User:Contribution/(31-Limit Factor Gradient) x (26 Tritaves Chain) x (Infinite Octaves)

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Purpose

The goal here is to cherry-pick a relevant Just Intonation subset such as every single note of a complex piece of music are all distant from each others by intervals of this subset.

31-Limit Factor Gradient

The set such as:

  • Intervals with primes from 4 to 8 are 3-max-factor-limit
  • Intervals with primes from 8 to 16 are 2-max-factor-limit
  • Intervals with primes from 16 to 32 are 1-max-factor-limit
  • Intervals with primes smaller than 4 or bigger than 32 are 0-max-factor-limit

is (5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit = {31/1; 29/1; 23/1; 19/1; 17/1; 13*13/1; 13/11; 13*11/1; 13/7; 13*7/1; 13/5; 13*5/1; 13/1; 11*11/1; 11/7; 11*7/1; 11/5; 11*5/1; 11/1; 7*7*7/1; 7*7/5; 7*7*5/1; 7/5*5; 7*5*5/1; 7*7/1; 7/5; 7*5/1; 7/1; 5*5*5/1; 5*5/1; 5/1; 1/1; 1/5; 1/5*5; 1/5*5*5; 1/7; 1/5*7; 5/7; 1/7*7; 1/5*5*7; 5*5/7; 1/5*7*7; 5/7*7; 1/7*7*7; 1/11; 1/5*11; 5/11; 1/7*11; 7/11; 1/11*11; 1/13; 1/5*13; 5/13; 1/7*13; 7/13; 1/11*13; 11/13; 1/13*13; 1/17; 1/19; 1/23; 1/29; 1/31}

26 Tritaves Chain

Every triple diminished and triple augmented intervals from a given reference note are covered through a chain of 26 negative and 26 positive octave-reduced tritaves (53 notes). 53 is also the number of octave-reduced tritaves whose the first and the last are distant from the Mercator's comma.

The set such as:

  • Intervals with 3 are 26-max-factor-limit
  • Intervals with primes smaller or bigger than 3 are 0-max-factor-limit

is (3;3;0;26)-limit = {3-26; 3-25; 3-24; 3-23; 3-22; 3-21; 3-20; 3-19; 3-18; 3-17; 3-16; 3-15; 3-14; 3-13; 3-12; 3-11; 3-10; 3-9; 3-8; 3-7; 3-6; 3-5; 3-4; 3-3; 3-2; 3-1; 1; 3; 32; 33; 34; 35; 36; 37; 38; 39; 310; 311; 312; 313; 314; 315; 316; 317; 318; 319; 320; 321; 322; 323; 324; 325; 326}

Infinite octaves

The set such as:

  • There is infinite octaves

is (2;2;0;+∞)-limit = {2n / n∈ℤ}

Cartesian Product

The Cartesian product between these three sets is ((5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit) × (3;3;0;26)-limit × (2;2;0;+∞)-limit.

Symmetry

Prime-bounded and factor-bounded limit

Let q be a number belonging to a given (pmin;pmax;fmin;fmax)-limit = pmin-min-prime-limit ∩ pmax-max-prime-limit ∩ fmin-min-factor-limit ∩ fmax-max-factor-limit.

- q belongs to pmin-min-prime-limit if and only if all primes of its factorization into primes are left-bounded to pmin. So, q-1 belongs to pmin-min-prime-limit too.

- q belongs to pmax-max-prime-limit if and only if all primes of its factorization into primes are right-bounded to pmax. So, q-1 belongs to pmax-max-prime-limit too.

- q belongs to fmin-min-factor-limit if and only if the sum of the exponent absolutes of its factorization into primes is left-bounded to fmin. So, q-1 belongs to fmin-min-factor-limit too.

- q belongs to fmax-max-factor-limit if and only if the sum of the exponent absolutes of its factorization into primes is right-bounded to fmax. So, q-1 belongs to fmax-max-factor-limit too.

So, if q belongs to a given (pmin;pmax;fmin;fmax)-limit, q-1 too.

Union of prime-bounded and factor-bounded limits

Let q be a number belonging to (pmin_1;pmax_1;fmin_1;fmax_1)-limit ∪ (pmin_2;pmax_2;fmin_2;fmax_2)-limit ∪ ... ∪ (pmin_n;pmax_n;fmin_n;fmax_n)-limit.

∃i (i∈ℕ ʌ i∈[1;n]) such as q ∈ (pmin_i;pmax_i;fmin_i;fmax_i)-limit. So, q-1 ∈ (pmin_i;pmax_i;fmin_i;fmax_i)-limit.

So, if q belongs to (pmin_1;pmax_1;fmin_1;fmax_1)-limit ∪ (pmin_2;pmax_2;fmin_2;fmax_2)-limit ∪ ... ∪ (pmin_n;pmax_n;fmin_n;fmax_n)-limit, q-1 too.

Center of symmetry

Let q be a strictly positive rational number such as q = q-1.

Its only solution is 1.

Prime-bounded and factor-bounded limit

Let (pmin;pmax;fmin;fmax)-limit be a set such as 1 belongs to it.

- 1 belongs to pmin-min-prime-limit whatever pmin is.

- 1 belongs to pmax-max-prime-limit whatever pmax is.

- 1 belongs to fmin-min-factor-limit if and only if fmin = 0.

- 1 belongs to fmax-max-factor-limit whatever fmax is.

So, 1 belongs to (pmin;pmax;0;fmax)-limit whatever pmin, pmax and fmax are.

Union of prime-bounded and factor-bounded limits

Let be a set (pmin_1;pmax_1;fmin_1;fmax_1)-limit ∪ (pmin_2;pmax_2;fmin_2;fmax_2)-limit ∪ ... ∪ (pmin_n;pmax_n;fmin_n;fmax_n)-limit such as 1 belongs to it.

1 belongs to this set if and only if ∃i (i∈ℕ ʌ i∈[1;n]) such as 1 ∈ (pmin_i;pmax_i;fmin_i;fmax_i)-limit, id est if and only if ∃i such as fmin_i = 0.

So, 1 belongs to (pmin_1;pmax_1;fmin_1;fmax_1)-limit ∪ (pmin_2;pmax_2;fmin_2;fmax_2)-limit ∪ ... ∪ (pmin_i;pmax_i;0;fmax_i)-limit ∪ ... ∪ (pmin_n;pmax_n;fmin_n;fmax_n)-limit, whatever pmin_1, pmax_1, fmin_1, fmax_1, pmin_2, pmax_2, fmin_2, fmax_2, ... pmin_i, pmax_i, fmax_i, ... pmin_n, pmax_n, fmin_n, fmax_n are.

Cartesian product of unions of prime-bounded and factor-bounded limits

((5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit) × (3;3;0;26)-limit × (2;2;0;+∞)-limit allows (1,1,1) as a center of symmetry between (a,b,c) and (a-1,b-1,c-1) whatever a ∈ (5;7;0;3)-limit ∪ (5;13;0;2)-limit ∪ (5;31;0;1)-limit), b ∈ (3;3;0;26)-limit and c ∈ (2;2;0;+∞)-limit are.

Tables

(31-Limit Factor Gradient) x (26 Tritaves Chain) x (Infinite Octaves)

See here.

No-Twos (31-Limit Factor Gradient) x (Infinite Tritaves)

See here.

Comparison

See here.