No-threes subgroup temperaments: Difference between revisions
Created page with "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; 41943..." |
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[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents | [[Tp tuning #T2 tuning|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|Gencom mapping]]: [{{val|1 0 1 3 1}}, {{val|0 0 7 -1 13}}] | |||
[[Mapping|Sval mapping]]: [{{val|1 1 3 1}}, {{val|0 7 -1 13}}] | |||
[[Tp tuning|POL2 generator]]: ~8/7 = 227.114 | |||
{{Val list|legend=1| 16, 21, 37 }} | |||
==== 2.5.7.11.13 ==== | |||
Subgroup: 2.5.7.11 | |||
[[Comma]]: 176/175, 640/639, 1308672/1294139 | |||
[[Gencom]]: [2 8/7; 176/175 640/639, 1308672/1294139] | |||
[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2}}, {{val|0 0 7 -1 13 9}}] | |||
[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2}}, {{val|0 7 -1 13 9}}] | |||
[[Tp tuning|POL2 generator]]: ~8/7 = 227.108 | |||
{{Val list|legend=1| 16, 21, 37 }} | |||
==== 2.5.7.11.13.17 ==== | |||
Subgroup: 2.5.7.11 | |||
[[Comma]]: 176/175, 221/200, 640/639, 833/832 | |||
[[Gencom]]: [2 8/7; 176/175 221/200, 640/639, 833/832] | |||
[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2 2}}, {{val|0 0 7 -1 13 9 11}}] | |||
[[Mapping|Sval mapping]]: [{{val|1 1 3 1 2 2}}, {{val|0 7 -1 13 9 11}}] | |||
[[Tp tuning|POL2 generator]]: ~8/7 = 227.242 | |||
{{Val list|legend=1| 16, 21, 37 }} | |||
=== Didacus === | === Didacus === | ||
Revision as of 14:19, 8 December 2021
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
Comma: 176/175, 640/639, 1308672/1294139
Gencom: [2 8/7; 176/175 640/639, 1308672/1294139]
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
Comma: 176/175, 221/200, 640/639, 833/832
Gencom: [2 8/7; 176/175 221/200, 640/639, 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