135edo - Revision history

FloraC: Partial reversal -- last edit removed the note on the edo's notability in the 2.3.7.11 subgroup

2026-05-04T20:16:17Z

Partial reversal -- last edit removed the note on the edo's notability in the 2.3.7.11 subgroup

← Older revision		Revision as of 20:16, 4 May 2026
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	== Theory ==		== Theory ==
	135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo\|270et]], 135et ~~supports an easy but extremely accurate equal tuning of the~~ [[2.3.7.11 subgroup\|2.3.7.11-]][[subgroup]] [[regular temperament\|temperament]] ~~[[gary]]. As an equal temperament~~, it is characterized by [[tempering out]] the [[garischisma]], the [[~~symbiotic comma~~]] ~~and~~ the [[~~argyria~~]], ~~which indirectly also tempers out~~ the [[~~septiennealimma~~]], the [[~~olympia~~]] and the [[~~chrysia~~]]. On top of this, it also has fairly good approximations to primes [[17/1\|17]], [[29/1\|29]], and [[31/1\|31]].		135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo\|270et]], 135et makes most sense to use as a [[2.3.7.11 subgroup\|2.3.7.11-]][[subgroup]] [[regular temperament\|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], the [[chrysia]], and the [[olympia]]. On top of this, it also has fairly good approximations to primes [[17/1\|17]], [[29/1\|29]], and [[31/1\|31]].

	If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.		If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

Eufalesio: Note to gary

2026-05-04T08:03:48Z

Note to gary

← Older revision		Revision as of 08:03, 4 May 2026
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	== Theory ==		== Theory ==
	135edo is [[consistent]] to the [[7-odd-limit]], but ~~there is a~~ large relative ~~delta~~ for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo\|270et]], 135et ~~probably makes more sense as a~~ [[2.3.7.11 subgroup\|2.3.7.11-]][[subgroup]] [[regular temperament\|temperament]], ~~where~~ it is characterized by [[tempering out]] the [[garischisma]], the [[~~septiennealimma~~]], the [[~~symbiotic comma~~]], the [[~~argyria~~]], and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1\|17]], [[29/1\|29]], and [[31/1\|31]].		135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo\|270et]], 135et supports an easy but extremely accurate equal tuning of the [[2.3.7.11 subgroup\|2.3.7.11-]][[subgroup]] [[regular temperament\|temperament]] [[gary]]. As an equal temperament, it is characterized by [[tempering out]] the [[garischisma]], the [[symbiotic comma]] and the [[argyria]], which indirectly also tempers out the [[septiennealimma]], the [[olympia]] and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1\|17]], [[29/1\|29]], and [[31/1\|31]].

	If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.		If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 135 factors into primes as {{nowrap\| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, provides good ~~correction~~ for the approximation to harmonics 5, 13, and 19.		Since 135 factors into primes as {{nowrap\| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==

Lucius Chiaraviglio: /* Subsets and supersets */ Link to 270edo

2026-02-21T06:33:49Z

Subsets and supersets: Link to 270edo

← Older revision		Revision as of 06:33, 21 February 2026
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 135 factors into primes as {{nowrap\| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.		Since 135 factors into primes as {{nowrap\| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: Expand

2026-02-11T10:57:04Z

Expand

← Older revision		Revision as of 10:57, 11 February 2026
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	== Theory ==		== Theory ==
	135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full 13-limit monster – [[270edo\|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it [[tempering ~~out\|tempers~~ out]] the [[garischisma]] ~~and~~ the [[symbiotic comma]]~~. If we consider~~ the ~~full 13-limit~~, the ~~flat-tending {{val~~\| ~~135 214 313 379 467 '''499''' }} (135f)~~ and ~~the sharp-tending {{val~~\| ~~135 214 '''314''' 379 467 500 }} (135c) are reasonable choices~~.		135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo\|270et]], 135et probably makes more sense as a [[2.3.7.11 subgroup\|2.3.7.11-]][[subgroup]] [[regular temperament\|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], and the [[chrysia]]. On top of this, it also has fairly good approximations to primes [[17/1\|17]], [[29/1\|29]], and [[31/1\|31]].

	~~Using~~ the ~~135f~~ val~~, it tempers out 32805/32768~~ (~~[[schisma]]~~) and {{~~monzo~~\| ~~-11 -15 15~~ }} (~~pentadecal comma~~) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.		If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

	Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 (magus comma) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.		Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo\| -11 -15 15 }} (pentadecal comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], [[2200/2187]], [[12005/11979]] and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.

			Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 ([[magus comma]]) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|135}}		{{Harmonics in equal\|135\|columns=11}}
			{{Harmonics in equal\|135\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 135edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 135 factors into {{~~factorization~~\|~~135~~}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.		Since 135 factors into primes as {{nowrap\| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\|-		\|-
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list~~\|Comma List~~]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \| [[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \| Optimal<br>8ve ~~Stretch~~ (¢)		! rowspan="2" \| Optimal<br>8ve stretch (¢)
	! colspan="2" \| Tuning ~~Error~~		! colspan="2" \| Tuning error
	\|-		\|-
	! [[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
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	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 214 -135 }}		\| {{Monzo\| 214 -135 }}
	\| {{~~mapping~~\| 135 214 }}		\| {{Mapping\| 135 214 }}
	\| −0.0843		\| −0.0843
	\| 0.0843		\| 0.0843
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	\| 2.3.7		\| 2.3.7
	\| 33554432/33480783, 40353607/40310784		\| 33554432/33480783, 40353607/40310784
	\| {{~~mapping~~\| 135 214 379 }}		\| {{Mapping\| 135 214 379 }}
	\| −0.0637		\| −0.0637
	\| 0.0747		\| 0.0747
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	\| 2.3.7.11		\| 2.3.7.11
	\| 19712/19683, 41503/41472, 43923/43904		\| 19712/19683, 41503/41472, 43923/43904
	\| {{~~mapping~~\| 135 214 379 467 }}		\| {{Mapping\| 135 214 379 467 }}
	\| −0.0328		\| −0.0328
	\| 0.0840		\| 0.0840
	\| 0.94		\| 0.94
			\|-
			\| 2.3.7.11.17
			\| 1089/1088, 2058/2057, 5832/5831, 19712/19683
			\| {{Mapping\| 135 214 379 467 552 }}
			\| −0.1100
			\| 0.1716
			\| 1.93
	\|}		\|}

	== Instruments ==		== Instruments ==
	[[Lumatone mapping for 135edo]]		* [[Lumatone mapping for 135edo]]

Lériendil: 27 was missed

2026-01-18T14:48:37Z

27 was missed

← Older revision		Revision as of 14:48, 18 January 2026
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 135 factors into {{factorization\|135}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.		Since 135 factors into {{factorization\|135}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, 27, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==

Yourmusic Productions: Add lumatone mapping link.

2025-03-12T12:07:59Z

Add lumatone mapping link.

← Older revision		Revision as of 12:07, 12 March 2025
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	\| 0.94		\| 0.94
	\|}		\|}

			== Instruments ==
			[[Lumatone mapping for 135edo]]

ArrowHead294 at 19:24, 20 February 2025

2025-02-20T19:24:44Z

← Older revision		Revision as of 19:24, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==
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	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
			\|-
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
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	\| {{monzo\| 214 -135 }}		\| {{monzo\| 214 -135 }}
	\| {{mapping\| 135 214 }}		\| {{mapping\| 135 214 }}
	\| -0.0843		\| −0.0843
	\| 0.0843		\| 0.0843
	\| 0.95		\| 0.95
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	\| 33554432/33480783, 40353607/40310784		\| 33554432/33480783, 40353607/40310784
	\| {{mapping\| 135 214 379 }}		\| {{mapping\| 135 214 379 }}
	\| -0.0637		\| −0.0637
	\| 0.0747		\| 0.0747
	\| 0.84		\| 0.84
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	\| 19712/19683, 41503/41472, 43923/43904		\| 19712/19683, 41503/41472, 43923/43904
	\| {{mapping\| 135 214 379 467 }}		\| {{mapping\| 135 214 379 467 }}
	\| -0.0328		\| −0.0328
	\| 0.0840		\| 0.0840
	\| 0.94		\| 0.94
	\|}		\|}

FloraC: /* Theory */ comma labeling

2024-05-26T11:16:38Z

Theory: comma labeling

← Older revision		Revision as of 11:16, 26 May 2024
Line 5:		Line 5:
	135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full 13-limit monster – [[270edo\|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it [[tempering out\|tempers out]] the [[garischisma]] and the [[symbiotic comma]]. If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.		135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full 13-limit monster – [[270edo\|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it [[tempering out\|tempers out]] the [[garischisma]] and the [[symbiotic comma]]. If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

	Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo\| -11 -15 15 }} (~~quintriyo~~ comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.		Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo\| -11 -15 15 }} (pentadecal comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.

	Using the 135c val, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.		Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 (magus comma) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: /* Theory */ rework

2024-05-26T11:13:20Z

Theory: rework

← Older revision		Revision as of 11:13, 26 May 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s.		135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1\|5th]] and [[13/1\|13th]] [[harmonic]]s. As every other step of the full 13-limit monster – [[270edo\|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it [[tempering out\|tempers out]] the [[garischisma]] and the [[symbiotic comma]]. If we consider the full 13-limit, the flat-tending {{val\| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val\| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

	Using the 135f [[val~~]] {{val\| 135 214 313 379 467 '''499''' }}~~, ~~which tends flat, 135et [[tempering out\|~~tempers out]] 32805/32768 ([[schisma]]) and {{monzo\| -11 -15 15 }} (quintriyo comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.		Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo\| -11 -15 15 }} (quintriyo comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], 2200/2187, 12005/11979 and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.

	Using the 135c val ~~{{val\| 135 214 '''314''' 379 467 500 }}, which tends sharp~~, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.		Using the 135c val, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.

	As every other step of the full 13-limit monster – [[270edo\|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it tempers out the [[garischisma]] and the [[symbiotic comma]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 15:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 135 factors into {{factorization\|135}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.		Since 135 factors into {{factorization\|135}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==

FloraC: +subsets and supersets

2024-05-26T11:10:36Z

+subsets and supersets

← Older revision		Revision as of 11:10, 26 May 2024
Line 13:		Line 13:
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|135}}		{{Harmonics in equal\|135}}

			=== Subsets and supersets ===
			Since 135 factors into {{factorization\|135}}, 135edo has subset edos {{EDOs\| 3, 5, 9, 15, and 45 }}. 270edo, which doubles it, provides good correction for the approximation to harmonics 5, 13, and 19.

	== Regular temperament properties ==		== Regular temperament properties ==