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CYCLE SUBJECTS / ASTRONOMY / LADMA ->
Vladimir Ladma
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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)).
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 |
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.
