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 ==
== Llywelyn aka shoe ==
{{See also| Chromatic pairs #Shoe }}
{{See also| Llywelyn clan #Llywelyn aka shoe }}
 
[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7


[[Comma list]]: 4194304/4117715
[[Comma list]]: 4194304/4117715


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


[[Gencom]]: [2 8/7; 4194304/4117715]
Mapping generators: 2, ~8/7


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


Optimal tuning ([[POTE]]): ~8/7 = 226.910
[[Gencom]]: [2 8/7; 4194304/4117715]


{{Val list|legend=1| 16, 37 }}
[[Optimal tuning]] ([[POTE]]): ~8/7 = 226.910


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
{{Val list|legend=1| 5, 11c, 16, 21, 37 }}


=== 2.5.7.11 ===
=== 2.5.7.11 subgroup ===
[[Subgroup]]: 2.5.7.11
Subgroup: 2.5.7.11


[[Comma list]]: 176/175, 1310720/1294139
Comma list: 176/175, 1310720/1294139


[[Sval]] [[mapping]]: [{{val| 1 1 3 1 }}, {{val| 0 7 -1 13 }}]
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]] [[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 }}]


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


{{Val list|legend=1| 16, 21, 37 }}
Optimal GPV sequence: {{Val list| 16, 21, 37 }}


=== 2.5.7.11.13 ===
=== 2.5.7.11.13 subgroup ===
Subgroup: 2.5.7.11.13
Subgroup: 2.5.7.11.13


[[Comma list]]: 176/175, 640/637, 1304576/1294139
Comma list: 176/175, 640/637, 847/845


[[Sval]] [[mapping]]: [{{val| 1 1 3 1 2 }}, {{val| 0 7 -1 13 9 }}]
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]] [[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 }}]


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


{{Val list|legend=1| 16, 21, 37 }}
Optimal GPV sequence: {{Val list| 16, 21, 37 }}


=== 2.5.7.11.13.17 ===
=== 2.5.7.11.13.17 subgroup ===
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


[[Comma list]]: 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 }}]
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]] [[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 }}]


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


{{Val list|legend=1| 16, 21, 37 }}
Optimal GPV sequence: {{Val list| 16, 21, 37 }}


== Didacus ==
== Didacus ==
Line 212: Line 215:


== Ostara ==
== Ostara ==
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  
'''Ostara''' is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  


Ostara can also refer to a collection of temperaments which temper out 16807/16796.
Ostara can also refer to a collection of temperaments which temper out 16807/16796.


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


Mapping: [{{Val|1 1 20 -49}}, {{Val|0 3 -39 119}}]
[[Comma list]]: 8589934592/8544921875, 53710650917/53687091200


Comma list: 8589934592/8544921875, 53710650917/53687091200
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


POTE Generator: ~5120/3773 = 529.003¢  
[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢  


Vals: 93, 431, 338, 524
{{Val list|legend=1| 93, 431, 338, 524 }}


=== 2.5.7.11.13 ===
=== 2.5.7.11.13 subgroup ===
Subgroup: 2.5.7.11.13
Subgroup: 2.5.7.11.13


Mapping: [{{Val|1 1 20 -49 35}}, {{Val|0 3 -39 119 -71}}]
Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125


Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125
Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


POTE Generator: ~1664/1225 = 529.003¢  
Optimal tuning (POTE): ~1664/1225 = 529.003¢  


=== 2.5.7.11.13.17 ===
=== 2.5.7.11.13.17 subgroup ===
Subgroup: 2.5.7.11.13.17
Subgroup: 2.5.7.11.13.17


Mapping: [{{Val|1 1 20 -49 35 42}}, {{Val|0 3 -39 119 -71 -86}}]
Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


POTE Generator: ~1664/1225 = 529.003¢  
Optimal tuning (POTE): ~1664/1225 = 529.003¢  


=== 2.5.7.11.13.17.19 ===
=== 2.5.7.11.13.17.19 subgroup ===
Subgroup: 2.5.7.11.13.17.19
Subgroup: 2.5.7.11.13.17.19


Mapping: [{{Val|1 1 20 -49 35 42}}, {{Val|0 3 -39 119 -71 -86}}]
Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


POTE Generator: ~19/14 = 529.003¢
Optimal tuning (POTE): ~19/14 = 529.003¢
 
<!-- name conflict
=== Jacobin ===
The 2.5.11.13 subgroup primarily contains temperaments developed for 1789edo, since it tempers out the jacobin comma 6656/6655, for which 2.5.11.13 is the subgroup, and the year 1789 is hallmark for the French revolution.
 
Subgroup: 2.5.11.13
 
Comma list: 6656/6655, {{monzo| -119 -46 15 47 }}
 
Mapping: [{{val| 1 74 3 74 }}, {{val| 0 -156 1 -153 }}]
 
Optimal tuning (POTE): ~11/8 = 551.370
 
Optimal GPV sequence: 37, 1789
-->
== Estates general ==
Named so because it is defined as the 1789 & 3125 temperament due to 3125 providing optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined for the 2.5.11.13.19 subgroup.
 
[[Subgroup]]: 2.5.11.13
 
[[Comma list]]: 6656/6655, {{monzo| 314 -78 14 -49 }}
 
[[Mapping]]: [{{val| 1 118 -107 -212 }}, {{val| 0 -133 127 248 }}]
 
[[Optimal tuning]] ([[POTE]]): ~995765040283203125/544935554911830016 = 1043.712
 
=== 2.5.11.13.19 subgroup ===
{{Todo|inline=1|comment=Weak extension, must use a different name. }}


=== 2.7.13.17.19 ===
Subgroup: 2.5.11.13.19
Subgroup: 2.7.13.17.19


Comma list: 16807/16796, 157339/157216, 47071232/47045881
Comma list: 6656/6655, 40960000000/40943078891, {{monzo| -133 50 -7 18 -6 }}


POTE Generator: 529.009¢ ~19/14
Mapping: [{{val| 1 118 -107 -212 450 }}, {{val| 0 -266 254 496 -1025 }}]


Vals: {{EDOs|93, 338, 431, 524, 617, 710}}
Optimal tuning (POTE): ~2588443885831192576/1914932769775390625 = 521.856


[[Category:Temperament collections]]
[[Category:Temperament collections]]

Revision as of 16:25, 30 August 2022

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

Llywelyn aka shoe

See also: Chromatic pairs § Shoe
See also: Llywelyn clan § Llywelyn aka shoe

Subgroup: 2.5.7

Comma list: 4194304/4117715

Sval mapping: [⟨1 1 3], ⟨0 7 -1]]

Mapping generators: 2, ~8/7

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

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

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

Template:Val list

2.5.7.11 subgroup

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

Optimal GPV sequence: Template:Val list

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 847/845

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

Optimal GPV sequence: Template:Val list

2.5.7.11.13.17 subgroup

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

Optimal GPV sequence: 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

Pakkanen (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

Frostburn

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

Ostara

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.

Ostara can also refer to a collection of temperaments which temper out 16807/16796.

Subgroup: 2.5.7.11

Comma list: 8589934592/8544921875, 53710650917/53687091200

Mapping: [⟨1 1 20 -49], ⟨0 3 -39 119]]

Optimal tuning (POTE): ~5120/3773 = 529.003¢

Template:Val list

2.5.7.11.13 subgroup

Subgroup: 2.5.7.11.13

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

Mapping: [⟨1 1 20 -49 35], ⟨0 3 -39 119 -71]]

Optimal tuning (POTE): ~1664/1225 = 529.003¢

2.5.7.11.13.17 subgroup

Subgroup: 2.5.7.11.13.17

Mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

Optimal tuning (POTE): ~1664/1225 = 529.003¢

2.5.7.11.13.17.19 subgroup

Subgroup: 2.5.7.11.13.17.19

Mapping: [⟨1 1 20 -49 35 42], ⟨0 3 -39 119 -71 -86]]

Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875

Optimal tuning (POTE): ~19/14 = 529.003¢

Estates general

Named so because it is defined as the 1789 & 3125 temperament due to 3125 providing optimal patent val for the jacobin comma, 3125 is 5 to the 5th power, and Estates General were called by Louis XVI on 5th May 1789 (05/05). Defined for the 2.5.11.13.19 subgroup.

Subgroup: 2.5.11.13

Comma list: 6656/6655, [314 -78 14 -49⟩

Mapping: [⟨1 118 -107 -212], ⟨0 -133 127 248]]

Optimal tuning (POTE): ~995765040283203125/544935554911830016 = 1043.712

2.5.11.13.19 subgroup

Todo:

Weak extension, must use a different name.

Subgroup: 2.5.11.13.19

Comma list: 6656/6655, 40960000000/40943078891, [-133 50 -7 18 -6⟩

Mapping: [⟨1 118 -107 -212 450], ⟨0 -266 254 496 -1025]]

Optimal tuning (POTE): ~2588443885831192576/1914932769775390625 = 521.856

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