Hemimean clan: Difference between revisions

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= Didacus =
= Didacus =
Subgroup: 2.5.7
Subgroup: 2.5.7


Line 48: Line 47:
{{Val list|legend=1| 31, 87, 118 }}
{{Val list|legend=1| 31, 87, 118 }}


Badness: 0.044284
[[Badness]]: 0.044284


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 65: Line 63:
: Eigenmonzos: 2, 11/7
: Eigenmonzos: 2, 11/7


{{Val list|legend=1| 31, 87, 118 }}
Vals: {{Val list| 31, 87, 118 }}


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 79: Line 76:
POTE generator: ~28/25 = 193.166
POTE generator: ~28/25 = 193.166


{{Val list|legend=1| 31, 56, 87, 118, 205d }}
Vals: {{Val list| 31, 56, 87, 118, 205d }}


Badness: 0.021738
Badness: 0.021738


= Misty =
= Misty =
{{see also| Misty family }}
{{see also| Misty family }}


Line 99: Line 95:
{{Val list|legend=1| 12, 87, 99 }}
{{Val list|legend=1| 12, 87, 99 }}


Badness: 0.036802
[[Badness]]: 0.036802


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 111: Line 106:
POTE generator: ~3/2 = 703.210
POTE generator: ~3/2 = 703.210


{{Val list|legend=1| 12, 87, 99e, 186e }}
Vals: {{Val list| 12, 87, 99e, 186e }}


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 125: Line 119:
POTE generator: ~3/2 = 703.303
POTE generator: ~3/2 = 703.303


{{Val list|legend=1| 12f, 87, 99ef, 186ef, 273ef }}
Vals: {{Val list| 12f, 87, 99ef, 186ef, 273ef }}


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 139: Line 132:
POTE generator: ~3/2 = 703.233
POTE generator: ~3/2 = 703.233


{{Val list|legend=1| 12f, 87, 99ef, 186efg }}
Vals: {{Val list| 12f, 87, 99ef, 186efg }}


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 153: Line 145:
POTE generator: ~3/2 = 703.253
POTE generator: ~3/2 = 703.253


{{Val list|legend=1| 12f, 87, 99ef, 186efgh }}
Vals: {{Val list| 12f, 87, 99ef, 186efgh }}


Badness: 0.019878
Badness: 0.019878


=== Mystic ===
=== Mystic ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 167: Line 158:
POTE generator: ~3/2 = 703.311
POTE generator: ~3/2 = 703.311


{{Val list|legend=1| 12, 87, 186e, 273e }}
Vals: {{Val list| 12, 87, 186e, 273e }}


Badness: 0.036872
Badness: 0.036872


== Hemimist ==
== Hemimist ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 181: Line 171:
POTE generator: ~11/8 = 551.527
POTE generator: ~11/8 = 551.527


{{Val list|legend=1| 24d, 87, 111, 198 }}
Vals: {{Val list| 24d, 87, 111, 198 }}


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 195: Line 184:
POTE generator: ~11/8 = 551.525
POTE generator: ~11/8 = 551.525


{{Val list|legend=1| 24d, 87, 111, 198 }}
Vals: {{Val list| 24d, 87, 111, 198 }}


Badness: 0.022923
Badness: 0.022923


= Semisept =
= Semisept =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 214: Line 202:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 230: Line 217:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 239: Line 225:
POTE generator: ~55/36 = 735.125
POTE generator: ~55/36 = 735.125


{{Val list|legend=1| 18e, 31, 80, 111, 364cd }}
Vals: {{Val list| 18e, 31, 80, 111, 364cd }}


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 253: Line 238:
POTE generator: ~55/36 = 735.126
POTE generator: ~55/36 = 735.126


{{Val list|legend=1| 31, 80, 111 }}
Vals: {{Val list| 31, 80, 111 }}


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 267: Line 251:
POTE generator: ~26/17 = 735.125
POTE generator: ~26/17 = 735.125


{{Val list|legend=1| 31, 80, 111 }}
Vals: {{Val list| 31, 80, 111 }}


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 281: Line 264:
POTE generator: ~26/17 = 735.116
POTE generator: ~26/17 = 735.116


{{Val list|legend=1| 31, 80, 111 }}
Vals: {{Val list| 31, 80, 111 }}


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 295: Line 277:
POTE generator: ~26/17 = 735.106
POTE generator: ~26/17 = 735.106


{{Val list|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
Vals: {{Val list| 31, 80, 111, 191cdh, 302cdgh }}


Badness: 0.014957
Badness: 0.014957


=== Semishly ===
== Semishly ==
 
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 309: Line 290:
POTE generator: ~13/10 = 464.980
POTE generator: ~13/10 = 464.980


{{Val list|legend=1| 31, 49f, 80f }}
Vals: {{Val list| 31, 49f, 80f }}


Badness: 0.028408
Badness: 0.028408
Line 327: Line 308:


[[Badness]]: 0.057978
[[Badness]]: 0.057978
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1344/1331, 3136/3125
Mapping: [{{val| 1 1 2 2 3 }}, {{val| 0 29 16 40 23 }}]
POTE generator: ~99/98 = 24.235
Vals: {{Val list| 49, 50, 99e }}
Badness: 0.053828
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 975/968, 1344/1331
Mapping: [{{val| 1 1 2 2 3 4 }}, {{val| 0 29 16 40 23 -15 }}]
POTE generator: ~99/98 = 24.181
Vals: {{Val list| 49, 50, 99e, 149e, 248eef, 397deeeff }}
Badness: 0.053531


= Subpental =
= Subpental =
Line 346: Line 353:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 355: Line 361:
POTE generator: ~56/45 = 378.440
POTE generator: ~56/45 = 378.440


{{Val list|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Vals: {{Val list| 19, 111, 130, 241, 371ce, 501cde, 872cde }}


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 369: Line 374:
POTE generator: ~56/45 = 378.437
POTE generator: ~56/45 = 378.437


{{Val list|legend=1| 19, 111, 130, 241, 371ce }}
Vals: {{Val list| 19, 111, 130, 241, 371ce }}


Badness: 0.023940
Badness: 0.023940
Line 391: Line 396:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 400: Line 404:
POTE generator: ~64/63 = 27.663
POTE generator: ~64/63 = 27.663


{{Val list|legend=1| 43, 87, 130, 217, 347e, 911cde }}
Vals: {{Val list| 43, 87, 130, 217, 347e, 911cde }}


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 414: Line 417:
POTE generator: ~64/63 = 27.660
POTE generator: ~64/63 = 27.660


{{Val list|legend=1| 43, 87, 130, 217, 347e, 564e }}
Val list: {{Val list| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.019504
Badness: 0.019504


= Emka =
= Emka =
== 5-limit ==
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 433: Line 435:


== 7-limit ==
== 7-limit ==
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 439: Line 440:


[[Mapping]]: [{{val| 1 14 6 12 }}, {{val| 0 -27 -8 -20 }}]
[[Mapping]]: [{{val| 1 14 6 12 }}, {{val| 0 -27 -8 -20 }}]
{{Multival|legend=1| 27 8 20 -50 -44 24 }}


[[POTE generator]]: ~48/35 = 551.782
[[POTE generator]]: ~48/35 = 551.782
Line 447: Line 450:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 456: Line 458:
POTE generator: ~11/8 = 551.765
POTE generator: ~11/8 = 551.765


{{Val list|legend=1| 37, 50, 87, 224d, 311d }}
Vals: {{Val list| 37, 50, 87, 224d, 311d }}


Badness: 0.054744
Badness: 0.054744


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 470: Line 471:
POTE generator: ~11/8 = 551.758
POTE generator: ~11/8 = 551.758


{{Val list|legend=1| 37, 50, 87, 224d, 311d, 398d }}
Vals: {{Val list| 37, 50, 87, 224d, 311d, 398d }}


Badness: 0.029741
Badness: 0.029741

Revision as of 00:27, 24 April 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

Template:Val list

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, and sycamore considered elsewhere.

A notable subgroup extension of didacus is roulette.

Hemithirds

Main article: Hemithirds
See also: Luna family § 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

Minimax tuning:

[[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

Template:Val list

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

See also: Misty family

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Template:Val list

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

Subpental

See also: Sensipent family

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

Template:Val list

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

See also: Escapade family

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

Template:Val list

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

Val list: Template:Val list

Badness: 0.019504

Emka

Subgroup: 2.3.5

Comma list: [-50 -8 27⟩

Mapping: [⟨1 14 6], ⟨0 -27 -8]]

POTE generator: ~9765625/7077888 = 551.784

Template:Val list

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

Template:Val list

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

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