Appassionato,

Thank you for your interest and for trying to contribute to music.

I'd better say immediately how Chas renews the age-old static approach. Chas Theory approach to temperament is dynamic.

Chas shifts the hub of the matter and, perhaps for the first time ever, describes how our semitonal scale can be ordered in a sound-whole, by beats. This new approach is then featuring three main “keys”: Beats, Proportions and Time.

During the whole history of temperaments, previous models have elected one single interval - the octave first and more recently the fifth (Cordier - 1981) and the 12th (Stopper - 1989) – fulcrum of the scale, in other words one of those intervals should sound “pure”, i.e. beat-less, and the frequencies were derived from this axiom.

Chas describes a dynamic system and a dual ratio: all intervals can beat, and their beat-ratio can define and order the scale-ratio and frequencies values.

Our semitonal scale, so far calculated on the basis of one pure interval, can now be represented as a “beating whole”, a pure whole in that sounds frequencies and beats find their closest interrelation, their biunivocal correspondence, the “in Time” proportion.

Chas is not anymore featuring a compromise but a fusion that can reach the absolute integer. In fact, all partials can share a “difference-ratio”, the 1:1 beat-proportion which now defines the scale exponential ratio. By modifying this beat proportion we can re-find past settings, or create new ones. This may prove how this modern theory is assumption-free. Our numerical generator - one (1) - can order our logarithmic scale together with primes 2 and 3, and still we could arbitrarily variate that “tare” (the 1:1 proportion) and calculate either an old or a new scale, ET or note by note.

From an “economic” point of view, it ends up that we loose one pure interval in favor of an integer and perfectly coherent whole, that we can dismiss a static, unjustifiable zero-beat approach in favor of a thoroughly dynamic one. We can dismiss a limited module - be it 7, 12 or 19 semitones - and go for an inter-modular, infinite set.

Conceptually, we can overtake the one-interval-supremacy idea and elaborate on the infinite beat-relations (you could read colors) of a dynamic sound whole. Beats can open to synergy and lead towards new degrees of harmonic resonance. More is linked below.

Stopper, you write:..."The fact that you want to ask other technicians about the work of three authors working on equal temperament improvements and not the authors directly (we are all three still alive), is very unusual for a scientific and objective documentation about theories and makes me assume that..."

Sure Cordier is still alive? Have you already posted your tuning sequence? How do you justify "beat-symmetry" and "zero-beat" 12ths? What is the meaning of a 19 semitones module? Let's tell each other, we are still alive, perhaps now it may be Time.

Regards, a.c.

Discussion in PW:

http://www.pianoworld.com/forum/ubbthreads.php/topics/1438536/1.html

http://www.pianoworld.com/forum/ubbthrea...DERN%20ETs.html

CHAS tuning practice in PW:

http://www.pianoworld.com/forum/ubbthreads.php/topics/1326050/1.html

CHAS MP3 - Amatorial recording on a Steinway S (5’ 1”, 155 cm)
http://www.box.net/shared/od0d7506cv

CHAS THEORY - RESEARCH REPORT BY G.R.I.M. (Department of Mathematics, University of Palermo, Italy):
http://math.unipa.it/~grim/Quaderno19_Capurso_09_engl.pdf

Would that be "the smallest positive integer not nameable in under eleven words"?

Kees

I do not think so. Nice paradox though, thanks.

Regards, a.c.

I saw someone display their "absolute integer" in heavy traffic the other day...

Signore Capurso,

we may take up further discussions if you can negate (or find someone else doing for you) the three falsification points i provided ealier in the CHAS thread about your CHAS paper: (link below)

http://www.pianoworld.com/forum/ubbthreads.php/topics/1340096/Re:%20CIRCULAR%20HARMONIC%20SYSTEM%20-.html#Post1340096


Bernhard Stopper

Bernhard, Stopper, Signor/Herr/Mr. Stopper, do you have a preference? For me it does not matter, you can call me Alfredo.

I shall reply in Chas thread.

Regards, a.c.

I don't think he's aware that he's posting on a forum where the people developing these temperaments discuss them. He's just recently heard of these temperaments, and is assuming that technicians in general will be able to tell him more, in the same way that they might explain ET.

just pledged! the digital download is nice for people like me living abroad, though i think the step of 35$ from 15$ to 50$ is a bit tough if you're interested in the bonus material.

Thank you everyone for providing the information and supporting the project! We have been shooting and are still shooting for the next few weeks; apologies for not posting earlier.