No-threes subgroup temperaments: Difference between revisions

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== Llywelyn ==
== Llywelyn ==
Subgroup: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma]]: 4194304/4117715
[[Comma list]]: 4194304/4117715
 
[[Sval]] [[mapping]]: [{{val| 1 1 3 }}, {{val| 0 7 -1 }}]


[[Gencom]]: [2 8/7; 4194304/4117715]
[[Gencom]]: [2 8/7; 4194304/4117715]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 1 3 }}, {{val| 0 0 7 -1 }}]


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
Optimal tuning ([[POTE]]): ~8/7 = 226.910
 
[[Tp tuning|POL2 generator]]: ~8/7 = 226.910


{{Val list|legend=1| 16, 37 }}
{{Val list|legend=1| 16, 37 }}
Line 19: Line 19:


=== 2.5.7.11 ===
=== 2.5.7.11 ===
Subgroup: 2.5.7.11
[[Subgroup]]: 2.5.7.11
 
[[Comma list]]: 176/175, 1310720/1294139


[[Comma]]: 176/175, 1310720/1294139
[[Sval]] [[mapping]]: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]


[[Gencom]]: [2 8/7; 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}}]
[[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
Optimal tuning ([[POTE]]): ~8/7 = 227.114


{{Val list|legend=1| 16, 21, 37 }}
{{Val list|legend=1| 16, 21, 37 }}
Line 36: Line 36:
Subgroup: 2.5.7.11.13
Subgroup: 2.5.7.11.13


[[Comma]]: 176/175, 640/637, 1304576/1294139
[[Comma list]]: 176/175, 640/637, 1304576/1294139
 
[[Sval]] [[mapping]]: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]


[[Gencom]]: [2 8/7; 176/175 640/637, 1304576/1294139]
[[Gencom]]: [2 8/7; 176/175 640/637, 1304576/1294139]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2}}, {{val|0 0 7 -1 13 9}}]
[[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}}]
Optimal tuning ([[POTE]]): ~8/7 = 227.108
 
[[Tp tuning|POL2 generator]]: ~8/7 = 227.108


{{Val list|legend=1| 16, 21, 37 }}
{{Val list|legend=1| 16, 21, 37 }}
Line 51: Line 51:
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


[[Comma]]: 176/175, 221/200, 640/637, 833/832
[[Comma list]]: 176/175, 221/200, 640/637, 833/832
 
[[Sval]] [[mapping]]: [{{val| 1 1 3 1 2 2 }}, {{val| 0 7 -1 13 9 11 }}]


[[Gencom]]: [2 8/7; 176/175 221/200, 640/637, 833/832]
[[Gencom]]: [2 8/7; 176/175 221/200, 640/637, 833/832]


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3 1 2 2}}, {{val|0 0 7 -1 13 9 11}}]
[[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}}]
Optimal tuning ([[POTE]]): ~8/7 = 227.242
 
[[Tp tuning|POL2 generator]]: ~8/7 = 227.242


{{Val list|legend=1| 16, 21, 37 }}
{{Val list|legend=1| 16, 21, 37 }}
Line 68: Line 68:
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]


Subgroup: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma]]: 3136/3125
[[Comma list]]: 3136/3125
 
[[Sval]] [[mapping]]: [{{val| 1 2 2 }}, {{val| 0 2 5 }}]


[[Gencom]]: [2 28/25; 3136/3125]
[[Gencom]]: [2 28/25; 3136/3125]


[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 2 2 }}, {{val| 0 0 2 5 }}]


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
Optimal tuning ([[POTE]]): ~28/25 = 93.772
 
[[Tp tuning|POL2 generator]]: ~28/25 = 93.772


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
Line 87: Line 87:
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]].


Subgroup: 2.5.7
[[Subgroup]]: 2.5.7
 
[[Comma list]]: [[2100875/2097152]]


[[Comma]]: [[2100875/2097152]]
[[Sval]] [[mapping]]: [{{val| 1 2 3 }}, {{val| 0 5 -3 }}]


[[Gencom]]: [2 256/245; 2100875/2097152]
[[Gencom]]: [2 256/245; 2100875/2097152]


[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 2 3 }}, {{val| 0 0 5 -3 }}]
 
[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
Optimal tuning ([[POTE]]): ~256/245 = 77.205


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
Line 108: Line 108:
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]].
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
[[Subgroup]]: 2.5.7


[[Comma list]]: 823543/819200
[[Comma list]]: 823543/819200
[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]


[[Gencom]]: [2 2744/2560; 823543/819200]
[[Gencom]]: [2 2744/2560; 823543/819200]


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
 
[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
Optimal tuning ([[POTE]]): ~343/320 = 116.291


{{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 ===
=== 2.5.7.13 ===
Subgroup: 2.5.7.13
[[Subgroup]]: 2.5.7.13


[[Comma list]]: 343/338, 640/637
[[Comma list]]: 343/338, 640/637
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 }}, {{val| 0 -7 -2 -3 }}]


[[Gencom]]: [2 14/13; 343/338 640/637]
[[Gencom]]: [2 14/13; 343/338 640/637]


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 }}, {{val| 0 0 -7 -2 -3 }}]


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
Optimal tuning ([[POTE]]): ~14/13 = 116.094
 
[[Tp tuning|POL2 generator]]: ~14/13 = 116.094


{{Val list|legend=1| 10, 21, 31}}
{{Val list|legend=1| 10, 21, 31}}


=== 2.5.7.13.17 ===
=== 2.5.7.13.17 ===
Subgroup: 2.5.7.13.17
[[Subgroup]]: 2.5.7.13.17


[[Comma list]]: 170/169, 224/221, 640/637
[[Comma list]]: 170/169, 224/221, 640/637
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 }}, {{val| 0 -7 -2 -3 1 }}]


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
[[Gencom]]: [2 14/13; 170/169 224/221 640/637]


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 }}, {{val| 0 0 -7 -2 -3 1 }}]


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
Optimal tuning ([[POTE]]): ~14/13 = 115.769
 
[[Tp tuning|POL2 generator]]: ~14/13 = 115.769


{{Val list|legend=1| 10, 21, 31}}
{{Val list|legend=1| 10, 21, 31}}


=== 2.5.7.13.17.19 ===
=== 2.5.7.13.17.19 ===
Subgroup: 2.5.7.13.17.19
[[Subgroup]]: 2.5.7.13.17.19


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Sval]] [[mapping]]: [{{val| 1 3 3 4 4 3 }}, {{val| 0 -7 -2 -3 1 13 }}]


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
[[Gencom]] [[mapping]]: [{{val| 1 0 3 3 4 4 3 }}, {{val| 0 0 -7 -2 -3 1 13 }}]


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
Optimal tuning ([[POTE]]): ~14/13 = 115.716


{{Val list|legend=1| 10, 21, 31, 52f}}
{{Val list|legend=1| 10, 21, 31, 52f }}


== Xeimty ==
== Xeimty (rank 3) ==
Subgroup: 2.5.7.11
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 625/616
[[Comma list]]: 625/616


TE tuned generators: ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653
Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653


{{Val list|legend=1| 13, 16, 22, 28, 35, 41, 47, 57, 63, 98c }}
{{Val list|legend=1| 13, 16, 22, 28, 35, 41, 47, 57, 63, 98c }}


=== Xeimtionic ===
== Xeimtionic ==
Subgroup: 2.5.7.11
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 245/242, 625/616
[[Comma list]]: 245/242, 625/616


TE tuned generators: ~2/1 = 1200.6817, ~28/25 = 205.0745
Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.6817, ~28/25 = 205.0745


{{Val list|legend=1| 29, 35, 41, 47, 88e}}
{{Val list|legend=1| 29, 35, 41, 47, 88e }}


== Yer (rank 3) ==
== Yer (rank 3) ==
Subgroup: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048
[[Comma list]]: 209/208, 2057/2048


[[Mapping|Sval mapping]]: [{{val|1 0 0 11 4}}, {{val|0 1 0 -2 -1}}, {{val|0 0 1 0 1}}]
[[Sval]] [[mapping]]: [{{val| 1 0 0 11 4 }}, {{val| 0 1 0 -2 -1 }}, {{val| 0 0 1 0 1 }}]


TE tuned generators: ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638
Optimal tuning ([[TE tuning|TE]]): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638


{{Val list|legend=1| 13, 24, 33, 37, 46, 57, 70, 127 }}
{{Val list|legend=1| 13, 24, 33, 37, 46, 57, 70, 127 }}


== Yamablu ==
== 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 [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].
Yamablu, with a generator of ~17/13, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


Subgroup: 2.11.13.17.19
[[Subgroup]]: 2.11.13.17.19


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Comma list]]: 209/208, 2057/2048, 83521/83486


[[Mapping|Sval mapping]]: [{{val|1 5 1 1 0}}, {{val|0 -4 7 8 11}}]
[[Sval]] [[mapping]]: [{{val| 1 5 1 1 0 }}, {{val| 0 -4 7 8 11 }}]


[[Tp tuning|POL2 generator]]: ~17/13 = 462.9606
Optimal tuning ([[POTE]]): ~17/13 = 462.9606


{{Val list|legend=1| 13, 44, 57, 70}}
{{Val list|legend=1| 13, 44, 57, 70}}

Revision as of 16:56, 12 August 2022

This is a collection of subgroup temperaments which omit the prime harmonic of 3.

Llywelyn

Subgroup: 2.5.7

Comma list: 4194304/4117715

Sval mapping: [⟨1 1 3], ⟨0 7 -1]]

Gencom: [2 8/7; 4194304/4117715]

Gencom mapping: [⟨1 0 1 3], ⟨0 0 7 -1]]

Optimal tuning (POTE): ~8/7 = 226.910

Template:Val list

RMS error: 0.5391 cents

2.5.7.11

Subgroup: 2.5.7.11

Comma list: 176/175, 1310720/1294139

Sval mapping: [⟨1 1 3 1], ⟨0 7 -1 13]]

Gencom: [2 8/7; 176/175 1310720/1294139]

Gencom mapping: [⟨1 0 1 3 1], ⟨0 0 7 -1 13]]

Optimal tuning (POTE): ~8/7 = 227.114

Template:Val list

2.5.7.11.13

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 1304576/1294139

Sval mapping: [⟨1 1 3 1 2], ⟨0 7 -1 13 9]]

Gencom: [2 8/7; 176/175 640/637, 1304576/1294139]

Gencom mapping: [⟨1 0 1 3 1 2], ⟨0 0 7 -1 13 9]]

Optimal tuning (POTE): ~8/7 = 227.108

Template:Val list

2.5.7.11.13.17

Subgroup: 2.5.7.11.13.17

Comma list: 176/175, 221/200, 640/637, 833/832

Sval mapping: [⟨1 1 3 1 2 2], ⟨0 7 -1 13 9 11]]

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]]

Optimal tuning (POTE): ~8/7 = 227.242

Template:Val list

Didacus

See also: Hemimean clan § Didacus

Related temperaments: roulette, hemithirds

Subgroup: 2.5.7

Comma list: 3136/3125

Sval mapping: [⟨1 2 2], ⟨0 2 5]]

Gencom: [2 28/25; 3136/3125]

Gencom mapping: [⟨1 0 2 2], ⟨0 0 2 5]]

Optimal tuning (POTE): ~28/25 = 93.772

Template:Val list

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

Comma list: 2100875/2097152

Sval mapping: [⟨1 2 3], ⟨0 5 -3]]

Gencom: [2 256/245; 2100875/2097152]

Gencom mapping: [⟨1 0 2 3], ⟨0 0 5 -3]]

Optimal tuning (POTE): ~256/245 = 77.205

Template:Val list

RMS error: 0.0586 cents

Mercy

See also: Quince clan § 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

Sval mapping: [⟨1 3 3], ⟨0 -7 -2]]

Gencom: [2 2744/2560; 823543/819200]

Gencom mapping: [⟨1 0 3 3], ⟨0 0 -7 -2]]

Optimal tuning (POTE): ~343/320 = 116.291

Template:Val list

2.5.7.13

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

Sval mapping: [⟨1 3 3 4], ⟨0 -7 -2 -3]]

Gencom: [2 14/13; 343/338 640/637]

Gencom mapping: [⟨1 0 3 3 4], ⟨0 0 -7 -2 -3]]

Optimal tuning (POTE): ~14/13 = 116.094

Template:Val list

2.5.7.13.17

Subgroup: 2.5.7.13.17

Comma list: 170/169, 224/221, 640/637

Sval mapping: [⟨1 3 3 4 4], ⟨0 -7 -2 -3 1]]

Gencom: [2 14/13; 170/169 224/221 640/637]

Gencom mapping: [⟨1 0 3 3 4 4], ⟨0 0 -7 -2 -3 1]]

Optimal tuning (POTE): ~14/13 = 115.769

Template:Val list

2.5.7.13.17.19

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 343/338, 640/637, 16384/16055

Sval mapping: [⟨1 3 3 4 4 3], ⟨0 -7 -2 -3 1 13]]

Gencom mapping: [⟨1 0 3 3 4 4 3], ⟨0 0 -7 -2 -3 1 13]]

Gencom: [2 14/13; 170/169 343/338 640/637 16384/16055]

Optimal tuning (POTE): ~14/13 = 115.716

Template:Val list

Xeimty (rank 3)

Subgroup: 2.5.7.11

Comma list: 625/616

Optimal tuning (TE): ~2/1 = 1200.6544, ~5/4 = 380.3004, ~11/8 = 551.9653

Template:Val list

Xeimtionic

Subgroup: 2.5.7.11

Comma list: 245/242, 625/616

Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745

Template:Val list

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]]

Optimal tuning (TE): ~2/1 = 1200.4457, ~11/8 = 548.4934, ~16/13 = 358.638

Template:Val list

Yamablu

Yamablu, with a generator of ~17/13, is named for its 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]]

Optimal tuning (POTE): ~17/13 = 462.9606

Template:Val list

RMS error: 0.4898 cents

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