Hemimage temperaments - Revision history

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:32:55Z

Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

← Older revision		Revision as of 12:32, 24 September 2026
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	Comma list: 385/384, 3388/3375, 9801/9800		Comma list: 385/384, 3388/3375, 9801/9800

	~~Mapping:~~ {{~~mapping~~\| 2 1 6 1 8 \| 0 8 -5 17 -4 }}		{{Mapping\|legend=0\| 2 1 6 1 8 \| 0 8 -5 17 -4 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 441/440, 3388/3375, 8019/8000		Comma list: 441/440, 3388/3375, 8019/8000

	~~Mapping:~~ {{~~mapping~~\| 2 3 4 5 6 \| 0 5 19 18 27 }}		{{Mapping\|legend=0\| 2 3 4 5 6 \| 0 5 19 18 27 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 196/195, 352/351, 729/728, 1001/1000		Comma list: 196/195, 352/351, 729/728, 1001/1000

	~~Mapping:~~ {{~~mapping~~\| 2 3 4 5 6 7 \| 0 5 19 18 27 12 }}		{{Mapping\|legend=0\| 2 3 4 5 6 7 \| 0 5 19 18 27 12 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 170/169, 196/195, 289/288, 352/351, 561/560		Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

	~~Mapping:~~ {{~~mapping~~\| 2 3 4 5 6 7 8 \| 0 5 19 18 27 12 5 }}		{{Mapping\|legend=0\| 2 3 4 5 6 7 8 \| 0 5 19 18 27 12 5 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 1331/1323, 1375/1372, 2200/2187		Comma list: 1331/1323, 1375/1372, 2200/2187

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 \| 0 1 2 3 3 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 \| 0 1 2 3 3 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 325/324, 352/351, 1001/1000, 1331/1323		Comma list: 325/324, 352/351, 1001/1000, 1331/1323

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 74 \| 0 1 2 3 3 0 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 74 \| 0 1 2 3 3 0 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000		Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 74 50 \| 0 1 2 3 3 0 1 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 74 50 \| 0 1 2 3 3 0 1 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475		Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 74 50 85 \| 0 1 2 3 3 0 1 0 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 74 50 85 \| 0 1 2 3 3 0 1 0 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399		Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 74 50 85 27 \| 0 1 2 3 3 0 1 0 2 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 74 50 85 27 \| 0 1 2 3 3 0 1 0 2 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405		Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

	~~Mapping:~~ {{~~mapping~~\| 20 0 -17 -39 -26 74 50 85 27 2 \| 0 1 2 3 3 0 1 0 2 3 }}		{{Mapping\|legend=0\| 20 0 -17 -39 -26 74 50 85 27 2 \| 0 1 2 3 3 0 1 0 2 3 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 3025/3024, 5632/5625, 10976/10935		Comma list: 3025/3024, 5632/5625, 10976/10935

	~~Mapping:~~ {{~~mapping~~\| 1 -8 1 -20 -21 \| 0 29 4 69 74 }}		{{Mapping\|legend=0\| 1 -8 1 -20 -21 \| 0 29 4 69 74 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 540/539, 896/891, 1331/1323		Comma list: 540/539, 896/891, 1331/1323

	~~Mapping:~~ {{~~mapping~~\| 1 0 -58 -21 -14 \| 0 1 38 15 11 }}		{{Mapping\|legend=0\| 1 0 -58 -21 -14 \| 0 1 38 15 11 }}

	Optimal tunings:		Optimal tunings:
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	Comma list: 169/168, 352/351, 364/363, 540/539		Comma list: 169/168, 352/351, 364/363, 540/539

	~~Mapping:~~ {{~~mapping~~\| 1 0 -58 -21 -14 -1 \| 0 1 38 15 11 8 }}		{{Mapping\|legend=0\| 1 0 -58 -21 -14 -1 \| 0 1 38 15 11 8 }}

	Optimal tunings:		Optimal tunings:

FloraC: Update

2026-09-15T08:16:23Z

Update

← Older revision		Revision as of 08:16, 15 September 2026
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	* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]		* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]
	* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic\|Magic family]]		* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic\|Magic family]]
	* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[~~Diaschismic family~~ #Echidna\|~~Diaschismic family~~]]		* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Stearnsmic clan #Echidna\|Stearnsmic clan]]
	* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]		* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]
	* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn\|Unicorn family]]		* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn\|Unicorn family]]

FloraC: Recategorize

2026-08-15T07:55:18Z

Recategorize

← Older revision		Revision as of 07:55, 15 August 2026
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	Badness (Sintel): 1.53		Badness (Sintel): 1.53

			[[Category:Hemimage temperaments\| ]] <!-- main article -->
	[[Category:Temperament collections]]		[[Category:Temperament collections]]
	[[Category:~~Hemimage temperaments\| ]] <!~~-~~- main article -->~~		[[Category:Catalogs of rank-2 temperaments]]
	~~[[Category:Rank~~ 2]]

FloraC: Update

2026-08-09T13:32:51Z

Update

← Older revision		Revision as of 13:32, 9 August 2026
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	* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]		* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]
	* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot\|Tetracot family]]		* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot\|Tetracot family]]
	* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[~~Ragismic microtemperaments~~ #Parakleismic\|~~Ragismic microtemperaments~~]]		* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Parakleismic\|Parakleismic family]]
	* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat\|Amity family]]		* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat\|Amity family]]
	* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]		* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]

FloraC at 15:11, 2 August 2026

2026-08-02T15:11:19Z

← Older revision		Revision as of 15:11, 2 August 2026
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	* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]		* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]

	Considered below are ~~degrees~~, bicommatic, ~~bisupermajor~~, squarschmidt, and leapmonth, in the order of increasing [[badness]].		Considered below are bisupermajor, bicommatic, degrees, squarschmidt, and leapmonth, in the order of increasing [[badness]].

	== Bisupermajor ==		== Bisupermajor ==

FloraC: - extensions beyond 31

2026-06-03T15:12:50Z

- extensions beyond 31

@@ Line 3: / Line 3: @@
 Temperaments discussed elsewhere are:
-* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
+* [[Quasisuper]] (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
 * ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
 * [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
@@ Line 128: / Line 128: @@
 Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.
-An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
+An obvious extension to the 23-limit exists by mapping [[23/20]] to 4\20 (1\5), [[69/56]] to 6\20 (3\10), and [[23/18]] to 7\20. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by mapping [[29/22]] to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out [[726/725]], which is the difference between [[33/25]] and [[29/22]], as well as [[784/783]] ({{S|28}}) and [[841/840]] ({{S|29}}). An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.
-By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.
+By equating [[37/28]] with 2\5 and more accurately [[85/74]] with 1\5 and [[44/37]] with 1\4 (among many other equivalences), we get an extension for prime [[37/1|37]] agreeing with many [[semiconvergent]]s, tempering out [[481/480]]. By mapping [[60/41]] and [[41/28]] to 11\20 or equivalently [[56/41]] and [[41/30]] to 9\20 and by mapping [[44/41]] to 1\10 (among many other equivalences), there is a very efficient extension for prime [[41/1|41]] tempering out [[451/450]].
-By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
+The 80-note generator chain is ideal, so [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].
 [[Subgroup]]: 2.3.5.7
@@ Line 240: / Line 240: @@
 Badness (Sintel): 1.13
 == Squarschmidt ==

FloraC: - chromat (more properly addressed in amity family). Sort links

2026-06-03T14:56:40Z

- chromat (more properly addressed in amity family). Sort links

Show changes

FloraC: + leapmonth (63 & 80)

2026-04-29T18:12:55Z

+ leapmonth (63 & 80)

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935). These include chromat, degrees, bicommatic, bisupermajor, and squarschmidt, considered below, as well as the following discussed elsewhere:
+This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).
 * ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
 * ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
@@ Line 16: / Line 18: @@
 * ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
 * ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]
 == Chromat ==
@@ Line 456: / Line 460: @@
 Badness (Sintel): 1.26
 [[Category:Temperament collections]]
 [[Category:Hemimage temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]

FloraC: /* Squarschmidt */

2026-04-28T13:06:26Z

Squarschmidt

← Older revision		Revision as of 13:06, 28 April 2026
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	: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''		: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''

	Squarschimidt may be described as {{nowrap\| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875. In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.		Squarschimidt may be described as {{nowrap\| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].

	~~=== 7-limit ===~~
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

FloraC: - classical squarschmidt (addressed in the 3et equivalence continuum)

2026-04-28T13:04:32Z

- classical squarschmidt (addressed in the 3et equivalence continuum)

← Older revision		Revision as of 13:04, 28 April 2026
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	== Squarschmidt ==		== Squarschmidt ==
	~~A generator for~~ the ~~squarschimidt temperament is the fourth root of~~ [[~~5/2]],~~ (5/2)<sup>1/4</sup>, tuned around 396.6 cents. The squarschimidt temperament can be described as {{nowrap\| 118 & 239 }} temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. ~~In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.~~		: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''

	~~[[Subgroup]]: 2.3.5~~		Squarschimidt may be described as {{nowrap\| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875. In the 11-limit, it tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

	~~[[Comma list]]:~~ {{~~monzo~~\| ~~61 4 -29~~ }}

	~~{{Mapping\|legend=1\| 1~~ -~~8 1 \| 0 29 4 }}~~
	~~: mapping generators: ~2~~, ~~~98304/78125~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1199.9653{{c}}, ~~~98304~~/~~78125 = 396.6094{{c}}~~
	~~: [[error map]]: {{val\| -0~~.~~099 +0.543 +0.029~~ -~~0.719 }}~~
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~~~98304~~/~~78125 = 396.6201{{c}}~~
	~~: error map: {{val\| 0.000 +0.653 +0.253 -0.552 }}~~

	~~{{Optimal ET sequence\|legend=1\| 118~~, ~~593~~, ~~711~~, ~~829, 947, 9588cc, 10535cc, 11482ccc }}~~

	~~[[Badness]] (Sintel): 5~~.12

	=== 7-limit ===		=== 7-limit ===

← Older revision		Revision as of 08:16, 15 September 2026
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	* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]		* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]
	* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic\|Magic family]]		* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic\|Magic family]]
	* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[~~Diaschismic family~~ #Echidna\|~~Diaschismic family~~]]		* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Stearnsmic clan #Echidna\|Stearnsmic clan]]
	* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]		* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]
	* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn\|Unicorn family]]		* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn\|Unicorn family]]

← Older revision		Revision as of 13:32, 9 August 2026
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	* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]		* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]
	* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot\|Tetracot family]]		* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot\|Tetracot family]]
	* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[~~Ragismic microtemperaments~~ #Parakleismic\|~~Ragismic microtemperaments~~]]		* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Parakleismic\|Parakleismic family]]
	* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat\|Amity family]]		* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat\|Amity family]]
	* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]		* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]