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

#### 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; 3^{2}; 3^{3}; 3^{4}; 3^{5}; 3^{6}; 3^{7}; 3^{8}; 3^{9}; 3^{10}; 3^{11}; 3^{12}; 3^{13}; 3^{14}; 3^{15}; 3^{16}; 3^{17}; 3^{18}; 3^{19}; 3^{20}; 3^{21}; 3^{22}; 3^{23}; 3^{24}; 3^{25}; 3^{26}}

## Infinite octaves

The set such as:

- There is infinite octaves

is (2;2;0;+∞)-limit = {2^{n} / 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.