Hemimean clan: Difference between revisions
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The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. | The '''hemimean clan''' tempers out the no-threes hemimean comma [[3136/3125]]. | ||
= Didacus = | == Didacus == | ||
Subgroup: 2.5.7 | Subgroup: 2.5.7 | ||
| Line 18: | Line 18: | ||
{{Val list|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c }} | {{Val list|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c }} | ||
== Extensions == | === Extensions === | ||
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, misty, semisept, sengagen, subpental, arch, and emka, considered below, as well as [[Meantone family #Septimal meantone|septimal meantone]], [[Archytas clan #Passion|passion]], [[Augmented family #Hexe|hexe]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], [[Ragismic microtemperaments #Parakleismic|parakleismic]], [[Kleismic family #Clyde|clyde]], [[Schismatic family #Bischismic|bischismic]], [[Stearnsmic clan #Decistearn|decistearn]], and [[Sycamore family #Sycamore|sycamore]] considered elsewhere. | 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, misty, semisept, sengagen, subpental, arch, and emka, considered below, as well as [[Meantone family #Septimal meantone|septimal meantone]], [[Archytas clan #Passion|passion]], [[Augmented family #Hexe|hexe]], [[Würschmidt family #Hemiwürschmidt|hemiwürschmidt]], [[Ragismic microtemperaments #Parakleismic|parakleismic]], [[Kleismic family #Clyde|clyde]], [[Schismatic family #Bischismic|bischismic]], [[Stearnsmic clan #Decistearn|decistearn]], and [[Sycamore family #Sycamore|sycamore]] considered elsewhere. | ||
A notable subgroup extension of didacus is [[Chromatic pairs #Roulette|roulette]]. | A notable subgroup extension of didacus is [[Chromatic pairs #Roulette|roulette]]. | ||
= Hemithirds = | == Hemithirds == | ||
{{main| Hemithirds }} | {{main| Hemithirds }} | ||
{{see also| Luna family #Hemithirds }} | {{see also| Luna family #Hemithirds }} | ||
| Line 49: | Line 49: | ||
[[Badness]]: 0.044284 | [[Badness]]: 0.044284 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 67: | Line 67: | ||
Badness: 0.019003 | Badness: 0.019003 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 80: | Line 80: | ||
Badness: 0.021738 | Badness: 0.021738 | ||
= Misty = | == Misty == | ||
{{see also| Misty family }} | {{see also| Misty family }} | ||
| Line 97: | Line 97: | ||
[[Badness]]: 0.036802 | [[Badness]]: 0.036802 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 110: | Line 110: | ||
Badness: 0.035932 | Badness: 0.035932 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 123: | Line 123: | ||
Badness: 0.027820 | Badness: 0.027820 | ||
==== 17-limit ==== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 136: | Line 136: | ||
Badness: 0.023908 | Badness: 0.023908 | ||
==== 19-limit ==== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 149: | Line 149: | ||
Badness: 0.019878 | Badness: 0.019878 | ||
=== Mystic === | ==== Mystic ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 162: | Line 162: | ||
Badness: 0.036872 | Badness: 0.036872 | ||
== Hemimist == | === Hemimist === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 175: | Line 175: | ||
Badness: 0.047439 | Badness: 0.047439 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 188: | Line 188: | ||
Badness: 0.022923 | Badness: 0.022923 | ||
= Semisept = | == Semisept == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 201: | Line 201: | ||
[[Badness]]: 0.630576 | [[Badness]]: 0.630576 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 216: | Line 216: | ||
[[Badness]]: 0.050472 | [[Badness]]: 0.050472 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 229: | Line 229: | ||
Badness: 0.022476 | Badness: 0.022476 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 242: | Line 242: | ||
Badness: 0.025204 | Badness: 0.025204 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 255: | Line 255: | ||
Badness: 0.019919 | Badness: 0.019919 | ||
=== 19-limit === | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 268: | Line 268: | ||
Badness: 0.016301 | Badness: 0.016301 | ||
=== 23-limit === | ==== 23-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19.23 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
| Line 281: | Line 281: | ||
Badness: 0.014957 | Badness: 0.014957 | ||
== Semishly == | === Semishly === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 294: | Line 294: | ||
Badness: 0.028408 | Badness: 0.028408 | ||
= Sengagen = | == Sengagen == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 309: | Line 309: | ||
[[Badness]]: 0.057978 | [[Badness]]: 0.057978 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 322: | Line 322: | ||
Badness: 0.053828 | Badness: 0.053828 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 335: | Line 335: | ||
Badness: 0.053531 | Badness: 0.053531 | ||
== Sengage == | === Sengage === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 348: | Line 348: | ||
Badness: 0.037416 | Badness: 0.037416 | ||
= Subpental = | == Subpental == | ||
{{see also|Sensipent family}} | {{see also|Sensipent family}} | ||
| Line 365: | Line 365: | ||
[[Badness]]: 0.054303 | [[Badness]]: 0.054303 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 378: | Line 378: | ||
Badness: 0.045352 | Badness: 0.045352 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 391: | Line 391: | ||
Badness: 0.023940 | Badness: 0.023940 | ||
= Arch = | == Arch == | ||
{{see also|Escapade family}} | {{see also|Escapade family}} | ||
| Line 408: | Line 408: | ||
[[Badness]]: 0.094345 | [[Badness]]: 0.094345 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 421: | Line 421: | ||
Badness: 0.036541 | Badness: 0.036541 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 434: | Line 434: | ||
Badness: 0.019504 | Badness: 0.019504 | ||
= Emka = | == 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. | [[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. | ||
| Line 449: | Line 449: | ||
[[Badness]]: 0.544752 | [[Badness]]: 0.544752 | ||
== 7-limit == | === 7-limit === | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 464: | Line 464: | ||
[[Badness]]: 0.144338 | [[Badness]]: 0.144338 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 477: | Line 477: | ||
Badness: 0.054744 | Badness: 0.054744 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 22:10, 2 June 2021
The hemimean clan tempers out the no-threes hemimean comma 3136/3125.
Didacus
Subgroup: 2.5.7
Comma list: 3136/3125
Sval mapping: [⟨1 0 -3], ⟨0 2 5]]
Mapping generators: ~2, ~28/25
Gencom mapping: [⟨1 0 0 -3], ⟨0 0 2 5]]
Gencom: [2 28/25; 3136/3125]
POTE generator: ~28/25 = 193.772
Extensions
Full 7-limit extensions 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, misty, semisept, sengagen, subpental, arch, and emka, considered below, as well as septimal meantone, passion, hexe, hemiwürschmidt, parakleismic, clyde, bischismic, decistearn, and sycamore considered elsewhere.
A notable subgroup extension of didacus is roulette.
Hemithirds
Subgroup: 2.3.5.7
Comma list: 1029/1024, 3136/3125
Mapping: [⟨1 4 2 2], ⟨0 -15 2 5]]
Wedgie: ⟨⟨ 15 -2 -5 -38 -50 -6 ]]
POTE generator: ~28/25 = 193.244
- [[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
- Eigenmonzos: 2, 7/6
- [[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
- Eigenmonzos: 2, 7/6
Badness: 0.044284
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 3136/3125
Mapping: [⟨1 4 2 2 7], ⟨0 -15 2 5 -22]]
POTE generator: ~28/25 = 193.227
Minimax tuning:
- 11-odd-limit
- [[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
- Eigenmonzos: 2, 11/7
Vals: Template:Val list
Badness: 0.019003
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 385/384, 625/624
Mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]
POTE generator: ~28/25 = 193.166
Vals: Template:Val list
Badness: 0.021738
Misty
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56], ⟨0 1 -4 -10]]
Wedgie: ⟨⟨ 3 -12 -30 -26 -56 -36 ]]
POTE generator: ~3/2 = 703.021
Badness: 0.036802
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4375/4356
Mapping: [⟨3 0 26 56 77], ⟨0 1 -4 -10 -14]]
POTE generator: ~3/2 = 703.210
Vals: Template:Val list
Badness: 0.035932
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 352/351, 364/363, 625/624
Mapping: [⟨3 0 26 56 77 92], ⟨0 1 -4 -10 -14 -17]]
POTE generator: ~3/2 = 703.303
Vals: Template:Val list
Badness: 0.027820
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2], ⟨0 1 -4 -10 -14 -17 3]]
POTE generator: ~3/2 = 703.233
Vals: Template:Val list
Badness: 0.023908
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Mapping: [⟨3 0 26 56 77 92 -2 8], ⟨0 1 -4 -10 -14 -17 3 1]]
POTE generator: ~3/2 = 703.253
Vals: Template:Val list
Badness: 0.019878
Mystic
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 441/440, 896/891, 1001/1000
Mapping: [⟨3 0 26 56 77 -46], ⟨0 1 -4 -10 -14 12]]
POTE generator: ~3/2 = 703.311
Vals: Template:Val list
Badness: 0.036872
Hemimist
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 3136/3125, 5120/5103
Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]
POTE generator: ~11/8 = 551.527
Vals: Template:Val list
Badness: 0.047439
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 676/675, 847/845, 3025/3024
Mapping: [⟨3 0 26 56 8 23], ⟨0 2 -8 -20 1 -5]]
POTE generator: ~11/8 = 551.525
Vals: Template:Val list
Badness: 0.022923
Semisept
Subgroup: 2.3.5
Comma list: 782757789696/762939453125
Mapping: [⟨1 12 6], ⟨0 -17 -6]]
POTE generator: ~192/125 = 735.146
Badness: 0.630576
7-limit
Subgroup: 2.3.5.7
Comma list: 1728/1715, 3136/3125
Mapping: [⟨1 12 6 12], ⟨0 -17 -6 -15]]
Wedgie: ⟨⟨ 17 6 15 -30 -24 18 ]]
POTE generator: ~75/49 = 735.155
Badness: 0.050472
11-limit
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 1331/1323
Mapping: [⟨1 12 6 12 20], ⟨0 -17 -6 -15 -27]]
POTE generator: ~55/36 = 735.125
Vals: Template:Val list
Badness: 0.022476
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 540/539, 1375/1372
Mapping: [⟨1 12 6 12 20 -11], ⟨0 -17 -6 -15 -27 24]]
POTE generator: ~55/36 = 735.126
Vals: Template:Val list
Badness: 0.025204
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Mapping: [⟨1 12 6 12 20 -11 -10], ⟨0 -17 -6 -15 -27 24 23]]
POTE generator: ~26/17 = 735.125
Vals: Template:Val list
Badness: 0.019919
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Mapping: [⟨1 12 6 12 20 -11 -10 -8], ⟨0 -17 -6 -15 -27 24 23 20]]
POTE generator: ~26/17 = 735.116
Vals: Template:Val list
Badness: 0.016301
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 18|, ⟨0 -17 -6 -15 -27 24 23 20 -22]]
POTE generator: ~26/17 = 735.106
Vals: Template:Val list
Badness: 0.014957
Semishly
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 176/175, 196/195, 275/273
Mapping: [⟨1 12 6 12 20 8], ⟨0 -17 -6 -15 -27 -7]]
POTE generator: ~13/10 = 464.980
Vals: Template:Val list
Badness: 0.028408
Sengagen
Subgroup: 2.3.5.7
Comma list: 3136/3125, 420175/419904
Mapping: [⟨1 1 2 2], ⟨0 29 16 40]]
Wedgie: ⟨⟨ 29 16 40 -42 -18 48 ]]
POTE generator: ~686/675 = 24.217
Badness: 0.057978
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1344/1331, 3136/3125
Mapping: [⟨1 1 2 2 3], ⟨0 29 16 40 23]]
POTE generator: ~99/98 = 24.235
Vals: Template:Val list
Badness: 0.053828
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 975/968, 1344/1331
Mapping: [⟨1 1 2 2 3 4], ⟨0 29 16 40 23 -15]]
POTE generator: ~99/98 = 24.181
Vals: Template:Val list
Badness: 0.053531
Sengage
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 196/195, 364/363, 625/624
Mapping: [⟨1 1 2 2 3 3], ⟨0 29 16 40 23 35]]
POTE generator: ~99/98 = 24.234
Vals: Template:Val list
Badness: 0.037416
Subpental
Subgroup: 2.3.5.7
Comma list: 3136/3125, 19683/19600
Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]
Wedgie: ⟨⟨ 14 18 45 -4 32 54 ]]
POTE generator: ~56/45 = 378.467
Badness: 0.054303
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]
POTE generator: ~56/45 = 378.440
Vals: Template:Val list
Badness: 0.045352
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: [⟨1 6 8 17 -6 16], ⟨0 -14 -18 -45 30 -39]]
POTE generator: ~56/45 = 378.437
Vals: Template:Val list
Badness: 0.023940
Arch
Subgroup: 2.3.5.7
Comma list: 3136/3125, 5250987/5242880
Mapping: [⟨1 2 2 2], ⟨0 -18 14 35]]
Wedgie: ⟨⟨ 18 -14 -35 -64 -106 -42 ]]
POTE generator: ~64/63 = 27.668
Badness: 0.094345
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 4000/3993
Mapping: [⟨1 2 2 2 3], ⟨0 -18 14 35 20]]
POTE generator: ~64/63 = 27.663
Vals: Template:Val list
Badness: 0.036541
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 3136/3125
Mapping: [⟨1 2 2 2 3 4], ⟨0 -18 14 35 20 -13]]
POTE generator: ~64/63 = 27.660
Vals: Template:Val list
Badness: 0.019504
Emka
Emka tempers out [-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 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: [-50 -8 27⟩
Mapping: [⟨1 14 6], ⟨0 -27 -8]]
POTE generator: ~9765625/7077888 = 551.784
Badness: 0.544752
7-limit
Subgroup: 2.3.5.7
Comma list: 3136/3125, 84035/82944
Mapping: [⟨1 14 6 12], ⟨0 -27 -8 -20]]
Wedgie: ⟨⟨ 27 8 20 -50 -44 24 ]]
POTE generator: ~48/35 = 551.782
Badness: 0.144338
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 2401/2376, 3136/3125
Mapping: [⟨1 14 6 12 3], ⟨0 -27 -8 -20 1]]
POTE generator: ~11/8 = 551.765
Vals: Template:Val list
Badness: 0.054744
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 196/195, 364/363, 385/384, 625/624
Mapping: [⟨1 14 6 12 3 6], ⟨0 -27 -8 -20 1 -5]]
POTE generator: ~11/8 = 551.758
Vals: Template:Val list
Badness: 0.029741