Posted by: alfredo capurso

## CIRCULAR HARMONIC SYSTEM - CHAS - 05/07/09 04:17 AM

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.

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.