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No-fives subgroup temperaments: Difference between revisions

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* [[Shoal17]]
* [[Shoal17]]


Music:
; Music
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)
* [https://www.youtube.com/watch?v=bwGjN56iMM0 ''Moody Improvisation in the Shoal Temperament''] by [[Budjarn Lambeth]] (2025)


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{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3


: sval mapping generators: ~17/12, ~3
[[Optimal tuning]]s:  
 
* [[WE]]: ~17/12 = 600.1471{{c}}, ~3/2 = 701.9563{{c}} (~17/16 = 101.8091{{c}})
: [[gencom]]: [17/12 3; 289/288]
: [[error map]]: {{val| +0.294 +0.295 -1.969 }}
 
* [[CWE]]: ~17/12 = 600.0000{{c}}, ~3/2 = 702.0260{{c}} (~17/16 = 102.0260{{c}})
[[Optimal tuning]] ([[CTE]]): ~17/12 = 1\2, ~3/2 = 702.3472 (~17/16 = 102.3472)  
: error map: {{val| 0.000 +0.071 -2.929 }}


{{Optimal ET sequence|legend=1| 12, 58, 70, 82, 94, 106, 118, 224g }}
{{Optimal ET sequence|legend=1| 10, 12, 58, 70, 82, 94, 106, 118, 224g }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2247 cents
[[Badness]] (Sintel): 0.0454


=== Gigapyth ===
=== Gigapyth ===
[[Subgroup]]: 2.3.85
[[Subgroup]]: 2.3.85


[[Comma list]]: 2.3.85 {{val|-40 1 6 }}
[[Comma list]]: 2.3.85 {{val| -40 1 6 }}


{{mapping|legend=2| 1 4 6 | 0 -6 1 }}
{{Mapping|legend=2| 1 -2 7 | 0 6 -1 }}
: mapping generators: ~2, ~128/85


: mapping generators: ~2, ~85/64
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.8260{{c}}, ~128/85 = 717.2589{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/85 = 716.8156{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.034
{{Optimal ET sequence|legend=1| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370***, 519b***** }}


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
<nowiki/>* Wart for 85
 
<nowiki>*</nowiki>Wart for 85


==== 2.3.7.85 subgroup ====
==== 2.3.7.85 subgroup ====
[[Subgroup]]: 2.3.7.85
Subgroup: 2.3.7.85
 
[[Comma list]]: 1029/1024, 7225/7203


{{mapping|legend=2| 1 4 2 6 | 0 -6 2 1}}
Comma list: 1029/1024, 7225/7203


: mapping generators: ~2, ~85/64
Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~85/64 = 483.031
Optimal tunings:
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}


[[Support]]ing [[ET]]s: 5, 47, 52, 57, 62, 67, 72, 77*, 82*, 87*, 92*, 139*, 149*, 159**
{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}


<nowiki>*</nowiki>Wart for 85
<nowiki/>* Wart for 85


=== Dog ===
=== 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]].
Dog 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 [[mavila]].


Subgroup: 2.3.19
[[Subgroup]]: 2.3.19


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


[[Gencom]]: [2 4/3; 81/76]
{{Mapping|legend=2| 1 0 -2 | 0 1 4 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 2 6}}, {{val|0 -1 -4}}]
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1203.3813{{c}}, ~3/2 = 680.5089{{c}}
[[Tp tuning|POL2 generator]]: ~4/3 = 521.403
: [[error map]]: {{val| +3.381 -18.065 +31.285 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 680.5856{{c}}
: error map: {{val| 0.000 -21.369 +24.8295 }}


{{Optimal ET sequence|legend=1| 5h, 7, 16, 23 }}
{{Optimal ET sequence|legend=1| 2, 5h, 7, 16, 23 }}


[[Tp tuning #T2 tuning|RMS error]]: 4.943 cents
[[Badness]] (Sintel): 0.491


=== Boethian ===
=== Boethian ===
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[[Subgroup]]: 2.3.19
[[Subgroup]]: 2.3.19


[[Comma list]]: [[513/512]]
[[Comma list]]: 513/512


{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~3/2 = 701.3288{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2498{{c}}, ~3/2 = 701.4958{{c}}
: [[error map]]: {{val| +0.250 -0.209 -0.501 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.3445{{c}}
: error map: {{val| 0.000 -0.610 -1.547 }}


{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296 }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 41, 53, 65, 77, 219, 296, 1557bhhhh, 1853bhhhh }}


[[Badness]] (Smith): 0.000374
[[Badness]] (Sintel): 0.0294


=== Lipsett ===
=== Lipsett ===
Lipsett temperament is a pleasantly melodic little temperament with a highly useable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director off the Canadian short film ’21-87’. Leia’s prison cell in Star Wars is numbered ‘2187’, as a nod to the influence the film had on George Lucas.
Lipsett is a pleasantly melodic little temperament with a highly usable 5-tone and 9-tone mos. It is audibly similar to [[semaphore]] temperament, so it could be thought of as semaphore but for the 23rd harmonic instead of the 7th. It is named for Arthur Lipsett, the director of the Canadian short film ''21-87''. Leia's prison cell in ''Star Wars'' is numbered 2187, as a nod to the influence the film had on George Lucas.


[[Subgroup]]: 2.3.23
[[Subgroup]]: 2.3.23


[[Comma list]]: [[2187/2116]]
[[Comma list]]: 2187/2116


{{Mapping|legend=2| 1 0 -1 | 0 2 7}}
{{Mapping|legend=2| 1 0 -1 | 0 2 7}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~46/27 = 948.526
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5339{{c}}, ~46/27 = 948.5629{{c}}
: [[error map]]: {{val| +0.534 -4.829 +11.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46/27 = 948.3272{{c}}
: error map: {{val| 0.000 -5.301 +10.016 }}


{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 43, 62i, 81i }}


[[Badness]] (Smith): 8.998 × 10<sup>-3</sup>
[[Badness]] (Sintel): 0.801


=== Porpoise ===
=== Porpoise ===
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