No-fives subgroup temperaments: Difference between revisions

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Created page with "This is a collection of subgroup temperaments which omit the prime harmonic of 5. == Semaphore == Subgroup: 2.3.7 Comma: 49/48 [[Gencom]..."
 
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Line 14: Line 14:
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385


{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
{{Val list|legend=1| 5, 14, 19, 24, 67d, 91d }}


[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.523 cents


== Bleu ==
== Bleu ==
Line 31: Line 31:
[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
[[Tp tuning|POL2 generator]]: ~54/49 = 139.848


{{Val list| 9, 17, 43, 60c }}
{{Val list| 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
Line 48: Line 48:
[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
[[Tp tuning|POL2 generator]]: ~12/11 = 140.005


{{Val list| 9, 17, 43, 60c }}
{{Val list| 9, 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
Line 65: Line 65:
[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
[[Tp tuning|POL2 generator]]: ~12/11 = 139.990


{{Val list|legend=1| 17, 43, 60c }}
{{Val list|legend=1| 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
Line 84: Line 84:
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321


{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}
{{Val list|legend=1| 5, 12, 17, 22, 27, 137bd }}


[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
Line 101: Line 101:
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192


{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
{{Val list|legend=1| 5, 12, 17, 39d, 56d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
Line 118: Line 118:
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137


{{Val list|legend=1| 5e, 12e, 17, 22, 39c, 56c }}
{{Val list|legend=1| 5f, 12f, 17, 22, 39d, 56d }}


[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
Line 135: Line 135:
[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
[[Tp tuning|POL2 generator]]: ~11/9 = 353.958


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
{{Val list|legend=1| 7, 10, 17, 44e, 61de, 78de }}


[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
Line 152: Line 152:
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
{{Val list|legend=1| 7, 10, 17, 44e, 61de, 78de }}


[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents


== Skwares ==
== Skwares ==
Related temperament: [[Meantone family #Squares|squares]]
Subgroup: 2.3.7
Subgroup: 2.3.7


Line 171: Line 169:
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365


{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}
{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bd, 347bd }}


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
Related temperament: [[Meantone family #Squares|squares]]


=== 2.3.7.11 ===
=== 2.3.7.11 ===
Line 188: Line 188:
[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
[[Tp tuning|POL2 generator]]: ~9/7 = 425.244


{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bde }}


[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents


=== 2.3.7.11.13 ===
==== 2.3.7.11.13 ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 205: Line 205:
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457


{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}
{{Val list|legend=1| 17, 48f, 65ef, 82d, 147df }}


[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents


=== Skwairs ===
==== Skwairs ====
Subgroup: 2.3.7.11.13
Subgroup: 2.3.7.11.13


Line 239: Line 239:
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544


{{Val list|legend=1| 12, 19, 31, 112b, 143b, 174b}}
{{Val list|legend=1| 12, 19, 31, 112b, 143b, 174b }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
 
Related temperament: [[Meantone family #Septimal meantone|meantone]]


== Leapfrog ==
== Leapfrog ==
Line 259: Line 263:


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
Related temperaments: [[Hemifamity temperaments #Leapday|leapday]], [[Sensamagic clan #Leapweek|leapweek]]


=== 2.3.7.11 ===
=== 2.3.7.11 ===
Line 353: Line 359:
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents


=== 2.3.7.13.17 ===
==== 2.3.7.13.17 ====
Subgroup: 2.3.7.13.17
Subgroup: 2.3.7.13.17


Line 369: Line 375:


== Hemif ==
== Hemif ==
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo
Subgroup: 2.3.7
Subgroup: 2.3.7


Line 386: Line 390:


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
Related temperaments: [[Breedsmic temperaments #Hemififths|hemififths]], namo


=== 2.3.7.11 ===
=== 2.3.7.11 ===
Line 400: Line 406:
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535


{{Val list|legend=1| 7, 17, 41, 58, 99d }}
{{Val list|legend=1| 7, 17, 41, 58, 99e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
Line 495: Line 501:
[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents


[[neutraltet7|Seven note albitonic scale]]
; Scales
 
* [[neutraltet7|Seven note albitonic scale]]
[[neutraltet10|Ten note chromatic scale]]
* [[neutraltet10|Ten note chromatic scale]]
 
* [[neutraltet17|Seventeen note mega chromatic scale]]
[[neutraltet17|Seventeen note mega chromatic scale]]


=== Namo ===
=== Namo ===

Revision as of 11:37, 22 January 2022

This is a collection of subgroup temperaments which omit the prime harmonic of 5.

Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

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

Sval mapping: [⟨1 2 3], ⟨0 -2 -1]]

POL2 generator: ~7/6 = 250.385

Template:Val list

RMS error: 2.523 cents

Bleu

Subgroup: 2.3.7

Comma: 17496/16807

Gencom: [2 54/49; 17496/16807]

Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]

Sval mapping: [⟨1 1 2], ⟨0 5 7]]

POL2 generator: ~54/49 = 139.848

Template:Val list

RMS error: 1.917 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

POL2 generator: ~12/11 = 140.005

Template:Val list

RMS error: 1.829 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 12/11; 78/77 99/98 144/143]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

POL2 generator: ~12/11 = 139.990

Template:Val list

RMS error: 1.752 cents

Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

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

Sval mapping: [⟨1 2 2], ⟨0 -1 2]]

POL2 generator: ~3/2 = 709.321

Template:Val list

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

Template:Val list

RMS error: 1.977 cents

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Gencom: [2 3/2; 64/63 78/77 99/98]

Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

POL2 generator: ~3/2 = 706.137

Template:Val list

RMS error: 2.095 cents

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Gencom: [2 11/9; 64/63 243/242]

Gencom mapping: [⟨1 1 0 4 2], ⟨0 2 0 -4 5]]

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

POL2 generator: ~11/9 = 353.958

Template:Val list

RMS error: 1.968 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 2 0 -4 5 -1]]

Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]

POL2 generator: ~11/9 = 353.775

Template:Val list

RMS error: 1.953 cents

Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]

Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]

POL2 generator: ~9/7 = 425.365

Template:Val list

RMS error: 1.149 cents

Related temperament: squares

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]

Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]

POL2 generator: ~9/7 = 425.244

Template:Val list

RMS error: 1.099 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [⟨1 3 0 6 7 9], ⟨0 -4 0 -9 -10 -15]]

Sval mapping: [⟨1 3 6 7 9], ⟨0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

Template:Val list

RMS error: 1.769 cents

Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Gencom: [2 9/7; 99/98, 144/143, 243/242]

Gencom mapping: [⟨1 3 0 6 7 3], ⟨0 -4 0 -9 -10 2]]

Sval mapping: [⟨1 3 6 7 3], ⟨0 -4 -9 -10 2]]

POL2 generator: ~9/7 = 424.702

Template:Val list

RMS error: 1.290 cents

Harrison

Subgroup: 2.3.7

Comma: 59049/57344

Gencom: [2 3/2; 59049/57344]

Gencom mapping: [⟨1 1 0 -3], ⟨0 1 0 10]]

Sval mapping: [⟨1 1 -3], ⟨0 1 10]]

POL2 generator: ~3/2 = 696.544

Template:Val list

RMS error: 1.226 cents

Related temperament: meantone

Leapfrog

See also: Gentle region

Subgroup: 2.3.7

Comma list: 14680064/14348907

Gencom: [2 3/2; 14680064/14348907]

Gencom mapping: [⟨1 1 0 -6], ⟨0 1 0 15]]

Sval mapping: [⟨1 0 -21], ⟨0 1 15]]

POL2 generator: ~3/2 = 704.721 cents

Template:Val list

RMS error: 0.6202 cents

Related temperaments: leapday, leapweek

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Gencom: [2 3/2; 896/891 1331/1323]

Gencom mapping: [⟨1 1 0 -6 -3], ⟨0 1 0 15 11]]

Sval mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]

POL2 generator: ~3/2 = 704.753 cents

Template:Val list

RMS error: 0.6047 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]

Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

Template:Val list

RMS error: 0.7541 cents

Music

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]

Sval mapping: [⟨1 0 -3], ⟨0 3 11]]

POL2 generator: ~81/56 = 633.525

Template:Val list

RMS error: 0.3519 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

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

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

POL2 generator: ~8/7 = 233.688

Template:Val list

RMS error: 0.3202 cents

Baladic

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Gencom: [91/64 8/7; 169/168 1029/1024]

Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]

POL2 generator: ~8/7 = 233.6044

Template:Val list

RMS error: 0.5452 cents

2.3.7.13.17

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Gencom: [17/12 8/7; 169/168 273/272 289/288]

Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

POL2 generator: ~8/7 = 233.6155

Template:Val list

RMS error: 0.5073 cents

Hemif

Subgroup: 2.3.7

Comma: 1605632/1594323

Gencom: [2 2187/1792; 1605632/1594323]

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

Sval mapping: [⟨1 1 -1], ⟨0 2 13]]

POL2 generator: ~2187/1792 = 351.485

Template:Val list

RMS error: 0.2344 cents

Related temperaments: hemififths, namo

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

POL2 generator: ~11/9 = 351.535

Template:Val list

RMS error: 0.6108 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

Template:Val list

RMS error: 0.7167 cents

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]

Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

Template:Val list

RMS error: 0.0383 cents

Parapyth (rank 3)

See also: Pentacircle temperaments § Parapyth

Subgroup: 2.3.7.11

Comma list: 896/891

Gencom: [2 3/2 28/27; 896/891]

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

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

POL2 tuning: ~3 = 1903.834, ~7 = 3369.872

Template:Val list

RMS error: 0.4149 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

Gencom mapping: [⟨1 1 0 1 4 6], ⟨0 1 0 3 -1 -4], ⟨0 0 0 1 1 1]]

Sval mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

Template:Val list

RMS error: 0.3789 cents

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Subgroup: 2.3.11

Comma: 243/242

Gencom: [2 11/9; 243/242]

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

Sval mapping: [⟨1 1 2], ⟨0 2 5]]

POL2 generator: ~11/9 = 350.525

Template:Val list

RMS error: 0.3021 cents

Scales

Namo

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Gencom: [2 11/9; 144/143 243/242]

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]

Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]

POL2 generator: ~11/9 = 351.488

Template:Val list

RMS error: 0.7038 cents

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