You may consider that even more work about different ET from the standard model from other people have been proposed, beside the book of Cordier.

You may read the article of Gary Schulze in the PTG journal from march 1982, where he explicitely describes a theoretical ET model based on the 31th root of six.

An ET based on the theoretical model of the 19th root of three has been proposed by me in euro-piano 3/1988. Inharmonicity consideration is targeted by using inharmonicity affected partials through the use of beats when tuning aurally.

A general mathematical model of tunings has been proposed by Guerino Mazzola 1989 in his book "Geometrie der Töne". In section 2.3.3,

a "convention dependant linear function of the form
Y = uX + v, where u and v are not constants,
with X = ln(f) " is proposed, what finally includes all possible theoretical ET and non ET models.

Thank you for your feed-backs,

I'm visiting my parents...and I'll soon be with you. a.c.

I don't believe my eyes!

So you guys can not come out with a definition of ET?

In this topic I've seen the most experienced people, I mean guys like Mr. Bremmer, Mr. Swafford, Mr. Scott, Certified Tuning Examinators who had developped new temperaments, new sequences for tuning ET and new techniques on tuning the hole piano or who had designed tuning software for ETDs, etc., and in spite all this knowledge, they get confused by Mr. Carpuso to the point of admiting they don't know what ET is for sure?

Mr. Carpuso doesn't give an answer with all that math stuff. He only plays with some clever equations which relate to nothing in piano tuning's real world.

His sequence is the same sequence up a fifth, down a fourth taught by Randy Potter in his course and he does not explain how to exactly temper fourths and fifths. It will suffice to give a look to his instructions:

"Step 2 – A4-A3 - tiny little narrow, just on the beating threshold"

"Step 3 – A3-D4-(A4) - wide, close to 1 beat/sec. – D4-(A4) faintly beating"


"Step 4 ...
...A3-E4 about 1,5 beat/3s - sensibly faster than D4-(A4)
E4-(A4) about 2 beats/1s - sensibly faster than A3-D4
"
The same "slightly faster than, or 3 beats in 5 seconds or one beat per second" of a ton of SBI's sequences. I feel as I was reading Braid-White book, written a century ago.

It is unbelievable that Mr. Carpuso dares to claim that he has constructed a new model in which there is a variable called 's' that can "calculate infinitesimal degrees of inharmonicity". And says in his first post: "Inharmonicity, which has always been calculated in an approximate way, can now be calculated with infinitesimal accuracy". (Mr. Carpuso confuses iH with stretch).

And it turns out in this thread that he doesn't even know what inharmonicity is, and how it affects beats!

And here we have gentlemen like Mr. Jeff Deutschle and Mr. Robert Scott explaining him what iH is!

Or brave Mr. Bremmer trying to find wide fifths and 12ths in a Master tuning in an effort to understand what Mr. Carpuso says! No way!

Don’t get confused by all this bla, bla, bla. At first I was impressed by this thread when reading Mr. Carpuso's paper, but now I am affraid that it is all about it. It impresses people by using confusing math, which leads nowhere. All that math isn't required to tune an octave slightly wide, a fifth beating 1.5 times in 3 seconds, etc...

Sorry Mr. Carpuso but I think you are an excellent illusionist.

Part of the problem, not just with this paper, is using terminology that varies from moment to moment. "All that math isn't required to tune an octave slightly wide,..." is ambiguous depending on whether one is talking about the interval's pitch numbers being wide or the interval's beats indicating that the interval is wide.

Rafael, the definition of ET is a temperament in which all intervals are tempered equally. It is as simple as that. I learned that from Kent Swafford. I also learned from Kent that it doesn't matter how wide or narrow the octave is. Since he had explained it to me, I heard it from other sources as well. With further work in techniques for constructing an ET, I learned that the scaling of the piano does not matter either.

Please, let us not mention "CTE" too much. No one who has that authority is allowed to "advertise" it. Only one member of any master tuning committee is required to have that title. PTG members who become inactive with tuning exams lose that title. It is only meant to be used for the purposes of conducting an official PTG tuning exam and is not meant to represent a level of skill. It has to do more with the administrative skills than it does with tuning skills. It is well known, however that to qualify to train in that area, the person must possess a higher standard of aural tuning skills than is required to become an RPT.

Therefore, there has been some suggestion within PTG that those who do possess these skills, whether they work with exams or not, be recognized for it. It may come about eventually but presently, there is no title which PTG bestows other than RPT.

So, let's be sensitive as well to those who are not PTG members, especially to those who may, in fact, have superior aural tuning skills. Anyone can use an ETD to test aural tunings by using the exam program which comes with them, if they care to learn how to do it. Anyone can set up a "reference" tuning either by themselves or with assistance from a more highly skilled technician, PTG member or not and regardless of whether that person ever worked as an examiner or not.

I can say, however that when an aural ET is being refined to its ultimate perfection such as during the construction of a Master or reference tuning, the people doing that look for the same amount of tempering in all intervals. There can be no "fudging", no favoring of any one kind of interval over another.
This is true regardless of octave type, size or width, however you may like to think about that. PTG does not have any specifications for that, even for the exams.

If anyone used Bernhard Stopper's model for a PTG exam and did the midrange aurally as required with sufficient accuracy, it would pass the exam. Any other octave width, even a slightly narrow width would work just as well. What matters is consistency, maintaining the same concept throughout. That is not to say that favoring one kind of interval over another may produce pleasing results in some circumstances, that is what I do every day. "To be or not to be ET, that is the question".

Regarding whether 5ths and 12ths become wide or not, they do, I am convinced of that. I learned that very long ago from Steve Fairchild who demonstrated it at a PTG convention. Now, I take what Kent said about my post to heart but the figures as I posted them still suggest as much.

Anyone can do what I do routinely in constructing octaves from F5 to the top and they will see that what I say is true. Beginning on F5 and with the ETD set on F5 (Partial 1), play the F3 and A#3. If you first stop the pattern when F3 is played, then play A#3, the pattern will roll slightly sharp. If you adjust F5 on the ETD so that the pattern rolls equally sharp and flat when F3 and A#3 are played alternately, you will find an ideal spot for F5 to be tuned. The double octave will be slightly wide and the 12th, slightly narrow, each by a very small amount, nearly imperceptible to the ear.

That is the basic "mindless octaves" concept. If by ear, the double octave is made to sound beatless, the 12th will beat noticeably. If the 12th is made beatless, the double octave will have a noticeable beat that may be considered unacceptable. However, when there is an exact compromise between both the double octave and the 12th, both intervals sound apparently in tune, the beat is so slight as to not be really noticeable, especially in a musical context.

That is why I dubbed the concept as "mindless" because if either the double octave or the 12th beats, it sounds "wrong" but when there is that exact compromise between the two, it sounds "right". This proved to be true for me even when tuning an unequal temperament.

Now, if you continue this technique upwards, you will inevitable find a point where both the double octave and the 12th will both stop the pattern and to the ear, both will sound perfectly in tune. When you continue upwards, you will find the exact opposite of what you found at F5. When the 12th stops the pattern, the double octave will be wide, when the double octave stops the pattern, the 12th will be narrow, still each by a very small amount.

This means (at least by my reasoning), that the 12th has become wide and therefore the 5th as well. However, at this point, the coincident partials for the 5th may well be out of hearing range and therefore, however wide they may be won't matter because they cannot be heard. In any case, a slightly wide 5th is not unpleasant to the ear, especially that high up where the sustain is so short. The same applies to single octaves: a slight or even slightly rapid beat in a single octave does not offend the ear, particularly in a true musical context.

I have now long taken to the practice of tuning pure double octaves and 5ths from F6 to the top. Sometimes, I don't start that until C7, it all depends on how wide the single octave sounds. If it is just too crazy, I go back to the double octave and 12th compromise until the single octaves can sound reasonable. So, I may start the pure double octave and 5th idea anywhere between F6 and C7 or maybe even a little higher but eventually, I get to 6:1 octaves in the high treble. I tune the low bass basically the same way. I let the piano tell me what it can take.

To me, this is a far better way to tune the extremes of the piano than to depend on a calculated stretch curve. (The totally advanced features of the Verituner notwithstanding). That curve is based upon assumptions instead of what the piano may really offer. Even if I use a calculated tuning for the middle, I change my partial selection and tune by direct interval at either extreme end. It doesn't take that much more time to do it that way but the results are certainly worth the time and effort.

That is one of the several reasons why I chose to use the SAT over the other choices of ETD which are available. For me, direct interval tuning is my preferred concept because I tell the ETD what I want, it does not tell me. I am in complete control of the results to the very highest and lowest notes and that is the way I want it. I do believe that for those who are used to using the other devices and software, a direct interval method of tuning the high treble and low bass are just as accessible as with an SAT.

I have met and had discussions with Bernhard Stopper and have also heard his tuning. It has a remarkably clear character to it. While I still do not fully understand it, I did gather from what he has said that the 12ths also become wide at some point in his tunings as well.

Others have long ago suggested pure triple octaves including Jim Coleman, Sr. and Virgil Smith. If one considers that 12ths and double octaves converge, then reverse at some point, then so would double octaves and 12th and triple octaves. I would suggest, as my own practice has told me that by the time one reaches the top end of the 7th octave, that is the point where double octaves and 12ths and triple octaves are both pure or at least so close to each other that the difference is insignificant.

I give Alfredo the benefit of the doubt that his concept lies somewhere within a structure that may be a composite of what I describe above. There is, after all, an obvious language barrier. Any manipulation of octave width and/or temperament will have some kind of effect. It cannot be said that one idea is right and the other is wrong, only that any two ideas are different and produce differing effects as a result.

We should all try to understand and be able to try and implement these differences depending on the goal of the tuning at hand. Passing the PTG tuning exam is one goal but creating a more pleasing effect for a different kind of circumstance is another.

Nicely put Bill.

Gadzar,

I do not recall ever saying that I did not understand the ET. ET is what I've used my entire life until I purchased RCT. Even now, the best tunings that I have saved on RCT are all my own tunings done using ET. I can tune a piano very well with that method thank you very much.

I am just NOW learning Bill's method or, trying too. That to me, is a bit more confusing, learning a completely new and different method after 40 years of tuning one way... Perhaps that is to what you are referring.

What does not matter to me at all, is the mathematical theoretical stuff (that I do not bother to read by the way) that is spewed back and forth in here. All that is to me, is one person trying to impress another and it does not impress me one bit. It doesn't mean that one can hear it... All that shows is one can talk it... Proving it is an entirely different matter.

Gadzar:

I think Alfredo is sincere, but thought he knew more than he actually did. He may be able to learn more here. By reviewing his paper and looking at the subject from a different angle I have learned some things. So I don’t think it is a waste.

My personal, practical definition of ET is where all M3s and M6s beat progressively faster. On poorly scaled pianos, I don’t think this is possible. One or the other (or both) will have a jump in beat speed across the break. Even on a well scaled piano, it may be difficult to achieve if the pinblock and rendering are poor. Each note must be within 0.2 cents of an ideal frequency!

Bill:

I am not sure if anyone has noticed, and I have delayed bringing up the subject because it may be moot, but your mindless octaves are a slightly different thing than Alfredo’s equal beating 12ths and 15ths. His have a common note on the bottom, your's are on the top. I am still mulling over what this might signify.

You’ve mentioned before that Steve Fairchild demonstrated that 5ths and 12ths become wide. How did he do this?

Ok. Guys, all of you know what is ET, but none of you tune it the same way. Saying that all intervalls must be tempered equally is easy but how to tune all intervalls equally tempered is not that clear!

What happens if I tune the fundamentals of all strings to their theoretical frequencies? The piano will sound untuned because of iH, but you can not say the intervalls are not equally tempered, in fact the intervlalls at the fundamentals will be equally tempered, thus it will be ET.

The question for me is to find a way to tune ET taking into account iH, and that's here exactly where beats come into play. Some say to tune a smooth progression of M3rds and M6ths (M10 and M17 where 3rds and 6ths are no more audible), others say to tune by octaves, double octaves or even triple octaves, using different types of octaves along the scale, others use to tune 12ths and 15ths (as I do), or a combination of the above, and all of us can get a well tuned piano, but what is then ET? Is there a true ET?

You can tune M3 and M6 progressively faster, but then what about 5ths, do they will beat also progressively slower? No way! You must include them in your tuning, as you must also include 4ths, it is not enough to test M6ths thinking of them as M6 = P4 + M3, you must control explicitly P4s. Do the P5s become beatless at some point and then become wide? Probably, but some of you say yes, some say who knows?

Nobody can for sure tell us how to spread the temperament accross the scale, each of you use a different way to do it. That is why I say you are not able to give a true definition of ET.

The pianos we all tune can sound great, no doubt, some of you are great tuners, but is there a true and unique ET? Or iH makes it impossible to find a UNIQUE WAY to tune those progressively faster beats?

Now, what is CHAS? Is it a mathematical model that can not be landed in a real piano tuning?

How does Mr. Carpuso effectivelly tune CHAS? The sequence he posts here, is as I've said before, one more of those sequences of 5ths and 4ths arbitrarily tempered to what sounds "slightly" wide and narrow.

That is not a new system! How is he choosing the values of the variables delta, s and s1? How is he puting these values in his way of tuning? He says nothing about that.

Mr. Melo,

Great post.

It is my position that theoretically there are an infinite number of equal temperaments, because any interval can be divided any number of times to create a temperament that qualifies an an equal temperament.

In practice, committees of 3 RPT piano techs regularly reach consensus on an optimum piano tuning for PTG's exam pianos.

I believe the practical application of equal temperament allows certain irregularities in the progressions of beat rates in order to preserve the overall progression of all the tuning intervals. In other words, the tuner makes the best compromise possible among all the tuning intervals, and the result is still an equal temperament. Inharmonicity-induced irregularities in the beat rate progressions, in practice, does not preclude calling a tuning equal temperament.

It is extremely interesting to me that as we go forward, it appears that talk of optimum stretch preferences seem to be giving way to talk of the optimum intervals on which to bass the underlying mathematical model of equal temperament. This is why the present discussion is of interest to me.

But either way, the underlying impetus for stretched tunings or tunings based on different equally-tempered intervals is the same -- inharmonicity.

I also await the answer to the question you ask, that is, "How does Mr. Carpuso effectivelly tune CHAS?" Presently, I doubt that "CHAS" actually exists as a successful piano tuning system, but, though I am skeptical, I believe I am still open to CHAS.

Gadzar:

I have had the same questions, but have only come up with some answers.

I have thought of a number of ways to define the stretch of an aural tuning. We can choose a non-beating intervals such as 2:1, 4:2, 4:1, 8:2, 3:1, 6:2 etc and let it define the stretch. Or we can choose one of these intervals to beat at a certain rate such as 2:1 @ 1/2bps. Or we can have a chosen interval beat progressively faster such as doubling every octave. Another scheme is equal beating between 2 intervals, such as mindless octaves, but then we also have to define an initial stretch. One that I like in the midsection is a beat ratio between fourths and fifths. And there can be a progression of a beat ratio between two intervals that changes. There can be hybrids of all these definitions, and any others that I have not mentioned. And if we know the iH of every string on a piano, we can calculate the frequencies of these intervals and then perform some kind of non-linear interpolation to calculate the other frequencies. When I think of doing this, I remind myself to call anyone that has written a tuning program “Mister”!

But can a tuning only be defined by some aural standard? I don’t think so. Each partial for the notes of a piano can be defined mathematically by a curve; even fractions of partials. Perhaps, even negative partials. A mathematical curve can be generated for any of these partials, or the difference between partials, or a ratio between partials. Calculations can then be performed to determine the other partials and a piano can be tuned either electronically or by using beats.

So where to go from here? I think back to the purpose of tuning to begin with: Preparing the piano to make music. And I think of real pianos and a story about rabbit stew. (True story.) A family was eating rabbit stew and someone asked who put a raison in it. It did not matter that there was only one “raison”, dinner was over! With pianos I think the important thing is that no intervals sound bad. That is the outside of the envelope. Everything else is OK, including non-ET.

Then what is ET? The broadest definition is that all keys sound the same. This would include, for most people, mild well-temperaments. But what about us folks that are going to listen to progressive beat rates? Can all intervals be progressive on a piano that has iH? I am not so sure. There are many descriptions of tunings where fifths become wide in the high treble. Well, unless they also become fast in the bass, this would not be progressive. Unless we want to say that a curve that goes from slow to fast to slow to stopped to wide is progressive…

And then there is the practicality of tuning a real piano. I asked a while back and it seems that the strings on a piano can only be set stabile within 0.3 cents. Why that is not even enough to guarantee that M3s and M6s beat progressively let alone 4ths! That is why I choose my practical definition of M3s and M6s beating progressively. If they do not, then an error that can probably be corrected can be pointed out.

Well, I’ve rambled enough. One person’s perfect ET is another’s poison. Because of iH, and personal preference, ET is region not a location.

And I have already asked questions about how a piano is tuned to CHAS and cannot understand the answers.

You guys are thinking about this too much. You tune a piano so that it sounds like it is equal tempered. That is all there is too it. Someone else can come along with a frequency counter and debate what it actually is, but as Duke Ellington put it, "If it sounds good, it is good."

I am not so sure. I tend to think that the piano is so popular because of the stretch that iH naturally gives to octaves. But also, additional stretch can be given in an attempt to satisfy the well-documented human ear's desire to hear stretched octaves. The use of 12ths, not necessarily pure, is a great tool in controlling this additional stretch.

Mr. Swafford, I’m with you, iH makes the difference.

Without iH all is easy, if we assume that

1 There is no iH
2 ET is defined as the division of the octave in 12 semitones equally tempered and
3 The octave's ratio is established to be 2:1

Then, the simple mathematical model of ET where

Semitone = 2^(1/12)


would perfectly do the work. Even if the “well-documented human ear's desire to hear stretched octaves” (by Tooner) is not satisfied.

With the presence of iH in pianos that model is no more applicable; we must tweak the frequencies calculated by this model in order to get acceptable tunings. That distorts what we understand by ET.

We lack a new mathematical model which includes iH and solves the problem of having several incompatible kinds of an interval. How can we tune an octave if there are 2:1, 4:2, 6:3, 8:4, 10:5, 12:6 octaves, and they are incompatible with each other? How can we tune a 5th if there are two distinct incompatible kinds of them?

Models like semitone = 3^(1/19) or even more complicated, like the formulas used by Mr. Capurso, don't solve the problem because iH is not directly addressed.

We need a new mathematical model where the octave's ratio is no more a constant but a variable value that will fit the piano's iH all along the scale. A new mathematical model where there will be only one type of each interval to tune, namely only one kind of octave, fifth, fourth, etc. A new mathematical model where the iH of each one of the strings in the piano will be taken into account.

Pianos are scaled in a discrete way, there is no continuous curve that can describe iH along the scale. We have sometimes six contiguous unisons using the same size of wire and then the next unison has another size, here we have a jump in iH. The same happens when we get to the wound strings, and to the doublewound strings. Our new mathematical model can not just ignore this, because our ears do not.

I can not think about such a new mathematical model without thinking at Virgil Smith's concepts explained in his book "New Techniques For Superior Aural Tuning".

In that book he says:

"Every note is composed of several different pitches called partials or overtones, but the ear does not normally hear these partials as separate pitches because of its unique ability to combine all the partials of a note into one sound and pitch.
However, many tuners have trained their ears to hear these partials as separate pitches, and to actually hear the beat between two specific partials. Many have found this helpful in learning to hear beats.
However, it is not necessary to hear the pitch of single matching partials to hear beats for aural tuning, because of the ability of the ear to combine all the partials of a note into one pitch.
When an interval is expanded or contracted to produce beats, the ear (when listening to the two notes normally) combines all the partials of both notes into two single pitches, just like it does with one note alone. In addition, it combines, all the beats between the partials into one beat. The beat then comes from all the partials instead of one set of partials.
This beat can be tuned to the desired speed or eliminated completely. This means that beats can be heard two different ways: between single matching partials, and between notes as the ear hears them naturally with all the partials of each note sounding. They both originate with partials, but they are heard in two different ways.
For clarity, one will be referred to as “partial beats”, and the other as “natural beats”. It is important that every tuner clearly understand this, for failure to understand this has lead to much confusion in the past.
Even though each note contains many different partials, the ear hears the note naturally as one pitch and sound. This sound is referred to as the “whole sound”.
Every tuner should be able to hear the beat by listening to the notes naturally – it is the way musicians and listeners hear them – and hearing them this way is the way the final quality of the tuning is judged. The finest quality aural tuning can be accomplished by dealing only with natural beats.
No mater what method of tuning is used, the final evaluation must be with the natural beat heard when listening to the whole sound of each note with all the partials of the notes contributing to the sound. In Some cases, the beat at the single matching partial level is different when all the partials are contributing to the one beat.”

------------- End of Mr. Virgil Smith's citation ---------------

So the “natural beat” concept can be applied to solve the problem of several kinds of intervals sounding simultaneously. It is then necessary to translate mathematically the ability of the human ear to combine all the partials into one sound and one pitch.

If we can translate to a mathematical formula this way of combining several “partial beats” into one “natural beat” there will be only one unique type of interval to tune. No more 2:1, 4:2, 6:3 octaves, nor 3:2, 6:4 fifths, nor 4:3, 8:6 fourths, but only one kind of octaves, fifths, etc.

In that respect I think Mr. Capurso's research although being a true effort to find a new model, missed the target by confusing iH with stretch. I think the solution to this problem is working on iH not stretching the width of the semitone in an arbitrary manner.

I think people like the late Dr. Albert Sanderson (Accutuner), Mr. Robert Scott (Tunelab), Mr. Dave Carpenter (Verituner), Mr. Dean Reyburn (Cybertuner), and Mr. Bernhard Stopper (Onlypure), who have developed the theory, algorithms, software and some of them even hardware of the most advanced ETDs available today, are aware of the lack of such a mathematical model.

The goal is to developpe a mathematical model which is able to combine all the inharmonic partials produced by two notes sounding together into one unique "natural beat". That will give us the ability to calculate real frequencies to tune the piano.

And then Tuning will be no more an Art but a Science.

Gadzar:

I understand how you are looking at the tuning puzzle, but do not know if it is workable. Here are some thoughts and observations on the subject.

First, thank you for quoting Mr. Smith’s book. I had read a number of discussions on this “natural beat” object, but since there was not agreement on what they were, I decided not to buy the book. By Mr. Smith’s description, it is something that can only be experienced, not calculated. But maybe it is not really an aural illusion. I have ideas what this might be.

Just as two frequencies produce a sum and difference beat (the difference beat is used for tuning) I believe that the two different beats also produce sum and difference beats. So if a fifth is beating at, say, 1 bps at the 3:2 partial match and 2.2 bps at the 6:4 partial match, there is also a beat at 1.2 bps and 3.2 bps. Would there be even more beats of beats of beats like a house of mirrors, I think so.

Something I have noticed is the smoothness of an octave when the wide 4:2 partials beats at exactly the same speed as the narrow 6:3 partials. (I can only get an octave tuned precisely this way by ghosting the partials directly.) The beats are there and can be determined with test notes, but I do not hear them. I believe this is because the sum of the beats is exactly twice the beat and the difference is zero. The beats of beats of beats are whole number multiples just like partials of non-iH tones.

I suspect that the purity of Mr. Stopper's tuning is due to getting as close to whole number beat of beat ratios as possible. And how he does this electronically without first sampling the iH of many notes is intriguing. Perhaps his tuning is the model you are looking for. After all, there is no choice for different stretch preferences.

But tuning to satisfy beats, or beats of beats of beats, will only produce a harmonically sounding tuning. There are other things that the human ear hears. For sure, it hears melodically and I believe also musically. The melodic octave seems to be 1220 cents wide, and is impossible to fully satisfy and still have good harmony. But compromises can, and I believe should, be made for it. And the musical way that the ear hears does not seem to be defined or discussed much. I believe it is simply whether a note is at the frequency that the ear expects it to be after hearing other notes. I have found that 8:2 double octaves work very well to satisfy my musical ear, but they are not always the best compromise harmonically. I believe the human ear does detect iH in single notes and expects other notes to be in the inharmonically correct place when listening musically.

I guess we can recognize each other as “math people” and I understand the need to quantify everything. When I think of my nautical background, the ship could only be in one place at one time, there could only be so many barrels of oil in a cargo tank, and if the ship displaced so much water it weighed so many tons. A place for everything and everything in its place. But is tuning the same thing? Like you, I want it to be, but it is not.

If nothing else, consider two identical pianos. One is tuned favoring harmonic tuning, the other favoring melodic. When sitting down and playing you may prefer the harmonic piano, but at a crowded restaurant (where the beats cannot be heard) you may prefer the melodic piano.

If everyone tuned the same and everyone liked the same tuning, this would indicate there is a universal optimum, and through empirical observation a model could be developed. But since this is not the case, this indicates that there is no mathematical model that will satisfy everyone.

So I have a question. Why do you think we are in need of a “new mathematical model”?

You are maybe right and we do not need a new model to tune, but when we talk about something like ET or the stretch on a piano or something else about piano tuning, we do need a reference in order to speak about the same thing. And the reference we have now is 2^(1/12) which does not correspond to what we are really doing.

The same thing happens with the Braid-White Sequence: everybody uses it to tune but each one in a different way, so the true BW sequence has now disapeared and what survives is a tweaked form of it.

Tooner,

I think a little: Why Einstein developped his Relativity theorie? Because there were things unexplained by Newton's theories.

In the same way, why a new tuning model? Because the existing one doesn't describe the real world as it is.

We human beings are like this. We must know!

Equal temperament is not 2^(1/12) except in some unworkable theory. For all practical purposes, it remains: Beatless octaves, and geometrically increasing beat rates of all other intervals.

If you really want to understand tuning, start by getting to know beats.

BDB, I don't want to be unpolit but let me tell you that I understand tuning and beats more than you can ever dream to understand yourself.

Tooner, here is the proof! When I said the model of ET is semitone = 2^(1/12) there was immediately someone who came tell me I was wrong. You see?

We definitely need a new model!

Gadzar:

So, we need a new model so that everyone will agree? It matters very little to me if people agree on anything…

I find that the BW beat rate model that is based on 2^(1/12) to be very workable because the beat rates, when applied to non-iH tones, produce a variable semitone ratio. BW did the right things for the wrong reasons. It is also interesting to consider that he used 6:3 octaves in the bass, 4:2 in the tenor and 4:1 in the treble and 2:1 in the high treble. This might be a bit conservative nowadays, but not anything like tuning to theoretical pitches.

Here is something you might find interesting. Maybe you already know this, but it still may spark something that we can discuss until Alfredo returns. The emphasis seems to be on the semitone ratio. A little while back I thought erroneously that any ratio larger than 2^(1/12) would produce a Railsback curve. I now realize that any fixed ratio will produce a straight line on a Railsback diagram, not a curve. The semitone ratio defines the slope of the curve at any point. So, for a tuning to produce a Railsback curve, the semitone ratio must be least at the midsection and greater at the extremes.

This makes me think more about Mr. Stopper’s use of a cent based on the 3^(1/19). It seemed silly at first to me to have an entirely new value for a cent, but if it is decided to base calculations on 3^(1/19) being a slope of zero, other calculations would be simpler and there may be less error when interpolating between 12ths.

Tooner,

When you say:



Aren't you in search of a new mathematical model that describes the correct way of tuning ET in a piano?

Of course you meant to be impolite. It is easier for you just to claim that you understand something better than me than it is for you to actually prove it.

The twelfth root of two is not a good model for equal temperament because nobody can hear a twelfth root of two. It is no more than a mathematical approximation. Tuning is about what you hear, not about numbers. There are other models of equal temperament that describe what you should be hearing when a scale is tuned that way, and they are equivalent. But the twelfth root of two is a derivation which is not useful in tuning practice, although it may be for setting frequencies.

BDB,

That is exactly what I am saying. ET is not described correctly by the model of semitone = 2^(1/12).

Now, we are searching a new model that does it! Haven't you red all what I've said in this thread?

BDB:

I consider the twelfth root of two as vital in aural tuning. Raising it to the power of 4 gives the ratio of CM3s, to the power of 2 - the difference in the beat rate for the M3-M6 test for a tempered 4th. And just like it is - the ratio of the beat rate of chromatic intervals. I admit that I do not hear this in decimal format, though.

Gadzar:

No, I am looking at different ways to describe tunings in general, not to define a correct tuning. I don’t believe there is such a thing, not even a “good” tuning. When someone says something is “correct” or “good” it just means it is what they prefer. It is not like the Deity declaring something to be “Correct” or “Good”.

Oh, and BDB may appear as “dumb as a fox” sometimes….

Jeff D.,

I am looking not for a "good" tuning. I'm sure BDB can do a good tuning without knowing what he is doing. But I am in search of a true standard for piano ET.

Jeff,



That's what I am talking about when saying that the model must take into account iH all along the scale. It can not be a constant ratio it must be a variable ratio. And this variable ratio will depend on the iH of the string being tuned in order to come out with Railsback curve. Such a model can be easyly implemented in an ETD like Verituner which already measures 8 partials of each string while tuning.

But the problem with Verituner is the lack of a mathematical model wich dictates the apropiate strech needed for each note. Verituner solves the problem by using what is called "Styles". One can define a tuning style by stating differents amounts of stretch all along the scale by means of defineing cents or beats in a given type of interval in a given point of the scale. I think there is a limit of 8 stretch points for a customized Style.

So, it is the responsability of the human tuner to design a Style which matches or fits exactly the piano's scale being tuned, if he fails to do a good match, the tuning won't be good.

That is where a new mathematical model can come to help the tuner, the human tuner, so he has not to tweak intervals, stretch, cents and beats.

Gadzar:

Then it seems that what you are looking for is an ETD that does things differently. It also seems that you see the selection of different Styles as being necessary due to different scalings, not for a preference of how a piano will sound. I am not sure what to think. (Was just given a large project with a deadline, so I am a bit distracted.)

But a tuning model does not have to use semitone ratios at all. From a Railsback curve, the cents deviation can be applied directly to the theoretical frequencies. It seems a much simpler way to calculate the frequencies. But then I wonder which partial should be used. The first partial is used for fewer intervals than the fourth partial. Why not just use the piano’s iH to create a Railsback curve for the 4th partial and go from there? I am just musing. This has been figured out by ETD programmers already.

So what do you think this new model would do or be like?

I am looking for a standard mathematical definition of ET which includes iH.

Of course, with that definition one can program an ETD.

I don't see the selection of different Styles as being necessary due to different scalings, it is necessary with the actual model because it is not perfect so we need to change our claculations for each different scale, it wouldn't be necessary with a better model.

The goal of course should be the way the piano will sound in ET.

I don't think that choosing a unique partial, the fourth partial for instance or the third partial, which is also used by 5ths, 8 ves, 12ths, etc., will work. iH can be estimated but there will be strings with irregularities in iH, even partials with negative iH where some partial may have a lower frecuency than its corresponding theoretical harmonic, so implementing a model based on estimations of the iH calculated from a single partial does not seem to be adecuate.

When tuning aurally we hear at the real partials, I think the model can do the same and it can determine tuning frecuencies from all the real partials being heard when playing a note. Just as the "natural beat" of Mr. Virgil Smith does, combining them mathematically in a one beat.

Let me put it that way, when tuning CM3s you are hearing at the 4th and 5th partials, but then you need to tune a 4th in order to get to the following set of CM3s, you need to hear to the 3rd partial. And when you hear at the F3-F4 and A3-A4 octaves, you need partials 1,2,3,4 and 6 at least to achieve good octaves. So when tuning aurally the temperament octave or tenth in this case you are hearing at the 1, 2, 3, 4, 5 and 6th partials. You can not ignore, nor estimate, none of them.

You can not privilegiate a one single partial of any note because the others will beat in an uncontrolled way. So you need a model which treats iH as a whole phenomenom combining all of these partials into one unique result to calculate the frecuency at which a given note will be tuned.

Gadzar:

Maybe there is a vocabulary problem. When we are using the word “model” I am thinking of this definition: “A system of postulates, data, and inferences presented as a mathematical description of an entity or state of affairs.” (Merriam-Webster online dictionary)

A model that I am familiar with is the Hydrostatic Tables for a ship. Before leaving port I would compare my calculations with the ship’s draft and then sometimes ask the Chief Engineer where he is hiding the extra 500 tons of fuel oil! (Actually, I would already have a pretty good idea.)

When using the word "model" are you thinking of an algorithm?

I am not sure that a standard definition of ET, or even a mathematical model of ET, needs to include iH. If we say that the beat rate of intervals should do such and such, and apply this to tuning a piano, the iH will be the determining factor for the frequency of the notes. But the iH is not in the definition or the model, but in the application. The end result, regardless of the piano, would be the same. The intervals would beat such and such.

But this is talking about a model of a tuning, not a model of a piano. A model of a piano must include iH, preferably tabular and not approximated.

And if we wanted to tune a piano by calculating the frequencies of individual notes, we would need both models. Then we could tune by whatever partial is convenient. We would know where the other partials will be and what the beat rates will be.

But what do you think this “natural beat” is and how would it be calculated?

I want all of the above. A model, an algorithm, a new ETD, etc.

The goal is to achieve accurate ET in pianos without having to tweak theoretical values when tuning a real piano because of iH.

About "natural beat", I am still wondering what Mr. Virgil Smith hears!

For me, there is a range, a vast range, when tuning an interval, where it sounds correct to my ears. It could be a little wider or narrower and still sound good. I must use a test to set it accurately where I think it must be.

For example when tuning A4 to the fork. Initialy I sound the fork and the center string of A4, and I tune to a clean, beatless sound. Then, I play F2-fork and compare the beats with F2-A4, and I tune equal beating. So where is the natural beat gone?

Again, for me it is way a large margin left, not accurate enough to make a tuning.

Maybe I am too young as a piano tuner to be able to identify such a "natural beat" with accuracy. But I am sure Mr. Virgil Smith can do it with the required accuracy.