181edo - Revision history

Overthink: /* Intervals */ delete everything beyond 23-limit

2026-04-04T00:52:31Z

Intervals: delete everything beyond 23-limit

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:29:59Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:29, 13 March 2026
Line 870:		Line 870:
	\| [[Cotritone]] (11-limit)		\| [[Cotritone]] (11-limit)
	\|}		\|}
	<nowiki />* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

FloraC: /* Regular temperament properties */ + cutefourths

2026-03-04T10:52:21Z

Regular temperament properties: + cutefourths

← Older revision		Revision as of 10:52, 4 March 2026
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	! rowspan="2" \| [[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
	\|-		\|-
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	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 287 -181 }}		\| {{Monzo\| 287 -181 }}
	\| {{~~mapping~~\| 181 287 }}		\| {{Mapping\| 181 287 }}
	\| −0.255		\| −0.255
	\| 0.255		\| 0.255
Line 777:		Line 777:
	\| 2.3.5		\| 2.3.5
	\| 2109375/2097152, {{monzo\| 14 -22 9 }}		\| 2109375/2097152, {{monzo\| 14 -22 9 }}
	\| {{~~mapping~~\| 181 287 420 }}		\| {{Mapping\| 181 287 420 }}
	\| +0.086		\| +0.086
	\| 0.525		\| 0.525
Line 784:		Line 784:
	\| 2.3.5.7		\| 2.3.5.7
	\| 2401/2400, 5120/5103, 390625/387072		\| 2401/2400, 5120/5103, 390625/387072
	\| {{~~mapping~~\| 181 287 420 508 }}		\| {{Mapping\| 181 287 420 508 }}
	\| +0.142		\| +0.142
	\| 0.465		\| 0.465
Line 791:		Line 791:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 385/384, 1375/1372, 2200/2187, 4000/3993		\| 385/384, 1375/1372, 2200/2187, 4000/3993
	\| {{~~mapping~~\| 181 287 420 508 626 }}		\| {{Mapping\| 181 287 420 508 626 }}
	\| +0.174		\| +0.174
	\| 0.421		\| 0.421
Line 798:		Line 798:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 325/324, 352/351, 385/384, 1375/1372, 1575/1573		\| 325/324, 352/351, 385/384, 1375/1372, 1575/1573
	\| {{~~mapping~~\| 181 287 420 508 626 670 }}		\| {{Mapping\| 181 287 420 508 626 670 }}
	\| +0.079		\| +0.079
	\| 0.439		\| 0.439
Line 805:		Line 805:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274		\| 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274
	\| {{~~mapping~~\| 181 287 420 508 626 670 740 }}		\| {{Mapping\| 181 287 420 508 626 670 740 }}
	\| +0.028		\| +0.028
	\| 0.425		\| 0.425
Line 812:		Line 812:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274		\| 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274
	\| {{~~mapping~~\| 181 287 420 508 626 670 740 769 }}		\| {{Mapping\| 181 287 420 508 626 670 740 769 }}
	\| +0.000		\| +0.000
	\| 0.404		\| 0.404
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	\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator		\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
	\|-		\|-
	! Periods<br />per 8ve		! Periods<br>per 8ve
	! Generator*		! Generator*
	! Cents*		! Cents*
	! Associated<br />ratio*		! Associated<br>ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 843:		Line 843:
	\| 39\181		\| 39\181
	\| 258.56		\| 258.56
	\| {{~~monzo~~\| -32 13 5 }}		\| {{Monzo\| -32 13 5 }}
	\| [[Lafa]]		\| [[Lafa]]
	\|-		\|-
Line 857:		Line 857:
	\| 49/40		\| 49/40
	\| [[Hemififths]] (7-limit)		\| [[Hemififths]] (7-limit)
			\|-
			\| 1
			\| 78\181
			\| 517.13
			\| 66/49
			\| [[Cutefourths]]
	\|-		\|-
	\| 1		\| 1
Line 862:		Line 868:
	\| 583.43		\| 583.43
	\| 7/5		\| 7/5
	\| [[Cotritone]] (~~181f~~)		\| [[Cotritone]] (11-limit)
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

Overthink: /* Intervals */ that's a bit overkill

2026-03-04T00:07:42Z

Intervals: that's a bit overkill

Show changes

Overthink: /* Intervals */ partially complete table

2026-02-21T05:38:09Z

Intervals: partially complete table

Show changes

Overthink: /* Intervals */ substitute

2026-02-21T05:18:56Z

Intervals: substitute

Show changes

Overthink: /* Theory */ expand on commas and no-5 interpretation

2026-01-18T22:27:24Z

Theory: expand on commas and no-5 interpretation

← Older revision		Revision as of 22:27, 18 January 2026
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	== Theory ==		== Theory ==
	181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval\|primes]] [[37/1\|37]] and [[43/1\|43]], and has unambiguous though not accurate approximations to [[29/1\|29]], [[31/1\|31]], and [[41/1\|41]].		181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval\|primes]] [[37/1\|37]] and [[43/1\|43]], and has unambiguous though not accurate approximations to [[29/1\|29]], [[31/1\|31]], and [[41/1\|41]]. However, the composite harmonics [[25/1\|25]], [[27/1\|27]], [[35/1\|35]], and [[39/1\|39]] cause inconsistencies, with harmonic 25 itself being inconsistent.

	As an equal temperament, 181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.		As an equal temperament, 181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the [[5-limit]]; [[2401/2400]], [[5120/5103]], and 390625/387072 in the [[7-limit]] ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the [[11-limit]]; and [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the [[13-limit]]. It tempers out [[375/374]], [[595/594]], and [[1275/1274]] in the [[17-limit]], [[400/399]] in the [[19-limit]], and [[300/299]] in the [[23-limit]].

			Because its harmonic [[5/1\|5]] causes some inconsistencies, and is less accurate than the other harmonics, 181edo can reasonably be treated as a no-5 system, where it is [[purely consistent]]{{idio}} (meaning all harmonics have under 25% [[relative error]]) up to the 23-odd-limit. It tempers out {{Monzo\|15 -13 2}} and {{Monzo\|-31 -7 15}} in the [[2.3.7 subgroup]]; 26411/26244, [[43923/43904]], and [[131072/130977]] in the [[2.3.7.11 subgroup]]; and [[352/351]], 20449/20412, [[31213/31104]], and 53361/53248 in the 2.3.7.11.13 subgroup. It tempers out [[833/832]] and [[1089/1088]] in the no-5 17-limit, [[343/342]], [[1729/1728]], and [[2432/2431]] in the no-5 19-limit, and [[392/391]] in the no-5 23-limit.

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

Overthink: + interval table

2025-11-30T05:14:20Z

+ interval table

← Older revision		Revision as of 05:14, 30 November 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	181edo is the 42nd [[prime edo]].		181edo is the 42nd [[prime edo]].

			== Intervals ==
			{{Interval table}}

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

FloraC: /* Theory */ don't bombard the readers with random prime numbers. 30% unsigned error isn't even special.

2025-09-26T11:27:05Z

Theory: don't bombard the readers with random prime numbers. 30% unsigned error isn't even special.

← Older revision		Revision as of 11:27, 26 September 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	181edo is only consistent to the [[7-odd-limit]], though ~~besides~~ [[10/9]] and [[~~9/5~~]] ~~all intervals are~~ consistent in the 21-odd-limit, ~~and~~ it ~~has low errors~~ on primes ~~all the way up to~~ 43~~. It even does well beyond that~~, ~~with less than 30% error on all primes up~~ to ~~137~~, ~~besides 47, 61, 73, 97~~, and ~~113~~. 181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.		181edo is only [[consistent]] to the [[7-odd-limit]], though except for [[9/5]], [[23/20]] and their [[octave complement]]s, it is consistent to the [[23-odd-limit]]. Beyond that, it does well on [[prime interval\|primes]] [[37/1\|37]] and [[43/1\|43]], and has unambiguous though not accurate approximations to [[29/1\|29]], [[31/1\|31]], and [[41/1\|41]].

			As an equal temperament, 181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|181}}{{Harmonics in equal\|181\|start=12~~}}{{Harmonics in equal~~\|~~181~~\|~~start~~=23}}		{{Harmonics in equal\|181\|columns=11}}
			{{Harmonics in equal\|181\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 181edo (continued)}}

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

Overthink at 04:46, 26 September 2025

2025-09-26T04:46:42Z

← Older revision		Revision as of 04:46, 26 September 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.		181edo is only consistent to the [[7-odd-limit]], though besides [[10/9]] and [[9/5]] all intervals are consistent in the 21-odd-limit, and it has low errors on primes all the way up to 43. It even does well beyond that, with less than 30% error on all primes up to 137, besides 47, 61, 73, 97, and 113. 181et [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo\| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|181}}		{{Harmonics in equal\|181}}{{Harmonics in equal\|181\|start=12}}{{Harmonics in equal\|181\|start=23}}

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