View source for 31-limit

{{Prime limit navigation|31}}
The '''31-limit''' consists of [[just intonation]] [[interval]]s whose [[ratio]]s contain no [[prime factor]]s higher than 31. It is the 11th [[prime limit]] and is a superset of the [[29-limit]] and a subset of the [[37-limit]]. 

The 31-limit is a rank-11 system, and can be modeled in a 10-dimensional lattice, with the primes 3 to 31 represented by each dimension. The prime 2 does not appear in the typical 31-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, an eleventh dimension is needed. 

Like the [[23-limit]], the 31-limit implies a substantial increment in its harmonic contents compared to previous prime limit. Specifically, these things are contained by the 31-limit, but not the 29-limit: 
* The [[31-odd-limit|31-]], [[33-odd-limit|33-]], and [[35-odd-limit]]; 
* Mode 16, 17, and 18 of the harmonic or subharmonic series; this means it completes the 5th octave of those series. 

The 31-limit intervals of the 2.3.31 subgroup are mainly [[interseptimal]], with [[32/31]] being a quartertone, [[36/31]] and [[31/27]] being semifourths, and [[31/24]] being a naiadic, with their [[octave complement]]s classified accordingly. While interseptimal intervals are abundant in lower limits, [[31/1|31]] is the first prime where such intervals occur without combining primes higher than 3. As such, the intervals of 31 are difficult to classify diatonically, but add a contrasting character to the lower limits, in particular [[11/1|11]] and [[13/1|13]], which are mainly neutral in quality. 

== Edo approximations ==
[[311edo]] is the smallest edo that is [[consistent]] to the [[31-odd-limit]]. [[1600edo]] is the smallest edo that is [[distinctly consistent]] to the 31-odd-limit. 

Edos with increasingly better approximations of the 31-limit ([[monotonicity limit]] ≥ 31 and decreasing [[TE error]]): {{EDOs| 99efk, 121ik, 130, 140, 149, 152fgj, 159, 183, 190g, 217, 243e, 270, 311, 388, 422, 525, 566gj, 571, 624jk, 639hj, 643ijk, 653, 692ik, 718, 742i, 863efgjk, 882, 908, 954hj, 1205g, 1289, 1308, 1395, 1578 }}, etc. For a more comprehensive list, see [[Sequence of equal temperaments by error]].

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "99efk" means taking the second closest approximations of harmonics 11, 13, and 31. }}

== Music ==
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=oTk3gWguv6I ''Soul Box''] (2020)
* [https://www.youtube.com/watch?v=HnFX-PdAK4U ''Echo and Narcissus''] (2020)
* [https://www.youtube.com/watch?v=li6MHeXWkJU ''Theme for a primate-like creature''] (2024)

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