Rhythm: The Music of the Spheres

"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."
Gottfried Wilhelm Leibniz

Rhythm is the basis of harmonic proportions and intervals, fundamental physical mechanisms. This harmonious equilibration was expounded upon by Pythagoras, father of Philosophy. Pythagorean thought formed the basis of the philosophy of Plato, and later Neo-Pythagoreans and Platonists, and greatly influenced the development of western science.

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):


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)


Anonymous said...

Hmmm.... #3 is a little out of place. 12 comes from the pythagorean tradition, but pythagoras used chains of fifths to produce all the other intervals, thus his major third was (3^4)/(2^4) or 81/64 instead of the shimmeringly pure 5/4. A tradition that limits itself to 12 semitones will always run into problems harmonically. This stems fromt the fact that no chain of a single interval (i.e. 3/2) ever maps onto a perfect octave. There's always a comma. Limiting notes to 12 usually makes the false assumption that there is only one major second and it works for all situations (ignoring distinction between 10/9 and 9/8). It leaves out all septimal intervals (including the blue third 7/6), except for an approximation of one of the tritones. And if there are only twelve notes, consonance will be compromised at some point and so will dissonance - all of the exotic, rational dissonances that give variety and flavor to music are ignored or at best, rudely "approximated." To truly respect the overtone series, harmony must be more flexible than 12 notes allow. (Or the 12 notes themselves must be flexible).

Hucbald said...

Actually David, Anon. is wrong.

Twelve Tone Equal Temperament (TTET from now on) is actually exactly what the harmonic overtone series implies is the most logical solution to the issue of temperament for fixed pitch instruments.

If you place a stack of seven perfect 2:1 octaves next to a stack of twelve 3:2 perfect fifths, you get the well known Pythagorean comma, which is slightly less than 24 cents (Less than 1/4 of a semitone after all of those "just" iterations).

Obviously, the pitch identity of all musical systems will require octave equivalance, so you can't stretch the octaves (Philosophically: Piano tuners do it all the time, which relates to the human perception of pitch, which is beyond the scope of this philosophical discussion), but contracting the fifths presents no problem at all, and 1/12 of the comma is less than two cents (This gives TTET).

The average limit of human perception of pitch differences is seven cents (Most trained musicians can discern three, exceptional ones can discern two), so contracting the fifths causes almost NO auditory difference. IOW, people who say that TTET destroyed music or that the harmonic series does not apply to TTET... ARE IDIOTS!

No human ear is decieved by the irrational numbers that represent intervals in TTET: They are perceived in virtually exactly the same way as the just ratios are, so there is little difference between TTET and 7-limit just. None at all in most practical senses... but not all.

Sure, in so-called well-tempered systems each key has a "character" and Baroque keyboard music probably ought to use those systems (I have a Davitt Moroney recording of Bach's Art of Fugue which uses them and is sublime), since Baroque keyboard music was composed within those systems, but the series itself implies that TTET is the best solution.

Look, the series implies that there are twenty-four possible tonics - twelve major and twelve minor - and that all of these tonics are available WITHIN A SINGLE COMPOSITION. So, in order for all of them to SOUND THE SAME relative to one another (TO BE EQUALS), TTET is the only solution: The most logical one which is perfectly equitable according to the implications of the series itself.

I wrote a book on this subject - The Musical Implications of the Harmonic Overtone Series - which I researched for over twenty years, so these on-the-verge mindless musings which often pollute contemporary musical thinking irritate me... to say the least. TTET did not ruin music: It is the long sought solution to the problem of equal pitch instruments... and has been since frets were first tied to a lute amost a milennium ago.

Oh... Happy New Year, David! ;^)

Anonymous said...

Listen -- ultrasound ionizes chemicals that then generate electromagnetic fields, light and even spacetime travel.


The Law of Pythagoras -- the Tetrad -- is based on asymmetry with C to G as 2:3 (YANG) and G to C as 3:4 (YIN).

The square root of two is approximated as the Tritone -- 9/8 cubed (professor Ernest McClain) -- and this is why it's the "devil's interval." The secret of free energy alchemy was lost.

I have a "book" of information online about the real meaning behind the overtone series -- infinite transduction.


drew hempel, MA