Hemimean clan: Difference between revisions

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Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]
Gencom mapping: [{{val| 1 0 0 -3 }}, {{val| 0 0 2 5 }}]


Gencom: [2 28/25; 3136/3125]
[[Gencom]]: [2 28/25; 3136/3125]


[[POTE generator]]: ~28/25 = 193.772
[[POTE generator]]: ~28/25 = 193.772
Line 22: Line 22:
{{see also|Misty family}}
{{see also|Misty family}}


== 7-limit ==
[[Comma list]]: 3136/3125, 5120/5103
Commas: 3136/3125, 5120/5103


[[POTE_tuning|POTE generator]]: ~3/2 = 703.021
[[Mapping]]: [{{val| 3 0 26 56 }}, {{val| 0 1 -4 -10 }}]


Map: [<3 0 26 56|, <0 1 -4 -10|]
{{Multival|legend=1| 3 -12 -30 -26 -56 -36 }}


Wedgie: <<3 -12 -30 -26 -56 -36||
[[POTE generator]]: ~3/2 = 703.021


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


== 11-limit ==
== 11-limit ==
Commas: 441/440, 3136/3125, 4375/4356
Comma list: 441/440, 3136/3125, 4375/4356


[[POTE_tuning|POTE generator]]: ~3/2 = 703.210
Mapping: [{{val| 3 0 26 56 77 }}, {{val| 0 1 -4 -10 -14 }}]


Map: [<3 0 26 56 77|, <0 1 -4 -10 -14|]
POTE generator: ~3/2 = 703.210


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


=== 13-limit ===
=== 13-limit ===
Commas: 196/195, 352/351, 364/363, 625/624
Comma list: 196/195, 352/351, 364/363, 625/624


[[POTE_tuning|POTE generator]]: ~3/2 = 703.303
Mapping: [{{val| 3 0 26 56 77 92 }}, {{val| 0 1 -4 -10 -14 -17 }}]


Map: [<3 0 26 56 77 92|, <0 1 -4 -10 -14 -17|]
POTE generator: ~3/2 = 703.303


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


==== 17-limit ====
==== 17-limit ====
Commas: 196/195, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 352/351, 364/363, 375/374, 595/594


[[POTE_tuning|POTE generator]]: ~3/2 = 703.233
Mapping: [{{val| 3 0 26 56 77 92 -2 }}, {{val| 0 1 -4 -10 -14 -17 3 }}]


Map: [<3 0 26 56 77 92 -2|, <0 1 -4 -10 -14 -17 3|]
POTE generator: ~3/2 = 703.233


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


==== 19-limit ====
==== 19-limit ====
Commas: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594
Comma list: 196/195, 210/209, 352/351, 364/363, 375/374, 595/594


[[POTE_tuning|POTE generator]]: ~3/2 = 703.253
Mapping: [{{val| 3 0 26 56 77 92 -2 8 }}, {{val| 0 1 -4 -10 -14 -17 3 1 }}]


Map: [<3 0 26 56 77 92 -2 8|, <0 1 -4 -10 -14 -17 3 1|]
POTE generator: ~3/2 = 703.253


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


=== Mystic ===
=== Mystic ===
Commas: 325/324, 441/440, 896/891, 1001/1000
Comma list: 325/324, 441/440, 896/891, 1001/1000


[[POTE_tuning|POTE generator]]: ~3/2 = 703.311
Mapping: [{{val| 3 0 26 56 77 -46 }}, {{val| 0 1 -4 -10 -14 12 }}]


Map: [<3 0 26 56 77 -46|, <0 1 -4 -10 -14 12|]
POTE generator: ~3/2 = 703.311


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


== Hemimist ==
== Hemimist ==
Commas: 3136/3125, 5120/5103, 3025/3024
Comma list: 3136/3125, 5120/5103, 3025/3024
 
Mapping: [{{val| 3 0 26 56 8 }}, {{val| 0 2 -8 -20 1 }}]


POTE generator: ~11/8 = 551.527
POTE generator: ~11/8 = 551.527


Map: [<3 0 26 56 8|, <0 2 -8 -20 1|]
{{Val list|legend=1| 24d, 87, 111, 198 }}
 
EDOs: {{EDOs|24d, 87, 111, 198}}


Badness: 0.0474
Badness: 0.0474


=== 13-limit ===
=== 13-limit ===
Commas: 352/351, 676/675, 847/845, 3025/3024
Comma list: 352/351, 676/675, 847/845, 3025/3024
 
Mapping: [{{val| 3 0 26 56 8 23 }}, {{val| 0 2 -8 -20 1 -5 }}]


POTE generator: ~11/8 = 551.525
POTE generator: ~11/8 = 551.525


Map: [<3 0 26 56 8 23|, <0 2 -8 -20 1 -5|]
{{Val list|legend=1| 24d, 87, 111, 198 }}
 
EDOs: {{EDOs|24d, 87, 111, 198}}


Badness: 0.0229
Badness: 0.0229


=Semisept=
= Semisept =
Comma: 782757789696/762939453125
= 5-limit =
[[Comma list]]: 782757789696/762939453125


POTE generator: ~192/125 = 735.146
[[Mapping]]: [{{val| 1 12 6 }}, {{val| 0 -17 -6 }}]


Map: [<1 12 6|, <0 -17 -6|]
[[POTE generator]]: ~192/125 = 735.146


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


Badness: 0.6306
[[Badness]]: 0.6306


==7-limit==
== 7-limit ==
Commas: 3136/3125, 1728/1715
[[Comma list]]: 3136/3125, 1728/1715


[[POTE_tuning|POTE generator]]: (~75/49) 735.155
[[Mapping]]: [{{val| 1 12 6 12 }}, {{val| 0 -17 -6 -15 }}]


Map: [<1 12 6 12|, <0 -17 -6 -15|]
{{Multival|legend=1| 17 6 15 -30 -24 18 }}


Wedgie: <<17 6 15 -30 -24 18||
[[POTE generator]]: ~75/49 = 735.155


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


==11-limit==
== 11-limit ==
Commas: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


[[POTE_tuning|POTE generator]]: (~55/36) 735.125
Mapping: [{{val| 1 12 6 12 20 }}, {{val| 0 -17 -6 -15 -27 }}]


Map: [<1 12 6 12 20|, <0 -17 -6 -15 -27|]
POTE generator: ~55/36 = 735.125


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


Badness: 0.0225
Badness: 0.0225


===Semishly===
=== 13-limit ===
Commas: 144/143, 176/175, 196/195, 275/273
Comma list: 176/175, 351/350, 540/539, 1375/1372


POTE generator: ~13/10 = 464.980
Mapping: [{{val| 1 12 6 12 20 -11 }}, {{val| 0 -17 -6 -15 -27 24 }}]


Map: [<1 12 6 12 20 8|, <0 -17 -6 -15 -27 -7|]
POTE generator: ~55/36 = 735.126


EDOs: 31, 49f, 80f
{{Val list|legend=1| 31, 80, 111 }}


Badness: 0.0284
Badness: 0.0252


==13-limit==
==== 17-limit ====
Commas: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


[[POTE_tuning|POTE generator]]: 735.126
Mapping: [{{val| 1 12 6 12 20 -11 -10 }}, {{val| 0 -17 -6 -15 -27 24 23 }}]


Map: [<1 12 6 12 20 -11|, <0 -17 -6 -15 -27 24|]
POTE generator: ~26/17 = 735.125


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


Badness: 0.0252
==== 19-limit ====
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


==17-limit==
Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 }}, {{val| 0 -17 -6 -15 -27 24 23 20 }}]
Commas: 176/175, 256/255, 351/350, 640/637, 715/714


[[POTE_tuning|POTE generator]]: (~26/17) 735.125
POTE generator: ~26/17 = 735.116


Map: [<1 12 6 12 20 -11 -10|, <0 -17 -6 -15 -27 24 23|]
{{Val list|legend=1| 31, 80, 111 }}


EDOs: 31, 80, 111
==== 23-limit ====
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


==19-limit==
Mapping: [{{val| 1 12 6 12 20 -11 -10 -8 18|, {{val| 0 -17 -6 -15 -27 24 23 20 -22 }}]
Commas: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


[[POTE_tuning|POTE generator]]: 735.116
POTE generator: ~26/17 = 735.106


Map: [<1 12 6 12 20 -11 -10 -8|, <0 -17 -6 -15 -27 24 23 20|]
{{Val list|legend=1| 31, 80, 111, 191cdh, 302cdgh }}


EDOs: 31, 80, 111
=== Semishly ===
Comma list: 144/143, 176/175, 196/195, 275/273


==23-limit==
Mapping: [{{val| 1 12 6 12 20 8 }}, {{val| 0 -17 -6 -15 -27 -7 }}]
Commas: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


[[POTE_tuning|POTE generator]]: 435.106
POTE generator: ~13/10 = 464.980


Map: [<1 12 6 12 20 -11 -10 -8 18|,
{{Val list|legend=1| 31, 49f, 80f }}


<0 -17 -6 -15 -27 24 23 20 -22|]
Badness: 0.0284
 
EDOs: 31, 80, 111, 191cdh, 302cdgh


=Sengagen=
=Sengagen=

Revision as of 17:44, 5 April 2021

The hemimean clan, tempering out the no-threes hemimean comma 3136/3125, contains the temperaments misty and semisept, considered below, as well as meantone, hemiwürschmidt, parakleismic, hemithirds, clyde, bischismic, passion, hexe and sycamore considered elsewhere.

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

Misty

See also: Misty family

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

11-limit

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

13-limit

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

17-limit

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

19-limit

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

Mystic

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

Hemimist

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: 0.0474

13-limit

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: 0.0229

Semisept

5-limit

Comma list: 782757789696/762939453125

Mapping: [⟨1 12 6], ⟨0 -17 -6]]

POTE generator: ~192/125 = 735.146

Template:Val list

Badness: 0.6306

7-limit

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

11-limit

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: 0.0225

13-limit

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: 0.0252

17-limit

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

19-limit

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

23-limit

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

Semishly

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: 0.0284

Sengagen

Commas: 3136/3125, 420175/419904

POTE generator: ~686/675 = 24.217

Map: [<1 1 2 2|, <0 29 16 40|]

Wedgie: <<29 16 40 -42 -18 48||

EDOs: 49, 50, 99, 248, 347, 446

Badness: 0.0580

Subpental

Commas: 3136/3125, 19683/19600

POTE generator: ~56/45 = 378.467

Map: [<1 6 8 17|, <0 -14 -18 -45|]

Wedgie: <<14 18 45 -4 32 54||

EDOs: 19, 111, 130, 929c, 1059c, 1189bc, 1319bc

Badness: 0.0543

11-limit

Commas: 540/539, 3136/3125, 8019/8000

POTE generator: ~56/45 = 378.440

Map: [<1 6 8 17 -6|, <0 -14 -18 -45 30|]

EDOs: 19, 111, 130, 241, 371ce, 501cde, 872cde

Badness: 0.0453

13-limit

Commas: 351/350, 540/539, 676/675, 3136/3125

POTE generator: ~56/45 = 378.437

Map: [<1 6 8 17 -6 16|, <0 -14 -18 -45 30 -39|]

EDOs: 19, 111, 130, 241, 371ce

Badness: 0.0239

Arch

Commas: 3136/3125, 5250987/5242880

POTE generator: ~64/63 = 27.668

Map: [<1 2 2 2|, <0 -18 14 35|]

Wedgie: <<18 -14 -35 -64 -106 -42||

EDOs: 43, 87, 130, 217, 347, 824c, 1171c, 1518cd

Badness: 0.0943

11-limit

Commas: 441/440, 3136/3125, 4000/3993

POTE generator: ~64/63 = 27.663

Map: [<1 2 2 2 3|, <0 -18 14 35 20|]

EDOs: 43, 87, 130, 217, 347e, 911cde

Badness: 0.0365

13-limit

Commas: 364/363, 441/440, 676/675, 3136/3125

POTE generator: ~64/63 = 27.660

Map: [<1 2 2 2 3 4|, <0 -18 14 35 20 -13|]

EDOs: 43, 87, 130, 217, 347e, 564e

Badness: 0.0195

Emka

Commas: 3136/3125, 84035/82944

POTE generator: ~48/35 = 551.782

Map: [<1 14 6 12|, <0 -27 -8 -20|]

EDOs: 37, 50, 87, 137d, 224d

Badness: 0.1443

11-limit

Commas: 385/384, 2401/2376, 3136/3125

POTE generator: ~11/8 = 551.765

Map: [<1 14 6 12 3|, <0 -27 -8 -20 1|]

EDOs: 37, 50, 87, 224d, 311d

Badness: 0.0547

13-limit

Commas: 196/195, 364/363, 385/384, 625/624

POTE generator: ~11/8 = 551.758

Map: [<1 14 6 12 3 6|, <0 -27 -8 -20 1 -5|]

EDOs: 37, 50, 87, 224d, 311d, 398d

Badness: 0.0297

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