No-threes subgroup temperaments: Difference between revisions
m →2.5.7.11.13: fix subgroup value |
m →2.5.7.11.13.17: Fix subgroup value |
||
| Line 49: | Line 49: | ||
=== 2.5.7.11.13.17 === | === 2.5.7.11.13.17 === | ||
Subgroup: 2.5.7.11 | Subgroup: 2.5.7.11.13.17 | ||
[[Comma]]: 176/175, 221/200, 640/637, 833/832 | [[Comma]]: 176/175, 221/200, 640/637, 833/832 | ||
Revision as of 13:51, 3 August 2022
This is a collection of subgroup temperaments which omit the prime harmonic of 3.
Llywelyn
Subgroup: 2.5.7
Comma: 4194304/4117715
Gencom: [2 8/7; 4194304/4117715]
Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]
Sval mapping: [⟨1 1 3], ⟨0 7 -1]]
POL2 generator: ~8/7 = 226.910
RMS error: 0.5391 cents
2.5.7.11
Subgroup: 2.5.7.11
Comma: 176/175, 1310720/1294139
Gencom: [2 8/7; 176/175 1310720/1294139]
Gencom mapping: [⟨1 0 1 3 1], ⟨0 0 7 -1 13]]
Sval mapping: [⟨1 1 3 1], ⟨0 7 -1 13]]
POL2 generator: ~8/7 = 227.114
2.5.7.11.13
Subgroup: 2.5.7.11.13
Comma: 176/175, 640/637, 1304576/1294139
Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]
Gencom mapping: [⟨1 0 1 3 1 2], ⟨0 0 7 -1 13 9]]
Sval mapping: [⟨1 1 3 1 2], ⟨0 7 -1 13 9]]
POL2 generator: ~8/7 = 227.108
2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17
Comma: 176/175, 221/200, 640/637, 833/832
Gencom: [2 8/7; 176/175 221/200, 640/637, 833/832]
Gencom mapping: [⟨1 0 1 3 1 2 2], ⟨0 0 7 -1 13 9 11]]
Sval mapping: [⟨1 1 3 1 2 2], ⟨0 7 -1 13 9 11]]
POL2 generator: ~8/7 = 227.242
Didacus
Related temperaments: roulette, hemithirds
Subgroup: 2.5.7
Comma: 3136/3125
Gencom: [2 28/25; 3136/3125]
Gencom mapping: [⟨1 0 2 2], ⟨0 0 2 5]]
Sval mapping: [⟨1 2 2], ⟨0 2 5]]
POL2 generator: ~28/25 = 93.772
RMS error: 0.2138 cents
Rainy
Three generators make an 8/7; five generators make a 5/4. This is the no-threes version of tertiaseptal.
Subgroup: 2.5.7
Gencom: [2 256/245; 2100875/2097152]
Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]
Sval mapping: [⟨1 2 3], ⟨0 5 -3]]
POL2 generator: ~256/245 = 77.205
RMS error: 0.0586 cents
Mercy
Two generators make an 8/7; seven generators make an 8/5. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of 31edo, and is the no-threes or elevens version of miracle.
Subgroup: 2.5.7
Comma list: 823543/819200
Gencom: [2 2744/2560; 823543/819200]
Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]
Sval mapping: [⟨1 3 3], ⟨0 -7 -2]]
POL2 generator: ~343/320 = 116.291
2.5.7.13
Subgroup: 2.5.7.13
Comma list: 343/338, 640/637
Gencom: [2 14/13; 343/338 640/637]
Gencom mapping: [⟨1 0 3 3 4], ⟨0 0 -7 -2 -3]]
Sval mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]
POL2 generator: ~14/13 = 116.094
2.5.7.13.17
Subgroup: 2.5.7.13.17
Comma list: 170/169, 224/221, 640/637
Gencom: [2 14/13; 170/169 224/221 640/637]
Gencom mapping: [⟨1 0 3 3 4 4], ⟨0 0 -7 -2 -3 1]]
Sval mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]
POL2 generator: ~14/13 = 115.769
2.5.7.13.17.19
Subgroup: 2.5.7.13.17.19
Comma list: 170/169, 343/338, 640/637, 16384/16055
Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]
Gencom mapping: [⟨1 0 3 3 4 4 3], ⟨0 0 -7 -2 -3 1 13]]
Sval mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]
POL2 generator: ~14/13 = 115.716
Yer (rank 3)
Subgroup: 2.11.13.17.19
Comma list: 209/208, 2057/2048
Sval mapping: [⟨1 0 0 11 4], ⟨0 1 0 -2 -1], ⟨0 0 1 0 1]]
TE tuned generators: ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Yamablu
Yamablu, with a generator of ~17/13, is named for it's tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The 13th Yamablu[13] scale is a linear-temperament version of Gjaeck.
Subgroup: 2.11.13.17.19
Comma list: 209/208, 2057/2048, 83521/83486
Sval mapping: [⟨1 5 1 1 0], ⟨0 -4 7 8 11]]
POL2 generator: ~17/13 = 462.9606
RMS error: 0.4898 cents