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. It is also 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 the [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.

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.

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=== 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.

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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 {{EDO intro|31920}}
-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. It is also 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 the [[Kirnberger's atom]], {{monzo| 161 -84 -12 }}.
 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.
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 === 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.
+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.