120/119: Difference between revisions
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== Temperaments == | == Temperaments == | ||
[[Tempering out]] this comma in the 17-limit leads to the rank-6 '''lynchismic | [[Tempering out]] this comma in the 17-limit leads to the rank-6 '''lynchismic''' temperament. In the 2.3.5.7.17 subgroup, tempering it out results in the rank-4 '''lynchic''' temperament. In either case, it allows you to assign [[10:12:15:17]] as the inverse of [[4:5:6:7]], an otonal chord that would otherwise be [[70:84:105:120]]. [[William Lynch's thoughts on septimal harmony and 22edo|William Lynch]] calls this the minor tetrad, and so equating it with the inverse of the major tetrad is quite useful. | ||
Since 120/119 factors as ([[225/224]])⋅([[256/255]]), it would make sense to temper them both out, so lynchic can be further tempered to a simple extension of [[marvel]] that adds prime 17 known as [[ | Since 120/119 factors as ([[225/224]])⋅([[256/255]]), it would make sense to temper them both out, so lynchic can be further tempered to a simple extension of [[marvel]] that adds prime 17 known as [[char]], though it loses accuracy when compared to marvel. | ||
=== Lynchic === | === Lynchic === | ||
Latest revision as of 14:40, 21 March 2026
| Interval information |
Suruyo comma
reduced
S18⋅S19⋅S20
120/119, the lynchisma, is a small 17-limit superparticular comma of about 14.49 cents. It is the difference between 20/17 and 7/6, 17/10 and 12/7, or 30/17 and 7/4.
Temperaments
Tempering out this comma in the 17-limit leads to the rank-6 lynchismic temperament. In the 2.3.5.7.17 subgroup, tempering it out results in the rank-4 lynchic temperament. In either case, it allows you to assign 10:12:15:17 as the inverse of 4:5:6:7, an otonal chord that would otherwise be 70:84:105:120. William Lynch calls this the minor tetrad, and so equating it with the inverse of the major tetrad is quite useful.
Since 120/119 factors as (225/224)⋅(256/255), it would make sense to temper them both out, so lynchic can be further tempered to a simple extension of marvel that adds prime 17 known as char, though it loses accuracy when compared to marvel.
Lynchic
Subgroup: 2.3.5.7.17
Subgroup-val mapping: [⟨1 0 0 0 3], ⟨0 1 0 0 1], ⟨0 0 1 0 1], ⟨0 0 0 1 -1]]
- mapping generators: ~2, ~3, ~5, ~7
- WE: ~2 = 1198.9483 ¢, ~3/2 = 702.1229 ¢, ~5/4 = 386.5249 ¢, ~7/4 = 973.6693 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8985 ¢, ~5/4 = 385.5601 ¢, ~7/4 = 973.5597 ¢
Optimal ET sequence: 10, 12, 19, 22, 27g, 31, 41, 53, 198ddggg
Badness (Sintel): 0.220
Lynchismic
Subgroup: 2.3.5.7.11.13.17
| [⟨ | 1 | 0 | 0 | 0 | 0 | 0 | 3 | ], |
| ⟨ | 0 | 1 | 0 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 1 | 0 | 0 | 0 | 1 | ], |
| ⟨ | 0 | 0 | 0 | 1 | 0 | 0 | -1 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 1 | 0 | 0 | ], |
| ⟨ | 0 | 0 | 0 | 0 | 0 | 1 | 0 | ]] |
- mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
- WE: ~2 = 1198.9483 ¢, ~3/2 = 702.1229 ¢, ~5/4 = 386.5249 ¢, ~7/4 = 973.6693 ¢, ~11/8 = 554.4584 ¢, ~13/8 = 843.6671 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8985 ¢, ~5/4 = 385.5601 ¢, ~7/4 = 973.5597 ¢, ~11/8 = 553.3440 ¢, ~13/8 = 842.6949 ¢
Optimal ET sequence: 19, 22, 26, 27eg, 31, 41, 49fg, 53, 65d, 84g, 92defg, 106g, 123dfgg, 128dg, 137gg, 145dgg, 171dgg
Badness (Sintel): 0.603