Interval family

Todo: merge articles Interval family → Interval

An interval family is a set that itself consists of multiple sets of intervals. For example all separate odd-limits taken together form an interval family.

Interval families are usually infinite with an obvious way of generating the interval sets belonging to the family. Most commonly interval families use interval sets that are based on some numerical limit where all intervals in the set are below that limit according to some metric.

The interval family and the metric often use the same name. For example odd-limit can either mean a metric on an interval or it can mean the family generated from that metric.

Catalog of interval families

Integer limit

Main article: Odd limit § Integer limit

For integer q, the q-integer-limit contains all rational intervals with numerator and denominator both less than or equal to q.

Odd limit

Main article: Odd limit

For the odd-limit all factors of 2 are removed from numerator and denominator before checking if they are below the chosen limit.

Larger than 1/1 integer limit

The larger-than-1/1-integer-limit uses the same metric as the integer-limit. The difference is that individual interval sets only contain intervals that are strictly larger than 1/1.

Odd prime sum limit

Main article: Odd prime sum limit

For the odd-prime-sum-limit all factors of two are removed from numerator and denominator. Then all remaining factors of the numerator and denominator are added together with their sum having to stay within the limit.

Integer Harmonics

Main article: Harmonic

Sets of integer harmonics contain intervals of the form a/1 with a being an arbitrary integer inside some limit.

Family table

Family	Example	Notes
Odd limit	6-Limit: 1/1, 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5, 8/1, 8/3, 8/5, 10/1, 10/3, 12/1, 12/5, 16/1, 16/3, 16/5, ... + inverse ratios 1/2, 1/3, 2/3, ...	Interval set is infinite, 6-odd-limit identical to 5-odd-limit
Integer limit	6-Limit: 1/1, 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5 + inverse ratios
Octave complemented integer limit [idiosyncratic term]	6-Limit: 1/1, 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5, 8/1, 8/3, 8/5, 10/1, 10/3, 12/1, 12/5 + inverse ratios	Interger-limit with one factor of 2 ignored from every interval
Larger-than-1/1-Integer-limit [idiosyncratic term]	6-Limit: 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5	Integer limit with only ratios larger than 1/1
Odd prime sum limit [idiosyncratic term]	6-Limit: 1/1, 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5, 8/1, 8/3, 8/5, 9/1, 9/2, 9/4, 9/5, 9/8, 10/1, 10/3, 10/9, 12/1, 12/5, 16/1, 16/3, 16/5, 16/9, ... + inverse ratios	Interval set is infinite
Harmonics	6-Limit: 1/1, 2/1, 3/1, 4/1, 5/1, 6/1
Prime harmonics	5-Limit: 2/1, 3/1, 5/1