# No-fives subgroup temperaments

This is a collection of subgroup temperaments which omit the prime harmonic of 5.

## Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

Gencom mapping: [1 2 0 3], 0 -2 0 -1]]

Sval mapping: [1 2 3], 0 -2 -1]]

POL2 generator: ~7/6 = 250.385

RMS error: 2.523 cents

## Bleu

Subgroup: 2.3.7

Comma: 17496/16807

Gencom: [2 54/49; 17496/16807]

Gencom mapping: [1 1 0 2], 0 5 0 7]]

Sval mapping: [1 1 2], 0 5 7]]

POL2 generator: ~54/49 = 139.848

RMS error: 1.917 cents

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [1 1 0 2 3], 0 5 0 7 4]]

Sval mapping: [1 1 2 3], 0 5 7 4]]

POL2 generator: ~12/11 = 140.005

RMS error: 1.829 cents

### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 12/11; 78/77 99/98 144/143]

Gencom mapping: [1 1 0 2 3 3], 0 5 0 7 4 6]]

Sval mapping: [1 1 2 3 3], 0 5 7 4 6]]

POL2 generator: ~12/11 = 139.990

RMS error: 1.752 cents

## Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [1 1 0 4], 0 1 0 -2]]

Sval mapping: [1 2 2], 0 -1 2]]

POL2 generator: ~3/2 = 709.321

RMS error: 1.856 cents

### Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [1 1 0 4 7], 0 1 0 -2 -6]]

Sval mapping: [1 0 6 13], 0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

RMS error: 1.977 cents

#### Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Gencom: [2 3/2; 64/63 78/77 99/98]

Gencom mapping: [1 1 0 4 7 9], 0 1 0 -2 -6 -9]]

Sval mapping: [1 0 6 13 18], 0 1 -2 -6 -9]]

POL2 generator: ~3/2 = 706.137

RMS error: 2.095 cents

### Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Gencom: [2 11/9; 64/63 243/242]

Gencom mapping: [1 1 0 4 2], 0 2 0 -4 5]]

Sval mapping: [1 1 4 2], 0 2 -4 5]]

POL2 generator: ~11/9 = 353.958

RMS error: 1.968 cents

#### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

Gencom mapping: [1 1 0 4 2 4], 0 2 0 -4 5 -1]]

Sval mapping: [1 1 4 2 4], 0 2 -4 5 -1]]

POL2 generator: ~11/9 = 353.775

RMS error: 1.953 cents

### Flutterpyth

Subgroup: 2.3.7.11.13.19

Comma list: 64/63, 209/208, 343/342, 364/363

Mapping: [1 1 4 10 15 9], 0 -1 -2 -11 -19 -8]]

Optimal tuning (CTE): ~3/2 = 713.459

Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-limit Ultrapyth. This temperament was created to yield blackdye tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.

## Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [1 3 6], 0 -4 -9]]

Sval mapping: [1 3 6], 0 -4 -9]]

POL2 generator: ~9/7 = 425.365

RMS error: 1.149 cents

Related temperament: squares

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [1 3 0 6 7], 0 -4 0 -9 -10]]

Sval mapping: [1 3 6 7], 0 -4 -9 -10]]

POL2 generator: ~9/7 = 425.244

RMS error: 1.099 cents

#### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [1 3 0 6 7 9], 0 -4 0 -9 -10 -15]]

Sval mapping: [1 3 6 7 9], 0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

RMS error: 1.769 cents

#### Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Gencom: [2 9/7; 99/98, 144/143, 243/242]

Gencom mapping: [1 3 0 6 7 3], 0 -4 0 -9 -10 2]]

Sval mapping: [1 3 6 7 3], 0 -4 -9 -10 2]]

POL2 generator: ~9/7 = 424.702

RMS error: 1.290 cents

#### Byhearted

For the full 19-limit version of this temperament, see Tetracot family #Byhearted.

Subgroup: 2.3.7.11.19

Comma list: 99/98, 243/242, 363/361

Gencom: [209/147 21/19; 99/98 243/242 363/361]

Sval mapping: [2 2 3 4 5], 0 4 9 10 12]]

POL2 generator: ~21/19 = 174.735

RMS error: 0.8727 cents

## Harrison

Subgroup: 2.3.7

Gencom: [2 3/2; 59049/57344]

Gencom mapping: [1 1 0 -3], 0 1 0 10]]

Sval mapping: [1 1 -3], 0 1 10]]

POL2 generator: ~3/2 = 696.544

RMS error: 1.226 cents

Related temperament: meantone

## Leapfrog

Subgroup: 2.3.7

Comma list: 14680064/14348907

Gencom: [2 3/2; 14680064/14348907]

Gencom mapping: [1 1 0 -6], 0 1 0 15]]

Sval mapping: [1 0 -21], 0 1 15]]

POL2 generator: ~3/2 = 704.721 cents

RMS error: 0.6202 cents

Related temperaments: leapday, leapweek, srutal

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Gencom: [2 3/2; 896/891 1331/1323]

Gencom mapping: [1 1 0 -6 -3], 0 1 0 15 11]]

Sval mapping: [1 0 -21 -14], 0 1 15 11]]

POL2 generator: ~3/2 = 704.753 cents

RMS error: 0.6047 cents

### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [1 1 0 -6 -3 -1], 0 1 0 15 11 8]]

Sval mapping: [1 0 -21 -14 -9], 0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

RMS error: 0.7541 cents

#### Skidoo

Subgroup: 2.3.7.11.13.23

Comma list: 169/168, 208/207, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [1 1 0 -6 -3 -1 0 0 1], 0 1 0 15 11 8 0 0 6]]

Sval mapping: [1 0 -21 -14 -9 -5], 0 1 15 11 8 6]]

POL2 generator: ~3/2 = 704.729 cents

RMS error: 0.6265 cents

##### 2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 169/168, 208/207, 232/231, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [1 1 0 -6 -3 -1 0 0 1 -11], 0 1 0 15 11 8 0 0 6 27]]

Sval mapping: [1 0 -21 -14 -9 -5 -38], 0 1 15 11 8 6 27]]

POL2 generator: ~3/2 = 704.729 cents

Music

## Doublehearted

Subgroup: 2.3.7

Comma list: 5764801/5668704

Gencom: [2 343/324; 5764801/5668704]

Sval mapping: [1 1 2], 0 8 11]]

POL2 generator: ~343/324 = 87.8304

RMS error: 0.5041 cents

Related temperaments: octacot

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 2401/2376

Gencom: [2 22/21; 243/242 2401/2376]

Sval mapping: [1 1 2 2], 0 8 11 20]]

POL2 generator: ~22/21 = 87.6512

RMS error: 0.7147 cents

Related temperaments: octacot

### 2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 133/132, 243/242, 343/342

Gencom: [2 19/18; 133/132 243/242 343/342]

Sval mapping: [1 1 2 2 3], 0 8 11 20 17]]

POL2 generator: ~19/18 = 87.6684

RMS error: 0.7065 cents

Related temperaments: octacot

## Magi

Subgroup: 2.3.7

Comma list: 537824/531441

Gencom: [2 243/196; 537824/531441]

Sval mapping: [1 0 -1], 0 5 12]]

POL2 generator: ~243/196 = 380.661

RMS error: 0.4277 cents

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 26411/26244

Gencom: [2 96/77; 896/891 26411/26244]

Sval mapping: [1 0 -1 6], 0 5 12 -8]]

POL2 generator: ~96/77 = 380.768

RMS error: 0.4262 cents

#### Balthazar

Subgroup: 2.3.7.11.13

Comma list: 169/168, 896/891, 26411/26244

Gencom: [2 143/128; 169/168 896/891 26411/26244]

Sval mapping: [1 0 -1 6 1], 0 10 24 -16 17]]

POL2 generator: ~143/128 = 190.407

RMS error: 0.6937 cents

#### Caspar

Subgroup: 2.3.7.11.13

Comma list: 144/143, 343/338, 729/728

Gencom: [2 26/21; 144/143 343/338 729/728]

Sval mapping: [1 0 -1 6 -2], 0 5 12 -8 18]]

POL2 generator: ~26/21 = 380.531

RMS error: 1.032 cents

#### Melchior

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363, 26411/26244

Gencom: [2 96/77; 352/351 364/363 26411/26244]

Sval mapping: [1 0 -1 6 11], 0 5 12 -8 -23]]

POL2 generator: ~96/77 = 380.766

RMS error: 0.3891 cents

### Hogwarts

Subgroup: 2.3.7.29

Comma list: 784/783, 5887/5832

Gencom: [2 36/29; 784/783 5887/5832]

Sval mapping: [1 0 -1 2], 0 5 12 9]]

POL2 generator: ~36/29 = 380.618

#### Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 144/143, 232/231, 343/338, 729/728

Sval mapping: [1 0 -1 6 -2 2], 0 5 12 -8 18 9]]

POL2 generator: ~26/21 = 380.526

## Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [1 0 0 -3], 0 3 0 11]]

Sval mapping: [1 0 -3], 0 3 11]]

POL2 generator: ~81/56 = 633.525

RMS error: 0.3519 cents

## Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

Gencom mapping: [1 1 0 3], 0 3 0 -1]]

Sval mapping: [1 1 3], 0 3 -1]]

POL2 generator: ~8/7 = 233.688

RMS error: 0.3202 cents

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Gencom: [91/64 8/7; 169/168 1029/1024]

Sval mapping: [2 2 6 7], 0 3 -1 1]]

POL2 generator: ~8/7 = 233.6044

Optimal ET sequence10, 26, 36, 154…, 190…, 226…, 262

RMS error: 0.5452 cents

#### 2.3.7.13.17

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Gencom: [17/12 8/7; 169/168 273/272 289/288]

Sval mapping[2 2 6 7 7], 0 3 -1 1 3]]

POL2 generator: ~8/7 = 233.6155

Optimal ET sequence10, 26, 36, 154…, 190…, 226

RMS error: 0.5073 cents

## Gigapyth

Subgroup: 2.3.85

Comma list: 2.3.85 -40 1 6]

Sval mapping[1 4 6], 0 -6 1]]

mapping generators: ~2, ~85/64

Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.034

Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*

*Wart for 85

### 2.3.7.85 subgroup

Subgroup: 2.3.7.85

Comma list: 1029/1024, 7225/7203

Sval mapping[1 4 2 6], 0 -6 2 1]]

mapping generators: ~2, ~85/64

Optimal tuning (CTE): ~2 = 1\1, ~85/64 = 483.031

Supporting ETs: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159*

*Wart for 85

## Hemif

Subgroup: 2.3.7

Comma: 1605632/1594323

Gencom: [2 2187/1792; 1605632/1594323]

Gencom mapping: [1 1 0 -1], 0 2 0 13]]

Sval mapping: [1 1 -1], 0 2 13]]

POL2 generator: ~2187/1792 = 351.485

RMS error: 0.2344 cents

Related temperaments: hemififths, namo

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [1 1 0 -1 2], 0 2 0 13 5]]

Sval mapping: [1 1 -1 2], 0 2 13 5]]

POL2 generator: ~11/9 = 351.535

RMS error: 0.6108 cents

#### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [1 1 0 -1 2 4], 0 2 0 13 5 -1]]

Sval mapping: [1 1 -1 2 4], 0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

RMS error: 0.7167 cents

#### Heartful

Subgroup: 2.3.7.11.19

Comma list: 243/242, 896/891, 1083/1078

Gencom: [2 21/19; 243/242 896/891 1083/1078]

Sval mapping: [1 1 -1 2 0], 0 4 26 10 29]]

POL2 generator: ~21/19 = 175.804

RMS error: 0.5360 cents

Related temperaments: bunya

## Hearts

Subgroup: 2.3.7

Comma list: 34451725707/34359738368 (trila-quadzo comma)

Gencom: [2 567/512; 34451725707/34359738368]

Sval mapping: [1 1 5], 0 4 -15]]

POL2 generator: ~567/512 = 175.433

RMS error: 0.0529 cents

Related temperaments: monkey, sesquiquartififths

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 65536/65219

Gencom: [2 256/231; 243/242 65536/65219]

Sval mapping: [1 1 5 2], 0 4 -15 10]]

POL2 generator: ~256/231 = 175.369

RMS error: 0.3224 cents

Related temperaments: monkey, sesquart

### 2.3.7.11.19

Subgroup: 2.3.7.11.19

Comma list: 243/242, 513/512, 1083/1078

Gencom: [2 21/19; 243/242 513/512 1083/1078]

Sval mapping: [1 1 5 2 6], 0 4 -15 10 -12]]

POL2 generator: ~21/19 = 175.341

RMS error: 0.3121 cents

Related temperaments: monkey, sesquart

## Navy

Subgroup: 2.3.7

Comma list: 282429536481/281974669312

Mapping[1 1 0], 0 5 24]]

POL2 generator: ~243/224 = 140.366

RMS error: 0.0296 cents

Related temperaments: tsaharuk, quanic

### 2.3.7.11

Subgroup: 2.3.7.11

Comma list: 1331/1323, 19712/19683

Mapping[1 1 0 1], 0 5 24 21]]

POL2 generator: ~88/81 = 140.407

RMS error: 0.3778 cents

Related temperaments: tsaharuk, quanic

### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 729/728, 1331/1323

Mapping[1 1 0 1 3], 0 5 24 21 6]]

POL2 generator: ~13/12 = 140.437

RMS error: 0.4044 cents

Related temperaments: tsaharuk, quanic

## Slendrismic

In slendrismic, the period (1\5) is given a very accurate interpretation of 147/128 = (3/2)/(8/7)2 = 8/7 * 1029/1024 = S7/S8, which is a significant interval as it is the "harmonic 5edostep" in that it's a rooted (/2^n) interval that approximates 1\5 very well. The generator is 1029/1024, the difference between 8/7 and 147/128 and therefore between 3/2 and (8/7)3. The temperament is named for the very "slender" generator as well as as a pun on "slendric" (which it shouldn't be confused with). One can consider this as a microtemperament counterpart to cloudy, which equates them.

Subgroup: 2.3.7

Comma list: 68719476736/68641485507

Mapping[5 8 14], 0 -2 1]]

POL2 generator: ~1029/1024 = 8.9906

RMS error: 0.0212 cents

Related temperaments: hemipental

## Hectosaros leap week

Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.

Subgroup: 2.3.7

Comma list: [-50 -746 439

Mapping: [1 313 532], 0 -439 -746]]

Optimal tuning (CTE): ~[17 343 143 = 851.248

RMS error: 0.0164 cents

### 2.3.7.13 subgroup

Subgroup: 2.3.7.13

Comma list: [-42 -2 -5 16, [10 -46 29 -5

Mapping: [1 313 532 208], 0 -439 -746 -288]]

Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248

### 2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 39337984/39328497, [0 -14 7 4 -3, [-18 -24 14 -1 5

Mapping: [1 313 532 208 58], 0 -439 -746 -288 -76]]

Optimal tuning (CTE): ~6144/3757 = 851.248

### 2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17.19

Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256

Mapping: [1 313 532 208 58 432], 0 -439 -746 -288 -76 -603]]

Optimal tuning (CTE): ~6144/3757 = 851.248

## Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [1 14/9 0 25/9 31/9], 0 2 0 2 1]]

Sval mapping: [9 0 11 24], 0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

RMS error: 0.0383 cents

## Parapyth (rank 3)

Subgroup: 2.3.7.11

Comma list: 896/891

Gencom: [2 3/2 28/27; 896/891]

Gencom mapping: [1 1 0 1 4], 0 1 0 3 -1], 0 0 0 1 1]]

Sval mapping: [1 0 0 7], 0 1 0 -4], 0 0 1 1]]

POL2 tuning: ~3 = 1903.834, ~7 = 3369.872

RMS error: 0.4149 cents

### 2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

Gencom mapping: [1 1 0 1 4 6], 0 1 0 3 -1 -4], 0 0 0 1 1 1]]

Sval mapping: [1 0 0 7 12], 0 1 0 -4 -7], 0 0 1 1 1]]

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

RMS error: 0.3789 cents

## Paralimmal

Subgroup: 2.3.11

Sval mapping[1 0 4], 0 3 -1]]

Optimal tuning (CTE): ~2 = 1/1, ~16/11 = 634.320

RMS error: 1.237 cents

## Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or neutrominant, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Subgroup: 2.3.11

Comma: 243/242

Gencom: [2 11/9; 243/242]

Gencom mapping: [1 1 0 0 2], 0 2 0 0 5]]

Sval mapping: [1 1 2], 0 2 5]]

POL2 generator: ~11/9 = 350.525

RMS error: 0.3021 cents

Scales

### Namo

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Gencom: [2 11/9; 144/143 243/242]

Gencom mapping: [1 1 0 0 2 4], 0 2 0 0 5 -1]]

Sval mapping: [1 1 2 4], 0 2 5 -1]]

POL2 generator: ~11/9 = 351.488

RMS error: 0.7038 cents

## Heartland (rank 3)

Heartland, with a generator of ~21/19, is named for its tempering of the heartlandisma, 3971/3969. Aside from the heartlandisma, the heartland temperament tempers out 243/242 (rastma) and 1083/1078 (bihendrixma), and slices the fifth in four (the number of chambers of the heart).

Subgroup: 2.3.7.11.19

Comma list: 243/242, 1083/1078

Gencom: [2 21/19 7; 243/242 1083/1078]

Sval mapping: [1 1 0 2 1], 0 4 0 10 3], 0 0 1 0 1]]

POL2 generator: ~21/19 = 175.2713, ~7 = 3369.3784

RMS error: 0.3066 cents

## Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

Mapping: [1 0 -6 -6], 0 2 12 13]]

POTE generator: ~19/11 = 945.4

### Semaerophore

Subgroup: 2.3.7.11.19

Comma list: 49/48, 77/76, 729/704

Mapping: [1 0 2 -6 -6], 0 2 1 12 13]]

POTE generator: ~7/4 = 944.667

## Superflat aka tridecimal

Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes 1053/1024 vanish, so 13/8 is a minor sixth, and 16/13 is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of flattone, although often even flatter (such as 40edo's fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of meantone and archy. Superflat diatonic scales have a character somewhere between neutral third scales (or mosh) and meantone diatonic scales.

Subgroup: 2.3.13

Sval mapping[1 1 6], 0 1 -4]]

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.939

RMS error: 1.591 cents

### 2.3.11.13

Subgroup: 2.3.11.13

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 692.247

## Ultraflat

Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. 27/26 is tempered out rather than 1053/1024, so 13/8 is a major sixth. These temperamenets intersect in 7edo, where major sixths and minor sixths are not distinguished.

Subgroup: 2.3.13

Sval mapping[1 1 2], 0 1 3]]

Optimal tuning (CTE): ~2 = 1/1, ~3/2 = 688.391

RMS error: 4.367 cents

## Threedic

Subgroup: 2.3.13

Sval mapping[1 0 0], 0 3 7]]

Optimal tuning (CTE): ~2 = 1/1, ~13/9 = 634.173

RMS error: 0.2054 cents

## Semitonic

Subgroup: 2.3.17

Sval mapping[2 0 5], 0 1 1]]

sval mapping generators: ~17/12, ~3
gencom: [17/12 3; 289/288]

Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)

RMS error: 0.2247 cents

## Dog

The dog temperament is based by 2L 5s or 7L 2s scale that makes 81/76 vanish, so 19/16 is a major third. It can be viewed as a 2.3.19 subgroup analogue of mavila.

Subgroup: 2.3.19

Comma list: 81/76

Gencom: [2 4/3; 81/76]

Sval mapping: [1 2 6], 0 -1 -4]]

POL2 generator: ~4/3 = 521.403

RMS error: 4.943 cents

## Boethian

Boethian is a diatonic-based temperament that makes 513/512 vanish, so 19/16 is a minor third. It can be viewed as a 2.3.19 subgroup analogue of schismic temperament.

Subgroup: 2.3.19

Mapping[1 0 9], 0 1 -3]]

Mapping generators: ~2, ~3

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 701.3288

## Porpoise

Subgroup: 2.3.29

Comma list: 24576/24389

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

CTE generator: ~32/29 = 166.067

## Sematology

This temperament tempers out 4107/4096 and thus equates 2 37/32's with 4/3.

Subgroup: 2.3.37

Comma list: 4107/4096

Gencom: [2 37/32; 4107/4096]

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

POTE generator: ~37/32 = 249.075

### 2.3.7.37 subgroup

Subgroup: 2.3.7.37

Comma list: 4107/4096, 259/256

Gencom: [2 37/32; 4107/4096 259/256]

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

POTE generator: ~37/32 = 247.782

### 2.3.5.37 subgroup

It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.

Subgroup: 2.3.5.37

Comma list: 4107/4096, 17592186044416/17562397269605

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605]

Mapping: [1 1 4 5], 0 -2 -8 1]]

POTE generator: ~37/32 = 251.393

#### 2.3.5.7.37 subgroup

Subgroup: 2.3.5.7.37

Comma list: 4107/4096, 17592186044416/17562397269605, 259/256

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]

Mapping: [1 1 4 1 5], 0 -2 -8 -1 1]]

POTE generator: ~37/32 = 251.204

## Reversed mavila

Subgroup: 2.3.37

Comma list: 81/74

Gencom: [2 4/3; 81/74]

Mapping: [1 1 0], 0 -1 12]]

POTE generator: ~4/3 = 521.397

## Reversed meantone

Subgroup: 2.3.41

Comma list: 82/81

Gencom: [2 4/3; 82/81]

Sval mapping: [1 2 7], 0 -1 -4]]

POL2 generator: ~4/3 = 494.509

### 2.3.7.41 subgroup

Subgroup: 2.3.7.41

Comma list: 64/63, 82/81

Gencom: [2 4/3; 64/63 82/81]

Sval mapping: [1 2 2 7], 0 -1 2 -4]]

POTE generator: ~4/3 = 490.0323

TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029

### 2.3.7.11.41 subgroup

Subgroup: 2.3.7.11.41

Comma list: 64/63, 82/81, 99/98

Gencom: [2 4/3; 64/63 82/81 99/98]

Sval mapping: [1 2 2 1 7], 0 -1 2 6 -4]]

POTE generator: ~4/3 = 492.1787

TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454

## Purpleheart

Subgroup: 2.3.7

Comma list: 2187/2048

Mapping[7 11 0], 0 0 1]]

mapping generators: ~9/8, ~7

Optimal tuning (CTE): ~9/8 = 1\7, ~7/4 = 968.826 (~64/63 = 59.746)

## Io aka undecimal

Io is a very low-complexity temperament which tempers out the undecimal quartertone 33/32, and with a generator representing both 3/2 and 16/11. It may be considered an exotemperament by some definitions and not one by others. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of flattone temperament.

The name Io was coined by CompactStar in 2024 based on the color name ilo, prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.

Subgroup: 2.3.11

Comma list: 33/32

Sval mapping[1 1 4], 0 1 -1]]

mapping generators: ~2/1, ~3/2

Optimal tuning (CTE): ~2/1 = 1\1, ~3/2 = 692.713