31920edo - Revision history

FloraC: Wrap the prime harmonics table

2025-07-30T14:05:20Z

Wrap the prime harmonics table

← Older revision		Revision as of 14:05, 30 July 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|31920\|prec=5\|columns=15}}		{{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 ===		=== 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.

Iwuqety at 11:14, 30 July 2025

2025-07-30T11:14:15Z

← Older revision		Revision as of 11:14, 30 July 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|31920}}		{{Harmonics in equal\|31920\|prec=5\|columns=15}}

	=== Subsets and supersets ===		=== 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.

ArrowHead294 at 14:38, 20 February 2025

2025-02-20T14:38:54Z

← Older revision		Revision as of 14:38, 20 February 2025
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	{{Infobox ET\|Consistency=41\|Distinct consistency=41}}		{{Infobox ET\|Consistency=41\|Distinct consistency=41}}
	{{~~EDO~~ intro~~\|31920~~}}		{{ED intro}}

	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.		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.

BudjarnLambeth at 07:16, 9 July 2023

2023-07-09T07:16:40Z

← Older revision		Revision as of 07:16, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET\|Consistency=41\|Distinct consistency=41}}		{{Infobox ET\|Consistency=41\|Distinct consistency=41}}
	{{EDO intro\|31920}}		{{EDO intro\|31920}}

BudjarnLambeth at 04:08, 9 July 2023

2023-07-09T04:08:48Z

← Older revision		Revision as of 04:08, 9 July 2023
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	{{Infobox ET\|Consistency=41\|Distinct consistency=41}}		{{novelty}}{{Infobox ET\|Consistency=41\|Distinct consistency=41}}
	{{EDO intro\|31920}}		{{EDO intro\|31920}}

FloraC: Note the enfactoring in the 3- and 5-limit; +links

2023-03-08T08:47:57Z

Note the enfactoring in the 3- and 5-limit; +links

← Older revision		Revision as of 08:47, 8 March 2023
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	{{EDO intro\|31920}}		{{EDO intro\|31920}}

	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.		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.
Line 10:		Line 10:

	=== Subsets and supersets ===		=== 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.

FloraC: Improve intro; subsectioning; +prime error table

2023-02-14T10:57:18Z

Improve intro; subsectioning; +prime error table

← Older revision		Revision as of 10:57, 14 February 2023
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	{{Infobox ET\|Consistency=41\|Distinct consistency=41}}		{{Infobox ET\|Consistency=41\|Distinct consistency=41}}
	The '''31920 division''' divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.		{{EDO intro\|31920}}

	~~Its~~ 41-limit ~~patent val~~ is ~~<31920 50592 74116 89611 110425 118118 130472 135594 144392 155067 158138 166286 171013~~\|.		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 }}.

	[[~~Category:Equal divisions~~ of ~~the octave\|#####~~]] ~~<!-- 5-digit number -->~~		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}}

			=== 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.

Plumtree: Infobox ET added

2022-10-04T22:27:30Z

Infobox ET added

← Older revision		Revision as of 22:27, 4 October 2022
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			{{Infobox ET\|Consistency=41\|Distinct consistency=41}}
	The '''31920 division''' divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.		The '''31920 division''' divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.

Fredg999 category edits: Categories

2022-07-04T01:45:07Z

Contribution: patent val

2021-01-06T15:28:56Z

patent val

← Older revision		Revision as of 15:28, 6 January 2021
Line 1:		Line 1:
	The 31920 division divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.		The 31920 division divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.

			Its 41-limit patent val is <31920 50592 74116 89611 110425 118118 130472 135594 144392 155067 158138 166286 171013\|.

← Older revision		Revision as of 01:45, 4 July 2022
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	The 31920 division divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.		The '''31920 division''' divides the octave into 31920 equal parts of 0.03759 cents each. It is distinctly consistent through the 41 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 temperament, tempering out the Kirnberger atom, \|161 -84 -12>. It is a very "smooth" number, with many divisors: 1, 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, 15960, 31920. These facts make it a good candidate for an [[Interval_size_measure\|interval size measure]], and one step of it may be called an [[imp\|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. 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.

	Its 41-limit patent val is <31920 50592 74116 89611 110425 118118 130472 135594 144392 155067 158138 166286 171013\|.		Its 41-limit patent val is <31920 50592 74116 89611 110425 118118 130472 135594 144392 155067 158138 166286 171013\|.

			[[Category:Equal divisions of the octave\|#####]] <!-- 5-digit number -->