Catalog of rank-4 temperaments: Difference between revisions

Revision as of 22:58, 1 September 2022

A rank-4 temperament has a period and three additional independent generators. Typical examples include 7-limit JI, full 11-limit temperament with a one-dimensional comma basis, and full 13-limit temperament with a two-dimensional comma basis.

Ptolemismic (100/99)

Subgroup: 2.3.5.7.11

Comma list: 100/99

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 -2], ⟨0 0 1 0 2], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803

Template:Val list *

* optimal patent val: 104

Badness: 0.0225 × 10^-6

Biyatismic (121/120)

Subgroup: 2.3.5.7.11

Comma list: 121/120

Mapping: [⟨1 0 1 0 2], ⟨0 1 1 0 1], ⟨0 0 -2 0 -1], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~11/10, ~7

Optimal tuning (POTE): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589

Template:Val list

Badness: 0.0345 × 10^-6

Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

Comma list: 176/175

Mapping: [⟨1 0 0 0 -4], ⟨0 1 0 0 0], ⟨0 0 1 0 2], ⟨0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~7

Optimal tuning (POTE): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113

Template:Val list

Badness: 0.0186 × 10^-6

Rastmic (243/242)

Subgroup: 2.3.5.7.11

Comma list: 243/242

Mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5], ⟨0 0 1 0 0], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464

Template:Val list

Badness: 0.0509 × 10^-6

Akua (352/351, 847/845)

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845

Mapping: [⟨1 0 0 10 0 5], ⟨0 1 0 -6 0 -3], ⟨0 0 1 1 0 0], ⟨0 0 0 0 1 1]]

Mapping generators: ~2, ~3, ~5, ~11

Optimal tuning (POTE): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538

Template:Val list

Badness: 2.550 × 10^-6

Werckismic (441/440)

Subgroup: 2.3.5.7.11

Comma list: 441/440

Mapping: [⟨1 0 0 0 -3], ⟨0 1 0 0 2], ⟨0 0 1 0 -1], ⟨0 0 0 1 2]]

Mapping generators:~2, ~3, ~5, ~7

Template:Val list

Commas 364/363, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440

Mapping: [⟨1 0 0 0 -3 -8], ⟨0 1 0 0 2 5], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 2 3]]

Mapping generators: ~2, ~3, ~5, ~7

Mapping to lattice: [⟨0 1 1 -1 -1 0], ⟨0 0 1 0 -1 -2], ⟨0 0 1 1 1 1]]

Lattice basis:

3/2 length = 1.2263, 14/11 length = 1.4629, 21/16 length = 1.4657

Minimax tuning:

13-odd-limit

[[1 0 0 0 0 0⟩, [5/3 0 1/3 -1/3 -1/3 1/3⟩, [1/6 0 5/6 2/3 -5/6 1/3⟩, [0 0 0 1 0 0⟩, [1/6 0 -1/6 2/3 1/6 1/3⟩, [0 0 0 0 0 1⟩]

Eigenmonzos: 2, 11/10, 8/7, 16/13

15-odd-limit

[[1 0 0 0 0 0⟩, [5/4 1/4 1/4 -1/4 -1/4 1/4⟩, [5/4 -3/4 5/4 -1/4 -1/4 1/4⟩, [17/8 -11/8 5/8 -1/8 3/8 1/8⟩, [5/2 -3/2 1/2 -1/2 1/2 1/2⟩, [17/8 -11/8 5/8 -9/8 3/8 9/8⟩]

Eigenmonzos: 2, 14/13, 6/5, 11/9

Template:Val list

Commas 351/350, 441/440

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 441/440

Mapping: [⟨1 0 0 0 -3 1], ⟨0 1 0 0 2 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 196/195, 352/351

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351

Mapping: [⟨1 0 0 0 -3 2], ⟨0 1 0 0 2 -1], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 2 2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Tannic

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1287/1280

Mapping: [⟨1 0 0 0 -3 11], ⟨0 1 0 0 2 -4], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 2 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 441/440, 561/560

Mapping: [⟨1 0 0 0 -3 11 7], ⟨0 1 0 0 2 -4 -3], ⟨0 0 1 0 -1 2 2], ⟨0 0 0 1 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 441/440, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 847/845

Mapping: [⟨1 0 0 0 -3 -3], ⟨0 1 0 0 2 2], ⟨0 0 1 1 1 1], ⟨0 0 0 2 4 5]]

Mapping generators: ~2, ~3, ~5, ~13/11

Template:Val list

Keenanismic (385/384)

Main article: Keenanismic family

Subgroup: 2.3.5.7.11

Comma list: 385/384

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 1], ⟨0 0 1 0 -1], ⟨0 0 0 1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35]

Minimax tuning:

11-odd-limit

[[1 0 0 0 0⟩, [0 1 0 0 0⟩, [7/3 1/3 2/3 -1/3 -1/3⟩, [7/3 1/3 -1/3 2/3 -1/3⟩, [7/3 1/3 -1/3 -1/3 2/3⟩]

Eigenmonzos: 2, 3, 7/5, 11/5

Template:Val list

Badness: 15.159 × 10^-9

Martwin

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384

Mapping: [⟨1 0 0 0 7 2], ⟨0 1 0 0 1 4], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Transpose: [2 3 5 7 385/35 324/25]

Lattice basis:

4/3 length = 1.0820, 6/5 length = 1.3935, 10/9 length = 1.6247

Minimax tuning: ^{[to be confirmed]}

13- and 15-odd-limit

[⟨1 0 0 0 0 0], ⟨0 1 0 0 0 0], ⟨2/3 4/3 1/3 0 0 -1/3], ⟨19/6 -1/6 -1/6 1/2 -1/2 1/6], ⟨19/6 -1/6 -1/6 -1/2 1/2 1/6], ⟨2/3 4/3 -2/3 0 0 2/3]]

Eigenmonzos: 2, 14/11, 13/10, 4/3

Template:Val list

Badness: 2.206 × 10^-6

Ancient

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 625/624

Mapping: [⟨1 0 0 0 7 -4], ⟨0 1 0 0 1 -1], ⟨0 0 1 0 -1 4], ⟨0 0 0 1 -1 0]]

Transpose: [2 3 5 7 385/35 625/48]

Template:Val list

Badness: 2.573 × 10^-6

Commas 351/350, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384

Mapping: [⟨1 0 0 0 7 1], ⟨0 1 0 0 1 -3], ⟨0 0 1 0 -1 2], ⟨0 0 0 1 -1 1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 352/351, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 -2], ⟨0 0 1 0 -1 -1], ⟨0 0 0 1 -1 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 364/363, 385/384

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 385/384

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 1 3], ⟨0 0 1 0 -1 -2], ⟨0 0 0 1 -1 -3]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 385/384, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 847/845

Mapping: [⟨1 0 0 0 7 7], ⟨0 1 0 0 1 1], ⟨0 0 1 1 -2 -2], ⟨0 0 0 2 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~13/11

Template:Val list

Swetismic (540/539)

Subgroup: 2.3.5.7.11

Comma list: 540/539

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 3], ⟨0 0 1 0 1], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 540/539, 847/845

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 847/845

Mapping: [⟨1 0 0 0 2 2], ⟨0 1 0 0 3 3], ⟨0 0 1 1 -1 -1], ⟨0 0 0 2 -4 -3]]

Mapping generators: ~2, ~3, ~5, ~13/11

Template:Val list

Commas 540/539, 625/624

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624

Mapping: [⟨1 0 0 0 2 -4], ⟨0 1 0 0 3 -1], ⟨0 0 1 0 1 4], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Commas 540/539, 676/675

Subgroup: 2.3.5.7.11

Comma list: 540/539, 676/675

Mapping: [⟨1 0 0 0 2 -1], ⟨0 2 0 0 6 3], ⟨0 0 1 0 1 1], ⟨0 0 0 1 -2 0]]

Mapping generators: ~2, ~26/15, ~5, ~7

Template:Val list

Pentacircle (896/891)

Subgroup: 2.3.5.7.11

Comma list: 896/891

Mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -4], ⟨0 0 1 0 0], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Template:Val list

Badness: 0.0658 × 10^-6

Commas 352/351, 364/363

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363

Mapping: [⟨1 0 0 0 7 12], ⟨0 1 0 0 -4 -7], ⟨0 0 1 0 0 0], ⟨0 0 0 1 1 1]]

Template:Val list

Badness: 3.375 × 10^-6

Topsy (847/845, 1001/1000)

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000

Mapping: [⟨1 0 0 2 0 1], ⟨0 1 0 0 0 0], ⟨0 0 1 1 1 1], ⟨0 0 0 4 -3 1]]

Mapping generators: ~2, ~3, ~5, ~13/10

Template:Val list

Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

Comma list: 3025/3024

Mapping: [⟨1 0 0 0 2], ⟨0 1 0 1 2], ⟨0 0 1 0 -1], ⟨0 0 0 2 1]]

Mapping generators: ~2, ~3, ~5, ~55/36

Template:Val list

Trimitone (8019/8000)

Subgroup: 2.3.5.7.11

Comma list: 8019/8000

Mapping: [⟨1 0 0 0 6], ⟨0 1 0 0 -6], ⟨0 0 1 0 3], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

Comma list: 9801/9800

Mapping: [⟨2 0 0 0 3], ⟨0 1 0 0 -2], ⟨0 0 1 0 1], ⟨0 0 0 1 1]]

Mapping generators: ~99/70, ~3, ~5, ~7

Template:Val list

Commas 1716/1715, 2080/2079

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079

Mapping: [⟨2 0 0 0 3 -7], ⟨0 1 0 0 -2 1], ⟨0 0 1 0 1 0], ⟨0 0 0 1 1 2]]

Mapping generators: ~99/70, ~3, ~5, ~7

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299

Minimax tuning:

13- and 15-odd-limit

[[1 0 0 0 0 0⟩, [7/10 4/5 0 -2/5 0 1/5⟩, [7/10 -1/5 1 -2/5 0 1/5⟩, [7/5 -2/5 0 1/5 0 2/5⟩, [11/5 -11/5 1 3/5 0 1/5⟩, [0 0 0 0 0 1⟩]

Eigenmonzos: 2, 6/5, 16/13, 9/7

Template:Val list

Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

Comma list: 14641/14580

Mapping: [⟨1 0 2 0 1], ⟨0 1 2 0 2], ⟨0 0 -4 0 -1], ⟨0 0 0 1 0]]

Mapping generators: ~2, ~3, ~18/11, ~7

Optimal tuning (POTE): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866

Template:Val list

Badness: 0.351 × 10^-6

Tridecimal semicanousmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 14641/14580

Mapping: [⟨1 0 2 0 1 -6], ⟨0 1 2 0 2 3], ⟨0 0 -4 0 -1 3], ⟨0 0 0 1 0 1]]

Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099

Optimal GPV sequence: Template:Val list

Badness: 17.1 × 10^-6

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1089/1088, 14641/14580

Mapping: [⟨1 0 2 0 1 -6 -4], ⟨0 1 2 0 2 3 6], ⟨0 0 -4 0 -1 3 -2], ⟨0 0 0 1 0 1 0]]

Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228

Optimal GPV sequence: Template:Val list

Badness: 34.0 × 10^-6

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444

Mapping: [⟨1 0 2 0 1 -6 -4 -4], ⟨0 1 2 0 2 3 6 7], ⟨0 0 -4 0 -1 3 -2 -4], ⟨0 0 0 1 0 1 0 0]]

Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734

Optimal GPV sequence: Template:Val list

Badness: 41.9 × 10^-6

Semiporwellismic (16384/16335)

See Semiporwellismic clan #Semiporwellismic.

Olympic (131072/130977)

Subgroup: 2.3.5.7.11

Comma list: 131072/130977

Mapping: [⟨1 0 0 0 17], ⟨0 1 0 0 -5], ⟨0 0 1 0 0], ⟨0 0 0 1 -2]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list

Tridecimal olympic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095

Mapping: [⟨1 0 0 0 17 12], ⟨0 1 0 0 -5 -2], ⟨0 0 1 0 0 -1], ⟨0 0 0 1 -2 -1]]

Mapping generators: ~2, ~3, ~5, ~7

Template:Val list *

* optimal patent val: 3044

@@ Line 2: / Line 2: @@
 == Ptolemismic (100/99) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[100/99]]
@@ Line 8: / Line 8: @@
 [[Mapping]]: [{{val| 1 0 0 0 2 }}, {{val| 0 1 0 0 -2 }}, {{val| 0 0 1 0 2 }}, {{val| 0 0 0 1 0 }}]
-[[POTE generator]]s: ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803
 {{Val list|legend=1| 7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e }} <nowiki>*</nowiki>
@@ Line 17: / Line 17: @@
 == Biyatismic (121/120) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[121/120]]
@@ Line 25: / Line 25: @@
 Mapping generators: ~2, ~3, ~11/10, ~7
-[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589
 {{Val list|legend=1| 14c, 15, 22, 31, 46, 53, 60e, 68, 77, 91e, 99, 130e, 159ee, 190ee }}
@@ Line 32: / Line 32: @@
 == Valinorsmic (176/175) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[176/175]]
@@ Line 40: / Line 40: @@
 Mapping generators: ~2, ~3, ~5, ~7
-[[POTE generator]]s: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
 {{Val list|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
@@ Line 47: / Line 47: @@
 == Rastmic (243/242) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[243/242]]
@@ Line 53: / Line 53: @@
 [[Mapping]]: [{{val| 1 1 0 0 2 }}, {{val| 0 2 0 0 5 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 0}}]
-[[POTE generator]]s: ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
+[[Optimal tuning]] ([[POTE]]): ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464
 {{Val list|legend=1| 7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202 }}
@@ Line 60: / Line 60: @@
 == Akua (352/351, 847/845) ==
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 352/351, 847/845
@@ Line 68: / Line 68: @@
 Mapping generators: ~2, ~3, ~5, ~11
-[[POTE generator]]s: ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538
 {{Val list|legend=1| 12f, 17c, 24d, 29, 41, 46, 53, 58, 87, 111, 140, 152f, 198, 350f, 437f, 490f }}
@@ Line 75: / Line 75: @@
 == Werckismic (441/440) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[441/440]]
@@ Line 86: / Line 86: @@
 === Commas 364/363, 441/440 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 364/363, 441/440
@@ Line 110: / Line 110: @@
 === Commas 351/350, 441/440 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 351/350, 441/440
@@ Line 121: / Line 121: @@
 === Commas 196/195, 352/351 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 196/195, 352/351
@@ Line 132: / Line 132: @@
 === Tannic ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 441/440, 1287/1280
@@ Line 143: / Line 143: @@
 ==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
+[[Subgroup]]: 2.3.5.7.11.13.17
 [[Comma list]]: 273/272, 441/440, 561/560
@@ Line 154: / Line 154: @@
 === Commas 441/440, 847/845 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 441/440, 847/845
@@ Line 167: / Line 167: @@
 {{Main| Keenanismic family }}
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[385/384]]
@@ Line 187: / Line 187: @@
 === Martwin ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 325/324, 385/384
-Mapping: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]
+[[Mapping]]: [{{val| 1 0 0 0 7 2 }}, {{val| 0 1 0 0 1 4 }}, {{val| 0 0 1 0 -1 -2 }}, {{val| 0 0 0 1 -1 0 }}]
 Mapping generators: ~2, ~3, ~5, ~7
@@ Line 210: / Line 210: @@
 === Ancient ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 385/384, 625/624
@@ Line 223: / Line 223: @@
 === Commas 351/350, 385/384 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 351/350, 385/384
@@ Line 234: / Line 234: @@
 === Commas 352/351, 385/384 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 352/351, 385/384
@@ Line 245: / Line 245: @@
 === Commas 364/363, 385/384 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 364/363, 385/384
@@ Line 256: / Line 256: @@
 === Commas 385/384, 847/845 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 385/384, 847/845
@@ Line 267: / Line 267: @@
 == Swetismic (540/539) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[540/539]]
@@ Line 278: / Line 278: @@
 === Commas 540/539, 847/845 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 540/539, 847/845
@@ Line 289: / Line 289: @@
 === Commas 540/539, 625/624 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 540/539, 625/624
@@ Line 300: / Line 300: @@
 === Commas 540/539, 676/675 ===
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 540/539, 676/675
@@ Line 311: / Line 311: @@
 == Pentacircle (896/891) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 896/891
@@ Line 317: / Line 317: @@
 [[Mapping]]: [{{val| 1 0 0 0 7 }}, {{val| 0 1 0 0 -4 }}, {{val| 0 0 1 0 0 }}, {{val| 0 0 0 1 1 }}]
-[[POTE generator]]s: ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722
 {{Val list|legend=1| 12, 17c, 19e, 22, 34d, 39d, 41, 58, 80, 87, 99e, 121, 145, 167, 208, 266e, 699bbcdeee }}
@@ Line 324: / Line 324: @@
 === Commas 352/351, 364/363 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 352/351, 364/363
@@ Line 335: / Line 335: @@
 == Topsy (847/845, 1001/1000) ==
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 847/845, 1001/1000
@@ Line 346: / Line 346: @@
 == Lehmerismic (3025/3024) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[3025/3024]]
@@ Line 357: / Line 357: @@
 == Trimitone (8019/8000) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[8019/8000]]
@@ Line 368: / Line 368: @@
 == Kalismic (9801/9800) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[9801/9800]]
@@ Line 379: / Line 379: @@
 === Commas 1716/1715, 2080/2079 ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 1716/1715, 2080/2079
@@ Line 398: / Line 398: @@
 == Semicanousmic (14641/14580) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: [[14641/14580]]
@@ Line 406: / Line 406: @@
 Mapping generators: ~2, ~3, ~18/11, ~7
-[[POTE generator]]s: ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
+[[Optimal tuning]] ([[POTE]]): ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866
 {{Val list|legend=1| 14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e }}
@@ Line 419: / Line 419: @@
 Mapping: [{{val| 1 0 2 0 1 -6 }}, {{val| 0 1 2 0 2 3 }}, {{val| 0 0 -4 0 -1 3 }}, {{val| 0 0 0 1 0 1 }}]
-POTE generators: ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
+Optimal tuning (POTE): ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099
 Optimal GPV sequence: {{val list| 87, 111, 181, 198, 323, 410 }}
@@ Line 432: / Line 432: @@
 Mapping: [{{val| 1 0 2 0 1 -6 -4 }}, {{val| 0 1 2 0 2 3 6 }}, {{val| 0 0 -4 0 -1 3 -2 }}, {{val| 0 0 0 1 0 1 0 }}]
-POTE generators: ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
+Optimal tuning (POTE): ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228
 Optimal GPV sequence: {{val list| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}
@@ Line 445: / Line 445: @@
 Mapping: [{{val| 1 0 2 0 1 -6 -4 -4 }}, {{val| 0 1 2 0 2 3 6 7 }}, {{val| 0 0 -4 0 -1 3 -2 -4 }}, {{val| 0 0 0 1 0 1 0 0 }}]
-POTE generators: ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
+Optimal tuning (POTE): ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734
 Optimal GPV sequence: {{val list| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}
@@ Line 455: / Line 455: @@
 == Olympic (131072/130977) ==
-Subgroup: 2.3.5.7.11
+[[Subgroup]]: 2.3.5.7.11
 [[Comma list]]: 131072/130977
@@ Line 466: / Line 466: @@
 === Tridecimal olympic ===
-Subgroup: 2.3.5.7.11.13
+[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: 2080/2079, 4096/4095

Catalog of rank-4 temperaments: Difference between revisions

Revision as of 22:58, 1 September 2022

Ptolemismic (100/99)

Biyatismic (121/120)

Valinorsmic (176/175)

Rastmic (243/242)

Akua (352/351, 847/845)

Werckismic (441/440)

Commas 364/363, 441/440

Commas 351/350, 441/440

Commas 196/195, 352/351

Tannic

17-limit

Commas 441/440, 847/845

Keenanismic (385/384)

Martwin

Ancient

Commas 351/350, 385/384

Commas 352/351, 385/384

Commas 364/363, 385/384

Commas 385/384, 847/845

Swetismic (540/539)

Commas 540/539, 847/845

Commas 540/539, 625/624

Commas 540/539, 676/675

Pentacircle (896/891)

Commas 352/351, 364/363

Topsy (847/845, 1001/1000)

Lehmerismic (3025/3024)

Trimitone (8019/8000)

Kalismic (9801/9800)

Commas 1716/1715, 2080/2079

Semicanousmic (14641/14580)

Tridecimal semicanousmic

17-limit

19-limit

Semiporwellismic (16384/16335)

Olympic (131072/130977)

Tridecimal olympic

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