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Vladimir Ladma

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Definition

Let rational number qεQ has prime partition:
q = product(p(j)^a(j)), p(j)εP (primes), a(j)εZ (integers).

We call natural period r the expression:
r = product(p(j)).

Congruences

Number of classes of expression
n^k mod k , n,kεNo (zero and positive integers),
is equal to natural period r of number k.

n\k	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	1	1	1	1	1	1	1	1	1	1
2	0	0	2	0	2	4	2	0	8	4	2	4
3	0	1	0	1	3	3	3	1	0	9	3	9
4	0	0	1	0	4	4	4	0	1	6	4	4
5	0	1	2	1	0	1	5	1	8	5	5	1
6	0	0	0	0	1	0	6	0	0	6	6	0
7	0	1	1	1	2	1	0	1	1	9	7	1
8	0	0	2	0	3	4	1	0	8	4	8	9
9	0	1	0	1	4	3	2	1	0	1	9	9
10	0	0	1	0	0	4	3	0	1	0	10	4
11	0	1	2	1	1	1	4	1	8	1	0	1
12	0	0	0	0	2	0	5	0	0	4	1	0

Dissonance amount

Let us have two tones, whose frequency ratio f1/f2 can be approximated by fraction q.
We assume, dissonance amount of this chord depends on natural period r (computed from prime partition of q).
A special case of this phenomenon is octave identity.

Consonance of ratio 2:1 (octave) is assumed to be the same as consonance of 4:1 (two octaves); similarly ratio 3:2 (fifth) is perceived like as 4:3 (fourth).
Prime 2 has special meaning in music: ratio 2:1 is also considered to be the same as ratio 1:1.