# Catalog of rank-4 temperaments

(Redirected from Rank-4 temperament)

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

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.9532, ~5/4 = 384.0675, ~7/4 = 970.8803

Optimal ET sequence7d, 8d, 10e, 12, 15, 19, 22, 27e, 34d, 41, 90e, 131e *

Badness: 0.0225 × 10-6

## Biyatismic (121/120)

Subgroup: 2.3.5.7.11

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): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466, ~7/4 = 966.9589

Badness: 0.0345 × 10-6

## Valinorsmic (176/175)

Subgroup: 2.3.5.7.11

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): ~2 = 1\1, ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113

Badness: 0.0186 × 10-6

## Rastmic (243/242)

Subgroup: 2.3.5.7.11

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

mapping generators: ~2, ~11/9, ~5, ~7

Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.5254, ~5/4 = 386.1653, ~7/4 = 968.6464

Optimal ET sequence7d, 10, 14c, 17c, 24, 27e, 31, 41, 58, 72, 130, 202

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): ~2 = 1\1, ~3/2 = 702.9075, ~5/4 = 387.0723, ~11/8 = 551.4538

Badness: 2.550 × 10-6

## Werckismic (441/440)

Subgroup: 2.3.5.7.11

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

Optimal ET sequence10, 12, 15, 19e, 26, 27e, 31, 41, 58, 72, 118, 130, 190, 248, 289, 320, 609d

### 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 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
[[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 (unchanged-intervals): 2, 11/10, 8/7, 16/13
[[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 (unchanged-intervals): 2, 14/13, 6/5, 11/9

Optimal ET sequence12f, 14cf, 15, 17c, 26, 29, 31f, 41, 46, 58, 72, 87, 130, 217, 289

### 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]]

### 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]]

Optimal ET sequence10, 12f, 17c, 19e, 27e, 29, 31, 41, 46, 58, 87, 118, 145, 232

### 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]]

#### 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]]

### 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

Optimal ET sequence12f, 16, 17c, 25e, 29, 41, 46, 58, 87, 103, 145, 149, 161, 190, 248, 438d

## Keenanismic (385/384)

Subgroup: 2.3.5.7.11

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]

[[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]
Eigenmonzo (unchanged-interval) basis: 2.3.7/5.11/5

Optimal ET sequence9, 10, 12e, 15, 19, 22, 31, 41, 53, 68, 72, 118, 159, 190, 212, 284, 330e, 402de

Badness: 15.2 × 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]]

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
[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]]
Eigenmonzo (unchanged-interval) basis: 2.3.11/7.13/5

Badness: 2.21 × 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]

Badness: 2.57 × 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]]

Badness: 2.98 × 10-6

### Zaxa

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]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 702.657, ~5/4 = 385.632, ~7/4 = 967.829

Badness: 3.35 × 10-6

### 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]]

Optimal ET sequence9, 15, 22, 26, 31f, 37, 41, 46, 63, 72, 87, 159

Badness: 3.32 × 10-6

### 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

Badness: 4.15 × 10-6

## Swetismic (540/539)

Subgroup: 2.3.5.7.11

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
• CTE: ~2 = 1\1, ~3/2 = 701.6167, ~5/4 = 386.0717, ~7/4 = 969.5334
• CWE: ~2 = 1\1, ~3/2 = 701.6950, ~5/4 = 386.1796, ~7/4 = 969.6366

Optimal ET sequence8d, 9, 10, 12e, 14c, 17c, 19, 22, 27e, 31, 41, 53, 58, 72, 130, 152, 224, 354, 506e, 578, 730de, 761d, 985d, 1115de, 1267dde

Badness: 0.0105 × 10-6

### Commas 540/539, 729/728

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 729/728

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

mapping generators: ~2, ~3, ~5, ~7
• CTE: ~2 = 1\1, ~3/2 = 701.6687, ~5/4 = 386.0441, ~7/4 = 969.5668
• CWE: ~2 = 1\1, ~3/2 = 701.7230, ~5/4 = 386.1818, ~7/4 = 969.6607

Badness: 1.73 × 10-6

### 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

Optimal ET sequence8d, 9, 12e, 17c, 32f, 33cd, 36ce, 41, 53, 58, 94, 103, 111, 152f, 255, 407f

Badness: 3.97 × 10-6

### 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]]

Badness: 3.59 × 10-6

### 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

Badness: 3.06 × 10-6

## 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]]

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8345, ~5/4 = 387.7585, ~7/4 = 969.8722

Badness: 0.0658 × 10-6

### Tridecimal pentacircle a.k.a. gentle

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]]

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

Optimal ET sequence16, 21, 24d, 29, 37, 45ef, 50, 53, 58, 87, 103, 111, 140, 190, 198, 301, 388, 689e

## Lehmerismic (3025/3024)

Subgroup: 2.3.5.7.11

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

Optimal ET sequence7d, 8d, 10, 15, 23de, 24d, 26, 31, 41, 65d, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 4002e, 5720e, 7168cee

## Trimitone (8019/8000)

Subgroup: 2.3.5.7.11

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
• CTE: ~2 = 1\1, ~3/2 = 701.5449, ~5/4 = 386.7538, ~7/4 = 968.8259
• CWE: ~2 = 1\1, ~3/2 = 701.4729, ~5/4 = 386.5374, ~7/4 = 968.6210

Optimal ET sequence12, 19, 26, 39d, 46, 53, 58, 72, 118, 130, 183, 190, 248, 255, 301, 373, 804, 876, 1177be

Badness: 0.0820 × 10-6

### Commas 729/728, 1001/1000

Subgroup: 2.3.5.7.11.13

Comma list: 729/728, 1001/1000

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

mapping generators: ~2, ~3, ~5, ~7
• CTE: ~2 = 1\1, ~3/2 = 701.5537, ~5/4 = 386.7680, ~7/4 = 968.8144
• CWE: ~2 = 1\1, ~3/2 = 701.4770, ~5/4 = 386.5437, ~7/4 = 968.6150

Badness: 3.33 × 10-6

## Kalismic (9801/9800)

Subgroup: 2.3.5.7.11

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

### 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]]

Lattice basis:

3/2 length = 1.1956, 7/4 length = 1.4506, 14/13 length = 1.8299
[[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]
Eigenmonzo (unchanged-intervals) basis: 2, 6/5, 16/13, 9/7

## Unisquary (12005/11979)

Subgroup: 2.3.5.7.11

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tuning (POTE): ~2 = 1＼1, ~3 = 1902.0307, ~5 = 2786.2325, ~11/7 = 783.5074

### Hensquary

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 1716/1715

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tuning (POTE): ~2 = 1＼1, ~3 = 1902.4103, ~5 = 2786.9909, ~11/7 = 783.9209

Optimal ET sequence9, 12f, 26, 37, 46, 49f, 58, 63, 72, 84, 118f, 121, 130

### Ekasquary

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 4459/4455

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

Mapping generators: ~2, ~3, ~5, ~11/7

Optimal tuning (POTE): ~2 = 1＼1, ~3 = 1901.9525, ~5 = 2786.1388, ~11/7 = 783.5477

Optimal ET sequence9, 12, 46f, 49f, 58, 60e, 63, 72, 118, 121, 130, 190, 193, 248, 311, 320, 383, 441

## Semicanousmic (14641/14580)

Subgroup: 2.3.5.7.11

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): ~2 = 1\1, ~3/2 = 702.2503, ~18/11 = 854.5421, ~7/4 = 968.6866

Optimal ET sequence14c, 17c, 24, 31, 63, 80, 87, 111, 118, 198, 212, 292, 323, 410, 851e

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): ~2 = 1\1, ~3/2 = 702.4931, ~18/11 = 854.6400, ~7/4 = 969.0099

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): ~2 = 1\1, ~3/2 = 702.4099, ~18/11 = 854.6338, ~7/4 = 969.0228

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): ~2 = 1\1, ~3/2 = 702.3413, ~18/11 = 854.6472, ~7/4 = 968.9734

Badness: 41.9 × 10-6

## Semiporwellismic (16384/16335)

Subgroup: 2.3.5.7.11

Comma list: 16384/16335

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

mapping generators: ~2, ~3, ~128/99, ~7

Badness: 0.219 × 10-6

## Symbiotic (19712/19683)

Subgroup: 2.3.5.7.11

Mapping[1 0 0 0 -8], 0 1 0 0 9], 0 0 1 0 0], 0 0 0 1 -1]]

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2681, ~5/4 = 386.4785, ~7/4 = 968.9552

Optimal ET sequence17c, 19e, 24, 34d, 41, 53, 58, 94, 99e, 118, 152, 270, 581, 733, 851, 1003, 1273, 1854, 2124b

Badness: 0.120 × 10-6

### Tridecimal symbiotic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 19712/19683

Mapping[1 0 0 0 -8 -13], 0 1 0 0 9 12], 0 0 1 0 0 -1], 0 0 0 1 -1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.2721, ~5/4 = 386.4790, ~7/4 = 968.9705

Badness: 3.31 × 10-6

## Olympic (131072/130977)

Subgroup: 2.3.5.7.11

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

### 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]]

Optimal ET sequence41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1699, 2280, 2774e *