# Consonant class scale

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A **consonant class scale** is a scale with a **consonant interval class**, that is, a number n such that all n-step intervals of the scale are "consonances" (of a certain odd limit).

The models for such scales are meantone[5] (in which the 2-step intervals are 5/4 and 4/3), and even better known, meantone[7] (in which the 2-step intervals are 6/5 and 5/4). Harmonizing a diatonic melody by parallel thirds is a commonly used formula that reliably produces beautiful harmony. Consonant class scales are an attempt to generalize this property of the diatonic scale.

## Rank-2 consonant class scales

### 3-limit

#### {4/3, 3/2}

- Augmented[6] (3-step intervals consist of 3 4/3's and 3 3/2's)

### 5-limit

#### {6/5, 5/4}

- Meantone[7] (2-step intervals consist of 4 6/5's and 3 5/4's)
- Porcupine[7] (2-step intervals consist of 5 6/5's and 2 5/4's)
- Mavila[7] (2-step intervals consist of 3 6/5's and 4 5/4's)
- Srutal[10] (3-step intervals consist of 4 6/5's and 6 5/4's)
- Negri[10] (3-step intervals consist of 3 6/5's and 7 5/4's)
- Blackwood[10] (3-step intervals consist of 5 6/5's and 5 5/4's)
- Sensi[11] (3-step intervals consist of 9 6/5's and 2 5/4's)
- Orson[13] (4-step intervals consist of 3 6/5's and 10 5/4's)
- Godzilla[14] (4-step intervals consist of 8 6/5's and 6 5/4's)
- Helmholtz[17] (5-step intervals consist of 8 6/5's and 9 5/4's)
- Superpyth[17] (5-step intervals consist of 9 6/5's and 8 5/4's)

#### {6/5, 4/3}

- Mavila[5] (2-step intervals consist of 1 6/5 and 4 4/3's)
- Hanson[7] (2-step intervals consist of 6 6/5's and 1 4/3)
- Srutal[8] (3-step intervals consist of 2 6/5's and 6 4/3's)
- Doublewide[10] (3-step intervals consist of 8 6/5's and 2 4/3's)

#### {5/4, 4/3}

- Meantone[5] (2-step intervals consist of 1 5/4 and 4 4/3's)
- Porcupine[8] (3-step intervals consist of 3 5/4's and 5 4/3's)
- Diminished[8] (3-step intervals consist of 4 5/4's and 4 4/3's)
- Mohajira/Neutrominant[10] (4-step intervals consist of 2 5/4's and 8 4/3's)
- Hanson[11] (4-step intervals consist of 6 5/4's and 5 4/3's)
- Tetracot[13] (5-step intervals consist of 4 5/4's and 9 4/3's)

### 7-limit

#### {8/7, 7/6}

- Slendric[5] (1-step intervals consist of 4 8/7's and 1 7/6)
- Superpyth[5] (1-step intervals consist of 3 8/7's and 2 7/6's)
- Orwell[9] (2-step intervals consist of 1 8/7 and 8 7/6's)
- Miracle[10] (2-step intervals consist of 8 8/7's and 2 7/6's)
- Pajara[10] (2-step intervals consist of 6 8/7's and 4 7/6's)
- Beatles/Neutrominant/Hemif[10] (2-step intervals consist of 6 8/7's and 4 7/6's)
- Hedgehog[14] (3-step intervals consist of 4 8/7's and 10 7/6's)
- Porcupine[15] (3-step intervals consist of 9 8/7's and 6 7/6's)
- Augene[15] (3-step intervals consist of 6 8/7's and 9 7/6's)
- Meantone[19] (4-step intervals consist of 9 8/7's and 10 7/6's)
- Sensi[19] (4-step intervals consist of 6 8/7's and 13 7/6's)

#### {8/7, 6/5}

- Keemun[4] (aka magical seventh chord)
- Gorgo[5] (1-step intervals consist of 4 8/7's and 1 6/5)
- Porcupine[8] (2-step intervals consist of 2 8/7's and 6 6/5's)
- Armodue[9] (2-step intervals consist of 4 8/7's and 5 6/5's)
- Negri[9] (2 step intervals consist of 7 8/7's and 2 6/5's)
- Astrology[10] (2-step intervals consist of 8 8/7's and 2 6/5's)
- Neutrominant[10] (2-step intervals consist of 6 8/7's and 4 7/6's)
- Flattone[12] (3-step intervals consist of 3 8/7's and 9 6/5's)
- Magic[16] (4-step intervals consist of 4 8/7's and 12 6/5's)
- Mohajira[17] (4-step intervals consist of 6 8/7's and 11 6/5's)

#### {8/7, 5/4}

- Hemithirds[7] (2-step intervals consist of 5 5/4's and 2 8/7's)
- Godzilla[9] (2-step intervals consist of 8 7/6's and 1 5/4)
- Magic[13] (4-step intervals consist of 1 8/7 and 12 5/4's)

#### {7/6, 6/5}

- Myna[4] (aka starling tetrad)
- Sensi[8] (2-step intervals consist of 2 7/6's and 6 6/5's)
- Hedgehog[8] (2-step intervals consist of 4 7/6's and 4 6/5's)
- Superpelog[9] (2-step intervals consist of 8 7/6's and 1 6/5)
- Negri[9] (2 step intervals consist of 7 7/6's and 2 6/5's)
- Meantone[12] (3-step intervals consist of 3 7/6's and 9 6/5's)
- Augene[12] (3-step intervals consist of 3 7/6's and 9 6/5's)
- Pajara[12] (3-step intervals consist of 6 7/6's and 6 6/5's)
- Valentine[16] (4-step intervals consist of 4 7/6's and 12 6/5's)
- Muggles[16] (4-step intervals consist of 4 7/6's and 12 6/5's)
- Miracle[21] (5-step intervals consist of 13 7/6's and 8 6/5's)

#### {7/6, 5/4}

- Deutone[7] (2-step intervals consist of 2 7/6's and 5 5/4's)
- Godzilla[9] (2-step intervals consist of 8 7/6's and 1 5/4)
- Superpyth[12] (3-step intervals consist of 9 7/6's and 3 5/4's)

#### {6/5, 9/7}

- Godzilla[9] (3-step intervals consist of 3 6/5's and 6 9/7's)
- Superpyth[12] (4-step intervals consist of 4 6/5's and 8 9/7's)

#### {5/4, 7/5}

- Keemun[7] (3-step intervals consist of 2 5/4's and 5 7/5's)

#### {4/3, 7/5}

- Meantone[7] (3-step intervals consist of 6 4/3's and 1 7/5; both 2-step intervals are also consonant)
- Beatles[7] (3-step intervals consist of 5 4/3's and 2 7/5')
- Negri[9] (4-step intervals consist of 5 4/3's and 4 7/5's)
- Sensi[11] (5-step intervals consist of 4 4/3's and 7 7/5's)
- Miracle[11] (5-step intervals consist of 5 4/3's and 6 7/5's)
- Magic[16] (7-step intervals consist of 11 4/3's and 5 7/5's)

#### {4/3, 10/7}

- Garibaldi[7] (3-step intervals consist of 6 4/3's and 1 10/7)
- Dominant[7] (3-step intervals consist of 6 4/3's and 1 10/7)
- Augene[9] (4-step intervals consist of 6 4/3's and 3 10/7's)
- Godzilla[9] (4-step intervals consist of 7 4/3's and 2 10/7's)
- Superpyth[12] (5-step intervals consist of 11 4/3's and 1 10/7)
- Muggles[13] (6-step intervals consist of 8 4/3's and 5 10/7's)
- Rodan[21] (9-step intervals consist of 18 4/3's and 3 10/7's)

#### {7/5, 10/7}

- Hemithirds[6] (3-step intervals consist of 3 7/5's and 3 10/7's)
- Miracle[10] (5-step intervals consist of 5 7/5's and 5 10/7's)
- Beatles[10] (5-step intervals consist of 5 7/5's and 5 10/7's)
- Negri[10] (5-step intervals consist of 5 7/5's and 5 10/7's)
- Blacksmith[10] (5-step intervals consist of 5 7/5's and 5 10/7's)
- Meantone[12] (6-step intervals consist of 6 7/5's and 6 10/7's)
- Augene[12] (6-step intervals consist of 6 7/5's and 6 10/7's)
- Muggles[16] (8-step intervals consist of 8 7/5's and 8 10/7's)
- Superpyth[22] (11-step intervals consist of 11 7/5's and 11 10/7's)

### 9-limit

#### {7/6, 9/7}

- Superpyth[7] (2-step intervals consist of 3 9/7's and 4 7/6's)

#### {6/5, 9/7}

- Sensamagic, Godzilla[9] (3-step intervals consist of 3 6/5's and 6 9/7's)
- Hedgehog[6] (2-step intervals consist of 2 6/5's and 4 9/7's)

#### {5/4, 9/7}

- Orwell[9] (3-step intervals consist of 6 5/4's and 3 9/7's)
- Negri[9] (3-step intervals consist of 6 5/4's and 3 9/7's)
- Meantone[12] (4-step intervals consist of 8 5/4's and 4 9/7's)
- Pajara[12] (4-step intervals consist of 8 5/4's and 4 9/7's)
- Miracle[21] (7-step interval consists of 14 5/4's and 7 9/7's)

#### {9/7, 4/3}

- Semaphore[5] (2-step intervals consist of 2 9/7's and 3 4/3's)
- Superpyth[5] (2-step intervals consist of 1 9/7 and 4 4/3's)
- Diminished[8] (3-step intervals consist of 4 9/7's and 4 4/3's)
- Sensi[8] (3-step intervals consist of 7 9/7's and 1 4/3)
- Hedgehog[8] (3-step intervals consist of 6 9/7's and 2 4/3's)
- Beatles/Hemif/Neutrominant[10] (4-step intervals consist of 2 9/7's and 8 4/3's)
- Pajara[10] (4-step intervals consist of 2 9/7's and 8 4/3's)
- Negri[10] (4-step intervals consist of 4 9/7's and 6 4/3's)
- Orwell[13] (5-step intervals consist of 7 9/7's and 6 4/3's)
- Keemun[15] (6-step intervals consist of 6 9/7's and 9 4/3's)
- Augene[15] (6-step intervals consist of 3 9/7's and 12 4/3's)
- Mohajira[17] (7-step intervals consist of 2 9/7's and 17 4/3's)

#### {9/7, 7/5}

- Deutone[7] (3-step intervals consist of 4 7/5's and 3 9/7's)
- Magic[13] (5-step intervals consist of 11 9/7's and 2 7/5's)

#### {9/7,10/7}

- Muggles[10] (5-step intervals consist of 8 9/7's and 2 10/7's)

#### {9/7, 3/2}

- Magic[7] (3-step intervals consist of 5 9/7's and 2 3/2's)

### 11-limit

#### {12/11, 9/8}

- Mohajira[7] (1-step intervals consist of 4 12/11's and 3 9/8s)

#### {6/5, 11/9}

- Mohajira[7] (2-step intervals consist of 1 6/5 and 6 11/9's)

#### {4/3, 11/8}

- Mohajira[7] (3-step intervals consist of 5 4/3's and 2 11/8's)
- Orwell[9] (4-step intervals consist of 2 4/3's and 7 11/8's)

### 13-limit

#### {13/11, 13/10}

- Shoe[6] (2-step intervals consist of 2 13/11's and 4 13/10's)

### 3.5.7-subgroup

#### {9/7, 7/5}

- Sirius[7] (2-step intervals consist of 2 9/7's and 5 7/5's)

#### {7/5, 5/3}

- Rank-2 Bohlen-Pierce[5] (2-step intervals consist of 2 7/5's and 3 5/3's)

#### {5/3, 9/5}

- Sirius[6] (3-step intervals consist of 3 5/3's and 3 9/5's)

## Rank-3 consonant class scales

### 7-limit

#### {8/7, 6/5, 5/4}

- Marvel, marvel11max7a and marvel11max7b, 7 notes (2-step intervals consist of 1 8/7, 2 6/5's, and 4 5/4's)

#### {7/6, 6/5, 5/4}

#### {5/4, 4/3, 7/5}

- Breed or Jove, 7 notes (3-step intervals consist of 2 5/4's, 1 4/3, and 4 7/5's)

#### {7/6, 6/5, 5/4}

- Portent; 4 5/4's 4 6/5's and 3 7/6's in the third class

## Rank-4 consonant class scales

### 5-limit

#### {6/5, 5/4, 4/3}

- 1-3-5-11 hexany, 6 notes (2-step intervals consist of 2 6/5's, 2 5/4's, and 2 4/3's)

## External links

- http://tech.groups.yahoo.com/group/tuning-math/message/6056
^{[dead link]} - http://tech.groups.yahoo.com/group/tuning-math/message/6060
- http://tech.groups.yahoo.com/group/tuning-math/message/6062
- http://tech.groups.yahoo.com/group/tuning-math/message/6064
- http://tech.groups.yahoo.com/group/tuning-math/message/6065
- http://tech.groups.yahoo.com/group/tuning-math/message/6067
- http://tech.groups.yahoo.com/group/tuning-math/message/6068
- http://tech.groups.yahoo.com/group/tuning-math/message/6075
- http://tech.groups.yahoo.com/group/tuning-math/message/6082
- http://tech.groups.yahoo.com/group/tuning-math/message/6086
- http://tech.groups.yahoo.com/group/tuning-math/message/6088
- http://tech.groups.yahoo.com/group/tuning-math/message/6089
- http://tech.groups.yahoo.com/group/tuning-math/message/6090
- http://tech.groups.yahoo.com/group/tuning-math/message/6092
- http://tech.groups.yahoo.com/group/tuning-math/message/6094
- http://tech.groups.yahoo.com/group/tuning-math/message/6097
- http://tech.groups.yahoo.com/group/tuning-math/message/6100
- http://tech.groups.yahoo.com/group/tuning-math/message/6102
- http://tech.groups.yahoo.com/group/tuning-math/message/6103
- http://tech.groups.yahoo.com/group/tuning-math/message/6104
- http://tech.groups.yahoo.com/group/tuning-math/message/6169
- http://tech.groups.yahoo.com/group/tuning-math/message/15082
- http://tech.groups.yahoo.com/group/tuning-math/message/15102
- http://tech.groups.yahoo.com/group/tuning-math/message/15103
- http://tech.groups.yahoo.com/group/tuning-math/message/15123
- http://tech.groups.yahoo.com/group/tuning-math/message/15483
- http://tech.groups.yahoo.com/group/tuning-math/message/15501
- http://tech.groups.yahoo.com/group/tuning-math/message/19390
- http://tech.groups.yahoo.com/group/tuning-math/message/19410
- http://launch.groups.yahoo.com/group/tuning/message/41360