Interval family
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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
For integer q, the q-integer-limit contains all rational intervals with numerator and denominator both less than or equal to q.
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
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
Sets of integer harmonics contain intervals of the form a/1 with a being an arbitrary integer inside some limit.
Primodality
In contrast to other interval sets individual primodes are not defined by a limit and do not necessarily contain all previous sets. Rather the denominator of all of the primode's intervals simply has to be equal to a specific integer while the numerator is free to vary.