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Subgroup temperaments: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}


{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}


{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}
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[[Optimal tuning]]:
[[Optimal tuning]]:
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}


{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}


{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}


{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
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[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
=== Seventeen-cot ===
Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026.
[[Subgroup]]: 2.3.11/5.13/5
[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}}
: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}}
: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }}
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161* }}
[[Badness]] (Sintel): 0.080


=== Blackweed ===
=== Blackweed ===
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
: error map: {{val| 0.000 +18.045 +2.475 }}
: error map: {{val| 0.000 +18.045 +2.475 }}


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}}


{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}}
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}}


{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}
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; Music
; Music
* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9]
* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9]
=== Seventeen-cot ===
Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.
====2.3.13/5 subgroup====
Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
edo join: 29 & 146
{{Mapping|legend=3| 1 1 1 | 0 17 11 }}
: mapping generators: ~2, ~2250/2197
Optimal tunings:
* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}
edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262
Badness (Sintel): 0.064
====2.3.11/5.13/5 subgroup====
Comma basis: 225000/224939, 43940/43923
edo join: 29 & 146
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165
Optimal tunings:
* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}
edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
Badness (Sintel): 0.080


=== Fiventeen ===
=== Fiventeen ===
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[[Subgroup]]: 2.3.19/7
[[Subgroup]]: 2.3.19/7


[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})
[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})


{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
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