# Marvel temperaments

(Redirected from Tritonic)

This page discusses miscellaneous rank-2 temperaments tempering out [-5 2 2 -1 = 225/224, the marvel comma or septimal kleisma.

Temperaments considered in families and clans are:

Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.

Since (5/4)2 = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.

The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.

Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)

## Wizard

For the 5-limit version of this temperament, see High badness temperaments #Wizard.

Subgroup: 2.3.5.7

Comma list: 225/224, 118098/117649

Mapping[2 1 5 2], 0 6 -1 10]]

mapping generators: ~1225/864, ~245/216

Optimal tuning (POTE): ~1225/864 = 1\2, ~5/4 = 383.256 (~245/216 = 216.744)

Wedgie⟨⟨12 -2 20 -31 -2 52]]

Scales: wizard22

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4000/3993

Mapping: [2 1 5 2 8], 0 6 -1 10 -3]]

Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.232 (~25/22 = 216.768)

Scales: wizard22

#### Lizard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 385/384

Mapping: [2 1 5 2 8 11], 0 6 -1 10 -3 -10]]

Optimal tuning (POTE): ~55/39 = 1\2, ~5/4 = 383.389 (~25/22 = 216.711)

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 273/272, 289/288, 351/350, 375/374

Mapping: [2 1 5 2 8 11 6], 0 6 -1 10 -3 -10 6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.381 (~17/15 = 216.619)

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342

Mapping: [2 1 5 2 8 11 6 2], 0 6 -1 10 -3 -10 6 18]]

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.477 (~17/15 = 216.523)

#### Gizzard

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1573/1568

Mapping: [2 1 5 2 8 -2], 0 6 -1 10 -3 26]]

Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.170 (~25/22 = 216.830)

##### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 289/288, 325/324, 375/374, 385/384

Mapping: [2 1 5 2 8 -2 6], 0 6 -1 10 -3 26 6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.175 (~25/22 = 216.825)

##### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594

Mapping: [2 1 5 2 8 -2 6 15], 0 6 -1 10 -3 26 6 -18]]

Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.138 (~17/15 = 216.862)

### Mage

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1331/1296

Mapping: [2 1 5 2 4], 0 6 -1 10 8]]

Optimal tuning (POTE): ~77/54 = 1\2, ~5/4 = 383.124 (~55/48 = 216.876)

## Tritonic

For the 5-limit version of this temperament, see High badness temperaments #Tritonic.

Subgroup: 2.3.5.7

Comma list: 225/224, 50421/50000

Mapping[1 4 -3 -3], 0 -5 11 12]]

Wedgie⟨⟨5 -11 -12 -29 -33 3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.286

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 441/440

Mapping: [1 4 -3 -3 2], 0 -5 11 12 3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 196/195, 275/273

Mapping: [1 4 -3 -3 2 -5], 0 -5 11 12 3 18]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.108

### Tritoni

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 27783/27500

Mapping: [1 4 -3 -3 17], 0 -5 11 12 -28]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389

## Septimin

For the 5-limit version of this temperament, see High badness temperaments #Septimin.

Subgroup: 2.3.5.7

Comma list: 225/224, 84035/82944

Mapping[1 4 1 5], 0 -11 6 -10]]

Wedgie⟨⟨11 -6 10 -35 -15 40]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.632

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 2401/2376

Mapping: [1 4 1 5 5], 0 -11 6 -10 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 245/242

Mapping: [1 4 1 5 5 7], 0 -11 6 -10 -7 -15]]

Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.700

## Merman

For the 5-limit version of this temperament, see High badness temperaments #Merman.

Subgroup: 2.3.5.7

Comma list: 225/224, 2500000/2470629

Mapping[1 5 -5 -5], 0 -7 15 16]]

Wedgie⟨⟨7 -15 -16 -40 -45 5]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.585

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1344/1331

Mapping: [1 5 -5 -5 2], 0 -7 15 16 3]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.606

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 225/224, 364/363, 441/440

Mapping: [1 5 -5 -5 2 12], 0 -7 15 16 3 -17]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.657

## Slender

Slender (31 & 32) tempers out the hewuermera comma in addition to the marvel comma. This temperament has a generator of 49/48, 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.

Subgroup: 2.3.5.7

Comma list: 225/224, 589824/588245

Mapping[1 2 2 3], 0 -13 10 -6]]

Wedgie⟨⟨13 -10 6 -46 -27 42]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.413

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 1331/1323

Mapping: [1 2 2 3 4], 0 -13 10 -6 -17]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.387

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 275/273, 385/384, 1331/1323

Mapping: [1 2 2 3 4 3], 0 -13 10 -6 -17 22]]

Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.314

## Triton

For the 5-limit version of this temperament, see High badness temperaments #Stump.

Subgroup: 2.3.5.7

Comma list: 225/224, 1029/1000

Mapping[1 0 6 7], 0 3 -7 -8]]

Wedgie⟨⟨3 -7 -8 -18 -21 1]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 568.865

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 1029/1000

Mapping: [1 0 6 7 4], 0 3 -7 -8 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 569.144

## Submajor

Subgroup: 2.3.5

Comma list: 69198046875/68719476736

Mapping[1 4 -1], 0 -8 11]]

Optimal tuning (POTE): ~2 = 1\1, ~10125/8192 = 362.321

### 7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 51200/50421

Mapping[1 4 -1 1], 0 -8 11 6]]

Wedgie⟨⟨8 -11 -6 -36 -32 17]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.255

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: [1 4 -1 1 11], 0 -8 11 6 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: [1 4 -1 1 11 4], 0 -8 11 6 -25 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105

### Interpental

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: [1 4 -1 1 -5], 0 -8 11 6 28]]

Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: [1 4 -1 1 -5 4], 0 -8 11 6 28 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402

## Marvolo

Subgroup: 2.3.5.7

Comma list: 225/224, 156250000/155649627

Mapping[1 2 1 1], 0 -6 19 26]]

Wedgie⟨⟨6 -19 -26 -44 -58 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.348

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4000/3993

Mapping: [1 2 1 1 2], 0 -6 19 26 21]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.340

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 441/440

Mapping: [1 2 1 1 2 3], 0 -6 19 26 21 10]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 364/363, 441/440

Mapping: [1 2 1 1 2 3 2], 0 -6 19 26 21 10 30]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440

Mapping: [1 2 1 1 2 3 2 3], 0 -6 19 26 21 10 30 18]]

Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330

## Enneaportent

Subgroup: 2.3.5.7

Comma list: 225/224, 40353607/40310784

Mapping[9 0 28 11], 0 2 -1 2]]

Wedgie⟨⟨18 -9 18 -56 -22 67]]

Optimal tuning (POTE): ~2592/2401 = 1\9, ~5/4 = 383.165

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12005/11979

Mapping: [9 0 28 11 24], 0 2 -1 2 1]]

Optimal tuning (POTE): ~121/112 = 1\9, ~5/4 = 383.146

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 364/363, 1716/1715

Mapping: [9 0 28 11 24 19], 0 2 -1 2 1 2]]

Optimal tuning (POTE): ~14/13 = 1\9, ~5/4 = 383.047

## Gracecordial

For the 5-limit version of this temperament, see High badness temperaments #Gracecordial.

Subgroup: 2.3.5.7

Comma list: 225/224, 781250000/771895089

Mapping[1 0 34 63], 0 1 -20 -38]]

Wedgie⟨⟨1 -20 -38 -34 -63 -32]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.824

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 236328125/234365481

Mapping: [1 0 34 63 -90], 0 1 -20 -38 59]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.834

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 831875/830466

Mapping: [1 0 34 63 -90 -66], 0 1 -20 -38 59 44]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913

Mapping: [1 0 34 63 -90 -66 -7], 0 1 -20 -38 59 44 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444

Mapping: [1 0 34 63 -90 -66 -7 9], 0 1 -20 -38 59 44 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842

#### 23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528

Mapping: [1 0 34 63 -90 -66 -7 9 -43], 0 1 -20 -38 59 44 7 -3 30]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.843

#### 29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493

Mapping: [1 0 34 63 -90 -66 -7 9 -43 -49], 0 1 -20 -38 59 44 7 -3 30 34]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842

#### 31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493

Mapping: [1 0 34 63 -90 -66 -7 9 -43 -49 -79], 0 1 -20 -38 59 44 7 -3 30 34 53]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.838

### Gracecord

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 109375/107811

Mapping: [1 0 34 63 89], 0 1 -20 -38 -54]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.885

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 364/363, 441/440, 6125/6084

Mapping: [1 0 34 63 89 113], 0 1 -20 -38 -54 -69]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.890

#### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989

Mapping: [1 0 34 63 89 113 -7], 0 1 -20 -38 -54 -69 7]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.870

#### 19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663

Mapping: [1 0 34 63 89 113 -7 9], 0 1 -20 -38 -54 -69 7 -3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.866

## Alphorn

Subgroup: 2.3.5.7

Comma list: 225/224, 5764801/5668704

Mapping[1 9 0 13], 0 -16 5 -22]]

Wedgie⟨⟨16 -5 22 -45 -10 65]]

Optimal tuning (POTE): ~2 = 1\1, ~48/35 = 556.221

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 12250/11979

Mapping: [1 9 0 13 3], 0 -16 5 -22 1]]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 556.144

## Misneb

For the 5-limit version of this temperament, see High badness temperaments #Misneb.

Subgroup: 2.3.5.7

Comma list: 225/224, 4194304/4117715

Mapping[1 3 1 3], 0 -15 14 -2]]

Wedgie⟨⟨15 -14 2 -57 -39 44]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.235

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1310720/1294139

Mapping: [1 3 1 3 1], 0 -15 14 -2 26]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323

#### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 176/175, 640/637, 847/845

Mapping: [1 3 1 3 1 2], 0 -15 14 -2 26 18]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323

### Musneb

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 66550/64827

Mapping: [1 3 1 3 6], 0 -15 14 -2 -27]]

Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.142

## Untriton

For the 5-limit version of this temperament, see High badness temperaments #Untriton.

Subgroup: 2.3.5.7

Comma list: 225/224, 125000000/121060821

Mapping[1 6 -7 -7], 0 -9 19 20]]

Wedgie⟨⟨9 -19 -20 -51 -57 7]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.641

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 22000/21609

Mapping: [1 6 -7 -7 1], 0 -9 19 20 5]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.626

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 225/224, 275/273, 1040/1029

Mapping: [1 6 -7 -7 1 -12], 0 -9 19 20 5 32]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.654

## Quintannic

Subgroup: 2.3.5.7

Comma list: 225/224, 9805926501/9765625000

Mapping[1 1 5 7], 0 5 -23 -36]]

Wedgie⟨⟨5 -23 -36 -48 -71 -19]]

Optimal tuning (POTE): ~2 = 1\1, ~10000/9261 = 139.838

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 43923/43750

Mapping: [1 1 5 7 8], 0 5 -23 -36 -39]]

Optimal tuning (POTE): ~2 = 1\1, ~320/297 = 139.827

### 13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 441/440, 1001/1000, 1188/1183

Mapping: [1 1 5 7 8 3], 0 5 -23 -36 -39 6]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.812

### 17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 375/374, 441/440, 891/884

Mapping: [1 1 5 7 8 3 7], 0 5 -23 -36 -39 6 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.815

## Gwazy

Subgroup: 2.3.5.7

Comma list: 225/224, 5971968/5764801

Mapping[2 1 6 4], 0 8 -5 6]]

Wedgie⟨⟨16 -10 12 -53 -26 56]]

Optimal tuning (POTE): 1\2, ~35/32 = 162.658

### 11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 65536/65219

Mapping: [2 1 6 4 8], 0 8 -5 6 -4]]

Optimal tuning (POTE): 1\2, ~11/10 = 162.592

## Tertiosec

For the 5-limit version of this temperament, see High badness temperaments #Tertiosec.

Subgroup: 2.3.5.7

Comma list: 225/224, 14495514624/13841287201

Mapping[3 7 5 9], 0 -8 7 -2]]

Wedgie⟨⟨24 -21 6 -89 -58 73]]

Optimal tuning (POTE): 1\3, ~15/14 = 112.283