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Hemimean clan: Difference between revisions

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-arch (addressed in escapade family)
Copy hemiwuerschmidt and spell here as strong extensions, along with hemithirds
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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, sengagen, subpental, mowglic, 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 hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, 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]]
* ''[[Hexe]]'', {50/49, 128/125} → [[Augmented family #Hexe|Augmented family]]
* ''[[Passion]]'', {64/63, 3125/3087} → [[Passion family #Septimal passion|Passion family]]
* ''[[Passion]]'', {64/63, 3125/3087} → [[Passion family #Septimal passion|Passion family]]
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* ''[[Sycamore]]'', {686/675, 875/864} → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Sycamore]]'', {686/675, 875/864} → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'', {2048/2025, 3136/3125} → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Bidia]]'', {2048/2025, 3136/3125} → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hemiwürschmidt]]'', {2401/2400, 3136/3125} → [[Würschmidt family #Hemiwürschmidt|Würschmidt family]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Parakleismic]], {3136/3125, 4375/4374} → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Septimal misty|Misty family]]
* [[Misty]], {3136/3125, 5120/5103} → [[Misty family #Septimal misty|Misty family]]
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Optimal GPV sequence: {{val list| 18, 19, 37, 93, 130 }}
Optimal GPV sequence: {{val list| 18, 19, 37, 93, 130 }}
== Hemiwürschmidt ==
{{See also| Würschmidt family #Hemiwürschmidt }}
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
Subgroup: 2.3.5.7
[[Comma list]]: 2401/2400, 3136/3125
[[Mapping]]: [{{val| 1 15 4 7 }}, {{val| 0 -16 -2 -5 }}]
{{Multival|legend=1| 16 2 5 -34 -37 6 }}
[[POTE generator]]: ~28/25 = 193.898
{{Val list|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}
[[Badness]]: 0.020307
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 3136/3125
Mapping: [{{val| 1 15 4 7 37 }}, {{val| 0 -16 -2 -5 -40 }}]
POTE generator: ~28/25 = 193.840
Vals: {{Val list| 31, 99e, 130, 650ce, 811ce }}
Badness: 0.021069
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 351/350, 441/440, 3584/3575
Mapping: [{{val| 1 15 4 7 37 -29 }}, {{val| 0 -16 -2 -5 -40 39 }}]
POTE generator: ~28/25 = 193.829
Vals: {{Val list| 31, 99e, 130, 291, 421e, 551ce }}
Badness: 0.023074
==== Hemithir ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 196/195, 275/273
Mapping: [{{val| 1 15 4 7 37 -3 }}, {{val| 0 -16 -2 -5 -40 8 }}]
POTE generator: ~28/25 = 193.918
Vals: {{Val list| 31, 68e, 99ef }}
Badness: 0.031199
=== Hemiwur ===
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 1375/1372
Mapping: [{{val| 1 15 4 7 11 }}, {{val| 0 -16 -2 -5 -9 }}]
POTE generator: ~28/25 = 193.884
Vals: {{Val list| 31, 68, 99, 130e, 229e }}
Badness: 0.029270
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 196/195, 275/273
Mapping: [{{val| 1 15 4 7 11 -3 }}, {{val| 0 -16 -2 -5 -9 8 }}]
POTE generator: ~28/25 = 194.004
Vals: {{Val list| 31, 68, 99f, 167ef }}
Badness: 0.028432
==== Hemiwar ====
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 105/104, 121/120, 1375/1372
Mapping: [{{val| 1 15 4 7 11 23 }}, {{val| 0 -16 -2 -5 -9 -23 }}]
POTE generator: ~28/25 = 193.698
Vals: {{Val list| 6f, 31 }}
Badness: 0.044886
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3025/3024, 3136/3125
Mapping: [{{val| 1 15 4 7 24 }}, {{val| 0 -32 -4 -10 -49 }}]
Mapping generators: ~2, ~147/110
POTE generator: ~147/110 = 503.0404
Vals: {{Val list| 31, 105be, 136e, 167, 198, 427c }}
Badness: 0.034814
=== Semihemiwür ===
<!-- Name revised from "Semihemiwürschimidt" on 11 September 2021 -->
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 3136/3125, 9801/9800
Mapping: [{{val| 2 14 6 9 -10 }}, {{val| 0 -16 -2 -5 25 }}]
Mapping generators: ~99/70, ~495/392
POTE generator: ~28/25 = 193.9021
Vals: {{Val list| 62e, 68, 130, 198, 328 }}
Badness: 0.044848
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125
Mapping: [{{val| 2 14 6 9 -10 25 }}, {{val| 0 -16 -2 -5 25 -26 }}]
POTE generator: ~28/25 = 193.9035
Vals: {{Val list| 62e, 68, 130, 198, 328 }}
Badness: 0.023388
===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625
Mapping: [{{val| 2 14 6 9 -10 25 19 }}, {{val| 0 -16 -2 -5 25 -26 -16 }}]
POTE generator: ~28/25 = 193.9112
Vals: {{Val list| 62e, 68, 130, 198, 328g, 526cfgg }}
Badness: 0.028987
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625
Mapping: [{{val| 2 14 6 9 -10 25 19 20 }}, {{val| 0 -16 -2 -5 25 -26 -16 -17 }}]
POTE generator: ~19/17 = 193.9145
Vals: {{Val list| 62e, 68, 130, 198, 328g, 526cfgg }}
Badness: 0.021707
===== Semihemiwürand =====
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224
Mapping: [{{val| 2 14 6 9 -10 25 -4 }}, {{val| 0 -16 -2 -5 25 -26 18 }}]
POTE generator: ~28/25 = 193.9112
Vals: {{Val list| 62eg, 68, 130g, 198g }}
Badness: 0.029718
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224
Mapping: [{{val| 2 14 6 9 -10 25 -4 -3 }}, {{val| 0 -16 -2 -5 25 -26 18 17 }}]
POTE generator: ~19/17 = 193.9428
Vals: {{Val list| 62egh, 68, 130gh, 198gh }}
Badness: 0.029545


== Hemithirds ==
== Hemithirds ==
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Badness: 0.021738
Badness: 0.021738
== Spell ==
{{See also| Magic family #Spell }}
Subgroup: 2.3.5.7
[[Comma list]]: 49/48, 3125/3072
[[Mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 10 2 5 }}]
{{Multival|legend=1| 10 2 5 -20 -20 6 }}
[[POTE generator]]: ~28/25 = 189.927
{{Val list|legend=1| 6, 19, 82dd }}
[[Badness]]: 0.080958
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 125/121
Mapping: [{{val| 1 0 2 2 3 }}, {{val| 0 10 2 5 3 }}]
POTE generator: ~11/10 = 190.285
Optimal GPV sequence: {{Val list| 6, 19, 44de, 63dee, 82ddee }}
Badness: 0.059791
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 78/77, 125/121
Mapping: [{{val| 1 0 2 2 3 4 }}, {{val| 0 10 2 5 3 -2 }}]
POTE generator: ~11/10 = 189.928
Optimal GPV sequence: {{Val list| 6, 19, 82ddeeff }}
Badness: 0.045591
==== Cantrip ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 125/121
Mapping: [{{val| 1 0 2 2 3 1 }}, {{val| 0 10 2 5 3 17 }}]
POTE generator: ~11/10 = 190.360
Optimal GPV sequence: {{Val list| 19, 44de, 63dee, 82ddee }}
Badness: 0.041603


== Semisept ==
== Semisept ==
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