Xenharmonic Wiki

No-fives subgroup temperaments: Difference between revisions

← Older editNewer edit →
VisualWikitext
m →Boethian: Fix grammar error left over from previous changes
→Temperaments with a 2.3.7 gene: adopt Sintel's badness, WE and CWE tunings (1/). + 2.3.7.11.13-subgroup flutterpyth and correction
Line 7: Line 7:


=== Bleu ===
=== Bleu ===
Bleu can be described as the 9 & 17 temperament in the no-5 13-limit.  
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit.  


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 16: Line 16:


{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
: mapping generators: ~2, ~54/49


: [[gencom]]: [2 54/49; 17496/16807]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3538{{c}}, ~54/49 = 139.848{{c}}
: [[error map]]: {{val| -0.646 -3.736 +8.293 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.848{{c}}
: error map: {{val| 0.000 -3.270 +9.333 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~54/49 = 139.848
{{Optimal ET sequence|legend=1| 8d, 9, 17, 43, 60d, 103d }}


{{Optimal ET sequence|legend=1| 9, 17, 43, 60d }}
[[Badness]] (Sintel): 2.48
 
[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents


==== 2.3.7.11 subgroup ====
==== 2.3.7.11 subgroup ====
Line 30: Line 33:
Comma list: 99/98, 864/847
Comma list: 99/98, 864/847


Sval mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}
Subgroup-val mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}


Gencom mapping: {{mapping| 1 1 0 2 3 | 0 5 0 7 4 }}
Gencom mapping: {{mapping| 1 1 0 2 3 | 0 5 0 7 4 }}


: gencom: [2 12/11; 99/98 864/847]
Optimal tunings:  
* WE: ~2 = 1198.6613{{c}}, ~12/11 = 139.8489{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.7839{{c}}


Optimal tuning]] (POTE): ~2 = 1\1, ~12/11 = 140.005
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


Optimal ET sequence: {{Optimal ET sequence| 9, 17, 43, 60d }}
Badness (Sintel): 0.624
 
RMS error: 1.829 cents


==== 2.3.7.11.13 subgroup ====
==== 2.3.7.11.13 subgroup ====
Line 47: Line 50:
Comma list: 78/77, 99/98, 144/143
Comma list: 78/77, 99/98, 144/143


Sval mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}
Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}


Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}
Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


: gencom: [2 13/12; 78/77 99/98 144/143]
Optimal tunings:  
 
* WE: ~2 = 1198.9768{{c}}, ~13/12 = 139.8704{{c}}
Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.990
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8166{{c}}


Optimal ET sequence: {{Optimal ET sequence| 17, 43, 60d }}
{{Optimal ET sequence|legend=0| 8d, 9, 17, 43, 60d }}


RMS error: 1.752 cents
Badness (Sintel): 0.400


; Music
; Music
Line 75: Line 78:


==== Flutterpyth ====
==== Flutterpyth ====
Restricted to 2.3.7.11, this temperament is a no-5 restriction of 11-limit [[ultrapyth]]. This temperament was created to yield [[blackdye]] tunings where [[aberrisma|aberrisma-altered]] 3-limit thirds become tempered [[13/11]][[~]][[19/16]] and [[14/11]].
Subgroup: 2.3.7.11.13
Comma list: 64/63, 364/363, 1078/1053
Mapping: {{mapping| 1 0 6 21 34 | 0 1 -2 -11 -19 }}
Optimal tunings:
* WE: ~2 = 1196.9412{{c}}, ~3/2 = 711.0195{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 713.0039{{c}}
{{Optimal ET sequence|legend=0| 32f, 69bf, 101beff }}
Badness (Sintel): 1.52
===== 2.3.7.11.13.19 subgroup =====
Subgroup: 2.3.7.11.13.19
Subgroup: 2.3.7.11.13.19


Comma list: 64/63, 209/208, 343/342, 364/363
Comma list: 64/63, 209/208, 343/342, 364/363


Mapping: {{mapping| 1 1 4 10 15 9 | 0 -1 -2 -11 -19 -8 }}
Mapping: {{mapping| 1 0 6 21 34 17 | 0 1 -2 -11 -19 -8 }}
 
Optimal tunings:
* WE: ~2 = 1197.4072{{c}}, ~3/2 = 711.2733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 712.9612{{c}}


Optimal tuning (CTE): ~3/2 = 713.459
{{Optimal ET sequence|legend=0| 32f, 69bf }}


Restricted to 2.3.7.11.13, this temperament is a no-5 restriction of 13-limit [[Ultrapyth]]. This temperament was created to yield [[blackdye]] tunings where aberrisma-altered 3-limit thirds become tempered 13/11~19/16 and 14/11.
Badness (Sintel): 1.28


=== Skwares ===
=== Skwares ===
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"