No-threes subgroup temperaments: Difference between revisions

* For no-fives, see [[#No-threes no-fives subgroup temperaments]].

== Temperaments with a 2.5.7 gene ==

Frostburn is the common [[restriction]] of [[magic family #Quadrimage|quadrimage]] and [[subgroup temperaments #Baldy|baldy]].

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

==== 2.5.7.11 subgroup ====

Subgroup-val mapping: {{mapping| 1 3 4 5 | 0 -4 -7 -9 }}

Optimal tunings:  

* WE: ~2 = 1200.6817{{c}}, ~28/25 = 205.0734{{c}}

{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}

Mabilic is the no-3 [[restriction]] 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.

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.5781{{c}}, ~80/49 = 842.6730{{c}}

: [[error map]]: {{val| -0.422 -0.569 +1.866 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 842.9136{{c}}

{{Optimal ET sequence|legend=0| 7c, 10, 27, 37, 84, 121 }}

Badness (Sintel): 1.87

Subgroup: 2.5.7.13

Comma list: 640/637, 10985/10976

Gencom mapping: {{mapping| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}

: mapping generators: ~2, ~13/8

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.9447{{c}}

{{Optimal ET sequence|legend=0| 7c, 10, 17, 27, 37, 84, 121, 279df, 400ddf }}

Badness (Sintel): 0.319

{{See also| Chromatic pairs #Silver }}

Silver can be described as the 10 & 37 temperament in the 2.5.7.13.17 subgroup.  

Comma list: 170/169, 640/637, 5525/5488

Subgroup-val mapping: {{mapping| 1 -4 0 3 9 | 0 9 4 1 -7 }}

Gencom mapping: {{mapping| 1 0 -4 0 0 3 9 | 0 0 9 4 0 1 -7 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 842.7143{{c}}

{{Optimal ET sequence|legend=0| 10, 27, 37, 47, 84, 131, 178g }}

Badness (Sintel): 0.504

=== Ostara ===

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale, interpreted in general no-3's 19-limit. It is a weak extension of the unnamed 2.5.7-subgroup 28 & 31 temperament, which tempers out 8589934592/8544921875.  

Comma list: 8589934592/8544921875, 30691800524/30517578125

 

Subgroup-val mapping: {{mapping| 1 1 20 -49 | 0 3 -39 119 }}

: mapping generators: ~2, ~5120/3773

Optimal tunings:  

* WE: ~2 = 1199.9115{{c}}, ~5120/3773 = 528.9650{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5120/3773 = 529.0037{{c}}

{{Optimal ET sequence|legend=0| 93, 245e, 338, 955c, 1386c }}

Badness (Sintel): 11.7

Comma list: 1001/1000, 34420736/34328125, 5670699008/5661858125

 

 

{{Optimal ET sequence|legend=0| 93, 245e, 338, 431, 1386c }}

Badness (Sintel): 3.42

==== 2.5.7.11.13.17 subgroup ====

Subgroup: 2.5.7.11.13.17

Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 | 0 3 -39 119 -71 -86 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0046{{c}}

{{Optimal ET sequence|legend=0| 93, 338, 431, 955c, 1386cg }}

Badness (Sintel): 1.99

Subgroup: 2.5.7.11.13.17.19

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

Subgroup-val mapping: {{mapping| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}

* WE: ~2 = 1199.9081{{c}}, ~19/14 = 528.9639{{c}}

Badness (Sintel): 1.29

=== French decimal ===

French decimal is conceived upon the fact that 1789edo has an excellent 5/4, and uses it as the generator. This rings particularly true for the French attempts to decimalize a lot more things than we are used to today. Using the maximal evenness method of finding rank-2 temperaments, a 1525 & 1789 temperament is obtained.

[[Subgroup]]: 2.5.7

[[Comma list]]: {{monzo| 372 -159 -1 }}

: mapping generators: ~2, ~5

* [[WE]]: ~2 = 1199.9901{{c}}, ~5/4 = 386.3562{{c}}

: [[error map]]: {{val| -0.010 +0.023 +0.000 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.3595{{c}}

: error map: {{val| 0.000 +0.046 +0.019 }}

{{Optimal ET sequence|legend=1| 205, 264, 733, 997, 2258, 3255, 7507, 10762 }}

==== 2.5.7.11 subgroup ====

Comma list: {{monzo| -49 8 17 -5 }}, {{monzo| 45 -27 10 -3 }}

Subgroup-val mapping: {{mapping| 1 0 372 1255 | 0 1 -159 -539 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.3611{{c}}

{{Optimal ET sequence|legend=0| 264, 997e, 1261e, 1525, 1789 }}

Comma basis: 28824005/28792192, 200126927/200000000, 6106906624/6103515625

 

 

{{Optimal ET sequence|legend=0| 261, 1261e, 1525, 1789 }}

Badness (Sintel): 10.5

{{Main| Bastille }}

Bastille is 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

[[Comma list]]: {{monzo| 1426 -596 -15 }}

: mapping generators: ~2, ~{{monzo| -380 159 4 }}

* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}

: [[error map]]: {{val| -0.009 +0.020 +0.001 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -380 159 4 }} = 505.7570{{c}}

: error map: {{val| 0.000 +0.041 +0.018 }}

{{Optimal ET sequence|legend=1| 382, 1025, 1407, 14452, 15859c, 17266c, …, 27115cd }}

[[Badness]] (Sintel): 7.18 × 10³

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 generators: ~2, ~5, ~77/50

: [[error map]]: {{val| +0.080 +0.507 -1.318 -0.643 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 386.7404{{c}}, ~77/50 = 745.4102{{c}}

: error map: {{val| 0.000 +0.427 -0.445 -0.808 }}

{{Optimal ET sequence|legend=1| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 388, 425, 652e, 689e, 1077de }}

[[Badness]] (Sintel): 1.86

==== 2.5.7.11.13 subgroup ====

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.6458{{c}}, ~20/13 = 745.4431{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}

Badness (Sintel): 0.179

Subgroup: 2.5.7.11.13.17

Comma list: 170/169, 221/220, 847/845

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1074{{c}}, ~17/11 = 745.0940{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 29g, 37, 66g, 87g, 124g }}

Badness (Sintel): 0.438

* Jacobin superfamily ([[6656/6655]]) → [[The Jacobins]]

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[astrology]] and [[wizard]].

[[Comma list]]: 15625/15488

{{Mapping|legend=2| 2 0 -7 | 0 1 3 }}

{{Mapping|legend=3| 2 0 4 0 5 | 0 0 1 0 3 }}

[[Optimal tuning]]s:

: [[error map]]: {{val| +0.366 -1.697 +1.252 }}

* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}

: error map: {{val| 0.000 -2.316 +0.675 }}

{{Optimal ET sequence|legend=1| 6, 16, 22, 28, 50, 122, 172, 222, 394c }}

[[Badness]] (Sintel): 0.266

=== Insect ===

[[Subgroup]]: 2.5.11

[[Comma list]]: 33275/32768

{{Mapping|legend=2| 1 0 5 | 0 3 -2 }}

: mapping generators, ~2, ~55/32

: [[error map]]: {{val| +1.124 -0.813 -2.700 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}

: error map: {{val| 0.000 -3.099 -6.975 }}

{{Optimal ET sequence|legend=1| 9, 13, 22, 97e, 119e, 141e, 163e, 304ceee }}

[[Badness]] (Sintel): 0.564

{{Mapping|legend=2| 1 1 3 | 0 3 1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1203.0339{{c}}, ~11/8 = 528.4296{{c}}

: [[error map]]: {{val| +3.034 +2.009 -13.787 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 528.1575{{c}}

: error map: {{val| 0.000 -1.841 -23.160 }}

 

=== Sephiroth ===

{{See also| Chromatic pairs #Sephiroth }}

Sephiroth is the no-7 [[restriction]] of [[muggles]].  

[[Comma list]]: 34375/32768

{{Mapping|legend=2| 1 0 15 | 0 1 -5 }}

{{Mapping|legend=3| 1 0 0 0 15 | 0 0 1 0 -5 }}

[[Optimal tuning]]s:

: [[error map]]: {{val| +3.329 -6.046 -2.722 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 372.1586{{c}}

: error map: {{val| 0.000 -14.155 -12.111 }}

{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29, 132cceee }}

[[Badness]] (Sintel): 1.85

==== 2.5.11.13 subgroup ====

Subgroup: 2.5.11.13

Comma list: 65/64, 6875/6656

Subgroup-val mapping: {{mapping| 1 0 15 6 | 0 1 -5 -1 }}

Gencom mapping: {{mapping| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}

* WE: ~2 = 1203.3825{{c}}, ~5/4 = 373.6318{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 372.1519{{c}}

Badness (Sintel): 0.410

Subgroup: 2.5.11.13.17

Subgroup-val mapping: {{mapping| 1 0 15 6 11 | 0 1 -5 -1 -3 }}

Gencom mapping: {{mapping| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 371.6773{{c}}

{{Optimal ET sequence|legend=0| 3, 10, 13, 16, 29g, 129ccceeffgggg }}

Badness (Sintel): 0.299

[[Comma list]]: 26/25

{{Mapping|legend=2| 1 2 3 | 0 1 2 }}

* [[WE]]: ~2 = 1198.0216{{c}}, ~5/4 = 410.2152{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 408.9029{{c}}

{{Optimal ET sequence|legend=1| 3, 20c, 23c, 26c }}

{{Mapping|legend=2| 1 -2 0 | 0 7 6 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 740.8058{{c}}

{{Optimal ET sequence|legend=1| 8, 13, 21, 34, 81, 115 }}

[[Badness]] (Sintel): 1.93

== No-threes no-fives subgroup temperaments ==

 

=== Amaranthine ===

{{See also| No-fives subgroup temperaments #Chrysanthemum }}

 

[[Subgroup]]: 2.7.11

[[Comma list]]: 5767168/5764801

: mapping generators: ~2, ~7

* [[WE]]: ~2 = 1199.9846{{c}}, ~7/4 = 968.9078{{c}}

: [[error map]]: {{val| -0.015 +0.051 -0.010 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 968.9174{{c}}

: error map: {{val| 0.000 +0.091 +0.021 }}

{{Optimal ET sequence|legend=1| 26, 83, 109, 135, 161, 296, 1641, 1937, 2233, 2529, 2825, 3121, 6538d, 9659d, 12780dd }}

=== Argument ===

Argument tempers out [[1372/1331]] in the 2.7.11 subgroup. It is the no-3 [[restriction]] of [[#Augment|augment]].

 

[[Subgroup]]: 2.7.11

 

[[Comma list]]: 1372/1331

 

: mapping generators: ~14/11, ~7

 

[[Optimal tuning]]s:

* [[WE]]: ~14/11 = 399.8041{{c}}, ~7/4 = 963.1666{{c}}

: [[error map]]: {{val| -0.588 -6.835 +10.281 }}

* [[CWE]]: ~14/11 = 400.0000{{c}}, ~4/4 = 962.7466{{c}}

: error map: {{val| 0.000 -6.079 +11.429 }}

 

{{Optimal ET sequence|legend=1| 6, 9, 15, 36, 51e, 66e }}

 

[[Badness]] (Sintel): 0.475

{{Mapping|legend=3| 1 0 0 1 3 1 | 0 0 0 4 1 6 }}

: mapping generators: ~2, ~11/8

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 540.5091{{c}}

{{Optimal ET sequence|legend=1| 9, 11, 20 }}

[[Badness]] (Sintel): 0.368

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup, tempering out [[1573/1568]] and [[15488/15379]].  

 

 

 

 

 

 

[[Badness]] (Sintel): 1.73

Subgroup: 2.7.11.13

Subgroup-val mapping: {{mapping| 1 0 1 3 | 0 8 7 2 }}

Gencom mapping: {{mapping|| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}

Optimal tunings:

* WE: ~2 = 1199.8668{{c}}, ~14/11 = 421.2623{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 421.2874{{c}}

{{Optimal ET sequence|legend=0| 17, 20, 37, 57, 94, 225 }}

Badness (Sintel): 0.307

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup. Among the notable tunings is pure-7 tuning, 7^1/4 of 842.2065{{c}}, which is also the CTC (constrained Tenney–Chebyshevian) tuning.  

[[Comma list]]: 28672/28561

{{Mapping|legend=2| 1 0 3 | 0 4 1 }}

{{Mapping|legend=3| 1 0 0 0 0 3 | 0 0 0 4 0 1 }}

: [[error map]]: {{val| -0.217 -0.143 +0.991 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/8 = 842.2568{{c}}

{{Optimal ET sequence|legend=1| 3, 7, 10, 27, 37, 47, 57, 104, 463f, 567f, 671ff, 775ff }}

[[Badness]] (Sintel): 0.115

{{Mapping|legend=2| 1 2 3 | 0 3 4 }}

* [[WE]]: ~2 = 1200.1379{{c}}, ~17/14 = 324.3440{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 324.3738{{c}}

{{Optimal ET sequence|legend=1| 4, 7, 11, 26, 37 }}

{{Mapping|legend=2| 1 0 1 | 0 10 11 }}

* [[WE]]: ~2 = 1199.9723{{c}}, ~17/14 = 336.8586{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 336.8621{{c}}

{{Optimal ET sequence|legend=1| 7, 18dg, 25, 32, 57, 488, 545, 602, 659, 716, 773, 830, 887, 1717g }}

{{Mapping|legend=2| 1 0 6 | 0 3 -1 }}]

: mapping generators, ~2, ~64/53

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.6967{{c}}, ~64/53 = 323.1839{{c}}

: [[error map]]: {{val| -0.303 +0.119 +1.491 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/53 = 323.1959{{c}}

: error map: {{val| 0.000 +0.762 +3.300 }}

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, tempers out [[1352/1331]], and is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.  

[[Comma list]]: 1352/1331

: mapping generators, ~2, ~143/128

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.8795{{c}}, ~143/128 = 182.5017{{c}}

: [[error map]]: {{val| +0.880 -1.174 -2.013 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~143/128 = 182.4386{{c}}

: error map: {{val| 0.000 -4.002 -5.405 }}

{{Optimal ET sequence|legend=1| 6, 7, 13, 33, 46, 79, 125f, 204ef, 329eeff }}

[[Badness]] (Sintel): 0.451

[[Comma list]]: 29282/28561

[[Comma list]]: 209/208, 2057/2048, 83521/83486

{{Mapping|legend=2| 1 1 8 9 11 | 0 4 -7 -8 -11 }}

: mapping generators: ~2, ~26/17

* [[WE]]: ~2 = 1200.4661{{c}}, ~26/17 = 737.3256{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/17 = 737.0014{{c}}

{{Optimal ET sequence|legend=1| 13, 44, 57, 70, 127, 197eh }}

Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.  

[[Subgroup]]: 2.11.13.17.19

{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~247/242 = 36.3676{{c}}, ~11/8 = 550.4246{{c}}

* [[CWE]]: ~247/242 = 36.3636{{c}}, ~11/8 = 550.4187{{c}}

{{Optimal ET sequence|legend=1| 33, 165, 198, 231, 495h }}

Berylic tempers out the [[berylisma]] in the 2.11.37 subgroup, representing the fact that [[44/37]] is a {{w|continued fraction}} convergent to 2^1/4 the fourth root of 2. Beryllic is an example of a temperament which has an astronomically low [[badness]], being a very high-accuracy [[microtemperament]] with low-to-average [[complexity]] for the harmonics in its [[subgroup]]. This also makes it simultaneously supported by edo systems as low as [[16edo]] and up into the tens of thousands. The tradeoff with this temperament, not captured within the metric of badness, is that it is defined within an obscure subgroup, 2.11.37.

If one wishes to explore harmony in this temperament, a great way is to use the 8-note [[4L 4s]] [[mos]], and use the [[32:37:44]] triad and its inversion [[296:352:407|1/(44:37:32)]] as the root chords. However, the consonance of the 37th harmonic is questionable.

[[Subgroup]]: 2.11.37

[[Comma list]]: 1874161/1874048

{{Mapping|legend=2| 4 0 7 | 0 1 1 }}

: mapping generators: ~44/37, ~11

[[Optimal tuning]]s:

* [[CWE]]: ~44/37 = 300.0000{{c}}, ~11/8 = 551.3237{{c}}

: error map: {{val| 0.000 +0.006 -0.020 }}

{{Optimal ET sequence|legend=1| 4, 16, 20, 24, 76, 100, 124, 148, 616, 764, 912, 1060, 3328, 4388, 5448 }}

[[Badness]] (Sintel): 0.00188

{{Mapping|legend=2| 1 1 2 | 0 6 5 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.8817{{c}}, ~26/19 = 539.9150{{c}}

: [[error map]]: {{val| -0.118 -1.156 +1.825 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/19 = 539.9280{{c}}

{{Optimal ET sequence|legend=1| 9, 11, 20 }}

 

{{Mapping|legend=2| 1 0 0 11 4 | 0 1 0 -2 -1 | 0 0 1 0 1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.4447{{c}}, ~11/8 = 548.4929{{c}}, ~13/8 = 841.3613{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 548.2193{{c}}, ~13/8 = 841.4707{{c}}