137edo - Revision history

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"

2026-03-13T13:32:22Z

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 13:32, 13 March 2026
Line 90:		Line 90:
	\| [[Emka]] (137d) / [[emkay]] (137)		\| [[Emka]] (137d) / [[emkay]] (137)
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|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

	== Diagrams ==		== Diagrams ==

ArrowHead294 at 17:01, 18 February 2025

2025-02-18T17:01:43Z

← Older revision		Revision as of 17:01, 18 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|137~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:02, 15 January 2025

2025-01-15T18:02:29Z

← Older revision		Revision as of 18:02, 15 January 2025
Line 26:		Line 26:
	\| {{monzo\| -217 137 }}		\| {{monzo\| -217 137 }}
	\| {{mapping\| 137 217 }}		\| {{mapping\| 137 217 }}
	\| 0.3865		\| +0.3865
	\| 0.3866		\| 0.3866
	\| 4.41		\| 4.41
Line 33:		Line 33:
	\| {{monzo\| -21 3 7 }}, {{monzo\| -13 17 -6 }}		\| {{monzo\| -21 3 7 }}, {{monzo\| -13 17 -6 }}
	\| {{mapping\| 137 217 318 }}		\| {{mapping\| 137 217 318 }}
	\| 0.3887		\| +0.3887
	\| 0.3157		\| 0.3157
	\| 3.60		\| 3.60
Line 90:		Line 90:
	\| [[Emka]] (137d) / [[emkay]] (137)		\| [[Emka]] (137d) / [[emkay]] (137)
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Diagrams ==		== Diagrams ==

ArrowHead294: Partial undo

2024-11-16T12:51:48Z

Partial undo

← Older revision		Revision as of 12:51, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3		\| 2.3
Line 27:		Line 36:
	\| 0.3157		\| 0.3157
	\| 3.60		\| 3.60
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
Line 73:		Line 89:
	\| 11/8		\| 11/8
	\| [[Emka]] (137d) / [[emkay]] (137)		\| [[Emka]] (137d) / [[emkay]] (137)
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Diagrams ==		== Diagrams ==

ArrowHead294 at 03:44, 16 November 2024

2024-11-16T03:44:17Z

← Older revision		Revision as of 03:44, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
Line 35:		Line 27:
	\| 0.3157		\| 0.3157
	\| 3.60		\| 3.60
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
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	\| 11/8		\| 11/8
	\| [[Emka]] (137d) / [[emkay]] (137)		\| [[Emka]] (137d) / [[emkay]] (137)
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Diagrams ==		== Diagrams ==

FloraC: /* Regular temperament properties */ +borwell

2024-05-26T10:57:17Z

Regular temperament properties: +borwell

← Older revision		Revision as of 10:57, 26 May 2024
Line 69:		Line 69:
	\| 6/5		\| 6/5
	\| [[Parakleismic]]		\| [[Parakleismic]]
			\|-
			\| 1
			\| 53\137
			\| 464.23
			\| 72/55
			\| [[Borwell]]
	\|-		\|-
	\| 1		\| 1

FloraC: +one-linear overview and note more 5-limit commas

2024-05-26T10:53:41Z

+one-linear overview and note more 5-limit commas

← Older revision		Revision as of 10:53, 26 May 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	137edo ~~provides the [[optimal patent val]] for~~ 7-limit [[~~orwell~~]] ~~temperament and for the planar temperament~~ [[~~tempering out~~]] [[~~2430/2401~~]]~~. It tempers out 2109375/2097152~~ ([[~~semicomma~~]]) in the 5-limit; [[225/224]], [[1728/1715]], 2430/2401 in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit.		137edo is a fairly accurate 5-limit temperament and also a strong no-7 19-limit temperament. The equal temperament [[tempering out\|tempers out]] 2109375/2097152 ([[semicomma]]), {{monzo\| -13 17 -6 }} ([[graviton]]), {{monzo\| 8 14 -13 }} ([[parakleisma]]), and {{monzo\| -29 -11 20 }} (gammic comma) in the 5-limit. Using the [[patent val]], it tempers out [[225/224]], [[1728/1715]], 2430/2401 in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit. It provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament [[tempering out]] [[2430/2401]].

	=== Prime harmonics ===		=== Prime harmonics ===
Line 62:		Line 62:
	\| 271.53		\| 271.53
	\| 75/64		\| 75/64
	\| [[Orwell]] (~~137~~) / [[sabric]] (137d)		\| [[Orwell]] (137e) / [[sabric]] (137d)
	\|-		\|-
	\| 1		\| 1

FloraC: Cleanup and set straight the rank-2 temps it supports

2024-05-26T10:05:15Z

Cleanup and set straight the rank-2 temps it supports

← Older revision		Revision as of 10:05, 26 May 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	137edo provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament tempering out [[2430/2401]]. It tempers out 2109375/2097152 ([[semicomma]]) in the 5-limit; [[225/224]] ~~and~~ [[1728/1715]] in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit.		137edo provides the [[optimal patent val]] for 7-limit [[orwell]] temperament and for the planar temperament [[tempering out]] [[2430/2401]]. It tempers out 2109375/2097152 ([[semicomma]]) in the 5-limit; [[225/224]], [[1728/1715]], 2430/2401 in the 7-limit; [[243/242]] in the 11-limit; [[351/350]] in the 13-limit; [[375/374]] and [[442/441]] in the 17-limit; and [[324/323]] and [[495/494]] in the 19-limit.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	~~Since 137~~ is the 33rd [[prime ~~number~~]], ~~137edo has no proper divisors aside from 1~~.		137edo is the 33rd [[prime edo]], following [[131edo]] and before [[139edo]]. [[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.

	~~[[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.~~		== 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]]
	! 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]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\|2.3		\| 2.3
	\|{{monzo\|-217 137}}		\| {{monzo\| -217 137 }}
	\|{{~~val~~\|137 217}}		\| {{mapping\| 137 217 }}
	\| 0.3865		\| 0.3865
	\| 0.3866		\| 0.3866
	\| 4.41		\| 4.41
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|-21 3 7}}, {{monzo\|-13 17 -6}}		\| {{monzo\| -21 3 7 }}, {{monzo\| -13 17 -6 }}
	\|{{~~val~~\|137 217 318}}		\| {{mapping\| 137 217 318 }}
	\| 0.3887		\| 0.3887
	\| 0.3157		\| 0.3157
	\| 3.60		\| 3.60
	\|}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{\| class="wikitable center-all left-5"		{\| class="wikitable center-all left-5"
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(reduced)~~		! Generator*
	! Cents~~<br>(reduced)~~		! Cents*
	! Associated<br>~~ratio~~		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|3\137		\| 3\137
	\|26.28		\| 26.28
	\|1594323/1562500		\| 1594323/1562500
	\|[[Sfourth]] (5-limit)		\| [[Sfourth]] (5-limit)
	\|-		\|-
	\|1		\| 1
	\|4\137		\| 4\137
	\|35.04		\| 35.04
	\|1990656/1953125		\| 1990656/1953125
	\|[[Gammic]]		\| [[Gammic]] (137d) / [[gammy]] (137)
	\|-		\|-
	\|1		\| 1
	\|31\137		\| 31\137
	\|271.53		\| 271.53
	\|75/64		\| 75/64
	\|[[~~Orson~~]]		\| [[Orwell]] (137) / [[sabric]] (137d)
	\|-		\|-
	\|1		\| 1
	\|36\137		\| 36\137
	\|315.33		\| 315.33
	\|6/5		\| 6/5
	\|[[Parakleismic]]		\| [[Parakleismic]]
	\|-		\|-
	\|1		\| 1
	\|59\137		\| 59\137
	\|516.79		\| 516.79
	\|27/20		\| 27/20
	\|[[~~Gravity~~]]		\| [[Marvo]] (137)
	\|-		\|-
	\|1		\| 1
	\|63\137		\| 63\137
	\|551.82		\| 551.82
	\|~~9765625~~/~~7077888~~		\| 11/8
	\|[[Emka]] (~~5-limit~~)		\| [[Emka]] (137d) / [[emkay]] (137)
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Diagrams ==		== Diagrams ==

Francium: +regular temperament properties

2023-11-03T21:18:55Z

+regular temperament properties

← Older revision		Revision as of 21:18, 3 November 2023
Line 12:		Line 12:

	[[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.		[[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.

			==Regular temperament properties==
			{\| class="wikitable center-4 center-5 center-6"
			! rowspan="2" \|[[Subgroup]]
			! rowspan="2" \|[[Comma list\|Comma List]]
			! rowspan="2" \|[[Mapping]]
			! rowspan="2" \|Optimal<br>8ve Stretch (¢)
			! colspan="2" \|Tuning Error
			\|-
			![[TE error\|Absolute]] (¢)
			![[TE simple badness\|Relative]] (%)
			\|-
			\|2.3
			\|{{monzo\|-217 137}}
			\|{{val\|137 217}}
			\| 0.3865
			\| 0.3866
			\| 4.41
			\|-
			\|2.3.5
			\|{{monzo\|-21 3 7}}, {{monzo\|-13 17 -6}}
			\|{{val\|137 217 318}}
			\| 0.3887
			\| 0.3157
			\| 3.60
			\|}
			=== Rank-2 temperaments ===
			{\| class="wikitable center-all left-5"
			\|+Table of rank-2 temperaments by generator
			! Periods<br>per 8ve
			! Generator<br>(reduced)
			! Cents<br>(reduced)
			! Associated<br>ratio
			! Temperaments
			\|-
			\|1
			\|3\137
			\|26.28
			\|1594323/1562500
			\|[[Sfourth]] (5-limit)
			\|-
			\|1
			\|4\137
			\|35.04
			\|1990656/1953125
			\|[[Gammic]]
			\|-
			\|1
			\|31\137
			\|271.53
			\|75/64
			\|[[Orson]]
			\|-
			\|1
			\|36\137
			\|315.33
			\|6/5
			\|[[Parakleismic]]
			\|-
			\|1
			\|59\137
			\|516.79
			\|27/20
			\|[[Gravity]]
			\|-
			\|1
			\|63\137
			\|551.82
			\|9765625/7077888
			\|[[Emka]] (5-limit)
			\|}

	== Diagrams ==		== Diagrams ==

FloraC: 274edo as a correction

2023-09-02T11:46:42Z

274edo as a correction

← Older revision		Revision as of 11:46, 2 September 2023
Line 10:		Line 10:
	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 137 is the 33rd [[prime number]], 137edo has no proper divisors aside from 1.		Since 137 is the 33rd [[prime number]], 137edo has no proper divisors aside from 1.

			[[274edo]], which doubles it, provides a correction for its approximation to harmonic 7.

	== Diagrams ==		== Diagrams ==