Xenharmonic Wiki

No-threes subgroup temperaments: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,033 edits
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,375 edits
→Bastille: again link
 
(10 intermediate revisions by 4 users not shown)
Line 21: Line 21:


=== Frostburn ===
=== Frostburn ===
{{See also| Magic family #Quadrimage }}
{{See also| Magic family #Quadrimage | Subgroup temperaments #Baldy }}
{{See also| Subgroup temperaments #Baldy }}


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 35: Line 34:


{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}
[[Badness]] (Sintel): 0.886


==== 2.5.7.11 ====
==== 2.5.7.11 ====
Line 47: Line 48:
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745
Optimal tuning (TE): ~2/1 = 1200.6817, ~28/25 = 205.0745


{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}
 
Badness (Sintel): 0.463


=== Mabilic ===
=== Mabilic ===
{{See also| Chromatic pairs #Mabilic }}
{{See also| Chromatic pairs #Mabilic }}{{Main|Mabilic and trismegistus}}Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.
 
Given below is the no-three version of [[Mavila family#Armodue|armodue]], [[Mabila family#Semabila|semabila]], and [[Magic family#Trismegistus|trismegistus]]. It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma.


[[Subgroup]]: 2.5.7
[[Subgroup]]: 2.5.7
Line 98: Line 99:
Comma basis: {{monzo|372 -159 -1}}
Comma basis: {{monzo|372 -159 -1}}


Sval mapping: [{{val| 1 2 54}}, {{val|0 1 -159}}]
Sval mapping: [{{val|1 2 54}}, {{val|0 1 -159}}]


Optimal tuning (CTE): ~5/4 = 386.360
Optimal tuning (CTE): ~5/4 = 386.360


{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
{{Optimal ET sequence|legend=1|205, 264, 469, 733, 997, 1261, 1525, 1789}}, ...
[[Badness]] (Sintel): 148.6


==== 2.5.7.11 subgroup ====
==== 2.5.7.11 subgroup ====
Line 113: Line 116:
Optimal tuning (CTE): ~5/4 = 386.361
Optimal tuning (CTE): ~5/4 = 386.361


{{Optimal ET sequence|legend=1|264, 733}}, ...
{{Optimal ET sequence|legend=0|264, 733}}, ...
 
Badness (Sintel): 52.150


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 124: Line 129:
Optimal tuning (CTE): ~5/4 = 386.361
Optimal tuning (CTE): ~5/4 = 386.361


{{Optimal ET sequence|legend=1|1525, 1789}}, ...
{{Optimal ET sequence|legend=0|1525, 1789}}, ...
 
Badness (Sintel): 10.518


=== Bastille ===
=== Bastille ===
{{Main| Bastille }}
{{Main| Bastille }}


Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an eponymous historical event which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].
Described as the 2.5.7 subgroup 1407 & 1789 temperament, and named after an [[wikipedia:Storming of the Bastille|eponymous historical event]] which took place on July 14, 1789 (14/07/1789). Extensions discussed elsewhere include [[The Jacobins#Double bastille|double bastille]].


Subgroup: 2.5.7
Subgroup: 2.5.7
Line 140: Line 147:


{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
{{Optimal ET sequence|legend=1|382, 1025, 1407, 1789, 3196}}, ...
[[Badness]] (Sintel): 7224.3


=== Augment ===
=== Augment ===
Line 173: Line 182:
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]
[[Mapping]]: [{{val| 1 1 20 -49 }}, {{val| 0 3 -39 119 }}]


[[Optimal tuning]] ([[POTE]]): ~5120/3773 = 529.003¢  
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000¢, ~5120/3773 = 529.003¢
* [[CWE]]: ~2 = 1200.000¢, ~5120/3773 = 529.004¢


{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
{{Optimal ET sequence|legend=1| 93, 431, 338, 524 }}
[[Badness]] (Sintel): 11.731


==== 2.5.7.11.13 subgroup ====
==== 2.5.7.11.13 subgroup ====
Line 184: Line 197:
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]
Sval Mapping: [{{val| 1 1 20 -49 35 }}, {{val| 0 3 -39 119 -71 }}]


Optimal tuning (POTE): ~1664/1225 = 529.003¢  
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.003¢


{{Optimal ET sequence|legend=1| 93, 245e, 338, 431, 1386c }}
{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}
 
Badness (Sintel): 3.415


==== 2.5.7.11.13.17 subgroup ====
==== 2.5.7.11.13.17 subgroup ====
Line 195: Line 212:
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251
Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251


Optimal tuning (POTE): ~1664/1225 = 529.003¢
Optimal tunings:  
* CTE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢
* CWE: ~2 = 1200.000¢, ~1664/1225 = 529.005¢


{{Optimal ET sequence|legend=1| 93, 338, 431, 955c, 1386cg }}
{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}
 
Badness (Sintel): 1.985


==== 2.5.7.11.13.17.19 subgroup ====
==== 2.5.7.11.13.17.19 subgroup ====
Subgroup: 2.5.7.11.13.17.19
Subgroup: 2.5.7.11.13.17.19


Sval Mapping: [{{val| 1 1 20 -49 35 42 }}, {{val| 0 3 -39 119 -71 -86 }}]
Sval Mapping: [{{val| 1 1 20 -49 35 42 21 }}, {{val| 0 3 -39 119 -71 -86 -38 }}]


Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875
Comma list: 1001/1000, 2128/2125, 3328/3325, 16807/16796, 147968/147875


Optimal tuning (POTE): ~19/14 = 529.003¢
Optimal tunings:
* CTE: ~2 = 1200.000¢, ~19/14 = 529.006¢
* CWE: ~2 = 1200.000¢, ~19/14 = 529.005¢
 
{{Optimal ET sequence|legend=0| 93, 338, 431 }}
 
Badness (Sintel): 1.285


=== Tricesimoprimal miracloid ===
=== Tricesimoprimal miracloid ===
Line 264: Line 291:
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5886 cents


=== Pakkanen (rank 3) ===
=== Pakkanen ===
[[Subgroup]]: 2.5.7.11
[[Subgroup]]: 2.5.7.11


Line 276: Line 303:


{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 29, 35, 41, 57, 63, 98c }}
[[Badness]] (Sintel): 0.573
=== No-threes naiad ===
{{See also| Wizardharry clan #Naiad | Werckismic temperaments #Seminaiad }}
This temperament can be described as the 21 & 29 & 37 temperament in no-threes subgroups. It expands [[Subgroup temperaments #Tridec|tridec]] and [[Subgroup temperaments #Naiadec|naiadec]].
[[Subgroup]]: 2.5.7.11
[[Comma list]]: 5021863/5000000
{{Mapping|legend=2| 1 0 2 0 | 0 1 1 1 | 0 0 -4 3 }}
: mapping generators: ~2, ~5, ~100/77
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.080¢, ~5 = 2786.820¢, ~100/77 = 454.618¢
* [[CWE]]: ~2 = 1200.000¢, ~5 = 2786.740¢, ~100/77 = 454.590¢
{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
[[Badness]] (Sintel): 1.862
==== 2.5.7.11.13 subgroup ====
Subgroup: 2.5.7.11.13
Comma list: 847/845, 1001/1000
Sval mapping: {{Mapping| 1 0 2 0 1 | 0 1 1 1 1 | 0 0 -4 3 1 }}
Optimal tunings:
* WE: ~2 = 1200.034¢, ~5 = 2786.678¢, ~13/10 = 454.569¢
* CWE: ~2 = 1200.000¢, ~5 = 2786.646¢, ~13/10 = 454.557¢
{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 50, 58, 66, 87, 103, 124 }}
Badness (Sintel): 0.179
==== 2.5.7.11.13.17 subgroup ====
Subgroup: 2.5.7.11.13.17
Comma list: 170/169, 221/220, 847/845
Sval mapping: {{Mapping| 1 0 2 0 1 1 | 0 1 1 1 1 1 | 0 0 -4 3 1 2 }}
Optimal tunings:
* WE: ~2 = 1200.407¢, ~5 = 2787.484¢, ~13/10 = 455.036¢
* CWE: ~2 = 1200.000¢, ~5 = 2787.107¢, ~13/10 = 454.906¢
{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 50, 58, 66g, 87g }}
Badness (Sintel): 0.438


== Higher 2.5 temperaments ==
== Higher 2.5 temperaments ==
Line 401: Line 481:


=== Amaranthine ===
=== Amaranthine ===
{{See also| No-fives subgroup temperaments #Chrysanthemum }}
Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].
Amaranthine is the high-accuracy 2.7.11 subgroup strong restriction of [[Gamelismic clan#11-limit 3|undecimal mothra]].


Line 413: Line 495:
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}
{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d }}


Badness (Dirichlet): 0.031
Badness (Sintel): 0.031


=== Score ===
=== Score ===
Line 559: Line 641:


=== Bluebirds ===
=== Bluebirds ===
{{Distinguish| Bluebird }}
{{See also| Chromatic pairs #Bluebirds }}
{{See also| Chromatic pairs #Bluebirds }}


Retrieved from "https://en.xen.wiki/w/No-threes_subgroup_temperaments"