Hemimean clan: Difference between revisions

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


Badness: 0.0443
Badness: 0.044284


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


Line 67: Line 66:
{{Val list|legend=1| 31, 87, 118 }}
{{Val list|legend=1| 31, 87, 118 }}


Badness: 0.0190
Badness: 0.019003


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


Line 81: Line 79:
{{Val list|legend=1| 31, 56, 87, 118, 205d }}
{{Val list|legend=1| 31, 56, 87, 118, 205d }}


Badness: 0.0217
Badness: 0.021738


= Misty =
= Misty =
Line 97: Line 95:


{{Val list|legend=1| 12, 87, 99 }}
{{Val list|legend=1| 12, 87, 99 }}
Badness: 0.036802


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


Line 109: Line 108:


{{Val list|legend=1| 12, 87, 99e, 186e }}
{{Val list|legend=1| 12, 87, 99e, 186e }}
Badness: 0.035932


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


Line 122: Line 122:
{{Val list|legend=1| 12f, 87, 99ef, 186ef, 273ef }}
{{Val list|legend=1| 12f, 87, 99ef, 186ef, 273ef }}


==== 17-limit ====
Badness: 0.027820


=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Line 134: Line 135:
{{Val list|legend=1| 12f, 87, 99ef, 186efg }}
{{Val list|legend=1| 12f, 87, 99ef, 186efg }}


==== 19-limit ====
Badness: 0.023908


=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 145: Line 147:


{{Val list|legend=1| 12f, 87, 99ef, 186efgh }}
{{Val list|legend=1| 12f, 87, 99ef, 186efgh }}
Badness: 0.019878


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


Line 157: Line 160:


{{Val list|legend=1| 12, 87, 186e, 273e }}
{{Val list|legend=1| 12, 87, 186e, 273e }}
Badness: 0.036872


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


Line 170: Line 174:
{{Val list|legend=1| 24d, 87, 111, 198 }}
{{Val list|legend=1| 24d, 87, 111, 198 }}


Badness: 0.0474
Badness: 0.047439


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


Line 184: Line 187:
{{Val list|legend=1| 24d, 87, 111, 198 }}
{{Val list|legend=1| 24d, 87, 111, 198 }}


Badness: 0.0229
Badness: 0.022923


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


Line 199: Line 200:
{{Val list|legend=1| 18, 31, 80, 111 }}
{{Val list|legend=1| 18, 31, 80, 111 }}


[[Badness]]: 0.6306
[[Badness]]: 0.630576


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


Line 214: Line 214:


{{Val list|legend=1| 18, 31, 80, 111 }}
{{Val list|legend=1| 18, 31, 80, 111 }}
[[Badness]]: 0.050472


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


Line 227: Line 228:
{{Val list|legend=1| 18e, 31, 80, 111, 364cd }}
{{Val list|legend=1| 18e, 31, 80, 111, 364cd }}


Badness: 0.0225
Badness: 0.022476
 
=== 13-limit ===


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


Line 241: Line 241:
{{Val list|legend=1| 31, 80, 111 }}
{{Val list|legend=1| 31, 80, 111 }}


Badness: 0.0252
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 254:
{{Val list|legend=1| 31, 80, 111 }}
{{Val list|legend=1| 31, 80, 111 }}


==== 19-limit ====
Badness: 0.019919


=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Line 267: Line 267:
{{Val list|legend=1| 31, 80, 111 }}
{{Val list|legend=1| 31, 80, 111 }}


==== 23-limit ====
Badness: 0.016301


=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17.19.23


Line 279: Line 280:
{{Val list|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Val list|legend=1| 31, 80, 111, 191cdh, 302cdgh }}


=== Semishly ===
Badness: 0.014957


== Semishly ==
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 291: Line 293:
{{Val list|legend=1| 31, 49f, 80f }}
{{Val list|legend=1| 31, 49f, 80f }}


Badness: 0.0284
Badness: 0.028408


= Sengagen =
= Sengagen =
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


Line 307: Line 308:
{{Val list|legend=1| 49, 50, 99, 248, 347, 446 }}
{{Val list|legend=1| 49, 50, 99, 248, 347, 446 }}


[[Badness]]: 0.0580
[[Badness]]: 0.057978


= Subpental =
= Subpental =
{{see also|Sensipent family}}


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 323: Line 325:
{{Val list|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
{{Val list|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}


[[Badness]]: 0.0543
[[Badness]]: 0.054303


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


Line 337: Line 338:
{{Val list|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
{{Val list|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}


Badness: 0.0453
Badness: 0.045352


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


Line 351: Line 351:
{{Val list|legend=1| 19, 111, 130, 241, 371ce }}
{{Val list|legend=1| 19, 111, 130, 241, 371ce }}


Badness: 0.0239
Badness: 0.023940


= Arch =
= Arch =
{{see also|Escapade family}}


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 367: Line 368:
{{Val list|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}
{{Val list|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}


[[Badness]]: 0.0943
[[Badness]]: 0.094345


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


Line 381: Line 381:
{{Val list|legend=1| 43, 87, 130, 217, 347e, 911cde }}
{{Val list|legend=1| 43, 87, 130, 217, 347e, 911cde }}


Badness: 0.0365
Badness: 0.036541


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


Line 395: Line 394:
{{Val list|legend=1| 43, 87, 130, 217, 347e, 564e }}
{{Val list|legend=1| 43, 87, 130, 217, 347e, 564e }}


Badness: 0.0195
Badness: 0.019504


= Emka =
= Emka =
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
Subgroup: 2.3.5.7


Line 409: Line 420:
{{Val list|legend=1| 37, 50, 87, 137d, 224d }}
{{Val list|legend=1| 37, 50, 87, 137d, 224d }}


[[Badness]]: 0.1443
[[Badness]]: 0.144338


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


Line 421: Line 431:
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: 0.0547
Badness: 0.054744


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


Line 435: Line 444:
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: 0.0297
Badness: 0.029741


[[Category:Regular temperament theory]]
[[Category:Regular temperament theory]]

Revision as of 13:00, 8 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

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

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

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

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

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

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

Template:Val list

Badness: 0.036872

Hemimist

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 5120/5103, 3025/3024

Mapping: [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]]

POTE generator: ~11/8 = 551.527

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

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: 3136/3125, 1728/1715

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

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

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

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

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

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

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

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

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

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

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

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]]

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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