Hemimage temperaments - Revision history

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
Line 424:		Line 424:
	: ''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
Line 422:		Line 422:

	== 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 ===

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-04-28T11:36:45Z

Switch to Sintel's badness, WE & CWE tunings

Show changes

FloraC: Units, misc. cleanup, - redundant category

2026-04-28T10:53:09Z

Units, misc. cleanup, - redundant category

Show changes

FloraC: Update links

2026-04-05T12:08:37Z

Update links

← Older revision		Revision as of 12:08, 5 April 2026
Line 10:		Line 10:
	* ''[[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) → [[Ragismic microtemperaments #Parakleismic\|Ragismic microtemperaments]]
	* ''[[Pluto]]'' (+4000/3969) → [[~~Mirkwai clan~~ #Pluto\|~~Mirkwai clan~~]]		* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]
	* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]		* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]
	* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]		* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]
	* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]		* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]

Overthink: /* Degrees */ formatting

2026-01-11T04:27:55Z

Degrees: formatting

← Older revision		Revision as of 04:27, 11 January 2026
Line 238:		Line 238:

	== Degrees ==		== Degrees ==
			{{About\|the regular temperament\|scale degrees\|degree}}
	{{ See also \| 20th-octave temperaments }}		{{ See also \| 20th-octave temperaments }}
	~~{{About\|the regular temperament\|scale degrees\|degree}}~~
	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.		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.

Overthink: /* Degrees */ + about template

2025-12-06T18:50:04Z

Degrees: + about template

← Older revision		Revision as of 18:50, 6 December 2025
Line 239:		Line 239:
	== Degrees ==		== Degrees ==
	{{ See also \| 20th-octave temperaments }}		{{ See also \| 20th-octave temperaments }}
			{{About\|the regular temperament\|scale degrees\|degree}}
	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.		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.

← Older revision		Revision as of 12:08, 5 April 2026
Line 10:		Line 10:
	* ''[[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) → [[Ragismic microtemperaments #Parakleismic\|Ragismic microtemperaments]]
	* ''[[Pluto]]'' (+4000/3969) → [[~~Mirkwai clan~~ #Pluto\|~~Mirkwai clan~~]]		* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto\|Octagar temperaments]]
	* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]		* ''[[Hendecatonic (temperament)\|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic\|Porwell temperaments]]
	* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]		* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths\|Kleismic family]]
	* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]		* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth\|Buzzardsmic clan]]