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No-fives subgroup temperaments: Difference between revisions

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Line 763: Line 763:
Subgroup: 2.3.7.13
Subgroup: 2.3.7.13


Comma list: {{monzo|-42  -2 -5 16}}, {{monzo|10 -46 29 -5}}
Comma list: {{monzo|-42 -2 -5 16}}, {{monzo|10 -46 29 -5}}


Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Mapping: [{{val|1 313 532 208}}, {{val|0 -439 -746 -288}}]
Line 774: Line 774:
Subgroup: 2.3.7.13.17
Subgroup: 2.3.7.13.17


Comma list: 39337984/39328497, {{monzo|0 -14  7  4 -3}}, {{monzo|-18 -24 14 -1  5}}
Comma list: 39337984/39328497, {{monzo|0 -14 7 4 -3}}, {{monzo|-18 -24 14 -1 5}}


Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Mapping: [{{val|1 313 532 208 58}}, {{val|0 -439 -746 -288 -76}}]
Line 847: Line 847:


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
== Paralimmic ==
[[Subgroup]]: 2.3.11
[[Comma list]]: [[4096/3993]]
{{Mapping|legend=2|1 0 4|0 3 -1}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
{{Optimal ET sequence|legend=1|11b, 13, 15, 17 }}
[[Tp tuning #T2 tuning|RMS error]]: 1.237 cents


== Neutral ==
== Neutral ==
Line 908: Line 921:
[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3066 cents


== Reversed meantone ==
== Aerophore ==
{{Main| Reversed meantone }}
[[Subgroup]]: 2.3.11.19
 
[[Comma list]]: 363/361, 729/704
 
[[Mapping]]: [{{val|1 0 -6 -6}}, {{val|0 2 12 13}}]
 
[[POTE generator]]: ~[[19/11]] = 945.4
 
{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}
 
=== Semaerophore ===
[[Subgroup]]: 2.3.7.11.19
 
[[Comma list]]: 49/48, 77/76, 729/704
 
[[Mapping]]: [{{val|1 0 2 -6 -6}}, {{val|0 2 1 12 13}}]
 
[[POTE generator]]: ~[[7/4]] = 944.667
 
{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
 
== Superflat aka tridecimal ==
Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.
 
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[1053/1024]]
 
{{Mapping|legend=2|1 1 6|0 1 -4}}
 
[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
 
{{Optimal ET sequence|legend=1|5f, 7, 12, 19, 45f, 64f, 147bfff }}
 
[[Tp tuning #T2 tuning|RMS error]]: 1.591 cents
 
=== 2.3.11.13 ===
[[Subgroup]]: 2.3.11.13
 
[[Comma list]]: [[144/143]], [[729/704]]
 
[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
 
{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
 
== Ultraflat ==
Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperamenets intersect in [[7edo]], where major sixths and minor sixths are not distinguished.
 
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[27/26]]
 
{{Mapping|legend=2|1 1 2|0 1 3}}
 
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
 
{{Optimal ET sequence|legend=1| 5, 7 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 4.367 cents
 
== Threedic ==
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[2197/2187]]
 
{{Mapping|legend=2|1 0 0|0 3 7}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
 
{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2054 cents
 
== Semitonismic ==
Subgroup: 2.3.17
 
[[Comma list]]: [[289/288]]
 
[[Gencom]]: [17/12 17/16; 289/288]
 
[[Mapping|Sval mapping]]: [{{val|2 3 8}}, {{val|0 1 1}}]
 
[[Tp tuning|POL2 generator]]: ~17/16 = 101.784
 
{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
 
== Dog ==
The dog temperament is based by [[2L 5s]] or [[7L 2s]] scale that makes [[81/76]] vanish, so [[19/16]] is a major third. It can be viewed as a 2.3.19 subgroup analogue of [[Pelogic family|mavila]].


Subgroup: 2.3.41
Subgroup: 2.3.19


[[Comma list]]: 82/81
[[Comma list]]: 81/76


[[Gencom]]: [2 4/3; 82/81]
[[Gencom]]: [2 4/3; 81/76]


[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]
[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]


[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
[[Tp tuning|POL2 generator]]: ~4/3 = 521.403


{{Optimal ET sequence|legend=1| 5, 12, 17 }}
{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}


=== 2.3.7.41 subgroup ===
[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
Subgroup: 2.3.7.41


[[Comma list]]: 64/63, 82/81
== Boethian ==
Boethian is a diatonic-based temperament that makes [[513/512]] vanish, so [[19/16]] is a minor third. It can be viewed as a 2.3.19 subgroup analogue of [[Schismatic family|schismic temperament]].


[[Gencom]]: [2 4/3; 64/63 82/81]
[[Subgroup]]: 2.3.19


[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
[[Comma list]]: [[513/512]]


[[POTE generator]]: ~4/3 = 490.0323
{{Mapping|legend=1| 1 0 9 | 0 1 -3 }}


[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
: Mapping generators: ~2, ~3


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}
[[Optimal tuning]] (CTE): ~2 = 1\1, ~3/2 = 701.3288


=== 2.3.7.11.41 subgroup ===
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
Subgroup: 2.3.7.11.41


[[Comma list]]: 64/63, 82/81, 99/98
[[Badness]]: 0.000374


[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
== Porpoise ==
[[Subgroup]]: 2.3.29


[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]
[[Comma list]]: 24576/24389


[[POTE generator]]: ~4/3 = 492.1787
[[Mapping]]: [{{val|1 2 5}}, {{val|0 3 -1}}]


[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
[[CTE tuning|CTE generator]]: ~32/29 = 166.067


{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}


== Sematology ==
== Sematology ==
Line 1,022: Line 1,124:
{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}
{{Optimal ET sequence|legend=1| 5l, 7l, 9, 16l }}


== Aerophore ==
== Reversed meantone ==
[[Subgroup]]: 2.3.11.19
{{Main| Reversed meantone }}


[[Comma list]]: 363/361, 729/704
Subgroup: 2.3.41


[[Mapping]]: [{{val|1 0 -6 -6}}, {{val|0 2 12 13}}]
[[Comma list]]: 82/81


[[POTE generator]]: ~[[19/11]] = 945.4
[[Gencom]]: [2 4/3; 82/81]


{{Optimal ET sequence|legend=1| 9eehh, 14, 19, 33 }}
[[Mapping|Sval mapping]]: [{{val|1 2 7}}, {{val|0 -1 -4}}]


=== Semaerophore ===
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
[[Subgroup]]: 2.3.7.11.19


[[Comma list]]: 49/48, 77/76, 729/704
{{Optimal ET sequence|legend=1| 5, 12, 17 }}


[[Mapping]]: [{{val|1 0 2 -6 -6}}, {{val|0 2 1 12 13}}]
=== 2.3.7.41 subgroup ===
Subgroup: 2.3.7.41


[[POTE generator]]: ~[[7/4]] = 944.667
[[Comma list]]: 64/63, 82/81


{{Optimal ET sequence|legend=1| 9eehh, 14, 33d, 47deh }}
[[Gencom]]: [2 4/3; 64/63 82/81]


== Superflat aka tridecimal ==
[[Mapping|Sval mapping]]: [{{val| 1 2 2 7 }}, {{val| 0 -1 2 -4 }}]
Superflat temperament, or alternatively, tridecimal temperament, is a diatonic-based temperament that makes [[1053/1024]] vanish, so [[13/8]] is a minor sixth, and [[16/13]] is a major third. The more accurate tunings for this temperament are generated by a fifth at least as flat as those of [[flattone]], although often even flatter (such as [[40edo]]'s fifth). Superflat can be viewed as a 2.3.13 subgroup analogue of [[meantone]] and [[archy]]. Superflat diatonic scales have a character somewhere between neutral third scales (or [[3L 4s|mosh]]) and meantone diatonic scales.


[[Subgroup]]: 2.3.13
[[POTE generator]]: ~4/3 = 490.0323


[[Comma list]]: [[1053/1024]]
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029


{{Mapping|legend=2|1 1 6|0 1 -4}}
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.939
=== 2.3.7.11.41 subgroup ===
Subgroup: 2.3.7.11.41


{{Optimal ET sequence|legend=1|5f, 7, 12, 19, 45f, 64f, 147bfff }}
[[Comma list]]: 64/63, 82/81, 99/98


=== 2.3.11.13 ===
[[Gencom]]: [2 4/3; 64/63 82/81 99/98]
[[Subgroup]]: 2.3.11.13


[[Comma list]]: [[144/143]], [[729/704]]
[[Mapping|Sval mapping]]: [{{val| 1 2 2 1 7 }}, {{val| 0 -1 2 6 -4 }}]


[[Optimal tuning]] (CTE): ~2 = 1\1, ~[[3/2]] = 692.247
[[POTE generator]]: ~4/3 = 492.1787


{{Optimal ET sequence|legend=1|7, 19, 26, 59b }}
[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454


== Ultraflat ==
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}
Ultraflat is the much more inaccurate cousin of superflat, with even flatter fifths. [[27/26]] is tempered out rather than [[1053/1024]], so [[13/8]] is a major sixth. These temperamenets intersect in [[7edo]], where major sixths and minor sixths are not distinguished.
 
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[27/26]]
 
{{Mapping|legend=2|1 1 2|0 1 3}}
 
[[Optimal tuning]] (CTE): ~2 = 1/1, ~[[3/2]] = 688.391
 
{{Optimal ET sequence|legend=1| 5, 7 }}
 
== Boethian ==
[[Subgroup]]: 2.3.19
 
[[Comma list]]: [[513/512]]
 
{{Mapping|legend=1| 1 0 9 | 0 1 - 3 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]] (CTE): ~2 = 1\1, ~3/2 = 701.3288
 
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
 
[[Badness]]: 0.000374
 
== Porpoise ==
[[Subgroup]]: 2.3.29
 
[[Comma list]]: 24576/24389
 
[[Mapping]]: [{{val|1 2 5}}, {{val|0 3 -1}}]
 
[[CTE tuning|CTE generator]]: ~32/29 = 166.067
 
{{Optimal ET sequence|legend=1| 7, 22, 29, 94, 123, 152j, 275jj, 427jjj }}
 
== Paralimmic ==
[[Subgroup]]: 2.3.11
 
[[Comma list]]: [[4096/3993]]
 
{{Mapping|legend=2|1 0 4|0 3 -1}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[16/11]] = 634.320
 
{{Optimal ET sequence|legend=1|11b, 13, 15, 17 }}
 
== Threedic ==
[[Subgroup]]: 2.3.13
 
[[Comma list]]: [[2197/2187]]
 
{{Mapping|legend=2|1 0 0|0 3 7}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1/1, ~[[13/9]] = 634.173
 
{{Optimal ET sequence|legend=1|11bff, 13f, 15, 17, 36, 53, 70, 123, 193, 316, 755f }}


[[Category:Temperament collections]]
[[Category:Temperament collections]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"