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I would like to make known a discovery that may be of interest to you.

My work as a piano tuner technician over the last 30 years has enabled me to carry out research on sound and beats, leading to the construction of a new model for temperament of the musical scale. The model, which I have called CHAS (Circular HArmonic System), overcomes the flaws inherent in equal temperament, and describes a powerful attractor, as well as a new concept of "purity".

The main results of the research are:

•a scale combining the prime numbers 2, 3 and 5, solving an age-old problem
•a self-contained physical and geometric entity, a perfectly symmetrical and synchronic attractor, determined by flows of beats
•an algorithm enabling construction of infinite microtonal scales and the rewriting of today’s musical scale
•an “s” variable that can produce infinite beats curves, calculate infinitesimal degrees of inharmonicity and predict developments within the system
•a new geometric average deriving from two proportional ratios: one linear and one exponential
•a resonance constant that may be adopted as a new reference standard in calculating the frequencies of partials
•two indications regarding the torsion of a plane and the helix

The system was presented in March 2007 at a conference in Messina, Italy: “Mandelbrot and Fractal Geometry Forty Years Later”. This February an article on CHAS was published by the Research Group for the Teaching and Learning of Mathematics of the University of Palermo (Italy), and can be accessed at: http://dipmat.math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

The CHAS model constitutes a key advance in music as well as in the physics of sound and of related phenomena.

In music, the model provides an algorithm enabling the construction of infinite microtonal scales. CHAS refines the equal semitone scale and solves the age-old problem of combining the prime numbers 2, 3 and 5, to deliver an extraordinarily euphonic and resonant set of sounds. All musical instruments may be tuned to this optimum scale. The system is accurate and natural because it draws on phenomena that are intrinsic to vibrating strings.

In physics and geometry, the CHAS model proves for the first time that flows of beats can determine a perfectly symmetrical and synchronic attractor in a dynamic system. Knowledge of the existence of this attractor may lead to further research into the nature of relationships between the energy of vibrating matter and the beats involved in resonance and interference phenomena. Inharmonicity, which has always been calculated in an approximate way, can now be calculated with infinitesimal accuracy.

I am seeking to promote wider understanding and application of the CHAS model. I would welcome any form of collaboration or support that you may be able to offer.

I hope you enjoy it. a.c.

Alfredo:

I will study the paper with great interest. I am particularly interested in how anything can be a "dynamic, stable and perfectly resonant system." as mentioned in the abstract. Also, I am wondering what a "synchronic attractor" might be.

The theory of piano tuning fascinates me, but lately I am realizing its usefulness is limited in aural tuning. Aural tuning is all about compromises, compromises that can be heard. They don’t need to be theorized to be heard, just listened to and accepted. I am thinking that the theory is really only necessary for designing a mathematical model so that ETDs can make the compromises without actually “hearing” them. And then there is the final limit on accuracy imposed by the pinblock and rendering points. Not to mention what the next passing thunderstorm may do to a tuning!

Thanks for posting!

Alfredo:

I have not finished reading your paper, but so that you have something to consider, check out This Topic . It is about what is called "Mindless Octaves" and may be what your algorithm "(3 − Δ)^ (1/19) = (4 + Δ)^ (1/ 24)" describes.

Alfredo:

I still have not finished reading your paper, and am having a harder and harder time understanding it. Perhaps there is a translation problem. But there is also a math error.

In paragraph 3.3 you state:

(3 − Δ)^ (1/19) = (4 + (Δ * s / s1))^ (1/ 24)
equals:
(3 − (Δ * s1))^ (1/19) = (4 + (Δ * s))^ (1/ 24)

This is no more true than stating:

4^ (1/2) = (4 * (4/2))^ (1/3)
equals:
(4*2)^ (1/2) = (4 * 4)^ (1/3)

Also, you make a big point of the numbers of 2, 3 and 5 being prime numbers, which is true, but do not explain why that is a problem. It is a moot point anyway, because 2, 3 and 5 are the numbers of partials, but are not frequency multipliers due to inharmonicity.

I’ll try to finish the paper and get what I can out of it.

Well observed Jeff.
Maybe a kind of scientific hoax of the category "cello scrotum":
http://www.timesonline.co.uk/tol/life_and_style/health/article5601050.ece

Bernhard:

Still waiting for your paper....

Jeff,

I'm having a hard time understanding the document as well, but the math error that you point out is not necessarily an error. He doesn't say that the two equations are always equal but only "if s is a fraction (s/s1)" and that "the denominator multiplies delta in the left-hand expression so that:" the two equations are equal.

Bob

Bob:

Not sure if I follow what you are saying, but I wondered also how "s" could be a fraction "s/s1" unless "s1" equals 1, in which case the equations would be equal, but why bother? I don't know if Alfredo will respond back or not. But, I don't understand what practical use his paper may have, let alone what he is saying.

Thanks for your interest. No need to rush, better you finish reading the paper, then we'll talk about anything you like. a.c.

I agree. Keep your ears in the mix and listen. In the day-to-day work of tuning and repairing everything from clunkers to concert instruments I find math theory unnecessary and distracting.

Alfredo:

I cannot understand your paper. Can you explain the basic concepts?

I'm always grateful for those who are willing to dive deeply into the math and try new things...its interesting to me, but not terribly relevant immediately to my work as a tuner. Those who are willing to take the time to try new processes and write on the results are, I think, bettering the profession and I thank them/you all!

For me though, I tend to stay with simple, tried methods that don't require my time in research...something there is precious little of in the crazy schedule.

RPD

Hi Alfredo,

I think most of us would be interested in knowing if your theories have changed the way you actually tune a piano, and if what you're trying to achieve in the process has changed at all.

Are there any problems in piano tuning and its practical musical application that your theories seek to remedy?

Jeff S.

Alfredo hasn’t elaborated on his paper. This Topic may die soon. Let me toss out something from the paper to see if anything productive comes from it.

The use of a Chas ratio of 2.0005312… is mentioned rather than the theoretic 2:1 octave ration. But should a fixed ratio be used in tuning at all? And if a fixed octave ratio, and therefore a fixed ratio for each interval, is used what are the results?

Tooner:

About theory and tuning - you may ignore theory and tune aurally in a casual or personal way, or use an ETD without careing what’s behind it. Maybe a simple question of knowledge and consciousness that is up to you.

About tuning and compromise - untill today we (aural tuners) could only think in terms of compromise becouse we had to get by with Equal Temperament and its unjustified premises, two unjustified assumptions that Chas model discards (section 3.0). Chas demontrates that the ratio 12th root of 2 is unsuitable, not only becouse of inharmonicity, but becouse it produces intervals incresingly narrow (12ths, 19ths and so on) together with intervals incresingly wide (10ths, 17ths ecc. - section 4.3 - graph 5). E.T. premises come out to be missleading.

Also the idea of “pure interval” is missleading. For example, the theoretical ratio 19th root of 3 (i.e. pure 12ths) is unprofitable, being an extreme case. Infact 19th root of 3 widens 3ths, octaves, 10ths, 17ths ecc. more than necessary and spoiles the symmetry of beats (sections 3.4 – 3.5).

About mindless octaves . I’ve had a look at the topic, then I visited Bill Bremmen’s site. I can not say whether he tunes Chas or not. Maybe he can tell you/us.

About Chas model’s relevance - Chas describes the precise form of a dynamic set. In section 2.0 you can read:
“Purity no longer derives from a single combination (refering to pure octaves) or from a pure ratio (refering to 2:1, 3:1, 5:1), but from a new set which is pure because it is perfectly congruent and coherent”.

In section 3.0:
“In this set the ratio must be identifiable both in the single elements (frequencies) forming the scale foreground, as well as in the differences (beats) arising from the infinite combinations of its elements, and forming the background. Each frequency or element in the scale must contain and bear witness to this bi-frontal ratio, which is pure in that it is natural, exactly proportional and perfectly synchronic”.

Chas ratio (1.0594865443501…) does not produce a pure interval nor a compromise, it leads to a pure set, pure in terms of beats and frequencies proportions. Tuning Chas you do not go for a compromise, you go for an optimum scale.

Stopper:

I’m sorry, I was not interested in your “cello scrotum”.
I understand you are commercializing an ETD device, at a cost of $ 600. Could I know on wich basis? Does reading about scrotum help?

Bob:

Thanks for your post.

RPD:

Thanks for your encouragement.

Jeff S.:

About problems in piano tuning - mainly, wrong teachings. For example, still today you learn 5ths must be narrow, Chas proves that this is not correct. In section 4.8 – graph 10, Chas model shows how and why the difference curve for ratio 3:2 (as for ratio 9:8) inverts its progression. From the middle-high register upwards, 5ths go purer and purer, to become wider and wider. The inversion of 5ths allow you to widen octaves in the correct mesure. You can immagine how Chas has changed the way I tune pianos.

Tooner:

what do you mean when you say: Alfredo hasn’t elaborated on his paper. This Topic may die soon.?

A fixed and correct ratio, as a standard reference, is more than productive, is the most precious figure that you could ever wish to find. a.c.

Alfredo:

I am glad you have now elaborated, and the Topic has not died.

You stated: “Chas ratio (1.0594865443501…) does not produce a pure interval nor a compromise, it leads to a pure set, pure in terms of beats and frequencies proportions. Tuning Chas you do not go for a compromise, you go for an optimum scale.”

I am gong to play the devil’s advocate to try to get to your basic concept.

The Chas ratio for an octave is only about 5/10,000 larger than 2. The difference between A5 being tuned to a theoretical 2:1 octave above A4 (A440) and a Chas octave is about ¼ Hz or about 1/2 cent. This is less stretch than you would have when tuning a 2:1 octave and taking into account inharmonicity. And speaking of inharmonicity, I have looked for how Chas applies a piano’s iH and have not found it. So, I have to assume that the Chas ratio is used only on theoretical tones and not tones with iH. And yet, one of your criticisms of ET is that it does not account for iH, but neither does Chas, so how is Chas superior? (Please do not take offense, as I said I am playing the devil’s advocate.)

So, how do you actually use Chas to tune?

Alfredo:

I know that you wrote the above in response to Jeff’s post, but since you made it public I hope you don’t mind me commenting on it.

The idea of fifths becoming wide fascinates me and so I worked out the math to see what the result would be with the Chas ratio.

F7 (note 81) is 32 semi-tones above A4/A440 (note 49) and C8 is 39 semi-tones above A4/A440 (note 49). So if we take 440Hz and multiply it by the Chas ratio to the power of the number of semi-tones and multiply that times the partial number we will arrive at the frequency of the partials in question.

440 * 1.0594865443501^32 * 3 = 8387.4
440 * 1.0594865443501^39 * 2 = 8379.2

Since the third partial of F7 is higher than the second partial of C8, even the highest fifth in the Chas system is still a narrow interval.

I'll let Jeff D. discuss the math with you, Alfredo.

For my part, I can say that the old teachings stated the beat rates of fifths should increase as one tuned up the scale. That model was based largely on a decades-old mathematical model, worked out before inharmonicity was understood.

Current understanding leans more toward the idea that the beat rates of fifths should either stay roughly the same as one moves up the scale or decrease, until the fifths become pure and possibly even wide (like you yourself seem to be saying).

My point is that, while your mathematic approach to this issue may be unique, the practical way fifths are understood and tuned -- at least here in America, among those in touch with current thought -- has already changed away from the old model. (My opinion and Jeff D.'s may not be quite the same on all aspects of this issue.)

Anyway, Alfredo, thanks for sharing and good luck.

Jeff S.

J. Smith:

Actually, I think I now understand how fifths could become wide and could probably come up with some math based scenarios incorporating iH to prove it. But I am not so sure that a fixed frequency ratio will cause this to happen, or even sticking to a specific interval type (which I don’t believe will produce a fixed ratio). For instance, if pure fifths are tuned, then of course the fifths never will become wide, even though this is a great deal of stretch and requires about an 8:4 octave in the temperament.

But what I would really like is for someone that says that fifths become wide to explain why and give an example. But something better than the usual ketchup-on-everything, because-of-inharmonicity platitude.

Thanks for bringing up my name. I was ill for 3 weeks in April and my computer also died. I finally got things going again.

I must first say that even though I have a university degree and some grad school, I studied music and foreign languages. I never had any math beyond 9th grade Algebra and 10th grade Geometry. However, I still do retain the basic concepts that I learned. Since I was a liberal arts student at the university, the entrance tests I took revealed that I needed no more math or science, so I never studied any.

As a piano technician, I have viewed the work I do as more of a mechanical skill than anything else. Yes, my musical training was helpful but there are excellent piano technicians who are not musicians at all and who also do not know any higher math.

I would say that I have often observed the very finest piano technicians who could not put into words or describe in writing just how they do what they do. They will tell you that tuning is an art and that excellence is achieved by practice and perseverance. Few of us would argue that.

What I know how to do has been the result of reading, watching and listening to many diverse sources. I have taken ideas from here and there and put them together to formulate the set of knowledge and skills I possess today.

The idea which I called "mindless octaves" is a very simple one indeed. At one PTG Convention, I observed Steve Fairchild RPT demonstrate it but I really didn't understand what he was doing at the time. Only later did I realize that the concept I had hit on purely by trial and error was what Steve had been doing.

I started tuning aurally in 1968 using a C Fork and a Braide White type of 4ths and 5ths sequence. The manual I had said that octaves were to be tuned "pure". The exception was that from on or about C6 to C8, about 2 beats per second should be put in each octave and that was what was called "stretching the octaves.

Obviously, the information I had was crude but from my perspective today, there have been countless others who have basically learned the same concepts. So, I too am interested when anyone can explore and discover any more about tuning than is generally known by most piano technicians. ETDs have been a great help in providing a tool for technicians to tune better than they could using aural skills but more and more people start out using them and know virtually nothing about how they work. They may know of inharmonicity but they rely on the device to solve the problem for them and don't have any idea whether the solution provided could be made better or not.

In 1985, I first heard of the concept of unequal temperament. I attended a demonstration by Professor Owen Jorgensen RPT. I was not at all influenced or persuaded to try any of that at the time. However, about 4 years later, I heard a pianist playing Brahms on a piano at a dealer while I was working on a piano in the workshop. Again and again, I heard beauty I had never experienced before. The experienced convinced me of the fact that there was something else to be tried. Once I did, I was hooked.

The "mindless octaves" idea is nothing more than making an exact compromise between the double octave and the octave and 5th (12th). I use the sostenuto pedal to do it. When I used the idea, I heard clarity and beauty that I had never heard before and the feedback from customers kept me at it. To me, it was such a simple technique that it became habitual, within my muscle memory to perform, I didn't really have to concentrate when doing it, I could be thinking about something else, it was just mindless yet the technique yielded such consistent and perfected results each and every time. I later discovered that the aural technique could be as accurate and consistent as any ETD would provide. I use it today when I set up a custom ETD program and when I conduct a PTG Exam Master Tuning.

The EBVT and its variants developed also entirely by ear. It was difficult for me to find a way to describe it in writing but I knew what I wanted to hear. I had rejected the idea of Equal Temperament (ET) 9 years before I started working on the EBVT. Basically the idea is that yes, ET provides for an absolute compromise that divides and distributes the comma equally and in small increments between all 24 Major and minor keys.

The problem with it is that as a compromise, it actually goes too far because it eliminates any distinction from one key or tonality to another. It ignores the idea that as musicians, we expect each key signature that we use to have its own purpose or distinct quality.

The modern piano is meant to play all kinds of music from all periods in all possible keys signatures. ET certainly allows for that but erases the color. The goal then is to find a compromise that retains a distinct character for each key but does not produce harshness that the contemporary sensibility (ear) cannot accept.

I became well aware that even though most piano technicians thing in terms of and firmly believe in ET and most often know of nothing other that ET, most aural tuners tune it imperfectly. That means therefore that everyone has a tolerance for deviation from ET that is acceptable.

The goal then was to work within that range of tolerance to create a mild version of a Well-Temperament, most often called Victorian style. Create a sequence from A within the F3 to F4 octave that most technicians today find familiar. Make it simple and easy to remember. I found that the Equal Beating concept helped with that: simplicity and the ability to replicate the idea accurately time and again. I also found a serendipitous bonus to that: Equal Beating M3s and M6s as well as other intervals have an uncanny ability to cancel themselves out. A triad with equal beating intervals can sound much "purer" than it really is. Intervals which might be considered too harsh out of context, get "swallowed" in the sound as a whole.

Thus the EBVT and the "mindless octaves" (which is also an equal beating concept) provide for an overall sound that is far more appealing than the most perfected ET can ever hope to be. It has this "crystal clear" sound to it. The "pipe organ" effect is another manifestation of it.

Of course, how good it sounds is my opinion. But it is the only way I have tuned most pianos (except for an occasional other kind of non ET) since I first began working with it in 1992. In my community, there are many fine technicians. Our PTG chapter has 21 members and 18 of them are RPTs. Several RPTs live within a 10 mile radius of me. So, people have their choice of whom to call and the customers of my local colleagues have their reasons for their loyalty. Having said that, I find time and again, year after year, the customers whose pianos I tune tell me that I have made the piano sound so much better than anyone else ever did. They offer descriptions such as "more musical", clarity, brilliance, pipe organ, and on and on.

It is the result of not accepting what I first leaned 40 years ago as the truth and whole truth and never seeking anything beyond it. It is the result of constantly looking for something better, a refinement of technique, applying other ideas, seeking limits, making something new and different from a combination of ideas and techniques.

So, while I have absolutely no idea of what Alfredo is talking about at this point, I say, go for it, I may find something I like after all and it could end up being a way to quantify and describe what could only have been called "art" or "instinct" in the past.

I must get back to work now. Cheers.