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Alfredo and Isaac:

I am not talking about how a tuning sounds. I am talking about the validity of tuning theories.

Fixed frequency ratios are not valid for inharmonic tones. That is why the Railsback curve is not a stratight line.

I have to say that I'm not so sure about that Scala file for Pianoteq: it's based on just one octave, with an offset of 1 cent per octave. In other words, it's a Scala file instead of a tuning.

I hear the beats on the octaves, but I'm not sure I would recognize the sound as similar to the recordings I've heard using CHas as the spine for the tuning.

One problem is that, using the scala file, the 3:1 checks on 12ths from A4 (which is A3 in PianoTeq) don't show an accurate result. And of course, since the Scala file just repeats the intervals with the subtraction or addition of a cent, the gradual widenings and narrowings of intervals are left out. Everything just has a cent subtracted or added to it at each octave.

It may be necessary to use PianoTeq Pro's Detune Note Edit pane and Spectrum profile NE pane, tuning by ear and by partial checks, along with the Unison detune NE pane, to get closer to the aural tuning that sounds like a CHas-based tuning.

Jake:

Are you listening to harmonic or inharmonic tones?

Jeff:

EDIT: You guys may have moved on to discuss other things, so I'm sorry if I'm responding to an earlier concern. But here's what I understand about the earlier concern:

I'd rather stay out of this battle, but I worry that there may be a mix-up. My impression is that you and Alfredo are sometimes talking about apples and oranges. His CHas is a theoretical variation on theoretical ET and focus, to me, seems to be on harmonic partials. He's not arguing for using fixed freqs in an actual tuning. In other words, my impression is that his temperament bears the same relation to a tuning as theoretical ET bears to a tuning--it's the map, but once he leaves the Mathematics Department, the trip is taken by listening to the piano to get the beats he wants and by doing partial checks. In the academic document, on the other hand, he's arguing for the general, theoretical point of departure. (If recordings of a piano tuned using the two octave approach had appeared earlier, would the entire discussion have gone better?)

In the version of the theory that I recently read, he doesn't claim that the temperament is a universal cure for IH. The claim is more along the lines of--conventional ET based on 12 repeating steps encourages inharmonicity, but if you use 24 steps, which sharps the notes just slightly from conventional ET, IH tends to naturally get reduced, and many partials beat in a sweet way. The math then demonstrates why the beating happens. I think the original doc may have started with more math, and then gone from there, so that the impression was that a formula was being imposed on a piano. In a way, it is--he's writing for a Math Department. But the math was arrived at from experience--it essentially explains why some partials match up and how some beat in a pleasing way if you use a different multiple for creating the theoretical intervals. (And then go in and tune the damned thing--but he can't say that in a math paper...Forgive me for this simplification, Alfredo. Is it roughly accurate?)

Sorry to wade into this entire battle.

Jeff,

We're cross-posting. Were you asking if I was listening to harmonic or enharmonic partials when listening to the sound based on the scala file in PianoTeq? Actually, I was going in and looking at the partial structure in the Spectrum note edit pane. (I don't know if you know PianoTeq. The pro version lets you repitch a note, or load a Scala file, and see the freqs of each partial. The older version of the program didn't always accurately show the freqs. The latest one, released about a week ago, does.)

That's really why I think that a manual tuning is best in PianoTeq--if you open the Detune NE pane and the Spectrum NE pane, you can change the pitch of each note and watch the effect on the freq of each partial. It's a little more complex, than that, really--since you can control the length of the "string," you can in a sense design the IH, and thus the effect of the pitch change, on each note.

Jake:

Thanks for the response. You probably do not want to get into this “battle.” There are disagreements on a number of levels…

I believe I understand what you are trying saying about a theoretical tuning being a map that is departed from when actually tuning. I call this doing the right thing for the wrong reason.

No, I do not have PianoTeq. And we should be careful to not confuse the word enharmonic with the word inharmonic. It was probably just a typo. If the frequencies of the upper partials that the pane display are exact multiples of the first partial, then the tones are harmonic. If they are not exact multiples I would think that somewhere the iH for each note is displayed.

The reason I ask about if the tones are inharmonic is what this can mean to the relationship between octave types and the justness of 12ths. The “more iH” (meaning either the value for a particular note is greater for a given logarithmic iH slope or the logarithmic iH slope is less steep for the given value of a particular note) the narrower the octave type can be with the result being just 12ths. For example: a “high iH” piano may produce beatless 12ths when tuned with 4:2 octaves in a particular part of the scale while a “low iH” would need to be tuned with 6:3 octaves to produce beatless 12ths in the same part of the scale.

The difficulty with using frequency ratios as a “map to take a departure from” is that taking the 19 root of 3, for example, and saying this will produce beatless 12ths does nothing in predicting the resulting octave type which will be a function of iH. And the same problem happens when using a “map” with equal beating 12ths and 15ths. The result depends on the piano’s iH, and the frequency ratio is really irrelevant and even misleading.

Yes, I meant "inharmonic," of course.

Pianoteq doesn't display the IH of a note, in the sense of determining the spread between some partials, or the difference between some partials and their idealized Fourier harmonic, and then performing a calculation that gives results such as IH = X. It instead just shows the freqs for each partial and lets you change the tension and\or length of the string. (I'd actually like to learn more about the way IH is calculated as a value--don't different people\companies\ETD's use different formulas for the calculation?)

Are you saying that he should discuss the details of IH --the way it varies with piano and string and octave and note--in the formal paper? I'll really have to look at paper again to review the discussion of IH.

Is it the use of the root that is worrisome, since it creates a formula that neglects IH as a variable? (But theoretical ET divides the octave up and ignores IH as a variable?)

If he spoke of a range of possible results, instead of absolutes, would his argument be stronger-a range of results for modern, well-designed pianos? I don't really know at all--I'm far from being qualified to evaluate the argument. (Let alone understand all of the math...)But let's remember--thinking aloud, here--that the formal paper states the argument only for the 24 note temperament, and not for the entire piano, so the variation might be fairly wide, but less wide than the variations caused by IH in ET over the same notes? That could become an entirely different paper--the testing of the theory on many, many pianos, instead of providing the mathematical proof required by a Mathematics Department (which is very different from the proofs required in physics, biology, etc...), which would generate the range of variations.

I don't know, really.

Jake:

I don't know, really.

No better place to start!

Sorry, I do not want to discuss Alfredo’s paper anymore. Everything I have to say about it has already been said. You can look it up if you care to.

Not too long I could also say, I don't know, really. Let me give you the path that I followed and how I look at the effects of iH.

Inharmonicity is explained in:

http://www.afn.org/~afn49304/youngnew.htm#ftnote8

And here is a link to the iH curves for a number of pianos:

http://www.goptools.com/gallery.htm

Wikipedia has some good pages on cent to frequency conversions and on inharmonicity.

Inharmonicity is the difference between what the fundamental pitch of a string should vibrate at when ignoring the stiffness and what it actually vibrates at in cents. Each partial is higher in pitch than exact multiples of the theoretical fundamental by the iH in cents times the square of the partial number. Higher partials are affected by iH much, much more than lower partials.

The iH of the strings of a piano are not all the same. For the unwound strings the iH doubles at around every 8 semi-tones for large pianos and less than doubles for smaller ones when going up the scale. For the wound strings the lowest notes have more iH than the higher ones.

There are three ways that iH effects tuning. First, the octaves are tuned wider than a 2:1 ratio. This also makes the cents between the fundamentals of intervals greater than theoretical. Second, the higher the nearly coincidental partials are in an interval the more iH causes them to beat as if they were narrower than they are. Third, since iH is a logarithmic curve, the effect is of wider intervals to beat as if they were more wider than narrower intervals. The combined effects are largely self cancelling on well scaled pianos.

I better stop here for now. But Let me also mention that the general beat rate of some intervals more than double each octave and others less than double.

Hello Jake, this occur (the correct application of the formula by "Nikkos" in the scala file) because of your interest in tunings and temperaments, so thank you for your opening of mind.

Philippe Guillaume, the pianoteq developper, was a piano technician before graduating to a doctorate in maths and begin to teach Applied maths.


ALfredo will answer on that later, but at first he said that Nikos (who made the file) correclty applied the theory, and understood the value of s (S=1).

The Chas ratio is provided on an octave, which is copied afterthat all along the scale.
If I understand well, when stretch = 1 in Pianoteq the inharmonicity is corrected at the 2:1 level. (makes a very compact ET, the stretch added afterthat apply out of the temperament zone , as in tuning, hence the little change in harmony. Listening to Rachmaninoff while passing from standard ET to Chas is an amazing experience ! I even believe that the scala file make the piano tone more realisitic.

If any difference exist they may be due to the non use of 3d and 4th partial. But immediately a first listening the Chas effect is noticeable, as a nice opening of the tuning (even with harpsichord tone).
To be complete, some colleagues are tuning with that kind of tone, probably not managed similar but in the same kind of harmony. For what I understand the ones that learned the Cordier "fall" in that kind of tone (while making the tuning with 3:2 5ths) because of the way the piano settle. This hypothesis is all possible and we will try to verify it in march.

Have a good day, all !

Isaac

Jake,

Your way to put Chas and my intentions into words is correct, right from the start when you write:

..."CHas is a theoretical variation on theoretical ET and focus, to me, seems to be on harmonic partials. He's not arguing for using fixed freqs in an actual tuning."...

Likewise, in writing:..."his temperament bears the same relation to a tuning as theoretical ET bears to a tuning..."

And..."conventional ET based on 12 repeating steps encourages inharmonicity, but if you use 24 steps, which sharps the notes just slightly from conventional ET, IH tends to naturally get reduced, and many partials beat in a sweet way. The math then demonstrates why the beating happens."...

..."But the math was arrived at from experience--it essentially explains why some partials match up and how some beat in a pleasing way if you use a different multiple for creating the theoretical intervals. (And then go in and tune the damned thing--but he can't say that in a math paper...)"...

For your ways and your open minded (and mind opening) approach, I appreciate very much your valuable contribute. Thank you for your analysis and your elaborations (not last, for all those question marks!).

Jeff, you write:

..."I am not talking about how a tuning sounds. I am talking about the validity of tuning theories."...

I kindly ask you to acknowledge some of my research's targets:

- how and why a tuning can sound better
- how Temperamental Theory and models can be improved
- how and why improved aural piano tuning practice can be related to modern theory

To me, it seems that you would prefere to talk about the non-validity of tuning theories.

Generally speacking, being prejudiced against theories and/or models can make impossible to share interest and efforts for further investigations (on theoretical and practical aspects), as it can make impossible to share even the practical evidencies that have led to the elaboration of one precise model, in this case Chas Temperamental model.

In my opinion, today Pianoteq may represent a great help for deepening some temperament and aural tuning issues, including your favorite questioning about iH.

You may then put your urgencies aside for a little while and, if you like, elaborate towards the application of Modern ETs models. How about recording and share some of your practical checks and tests?

Isaac, thanks a lot for the amount of energy and elaborations you are devoting. I hope to be able to contribute on Pianoteq settings and evidencies too. What can I do?

Regards, a.c.

.

Only when the theories are invalid.

Jeff, it was about three months ago when I addressed you to some simple reading:

http://en.wikipedia.org/wiki/Theory

...“Theories are abstract and conceptual, and to this end they are never considered right or wrong.

Instead, they are supported or challenged by observations in the world. They are 'rigorously tentative', meaning that they are proposed as true but expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation... 

...Sometimes two theories have exactly the same explanatory power because they make the same predictions. A pair of such theories is called indistinguishable, and the choice between them reduces to convenience or philosophical preference...

...The form of theories is studied formally in mathematical logic, especially in model theory...

...Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number),geometry (concepts of space), and probability (concepts of randomness and likelihood)...

...In science, generally, theories are constructed from elementary theorems that consist in empirical data about observable phenomena. A scientific theory is used as a plausible general principle or body of principles offered to explain a phenomenon...

...A scientific theory is a deductive theory, in that, its content is based on some formal system of logic and that some of its elementary theorems are taken as axioms. In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory.

...Theories are intended to be an accurate, predictive description of the natural world.

Theories as models
Main article: Scientific model

Theories are constructed to explain, predict, and master phenomena (e.g., inanimate things, events, or behavior of animals). A scientific theory can be thought of as a model of reality, and its statements as axioms of some axiomatic system. The aim of this construction is to create a formal system for which reality is the only model. The world is an interpretation (or model) of such scientific theories, only insofar as the sciences are true.

Theories in physics
In physics the term theory is generally used for a mathematical framework—derived from a small set of basic postulates (usually symmetries—like equality of locations in space or in time, or identity of electrons, etc.)—which is capable of producing experimental predictions for a given category of physical systems. A good example is classical electromagnetism, which encompasses results derived from gauge symmetry(sometimes called gauge invariance) in a form of a few equations called Maxwell's equations.

Intertheoretic reduction and elimination
Main article: intertheoretic reduction

If there is a new theory which is better at explaining and predicting phenomena than an older theory (i.e. it has more explanatory power), we are justified in believing that the newer theory describes reality more correctly. This is called an intertheoretic reduction because the terms of the old theory can be reduced to the terms of the new one.

For instance, our historical understanding about "sound," "light" and "heat" have today been reduced to "wave compressions and rarefactions," "electromagnetic waves," and "molecular kinetic energy," respectively. These terms which are identified with each other are called intertheoretic identities. When an old theory and a new one are parallel in this way, we can conclude that we are describing the same reality, only more completely.

In cases where a new theory uses new terms which do not reduce to terms of an older one, but rather replace them entirely because they are actually a misrepresentation it is called an intertheoretic elimination. For instance, the obsolete scientific theory that put forward an understanding of heat transfer in terms of the movement of caloric fluid was eliminated when a theory of heat as energy replaced it."

May I ask you now not to be misleading? Or else, would you start one more Topic on your personal outlooks? You know, I'd rather think of you as a reliable poster.

(O): Keep on tuning the way you do, you can be happy. Keep on wondering, you can still be happy.

a.c.

.

Alfredo:

As a young man I played trombone in circus bands. Whenever the magician wanted to hide what he was really doing he used smoke and mirrors. That is what your diversions on just what a theory is and how your tuning sounds are. They are just diversions from what I am saying, which is what you want to hide.

Your mathematical explanation of how you tune does not work due to iH. We both know it but you try to hide it.

But there is very good news. Many great tuners have been unable to explain how they tune, mathematically or otherwise. You may be one of them.

(O). Many of us might be unable to explain anything, some of us are able to doubt and share.

I've found interesting this short reading, style and format too.

http://books.google.com/books?id=4V...mp;as_brr=1#v=onepage&q=&f=false

It is a 1835 Essay on Temperament, you may like it too.

a.c.

.

Thanks, Alfredo:

I am not sure when I will get a chance to read though it. I like the idea that it was written by the head assistant on the nautical almanac establishment. From a quick glance, it is written much like the many navigation treatises I have read. I will probably enjoy it for that reason if for no other.

Seems to be a defense of straight ET, but I'll have to read more. Love the opening of chapter one:

"Sound is an affection of the mind"

Hello.

Yes Jake, I liked that too.

By reading also Claude Montal's work (Paris, 1836)

http://books.google.fr/books?id=PIw...r&cad=4#v=onepage&q=&f=false

kindly linked for us by a French colleague, again we can conferm our understanding about why, when we say "temperament", we still think about a suffered "compromise".

Also in Wikipedia,

http://en.wikipedia.org/wiki/Musical_tuning#Tuning_systems

"Systems for the twelve-note chromatic scale" (scroll down)

We still read: “It is impossible to tune the twelve-note chromatic scale so that all intervals are "perfect"; many different methods with their own various compromises have thus been put forward.”

Claude Montal writes (page 41): "L'octave seule doit etre rigoureusement juste...".

More recently, Serge Cordier would have written the equivalent of: "La quinte seule doit etre rigoureusement juste".

Six years later Bernhard Stopper would have written: "La douzième seule doit etre rigoureusement juste".

Today Chas temperamental theory does not favor any interval and does not limit the temperament's module to 13 (pure octaves), 8 (pure 5ths) or 20 (pure 12ths) notes.

Simply out of consciousness, Chas can be acknoledged as a geometric entity where all intervals are in reciprocal function, a self-bearing beats-architecture.

The "pure" concept is so renewed and the approach to the sound scale is shifted from a static ratio, like the one for any “beatless” interval, to a dynamic beat-ratio for an infinite set, stable and yet evolvable.

Then, our practical tempering does not need to be that kind of compromise, it can be the weaving of an ideal web.

Regards, Alfredo.

Reading further in the Woolhouse book, I see that he starts off by explaining an ideal ET, but then elides towards a well tempered tuning, with even a chart listing the best keys. Still haven't finished, however.

Odd to see how the various intervals have been favored.

A bit off-subject, but while we're discussing older theory: Here's a link to Barbour's Tuning and Temperament at the Internet Archive (registration may be required to download the pdf). Looking over the table of contents, I see that he doesn't even use the term "well tempered," instead devoting a chapter to "irregular systems":

http://www.archive.org/details/turningandtemper027139mbp

(Whoever originally uploaded it didn't proofread.)

To listen to correct use of HT on modeled historical pianos (forte and such).

Some of the recordings are convincing, despite some predictability in tone.


http://www.pianoteq.com/discover_pianoteq3?type=pianoforte

Thanks for your comments.

Generally speaking, I see tuning a piano like this :

On one side you have the piano harmony and resonance.

On the other side you have the need to obtain a congruent system to play in all keys.

(ther is a 3d aspect which is the room acoustics, but I will lay it aside)

If you tune the piano and the tempering method used is very strong, acoustically speaking, it may fight or modify somehow the harmony of the instrument (find it was the case with the Cordier tuning, where the relation between intervals is more heard than the piano resonance itself, probably the same occur when I listen to a Stopper tuning.

It is the same with a standard tuning which is highly stretched , at the edge of the octave beat, the stretching method is to me more "in front" than the natural harmony, that may be help the tone to be more crisp but it fight the quietness of listening.

The same happens with Chas, but by chance the resonance point and the congruence of intervals seem to match more in that case, and the crispness of tone, be it due to the Chas or to the extra tension given to the tuning pin and small wire segment , is felt as natural and "just" (musicians opinion, my brother wife , harpist even told me "at last a piano that sound tuned!")

We have no idea on how other instrumentalist perceive the pianos justness, they seem to accept what they find and try to match it.

I guess that some of the "mystery" of CHAS lies in the wy the 4th and the 5ths are reconciled thru the double octave.

I've find that the RBI intervals differs in beat rate depending of the instrument iH, as on other tuning methods, so it is not a method that use a "standard" beat rates set"

What I find funny is that the Chas is said to provide a better progression "at the expense of the octave justness". in fact I find that the octave is enhanced by the tuning, at last at the piano. it is probably slightly less stretched in the mediums but the progressiveness is then nicely going up and down. the double octave is at 0.98 cts higher (plus the ih correction automatically used when tuning).

The drawback is that resonance is very present, as if the sustain pedal was used more. the harmony is then sounding slightly closer, or smoother, the correct "charging" of the pin-block may well be necessary to keep the correct enlightening of the melody (but may be not, I will experiment with close unisons whenever possible)