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.

Table

Family	Description	Example	Notes
Integer limit	Rational intervals with both the numerator and denominator less than or equal to the 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	Inverses also part of the set (1/2, 1/3, 2/3, ...)
Odd limit	Same as integer limit but with all factors of 2 removed from numerator and denominator before comparing them to the 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, ...	Inverses also part of the set, 6-odd-limit identical to 5-odd-limit, interval set is infinite
Ignore-a-factor-of-two-integer-limit [idiosyncratic term]	Same as integer limit but with a single factor of 2 removed from numerator or denominator before comparing them to the 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	Inverses also part of the set
Larger-than-1/1-Integer-limit [idiosyncratic term]	Same as integer limit but only ratios larger than 1/1	6-Limit: 2/1, 3/1, 3/2, 4/1, 4/3, 5/1, 5/2, 5/3, 5/4, 6/1, 6/5
Odd prime sum limit [idiosyncratic term]	Rational intervals where the added factors (except 2) of the numerator and denominator are both independently less than or equal to the 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, 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, ...	Inverses also part of the set, interval set is infinite
Harmonics	Intervals with a denominator of 1 and a whole number numerator that is less than or equal to the limit	6-Limit: 1/1, 2/1, 3/1, 4/1, 5/1, 6/1
Prime harmonics	Same as harmonics but with the numerator being limited to prime numbers	6-Limit: 2/1, 3/1, 5/1	6-limit identical to 5-(prime)-limit