31920edo: Difference between revisions

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31920edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out ~~the~~ [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.

31920edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.

It is also the smallest multiple of [[12edo]] to be [[purely consistent]] in the 31-odd-limit (i.e. all odd harmonics up to and including 31 are approximated with no greater than 25% relative error).

The simplest of the commas under the 43-limit it tempers out are 47916/47915, 52480/52479, 58311/58310, 60516/60515, 67600/67599, 68783/68782, 72501/72500, 75141/75140, 76875/76874, 81549/81548, 81796/81795, 82944/82943, 88320/88319, 93093/93092, 93500/93499, 96876/96875 and 98736/98735.

=== Prime harmonics ===

{{Harmonics in equal|31920|prec=5|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31920edo (continued)}}

=== Subsets and supersets ===

31920 is a very composite number, with many divisors: 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, and 15960. These facts make it a good candidate for an [[interval size measure]], and one step of it may be called an [[imp]], so that the cent is 26.6 imps, and a [[12edo]] semitone is 2660 imps. A single step of [[15edo]] is 2128 imps, of [[19edo]] 1680 imps, of [[84edo]] 380 imps, of [[140edo]] 228 imps, of [[152edo]] 210 imps, of [[190edo]] 168 imps, and of 665edo 48 imps.

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 {{Infobox ET|Consistency=41|Distinct consistency=41}}
-{{EDO intro|31920}}
+{{ED intro}}
-edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out the [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.
+edo is distinctly [[consistent]] through the 41-odd-limit, with a smaller 41-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any smaller distinctly consistent division. Its 3rd harmonic derives from [[665edo]]. It is also [[Enfactoring|enfactored]] in the 5-limit, with the same tuning as 15960edo, which is an [[atomic]] tuning, tempering out [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.
+It is also the smallest multiple of [[12edo]] to be [[purely consistent]] in the 31-odd-limit (i.e. all odd harmonics up to and including 31 are approximated with no greater than 25% relative error).
 The simplest of the commas under the 43-limit it tempers out are 47916/47915, 52480/52479, 58311/58310, 60516/60515, 67600/67599, 68783/68782, 72501/72500, 75141/75140, 76875/76874, 81549/81548, 81796/81795, 82944/82943, 88320/88319, 93093/93092, 93500/93499, 96876/96875 and 98736/98735.
 === Prime harmonics ===
-{{Harmonics in equal|31920}}
+{{Harmonics in equal|31920|prec=5|intervals=prime|columns=9}}
+{{Harmonics in equal|31920|prec=5|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 31920edo (continued)}}
 === Subsets and supersets ===
 is a very composite number, with many divisors: 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 19, 20, 21, 24, 28, 30, 35, 38, 40, 42, 48, 56, 57, 60, 70, 76, 80, 84, 95, 105, 112, 114, 120, 133, 140, 152, 168, 190, 210, 228, 240, 266, 280, 285, 304, 336, 380, 399, 420, 456, 532, 560, 570, 665, 760, 798, 840, 912, 1064, 1140, 1330, 1520, 1596, 1680, 1995, 2128, 2280, 2660, 3192, 3990, 4560, 5320, 6384, 7980, 10640, and 15960. These facts make it a good candidate for an [[interval size measure]], and one step of it may be called an [[imp]], so that the cent is 26.6 imps, and a [[12edo]] semitone is 2660 imps. A single step of [[15edo]] is 2128 imps, of [[19edo]] 1680 imps, of [[84edo]] 380 imps, of [[140edo]] 228 imps, of [[152edo]] 210 imps, of [[190edo]] 168 imps, and of 665edo 48 imps.