List of superparticular intervals: Difference between revisions

[[superparticular|Superparticular]] numbers are ratios of the form (n+1)/n, or 1+1/n, where n is a whole number other than 1. They appear frequently in [[Just_intonation|Just Intonation]] and [[OverToneSeries|Harmonic Series]] music. Adjacent tones in the harmonic series are separated by superparticular intervals: for instance, the 20th and 21st by the superparticular ratio [[21/20|21/20]]. As the overtones get closer together, the superparticular intervals get smaller and smaller. Thus, an examination of the superparticular intervals is an examination of some of the simplest small intervals in rational tuning systems. Indeed, many but not all common [[Comma|comma]]s are superparticular ratios.

The list below is ordered by [[Harmonic_Limit|harmonic limit]], or the largest prime involved in the prime factorization. [[36/35|36/35]], for instance, is an interval of the [[7-limit|7-limit]], as it factors to (2<span style="font-size: 70%; vertical-align: super;">2</span>*3<span style="font-size: 70%; vertical-align: super;">2</span>)/(5*7), while 37/36 would belong to the 37-limit.

[http://en.wikipedia.org/wiki/St%C3%B8rmer%27s_theorem Størmer's theorem] guarantees that, in each limit, there are only a finite number of superparticular ratios. Many of the sections below are complete. For example, there is no 3-limit superparticular ratio other than 2/1, 3/2, 4/3, and 9/8. OEIS [http://oeis.org/A145604 A145604] gives the number of superparticular ratios in each prime limit, and [http://oeis.org/A117581 A117581] the largest numerator for each prime limit (with some exceptions, such as the 23-limit, where the largest value is smaller than that of a smaller prime limit, in this case the 19-limit).

See also: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]. Many of the names below come from [http://www.huygens-fokker.org/docs/intervals.html here].