No-threes subgroup temperaments: Difference between revisions
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This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3. | This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 3. | ||
== Llywelyn == | |||
Subgroup: 2.5.7 | Subgroup: 2.5.7 | ||
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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 | Subgroup: 2.5.7.11 | ||
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{{Val list|legend=1| 16, 21, 37 }} | {{Val list|legend=1| 16, 21, 37 }} | ||
=== 2.5.7.11.13 === | |||
Subgroup: 2.5.7.11 | Subgroup: 2.5.7.11 | ||
| Line 48: | Line 48: | ||
{{Val list|legend=1| 16, 21, 37 }} | {{Val list|legend=1| 16, 21, 37 }} | ||
=== 2.5.7.11.13.17 === | |||
Subgroup: 2.5.7.11 | Subgroup: 2.5.7.11 | ||
| Line 63: | Line 63: | ||
{{Val list|legend=1| 16, 21, 37 }} | {{Val list|legend=1| 16, 21, 37 }} | ||
== Didacus == | |||
{{See also| Hemimean clan #Didacus }} | {{See also| Hemimean clan #Didacus }} | ||
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[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents | [[Tp tuning #T2 tuning|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]]. | Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]]. | ||
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[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents | [[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents | ||
== Mercy == | |||
{{See also| Quince clan #Mercy }} | {{See also| Quince clan #Mercy }} | ||
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{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }} | {{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }} | ||
=== 2.5.7.13 === | |||
Subgroup: 2.5.7.13 | Subgroup: 2.5.7.13 | ||
| Line 138: | Line 137: | ||
{{Val list|legend=1| 10, 21, 31}} | {{Val list|legend=1| 10, 21, 31}} | ||
=== 2.5.7.13.17 === | |||
Subgroup: 2.5.7.13.17 | Subgroup: 2.5.7.13.17 | ||
| Line 153: | Line 152: | ||
{{Val list|legend=1| 10, 21, 31}} | {{Val list|legend=1| 10, 21, 31}} | ||
=== 2.5.7.13.17.19 === | |||
Subgroup: 2.5.7.13.17.19 | Subgroup: 2.5.7.13.17.19 | ||
| Line 172: | Line 171: | ||
[[Category:Subgroup temperaments]] | [[Category:Subgroup temperaments]] | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
{{Todo| cleanup | review }} | |||
Revision as of 15:21, 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