31-limit

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In 31-limit Just Intonation, all ratios in the system will contain no primes higher than 31.

Intervals

Here is a table of all 31-limit intervals with at most two primes from 5 to 31 in the ratio (excepting 5-limit intervals, which have at most three) — vertical axis, on a chain of 53 harmonic fifths — horizontal axis.

The rules to get this table are:

  1. 1/1 is the center of symmetry between reciprocals.
  2. The positive abscissa is the positive Pythagorean scale. The negative abscissa is deducted by rule 1.
  3. The positive ordinate is the positive quasi-prime scale (quasi-prime means one or two primes in the ratio, positive means the highest prime is in the numerator) from 5 to 31, in this logical order:
  • a ; a*a ; 5-limit allows also a*a*a to show in its reciprocal the worth mentioning diesis ;
  • b ; a*b ; a-1*b ; b*b ;
  • c ; a*c ; a-1*c ; b*c ; b-1*c ; c*c ;
  • d ; a*d ; a-1*d ; b*d ; b-1*d ; c*d ; c-1*d ; d*d ;
  • e ; a*e ; a-1*e ; b*e ; b-1*e ; c*e ; c-1*e ; d*e ; d-1*e ; e*e ;
  • etc...
The negative ordinate is deducted by rule 1.

For each interval, you'll find: its factorization (ignoring octaves), its just ratio, its Functional Just System notation, and its cents value (rounded to 9 decimal places).

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