Hemimean clan: Difference between revisions

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The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit [[extension]]s of didacus, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic~~, quitagar~~, and undetrita, considered below, as well as these considered elsewhere:

The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit [[extension]]s of didacus, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, and undetrita, considered below, as well as these considered elsewhere:

* ''[[Spell]]'', {49/48, 3125/3072} → [[Magic family #Spell|Magic family]]

* ''[[Hexe]]'', {50/49, 128/125} → [[Augmented family #Hexe|Augmented family]]

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== Semisept ==

~~Subgroup~~: 2.~~3.5~~

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''

~~[[Comma list]]: 782757789696/762939453125~~

~~[[Mapping]]: [{{val| 1 12 6 }}, {{val| 0 -17 -6 }}]~~

~~[[POTE generator]]: ~192/125 = 735.146~~

~~{{Val list|legend=1| 18, 31, 80, 111 }}~~

~~[[Badness]]: 0.630576~~

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

Subgroup: 2.3.5.7

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== Emka ==

: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''

[[Emka]] tempers out {{monzo|-50 -8 27}} in the 5-limit. This temperament can be described as 37&50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87&224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.

~~Subgroup: 2.3.5~~

~~[[Comma list]]: {{monzo| -50 -8 27 }}~~

~~[[Mapping]]: [{{val| 1 14 6 }}, {{val| 0 -27 -8 }}]~~

~~[[POTE generator]]: ~9765625/7077888 = 551.784~~

~~{{Val list|legend=1| 37, 50, 87, 137, 224 }}~~

~~[[Badness]]: 0.544752~~

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

Subgroup: 2.3.5.7

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== Decipentic ==

The generator for the ''decipentic'' temperament (43&56) is tenth root of fifth harmonic ([[5/1]], pentave), 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to 27/23). Aside from the hemimean comma, this temperament tempers out the [[Bronzismic temperaments|bronzisma]], 2097152/2083725 (satriru-agugu). [[99edo|99EDO]] is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.

Subgroup: 2.3.5.7

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== Mowglic ==

The ''mowglic'' temperament (19&161) is an extension of the [[Syntonic-enneadecal equivalence continuum|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.

Subgroup: 2.3.5.7

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Vals: {{Val list| 19, 123defgk, 142fk, 161 }}

Badness: 0.019331~~<br><br>~~

Badness: 0.019331

== Undetrita ==

The ''undetrita'' temperament (111&118) tempers out the hemimean comma (3136/3125) and [[skeetsma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt|11EDT]] tuning, and the name ''undetrita'' is a play on the words "undecimus" (Latin for "eleventh") and "[[tritave]]" (third harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111&118.

Subgroup: 2.3.5.7

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== Isra ==

Isra results from taking every other generator of [[septimal meantone]]. It is named after the Isrāʾ (''iss-RAH'') night journey in the Qur'an, because it's similar to [[luna]].

Subgroup: 2.9.5.7

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-The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit [[extension]]s of didacus, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, quitagar, and undetrita, considered below, as well as these considered elsewhere:
+The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. The head of this clan is the 2.5.7 subgroup temperament didacus. Full 7-limit [[extension]]s of didacus, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). These include hemithirds, semisept, emka, decipentic, arch, sengagen, subpental, mowglic, and undetrita, considered below, as well as these considered elsewhere:
 * ''[[Spell]]'', {49/48, 3125/3072} → [[Magic family #Spell|Magic family]]
 * ''[[Hexe]]'', {50/49, 128/125} → [[Augmented family #Hexe|Augmented family]]
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 == Semisept ==
-Subgroup: 2.3.5
+: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''
-[[Comma list]]: 782757789696/762939453125
-[[Mapping]]: [{{val| 1 12 6 }}, {{val| 0 -17 -6 }}]
-[[POTE generator]]: ~192/125 = 735.146
-{{Val list|legend=1| 18, 31, 80, 111 }}
-[[Badness]]: 0.630576
-=== 7-limit ===
 Subgroup: 2.3.5.7
@@ Line 198: / Line 187: @@
 == Emka ==
+: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
 [[Emka]] tempers out {{monzo|-50 -8 27}} in the 5-limit. This temperament can be described as 37&amp;50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87&amp;224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
-Subgroup: 2.3.5
-[[Comma list]]: {{monzo| -50 -8 27 }}
-[[Mapping]]: [{{val| 1 14 6 }}, {{val| 0 -27 -8 }}]
-[[POTE generator]]: ~9765625/7077888 = 551.784
-{{Val list|legend=1| 37, 50, 87, 137, 224 }}
-[[Badness]]: 0.544752
-=== 7-limit ===
 Subgroup: 2.3.5.7
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 == Decipentic ==
 The generator for the ''decipentic'' temperament (43&amp;56) is tenth root of fifth harmonic ([[5/1]], pentave), 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to 27/23). Aside from the hemimean comma, this temperament tempers out the [[Bronzismic temperaments|bronzisma]], 2097152/2083725 (satriru-agugu). [[99edo|99EDO]] is a good tuning for decipentic, with generator 23\99, and MOS of 9, 13, 17, 30, 43 or 56 notes are available.
 Subgroup: 2.3.5.7
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 == Mowglic ==
 The ''mowglic'' temperament (19&amp;161) is an extension of the [[Syntonic-enneadecal equivalence continuum|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
 Subgroup: 2.3.5.7
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 Vals: {{Val list| 19, 123defgk, 142fk, 161 }}
-Badness: 0.019331<br><br>
+Badness: 0.019331
 == Undetrita ==
 The ''undetrita'' temperament (111&amp;118) tempers out the hemimean comma (3136/3125) and [[skeetsma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt|11EDT]] tuning, and the name ''undetrita'' is a play on the words "undecimus" (Latin for "eleventh") and "[[tritave]]" (third harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111&amp;118.
 Subgroup: 2.3.5.7
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 == Isra ==
 Isra results from taking every other generator of [[septimal meantone]]. It is named after the Isrāʾ (''iss-RAH'') night journey in the Qur'an, because it's similar to [[luna]].
 Subgroup: 2.9.5.7