"Music is a hidden arithmatic exercise of the soul, which does not know that it is dealing with numbers, because it does many things by way of unnoticed conceptions which with clear conception it could not do. Those who believe that nothing can happen in the soul of which the soul is not conscious are wrong. For this reason the soul, although not realizing that it is involved in mathematical computation, still senses the effect of this unnoticeable forming of numbers either as a resultant feeling of well-being in the case of harmonies or as discomfort in the case of disharmonies."
Pythagoras laid the foundation for a holistic science which integrated spiritual, ethical, and metaphysical, as well as practical techniques. Pythagoras is famous for his axiomatic viewpoint that "there is geometry in the humming of the strings. There is music in the spacing of the spheres." From Plotinus we hear, "All music based upon melody and rhythm, is the earthly representative of heavenly music."
And from Sufi Hazrat Inayat Khan,
"When one looks at the cosmos, the movements of the stars and planets, the law of vibration and rhythm, all perfect and unchanging, it shows that the cosmic system is working by the law of music, the law of harmony; and whenever that harmony in the cosmic system is lacking in any way, then in proportion disaster comes to the world, and its influence is seen in the many destructive forces which are manifest there. The whole of astrological law and the science of magic and mysticism behind it, are based upon music."
Pythagoras systematized the laws which allow the creation of stringed instruments: musical scale intervals (octaves, fifths, fourths, thirds). He recognized that these fundamentally abstract relationships pervade all creation--even matter irself. In music, as in nature, a wave is a shape in motion. Each note has a wave-shape.
The octave comes from exactly doubling, or halving the string length, that is in 1:2 proportion, while the harmonious fifth has a 2:3 ratio and the fourth 3:4. There is also the less obvious 4:5 interval of the third, and even less obvious consonances.
Any tone in the overtone scale is higher than the preceding tone by precisely one whole number. These are the so-called harmonics. The lower the proportions of the numbers, the stronger the consonance, the more harmonious the sound of the two tones together. The primal polarity ratio of 1:2 is the most harmonious to our ears which are biologically geared to seven basic laws of harmony based on the primal law of whole-number quanta (which prevails in physics as well as music):
THE SEVEN LAWS OF HARMONY
1. the overtone scale
2. the interval proportions
3. the division of the octave into twelve semitones
4. the difference between consonance and dissonance, the consonance growing as the proportion of the numbers gets smaller
5. the difference between major and minor, the major proportion being the most frequent by far
6. the predominance of the 1:2 polarity, the octave
7. the law of the Lambdoma (a column of numbers written in the form of the Greek letter lambda, whose right leg consists of whole numbers going from one to infinity while the left leg contains the fractions of these same whole numbers, so that the coordinates of the open isosceles triangle follow the scale of overtones or undertones). There are correspondences in physics, acoustics, arithmatic, geometry, crystallography, cybernetics, theology and philosophy, the genetic code and I Ching.
The Mysticism of Sound: Music, The Power of the Word, and Cosmic Language (Sufi Message of Hazrat Inayat Khan Ser: Vol. 2)