31-limit: Difference between revisions

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== 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 [[3-limit|harmonic fifths]] — horizontal axis.

The rules to get this table are:

# 1/1 is the center of symmetry between reciprocals.

# The positive abscissa is the positive [[3-limit|Pythagorean scale]]. The negative abscissa is deducted by rule 1.

# The positive ordinate is the positive quasi-primes scale (one or two primes in the ratio), in this logical order:

::* a ; a*a ;

::* 5-limit allows also a*a*a to show the worth mentioning [[128/125|diesis]] in its symmetrical reciprocal

::* 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 intervals, you'll find: its factorization (ignoring octaves), its just ratio, its [[Functional Just System|Functional Just System]] notation, and its cents value (rounded to 9 decimal places).

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 == 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 [[3-limit|harmonic fifths]] — horizontal axis.
+The rules to get this table are:
+# 1/1 is the center of symmetry between reciprocals.
+# The positive abscissa is the positive [[3-limit|Pythagorean scale]]. The negative abscissa is deducted by rule 1.
+# The positive ordinate is the positive quasi-primes scale (one or two primes in the ratio), in this logical order:
+::* a ; a*a ;
+::* 5-limit allows also a*a*a to show the worth mentioning [[128/125|diesis]] in its symmetrical reciprocal
+::* b ; a*b ; a<sup>-1</sup>*b ; b*b ;
+::* c ; a*c ; a<sup>-1</sup>*c ; b*c ; b<sup>-1</sup>*c ; c*c ;
+::* d ; a*d ; a<sup>-1</sup>*d ; b*d ; b<sup>-1</sup>*d ; c*d ; c<sup>-1</sup>*d ; d*d ;
+::* e ; a*e ; a<sup>-1</sup>*e ; b*e ; b<sup>-1</sup>*e ; c*e ; c<sup>-1</sup>*e ; d*e ; d<sup>-1</sup>*e ; e*e ;
+::* etc...
+: The negative ordinate is deducted by rule 1.
 For each intervals, you'll find: its factorization (ignoring octaves), its just ratio, its [[Functional Just System|Functional Just System]] notation, and its cents value (rounded to 9 decimal places).