then it is E.T.

Happy?

Gadzar:

Theoretically F3-G#3 beats at nearly the same rate as A#3-D4. When a typical stretch is used on a well scaled piano, F3-G#3 will beat more nearly like B3-D#4.

I question everything I think, do or say when it comes to tuning. I continue to look for those that I can discuss the math with. I keep hoping that you may be one, but I guess not.

Are you saying that in a well scaled piano the beat reat of F3-G#3 won't be between those of C#4-F4 and F4-A4?

Or maybe that in a well scaled piano F3-G#3 woul beat faster than F4-A4?

And, in a well scaled piano, C#4-F4 would beat faster than F4-A4?

I don't get your point. You say:



Please, tell me what iH makes to beat rates of F3-G#3, C#4-F4 and F4-A4?

Are their beat rates affected in a way that couterdicts what I have said?

i.e.: C#4-F4 < F3-G#3 < F4-A4

Please answer that!

Gadzar:

My last post said exactly what I meant, including an ackonwledgement that we cannot discuss the math. Your last post confirms this.

Tooner,

Would you please include iH in calculating beat rates of M3s and m3s and post the results here so I can work with them? Can you?

I don't know how to make such calculations. (you say iH affects beat rates in an unexpected way).

I don't know of a formula that calculates accurately iH of piano's strings. I know a lot of formulas that intend to do it, but none of them, to my knowledge is accurate enough to actually tune a piano with its predictions.

I don't know a single ETD which can calculate a decent tuning without intervention and adjusting of stretch at different points by the human tuner. (I am not sure of this one, in regard of Mr. Stopper's Onlypure software, I have heard the user makes an adjustment at the beginning of the tuning but I don’t know what he adjusts for sure.)

Some ETD programmers take empirical data directly measured in pianos and put them into their software.

For example, in Tunelab, they use an empirical formula, with an empirical table of data to calculate iH constants for a given set of notes and from there calculate a tuning curve.

I don't know how Accutuner calculates its FAC tunings, but they say they only measure three notes and from there they calculate the entire tuning curve.

Verituner takes readings of all partials as we tune each string, so it works on actual partials and thus has not to use an expression of iH in its calculations. (I guess)

I confess I really don't know how they make their calculations.

Maybe Young? Have I to measure string's diameters, lengths, tensions, mass, rigidity, etc...before tuning the piano? Do partials duplicate in frequence every eight notes? In every piano?

This will be usefull maybe to design a piano, but can you calculate iH beat rates of a real piano with it?

Do you really want me to put here the inharmonic beat rates of M3s and m3s? Anything else?

Are you really aware of what you are asking me?

But even if I could include here an accurate expression of iH and calculate "inharmonic" beat rates of any given piano (not an easy task!), the resulting tuning would be exactly the same I already described.

The only difference would be that this tuning will be expressed numerically in terms of Hertz and cents, instead of "aurally" expressed: i.e.: in terms of uncalculated beat rates ratios and progressions, which in spite of not being "calculated" they are tunable for sure.

It is the tuner which tunes the piano and he imposes the beat rates. Math's only express numerically what is done by the tuner.

The fact that beat rates are affected by iH is not a recent discovery. Even Braid White knew it 100 years ago.

But you focus your attention in the fact that I took theoretical beat rates without iH, which surely will turn out being valid in this particular case. And you say nothing about the sequence I propose.



For a long time I have been searching a sequence to set the temperament in a piano, where I have no guesses to make. Where the piano has no surprises reserved when I went to tune it. A sequence where I am in absolute control over the tuning, knowing at each step where I am and where I am going to.

I think I have finally found such a sequence. All I have to decide by myself is the width of the temperament octave, the remaining is only appreciating and adjusting beat rates, no more decisions to make. I divide the initial octave A3-A4 in equal M3s of whatever width is needed, then I divide it again in equal m3s of the exact width to build up that octave. All this while having absolute control over the other kind of intervals involved in the tuning: fifths, fourths, sixths and octaves, which are the result of the only one decision I took at the beggining, namely: the width of the initial A3-A4 octave.

This allows me to harmoniously balance all of the intervals in the temperament. There are no surprises, there is no guessing, there are no subjective ways of tempering fifths and fourths. It is all pure, simple and objective beat rate appreciating and setting.

I thank Bill Bremmer for being the one who found the way to successfully divide the octave in Cm3s: tuning G#3 and D4 in the first place and wrote here the way of doing it. “El resto es pan comido” as we say in spanish.

As I said, I have tried to do it in several ocasions but I failed to find the right way to tune the Cm3s.

Rafael,

I hope I'm not making comment that is too stupid or obvious, but since we're leaving the "math" and going back to "aural" for now, I've been thinking about how to decide on the correct octave width for the initial temperament.

To me, the ideal temperament really is an 88-note temperament, not a 13-note or 17-note temperament. (Of course, typing it and accomplishing it are two very different things. ) All intervals compromised as far as what is reasonable, including octaves.

When deciding about the A4 to A3 octave, why not also temporarily tune the other A's (A1, A2 A5, A6) and temper them all to achieve your best compromise between each other (A4:A2, A1:A3, A3:A5, etc.) and let that be your guide on the width of A4:A3? You could also do the same on the F3:F4. Then use your CM3's and Cm3's.

Again, I'm sorry if this seems obvious. It's just that I never seem to read about tempering octaves. I do read about stretch. I do read about octave types. I do read about using other intervals to check octaves. But, I never seem to read much about using other octaves and octave multiples to temper octaves while tuning aurally (in addition to the other checks).

Joe, I wrote about the idea of tempered octaves long ago: http://web.archive.org/web/20031203194817/www.billbremmer.com/TemperedOctaves.htm

It's actually very simple but the very idea that octaves are not pure freaks some people out. I agree completely with you however that the entire range of the piano should reflect the temperament. None of the ETD programs view the outer octaves as a compromise between the double octave and octave and 5th (12th). In ET, they may very well approximate it but by default, not design. When the temperament is slightly unequal as is my usual practice, the ETD programs miss the boat entirely.

I often see ideas that want to create a "temperament" over a larger area than one octave. However, I have never seen that as having any particular advantage. Once the temperament octave is refined, expanding it beyond one octave, replicating it both higher and lower is simply the next task. No serious tuner ever tunes a temperament octave and merely tunes octaves up and down from that. Changing inharmonicity in various parts of the scale make what would be theoretically possible quite impossible.

Rafael: you have reached the usual impasse with Tooner. Once you do not agree that he is right, there is no further basis for discussion. He contradicts himself: inharmonicity has no significant effect on beat rates in one post but in another, he demands to know what effects it has and demands that you show how much effect there is in all of the infinite possibilities there are. It is a game. It is a game that only he wins. It has nothing to do with any practical application, it is only a game designed to prove that no piano can actually ever be tuned according to theory. We already all know that.

Only Braide-White's book has the answers, no others do and no other method is valid. CM3s or Cm3s cannot be accurate. They were not in Braide-White's book, so they cannot work. Tooner and only Tooner knows how to tune a piano properly. You must agree with that or there is no basis for discussion.

Yes, I suppose I would better not get upset by what Tooner says. In this same thread at the beggining he was saying that changes in beat rates due to iH were insignificant.











He uses theoretical beat rates:



But, when I do use theoretical beat rates, in the very same intervals, in the very same way he did, then he says:




And finally he doesn't want to discuss:




Well, I hope next time I will resist the temptation of playing his game again.


I have tuned the two pianos I have mentioned before. A spinet and a Baby Grand.

In both of them I made a pitch raise with my Verituner and then a second pass fine tuning using the CM3-Cm3 Sequence.

In setting the temperament I had no problem at all. It worked out as expected, with no surprises.

For the baby grand, I did not like the tuning calculated by Verituner in the bass section, so I tweaked it a little.

I think I am on the correct way.

About the hole piano being an 88 notes temperament: I agree, that is the ideal.

When starting to tune aurally, I used to tune from A3-A4 out by octaves up to the top and down to the bottom. And testing each octave with fifth and fourth (where audible). Then I played a run of 10ths and/or 17ths looking for uneveness in the progression of the beat rates.

Now, I've abandoned the tuning of octaves and I tune, ASAP, a compromise between double octaves and 12ths, the "mindless octaves" of Bill Bremmer in both ways going up to the treble and down to the bottom. I follow this procedure: First I play the octave, say F4-F5 and tune the upper note first to a beatless octave, then I go sharp from beatless just a little and then I play the 15th (F3-F5) and depress the sustain pedal while flattening and playing the upper note (F5). Then I compare with 12th (A#3-F5), depress the pedal and adjust F5 until I have equal beating between the 15th F3-F5 and the 12th A#3-F5.

I believe one must compare and tune bass and treble notes to the central octave, using wider intervals as one reaches the extremes of the keyboard, using 10ths, 17ths, 19ths, etc... But, I confess I don't tune this way!

Until now, what has worked better for me is the "mindless" 15th-12th equal beating compromise. It is really "mindless". I mean: if you check your tuning going down to the bottom by playing a run of 10ths, then 17ths, etc. there is a progression of beat rates, which I've found often ends with pure, beatless intervals. And when checking you look for uneveness in that progression to correct the tuning of notes in fault.

But when you tune the bass or the treble with an equal beating 15th-12th, then there is no progression, you do not bother of beat rates accelerating or decelerating, you only strive for equal beating intervals, it becomes "mindless".

And the most amazing of that is when you are finished. The bottom meets (or fits?) the treble perfectly as if you were tuning directly the one to the other by means of double, triple or quadruple octaves!

I have never tried to tune A2-A3 or A4-A5 octaves as if it was the temperament octave. Except when tuning the double octave A2-A4 temperament of Dr. Albert Sanderson. I don't know if it is possible or desirable at all.

Bill,

Great article. You most certainly did discuss this years ago. Sorry... This is the first time I saw it.

My thinking is that ideally, the entire range of the piano should also influence, but not determine absolutely, the width of temperament octave. A "dead octave" allows more wiggle room than most people realize (as you know). If the ideal width of the temperament octave can be determined first, there will be less work involved in expanding the temperament outwards... at least on quality pianos.

Or, maybe there is something I'm not seeing?

Thanks!



Rafael,

I was not thinking of having the outer octaves determine the temperament octave as much as having them inform how the temperament octave might be finessed.

But, you have to use what gets you the results you seek. The proof of your map is if it gets you to your desired destination. I absolutely respect this.

-Joe

Actually, I am starting to realize that the change in beat rates due to iH, separate from additional stretch, may be significant after all (for me that is 0.2 cents or more.) But this is through mathematical investigation. A blanket statement with no support is not convincing. But if it is greater than the error from tuning an initial set of CM3s may have is another question.

It is easy to imagine the effects of iH as stretching a rubber band with marks on it. This analogy is true for additional octave stretch when the iH does not change, such as the difference between tuning 4:2 or 6:3 octaves. It does not hold true when going from harmonic to inharmonic tones nor from a piano with less iH to one with more.

Gadzar, you have shown that you have the tools and knowledge to investigate this but may not realize it. Consider the difference between the theoretical beat rate of F3-A3 and what your algorithm calculated. It was a little bit slower, not faster as would be expected. Best of luck.

Joe:

I like to set my temperament octave by listening to the fourths and fifths within it. The slightest adjustment changes the “color” of the fifths and is important to me. This is always somewhere between a 4:2 and 6:3 octave and also leads to a very smooth sounding 8:2 double octave, which in turn makes for a barely audibly wide triple octave. The twelfths may beat barely audibly in some octaves and not in others.

Another contradiction: in one post, Tooner says he likes "pure" octaves, in the last, he says 4ths and 5ths tuning creates a compromise between a 4:2 and 6:3 octave which it does not. In the above, I see exactly what John Travis identified as the "tendency of the tuner (Tooner) to err towards the just 5th" which will create uneven M3s and M6s. In the post previous to that, Tooner once again makes the claim that CM3s are inaccurate. No matter what kinds of proofs are offered, no matter how many modern methods use CM3s to avoid uneven rapidly beating intervals, no matter how many PTG master tunings use CM3s as the ultimate control, no matter how many PTG tuning exam aural verifications use CM3s to identify error, the answer is still and always will be, "I still don't believe CM3s can be accurate".

The reason he believes this is the dependency and addiction to the 4ths and 5ths method of tuning. Such a method will inevitably produce CM3s which do not conform to the expected 4:5 ratio. The conclusion therefore is that CM3s are inaccurate and do not work. The result is virtually always what is not intended, an unequal temperament.

See my article written long ago about the most common error that results from the practice of 4ths & 5ths temperament construction: http://web.archive.org/web/20040407144904/http://www.billbremmer.com/ReverseWell.html

Bremmer must have incredible hearing to be able to tell if my M3s and M6s are uneven from so many miles away.

I think we're all aiming at slightly different targets...
Which is perfectly legitimate.

Jeff,
It sounds to me like you have a very special sensitivity to 4ths and 5ths. Fortunately, you discovered it and developed it to tune pianos.

We're all built a little differently. I have a customer that has a special sensitivity to F#'s. It's the way her ear is made.



Rafael,
I see your point about the limitations of pure math as a predictor. As you pointed out, I can easily see on my Verituner that the partials measured are obviously not what a pure math model would predict that they would be... proven by some negative partial numbers... sometimes seemingly random partial numbers. The theoretical motion of the string cannot be taken as an absolute when predicting beat speeds, even when factoring in inharmonicity. Impedance and likely other factors significantly affect the partial series as well.

It still comes back to the ear.

Thank you everyone!!!!

I have learned much here!!!

-Joe

Thanks, Joe:

I now usually tune all the octaves in the tenor using fourths and fifths.

I continue to resist the temptation to write an Access database application to calculate beat rates given a piano’s iH, chosen octave types, and other criteria. It would be a bit of a challenge, but not overwhelming. I guess I am concerned that it might be a “trivial pursuit.”

Nice articles Bill. I didn't know about them. I suppose these are articles you have published in the PTG Journal, aren't they?

Thanks for the links and addresses. Have you more writings on tuning?

I like the distorded keyboard that appears in the "Reverse Well Temperament" article. It is indeed a good image for illustrating how a reverse well temperament may sound.

Tooner,

Do you think it will take you long to make your dababase? I am waiting for the inharmonic beat rates of M3s and m3s in order to correct the figures I use for tests in tuning CM3s-Cm3s Temperament. In fact, I believed that you had already made such calculations, judging from what you said about how iH affects beat rates of M3s and m3s in an unexpected way...

Gadzar:

I have not decided if I will go through with the database project. I keep thinking of various “whistles and bells” to include and am reluctant to get involved with a “tar-baby” of my own invention. If I do work on it, I have to decide early on if it will be for my own knowledge or a tool to share with others. For example, a table with theoretical frequencies and partials is needed. But if it is accidentally altered, it would be nice if it could be re-built programmically for other users. And if it is also for other users, it really should be in Excel for their convenience. This would be a learning curve for me.

I guess it would take me a month to fit it in the cracks in my schedule. But I wonder what my motivation would really be. There does not seem to be much interest in this. For myself, I notice that something odd happens in the 5th and 6th octaves and kind of wonder just what it is. But since I only tune aurally, the important thing is how it sounds, which is always some sort of compromise anyway.

I had realized the m3 / M3 “anomaly” with a method that uses cents rather than frequencies a while ago, and it agreed with what I heard aurally. It can also be understood by considering a M6 outside M3 inside test, and then lowering the upper note an octave to produce a m3 instead of a M6. Unless this octave is 6:3 or wider, the m3 will beat faster than the M6 did. So you can expect the m3 to also beat faster than the M3. Again by using the “cents method”, I have found that iH affects M3s and M6s in a way that the test is more accurate with iH than without.

I see no reason that you cannot work out an algorithm for Cm3s like you did with CM3s to see what the difference is between the theoretical beat rates of m3s and those calculated with iH. I think the results of your CM3 algorithm are correct: to have the exact same octave type, the CM3 ratio has to change, and although the expected result of having a wider octave ratio due to iH would be for M3s to beat faster than theoretical, the opposite happened. Now I acknowledge that these effects are very small in this case, but they do exist.

OK, just got around to trying out Rafael's Cm3 sequence, given in detail in an earlier post on this thread, after he pointed me to it from a more recent thread here. I thought it'd be appropriate to post here for future reference.

It is very interesting. I think there are a couple of typos to be corrected: in step 6 I believe the first F4-A4 reference should be to C#4-F4, and the second reference in the same step should be to A3-C#4; then in step 9 the reference to G3-B3 should be to G#3-B3.

I like the initial step after the CM3rds ladder suggested by Bill Bremmer, the interlocking method of getting to G#3, but I am still a little uncomfortable that the only complement to the G#3-C#4 "about 1bps," is the estimate between three sets of theoretical beat rates, that of the m3 being set, and the two CM3rds that happen to bracket the theoretical rates of A3-C#4-F4. However, the construction of a Cm3rds ladder to D4 seems a worthy payoff for the effort. I appreciate that this gives me another method to approach the problem of what to do after constructing a CM3rds ladder. My next step is to move the approach down a M3rd to my C#3-C#4 temperament sequence

Jim,

About the typos you are absolutly right. Before posting I have double checked my writing and have seen no errors, but there they are!

Step 6 must be:

6. Estimate G#3. Minor third F3-G#3 has a theoretical beat rate of 9.4 bps, it should then beat faster than A3-C#4 (8.7 bps) and slower than C#4-F4 (11.0 bps), so it is really easy to tune it by first making a beatless fourth G#3-C#4 and then flattening G#3 until F3-G#3 beats at a mean rate between the beat rates of A3-C#4 and C#4-F4 M3s. In addition we can check the tempering of G#3-C#4 fourth to beat at near 1 bps.


And step 9 must be:

9. Tune B3. All we have to do is tune B3 to make G3#-B3 m3 beat slower than B3-D4 m3, but faster than F3-G#3 m3. For that we can tune first a pure m3 G#3-B3 and then flatten B3 until G#3-B3 beats faster than F3-G#3 but slower than B3-D4. Check for a smooth progression between the Cm3s F3-G#3-B3-D4-G#4. Theoretical ratio 5:6.


Thank you for your attentive reading!




The theoretical rates are not important at all, what matters is the fact that the actual rates you hear when tuning must respect the progression:

A3-C#4 slower than F3-G#3 slower than C#4-F4

and final check G#3- C#4 must beat "at about 1 bps" not exactly at 1 bps.

And don't forget that the tuning of G#3 in step 6 is only an estimate which will be refined in step 9.

Just as we do an estimate for F3 when tuning the ladder of CM3 to beat with A3 at about 7 bps, and refining it once we have our set of four contiguous major thirds. It is exactly the same approach: We make an estimate: m3 bracketed between two M3s (instead of estimating a rate of 9.4 bps which would be arbitrary and inaccurate) then once we have our set of Cm3s we refine the tuning of G#3 and D4, letting the piano tell us...!