Consonant class scale

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/Maqamic[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/Maqamic/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)
• Maqamic[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/Maqamic[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)

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}

• Hewuermity or Zeus, zeus7tri, 7 notes (2-step intervals consist of 2 7/6's, 1 6/5, and 4 5/4's)

{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)

Links

http://tech.groups.yahoo.com/group/tuning-math/message/6056

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