A musical animation of the first 16 elements of the harmonic series.
NB! Most computer speaker systems cannot render the deep tones, so use headphones, but be careful as the volume setting is quite high!

Here the tonal points are presented within the frames of an octave spiral structure which illustrates a fundamental principle in music:
The octave is a repetition of an already experienced quality on a new level.
The essence of the graphic spiral is likewise:

1-2-4-8 are octaves of the fundamental;
3-6-12 are octaves of the perfect fifth;
5-10 are octaves of the just major third etc.

The number figures are an attempt to develope a natural and intuitive graphic representation of the inherent structure of number based on factorization, here from 1 to 16.
It is not a 'number system' like our decimal system / 'base 10' (really based on the primes 2 and 5) or a sexigesimal system (primes 2, 3 and 5), but an attempt to come close to the essence of the language inherent in the concept of numbers, a language of primes.and compounds - arithmethic (and musical) atoms and molecules.

Proceeding through the spiral no. 1-16 the neighbouring intervals grow successively smaller:
1:2=octave;
2:3=perfect fifth;
3:4=perfect fourth;
4:5=just major third;
5:6=just minor third;
6:7=septimal third;
7:8=septimal second;
8:9=major wholetone;
9:10=minor wholetone,....;
15:16=just halftone.

The sounds are sine wave tones. Please notice, that this is not common tempered tuning values but frequencies from the harmonic series with a fundamental of C=64 Hz (no. 1).
The other frequencies are multipla of this tone (nx64 Hz).
The tonal axises represent divisions of a tonal circle of 360 degrees = 1200 musical cents.
Example: The 5-10 axis is situated at 115.8 degrees from the vertical axis corresponding to 386 cents (whereas the equaltempered major third measures 400 cents, corresponding to 120 degrees). More to be found (in danish) on [ Ссылка ]

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