The C.ha.s Temperament

micamahler wrote:

For me, yes; I am interested in the subject and even though I have already read the links you proposed, I am happy to hear about it (read: write) firsthand from its creator.

Quoting, then I also liked ...

Thank you for the encouragement.

What is observed when considering two sounds is that every distance from a "pure" ratio (1:1, 2:1, 3:2 etc.), i.e., every "difference" (between a given value and its equal or multiple) translates into a pulsation: in fact, two "different" waves can add and subtract their amplitude periodically, thus giving rise to the infamous "beat". Beats have always been (throughout the centuries) associated with something unpleasant, something to be avoided, a hateful phenomenon because two frequencies (two sounds) do not "merge" as they might in absolute consonance.

I would ask you: if the problem is represented by beats, how can the right solution ever be found without managing those very differences? This is what I asked myself: why try to find a correct scale structure by "cutting out" frequencies, why not instead expect a congruent and "relevant" rule from a formula that can act on the "differences"?

The fact is that historical temperaments, developed before E.T. [Equal Temperament], attempt to solve (or bypass) the problem pragmatically: either through the use of a greater number of notes (and keys) to better reduce and spread the "differences", or by accumulating the differences in "remote" keys to obtain a higher degree of consonance in the main keys, those most commonly used (e.g., C). More than a hundred historical temperaments are counted, each with different weights but sharing a dramatic fact: some interval significantly far from a "pure" ratio, an interval that could be defined as cacophonous or more simply out of tune.

It is well understood that, wanting to modulate in all keys, it was deemed necessary to zero out (at least in theory) every disparity between individual keys, and for this purpose, algebra could be used to distribute the differences across the twelve keys indiscriminately. Upon closer inspection, this might be the meaning of "equal" for the temperament formulated with the "twelfth root of two" (2^(1/12)). The E.T. formula uses a refined mathematical tool to design a "perfect" exponential progression and to "dilute" all differences within the 12-semitone module/measure, preserving the 2:1 ratio between the first and eighth degrees of the scale.

One day I also asked myself: if three thirds (e.g., C-E, E-G#, G#-C) compose an octave (C-C), how is it possible to order the beats of the thirds while simultaneously leaving the octave pure? What is the meaning of the octave in a 2:1 ratio? Perhaps an octave that is a tiny bit increasing disturbs the ear?

With these questions, I undertook my research, working on beats with the idea that it is precisely the differences that give character to every interval, with the idea that temperament did not necessarily have to refer to a single pure interval as a "whole", but rather to a strong set of a Reason that could be called "dynamic" and that holds true as much for scale frequencies as for differences.

I thus began to expand the succession of octaves progressively, trying to maintain also the progression of other intervals, one ear tuned to intonation, the other to the "coherence" of the beats—that is, the fact that for each interval "x", in the chromatic succession x, x1, x2... I could obtain a beat such that (B)x:(B)x1 = (B)x1:(B)x2. By doing so, I mentally drew the "beat curves" relative to each interval, wanting to identify and distinguish each curve.

Then, in fact, something did not add up: the theory told me "descending fifths", but in the high registers, my ear demanded apparently "pure" fifths. One day I understood that the beats of the fifths—that is, the beat curve of the fifths—could not be monotonic; it had to be dual. In other words, at a certain point in the scale, I had to invert that progression, making it so that from "progressively descending", the fifths became, to the ear, "progressively correct". There, finally, I had set aside every arbitrary assumption and could make tunings very similar to each other, even if I could not yet glimpse a new, plausible scale "constant".

I knew (because I felt it) that beats could well be ordered in a geometric progression, and I also knew that the eventual "difference constant" would have to imply two intervals; I also found confirmation that all intervals are linked to each other, interdependent, and that modifying the beat of a single interval is sufficient to modify the whole. And yet... something was still eluding me.

In fact, it happened that my tuning would change, even if only slightly, at the same time I was performing it—more precisely, in the time elapsed between the tuning of the central strings and the unisons of each note. That tiny fluctuation was enough to reduce the beauty of the final form, disordering what just a moment before

it could have seemed like an ideal order.

Thus I learned to manage those small variations, anticipating each time the eventual decline with slightly steeper curves, that is, by stretching the intervals a bit more. For years I consolidated the form both in abstract terms (defining the curves) and by subjecting the beats to a punctual and severe "rhythmic" re-examination. Then, aside from a little study necessary to orient myself in a babel of terms, the rest might already be known to you.

The formula for Chas's harmonic temperament is (3 - ∆)^(1/19) = (4 + s*∆)^(1/24)

The differences relative to two intervals, the octave+fifth and the double octave, are translated into a single difference-value ∆ (delta) which affects both intervals. The ∆ acts as a spreader; in fact, it widens the fourth that we find between those two intervals;

"s" represents the arbitrary variable, a "rational" regulator that allows us to modify the development curves of the form and to find in the theoretical model what we can implement in practice.

A question for Mr. Capurso.

How could today's Bach enhance this temperament? To explain myself better, what instruments would a composer have at their disposal if they wanted to get their hands dirty and write pieces following the laws of this new temperament?

I am also speaking in terms of notation... aside from retuning the entire orchestra... where possible.

How would the wind instruments be managed?

It seems incredible to me to be living in a moment where temperament is being revisited... the last time was many centuries ago.

Thank you Capurso for being among us, I too, like Eagle, am curious to understand how instruments (even electronic ones... keyboards, organs, etc., who knows) will adapt to this new logic.

Eagle wrote:

A question for Mr. Capurso.

How could today's Bach leverage this temperament? To explain myself better, a composer who wanted to get their hands dirty and write pieces following the laws of this new temperament, what instruments would they have at their disposal?

I am also speaking in terms of notation... aside from retuning the entire orchestra... where possible.

How would wind instruments be managed?

Hi Eagle,

the Chas harmonic temperament does not change the structure of the scale; it does not add or remove anything from the classic succession of semitones, so within the scope of "traditional" composition there is no need to introduce new instruments or changes to notation. H.T. optimizes resonance and renews the principle and practice of intonation, whether it involves a single instrument or an ensemble.

In other words, this temperament reorders the pitch so that individual frequencies and the "tensions" of all chords can—uniquely—faithfully lead back to a "natural" order (referring to linear and logarithmic proportions) both melodic and harmonic (referring to the hierarchy of intervals). Little (or nothing) more than what singers and musicians with a good ear—including wind players—are not already able to do and, at least in part, share.

In this sense, reversing the terms, it is perhaps the correct intonation (and thus the correct order of sounds in a scale) that can enhance the composition, much like what happens in both solo and orchestral settings.

The "non-traditional" composer, including microtonal ones, can still seek infinite "variations" in distance between one note and the next through the use of the variable "s."

This link refers to Rachmaninov's Piano Concerto No. 3, for piano and orchestra:

http://www.youtube.com/watch?v=SQ9BYCbJOfs

Aspirante wrote:

It seems incredible to me to live in a moment where temperament is being revisited... the last time was many centuries ago. Thank you Capurso for being among us; like Eagle, I am also curious to understand how instruments (including electronic ones... keyboards, organs, etc., for all I know) will adapt to this new logic.

Hi Aspirante, thanks for the welcome.

It seemed incredible to me for a long time too... to the point that I hesitated quite a bit (for years) before venturing into all of this.

The frequencies of H.T. are in logarithmic order, just like E.T.; keyboards, organs, and so on will not have to change their tuning much, at least in the middle register.

Open content (youtube)

alfcap wrote:

keyboards, organs and more shall not significantly modify the tuning, at least in the middle register.

And in the bass/treble registers?

alfcap wrote:

the Chas harmonic temperament does not modify the structure of the scale, it does not add or remove anything from the classic succession of semitones

Then why call it a temperament?

alfcap wrote:

therefore within the scope of "traditional" composition there is no need to introduce new instruments or changes to notation. T.A. optimizes resonance and renews the principle and practice of intonation, whether it be for a single instrument or an ensemble.

Unfortunately I don't have enough knowledge to understand you; I am talking about the "optimizes resonance" part

alfcap wrote:

In other words, this temperament reorders the pitch of the notes so that the individual frequencies and the "tensions" of all chords can - in a unique way - faithfully lead back to a "natural" order (referring to linear and logarithmic proportions) both melodic and harmonic (referring to the hierarchy of intervals). Little (or nothing) more than what singers and musicians gifted with a good ear - including wind players - are not already able to do and, at least in part, share.

In this sense, reversing the terms, it is perhaps the correct intonation (therefore the correct order of sounds in a scale) that can enhance the composition, much like what happens in both solo and orchestral settings.

A practical example with the notes ... at this point?

alfcap wrote:

The "non"-traditional composer, including microtonal ones, can nonetheless seek infinite "variations" of distance between one note and the next, through the use of the variable "s".

Which would that be?

Aparte from the variable "s", when talking about microtonality, one is talking about a different temperament... that's why I was wondering why to define CHAS as a temperament; in short, it seems to me like a tuning method (is it right or wrong to say so) that can be applied to existing temperaments... am I wrong?"}

Hi Eagle.

@ So why call it temperament?

Just to be clear: a scale can contain a variable number of notes; the semitone scale consists of (precisely) 12 semitones.

A temperament defines the distances between every note. In other words, the notes that make up a (any) scale can be "tempered" in different ways, depending on the method/calculation used to define the ratios between the various notes.

Tuning applies some kind of temperament to the instrument.

By extension, it also happens that the terms "scale" and "temperament" are used interchangeably, or that "temperament" is referred to as equal. English literature also distinguishes "Historical Temperaments", "Unequal Temperaments" (UT's), "Well temperaments" (WT's), "Victorian Temperaments", and naturally "Equal temperament" (ET).

Not long ago I realized how widespread the (erroneous) idea is that E.T. leaves no room for further improvements, so I decided to talk about it (in my English) in a Topic that I titled "Historical ET and Modern ETs":

http://www.pianoworld.com/forum/ubbthreads.php/topics/1724139/1.html

With "optimize resonance" I was referring to "sympathetic" resonance, to say that Chas frequencies are relatable to the partial sounds of every fundamental note. In fact, the partial sounds respond (actually) to a "non-linear" increase, meaning that if we consider any fundamental sound, the harmonic corresponding to its octave is not in a 2:1 ratio but rather a tiny bit higher, and so on for all the others.

@ A practical example with the notes ...?

This link refers to Rachmaninov's concerto no.3, for piano and orchestra:

http://www.youtube.com/watch?v=SQ9BYCbJOfs

Five other recordings can be found here:

http://chas.it/index...emid=44&lang=en

The last one at the bottom of the page is a "homemade" recording on a Steinway A, 1.55 m long, with a laptop.

@ ...through the use of the variable "s". ..."What would that be?

"

The meaning of the variable "s" is described in section 3.3 of the publication:

A NEW INTERPRETATIVE MODEL

OF SOME ACOUSTIC PHENOMENA:

THE CIRCULAR HARMONIC SYSTEM

(CIRCULAR HARMONIC SYSTEM – C.HA.S.)

http://math.unipa.it..._Capurso_09.pdf

@ ..."...in short, it seems to me a tuning method (is it right or wrong to say so) that can be applied to existing temperaments... am I wrong?"

I hope I have clarified (above) the difference between "temperament" and tuning. To be honest, saying that Chas is a temperament might not tell the whole story (but that's not a problem... ). This theoretical model shifts the entire axis (alas) around which dozens and dozens of temperaments have been developed (over the centuries); in fact, it moves from a raw "cutting" of frequencies to the subtle "proportion" of differences, and is finally free from any arbitrary assumption.

http://www.pianoworld.com/forum/ubbthreads.php/topics/1724139/1.html



Open content (youtube)

http://chas.it/index...emid=44&lang=en

Five more recordings can be found here:

...

I saw that the link didn't open the page, I'll try again:

http://www.chas.it/index.php?option=com_content&view=article&id=64&Itemid=44&lang=en

The last one at the bottom (on that page) is a "homemade" recording on a Steinway A, 1.55 m long, with a laptop.

Question, can CHAS also be applied in the context of quarter-tone tunings?

Xenakis wrote:

Question, can CHAS also be applied within the scope of quarter-tone tunings?

Hi Xenakis, the Chas algorithm can calculate both a "scale ratio", i.e., a fixed multiplier that will space the sounds regularly, and single ratios referring to successive "steps" that are not necessarily equal to each other.

From the basic formula (s = 1), the scale ratio 1.0594865443501... is obtained; the scale frequencies are obtained by multiplying (or dividing) a value/frequency (and subsequent ones) by that number.

Referring to quarter-tone tunings which we call 24 TET, i.e., a logarithmic (equal temperament) scale in which the octave is found not after twelve semitones but after 24 quarter tones, it is simple to apply the Chas principle (3 - ∆) = (4 + ∆); one just needs to replace the 19th and 24th roots with the 38th and 48th roots.

The basic formula,

(3 - ∆)^(1/19) = (4 + s*∆)^(1/24) therefore becomes:

(3 - ∆)^(1/38) = (4 + s*∆)^(1/48)

in the case s = 1

we find (naturally) the same value ∆ = 0.0021253899646… but (obviously) a different

incremental ratio = 1.029313627788…

Let me know if I understood your question correctly and/or if something doesn't add up.

Happy Sunday

Thanks alfcap, interesting.

I also have a curiosity, what are the basic questions that drove your research? What was it that triggered the "spark"?

Xenakis wrote:

Thanks alfcap, interesting.

I also have a curiosity, what are the basic questions that drove your research? What was it that triggered the "spark"?

Hi Xenakis,

perhaps the spark was the strong disappointment I felt the first time, after tuning thirteen notes… I thought I had a good ear… I used to tune my guitar without too many problems but, one thing after another, the result on the piano was a disaster.

And you, do you write music?

Alfredo

I am also reporting this document

http://www.luciocadeddu.com/tesi/Cannas_triennale.pdf

Very interesting document, which I will study as soon as possible.

To tell the truth, I am allergic to mathematical formulas even though I understand their importance. But I am here to ask, in practice, if today I agree to recognize, for example, 7 beats per second for thirds, sixths, and tenths, with a pure octave and one beat per second for fourths and fifths, using this C.H.A.S. method, how many beats must I identify in the proposed intervals: third, fourth, fifth, sixth, and octave?

Francesca wrote:

To tell you the truth, I am allergic to mathematical formulas even though I understand their importance. But I am here to ask, in practice, if today I agree to recognize, for example, 7 beats per second for thirds, sixths, and tenths, with a pure octave and one beat per second for fourths and fifths, using this C.H.A.S. method, how many beats must I identify in the proposed intervals: third, fourth, fifth, sixth, and octave?

The issue of beats is addressed in this document, I don't know if you have already had a chance to read it....

Trade1 wrote:

I also point out this document

http://www.luciocadeddu.com/tesi/Cannas_triennale.pdf

Anyway, I sent a message to Alfredo Capurso

Trade1 wrote:

The issue of beats is addressed in this document, I don't know if you have already had the chance to read it....

Anyway, I sent a message to Alfredo Capurso

I have read this document but I am struggling to understand its practical application, and this is certainly my own limitation, but since listening to the recordings I liked this tuning very much because I find it extremely rich in harmonics without causing disturbance at the same time, I would like to understand how to apply it. I would like to be able to experiment on this path. I await news from the Maestro. Thank you for your promptness in responding!

I too, I confess, am allergic to formulas. Or rather, I am sure of the demonstrated theoretical validity and I respect the solutions. I believe, however, that to learn how to tune, after a good theoretical and musical foundation, one must go into the field and observe that two instruments, even with very close serial numbers, can have different inharmonicities. Even the bridge cut or the "climbing" of the same can cause inharmonicity. False beats are also a curse for the Tuner and are very frequent. In a high-end grand piano, the A 440 can be inharmonic, while in the upright of the same house... no. Experience teaches much and is difficult to transmit. One can transmit notion and not knowledge. Knowledge is notion + experience. Sometimes... actually often, experience corrects many enunciations of notions significantly, and are therefore precious and indispensable. I have already told many anecdotes about Riccardo Orsini, a great Tuner and my Master. He did not rely on any scientific theory, but Rubinstein, when playing in Rome, always wanted him backstage. He had saved the recording of Chopin's Mazurkas, by tuning the Steinway at the RCA on via Tiburtina in Rome the right way. That recording exists, made in one afternoon without interruptions!!!

So, within my limits, and with respect for theories, I believe that the ear is still the severest judge regarding relative frequencies. Every piano, unfortunately, has different problems regarding inharmonicity. Also often some major design errors that must be compensated for.

For this reason, I believe that, really, one can very well teach how to regulate, voice, and repair a piano... but less so how to tune. In any case, everyone can and must try to realize it. After a certain number of instruments worked on, close to, as said, the basic notions, results will be obtained. It will be useful to compare with other scholars and submit the tuned instrument to their judgment. It's a pity that this happens little or almost not at all. Listening must be done live and not via recording. This I think... as a man of the twentieth century.

Francesca wrote:

To tell you the truth, I am allergic to mathematical formulas even though I understand their importance. But I am here to ask, in practice, if today I tune recognizing for example 7 beats per second for thirds, sixths, and tenths, with a perfect octave and one beat per second for fourths and fifths, using this C.H.A.S. method, how many beats must I identify in the proposed intervals: third, fourth, fifth, sixth, and octave?

Thank you, Trade 1.

Hi Francesca,

I need a tip: what instrument do you play, or... which instrument would you like to tune? Do you already have some experience?

Hi pianoexpert,

I agree, theory 'in itself' is not enough, and experience and a good 'ear' can lead to excellent results.

In my case, (along with practice) I thought I had to 'go through' the theory in order to be able to share a different and numerically describable approach.

You know how it goes, our 'feeling' remains debatable, numbers... less so.

A cordial greeting, a.c.