Let rational number qεQ has prime factorization:
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 factorization 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.