Natural period

Definition

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)).

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 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.

Formal resonance


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