No-threes subgroup temperaments: Difference between revisions

== Temperaments with a 2.5.7 gene ==

In rainy, three generators make an [[8/7]]; five generators make a [[5/4]]. It is the no-3's [[restriction]] of [[tertiaseptal]] (and [[valentine]]), notable theoretically as it equates ([[2/1]])/([[5/4]])³ (128/125, the lesser diesis) with ([[2/1]])/([[8/7]])⁵ (the 2.7-subgroup [[cloudy comma]], which is similar to the 2.5-subgroup lesser diesis in that tempering it out tunes the 8/7 about 8.8{{cent}} sharp, while tempering out 128/125 similarly sharpens the 5/4 by about 13.7{{cent}}). By tempering out their difference, stacked 5's and stacked 7's become easier to navigate, using the general-purpose diesis to simplify clusters.  

A highly notable tuning of rainy not shown here is [[311edo]], which is 140 + 171 so tuned between them.

 

 

[[Optimal tuning]]s:

: [[error map]]: {{val| +0.094 -0.072 -0.176 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/245 = 77.2093{{c}}

: error map: {{val| 0.000 -0.267 -0.454 }}

 

{{Optimal ET sequence|legend=1| 15, 16, 31, 109, 140, 171, 373, 544, 1259, 1803d }}

 

 

 

Augment is related to [[augmented]], but for 2.5.7 instead of 2.3.5.

 

 

 

 

 

* [[CWE]]: ~5/4 = 400.0000{{c}}, ~7/4 = 974.3418{{c}}

: error map: {{val| 0.000 +13.686 +5.516 }}

 

{{Optimal ET sequence|legend=1| 3, 6, 15, 21, 27, 102ccd, 129ccd }}

 

[[Badness]] (Sintel): 0.296

 

 

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

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

Optimal tunings:  

* CWE: ~5/4 = 400.0000{{c}}, ~7/4 = 968.4397{{c}}

{{Optimal ET sequence|legend=0| 3, 6, 15, 21 }}

Badness (Sintel): 0.196

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 ===

 

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.

 

 

 

 

 

: [[error map]]: {{val| +1.254 -1.698 -1.491 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 527.2041{{c}}

 

 

 

 

Huntington may be described as the 10 & 37 temperament in the 2.5.7.13 subgroup.  

[[Subgroup]]: 2.5.7

[[Comma list]]: 40960000/40353607

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

: mapping generators: ~2, ~80/49

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

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

: error map: {{val| 0.000 -0.091 +2.828 }}

{{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

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

: mapping generators: ~2, ~13/8

Optimal tunings:  

* WE: ~2 = 1199.4788{{c}}, ~13/8 = 842.6318{{c}}

* 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 }}

 

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 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.  

Subgroup: 2.5.7.11

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:  

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

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

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

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~1664/1225 = 529.0036{{c}}

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

 

 

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

Optimal tunings:  

* WE: ~2 = 1199.9054{{c}}, ~1664/1225 = 528.9628{{c}}

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

Optimal tunings:  

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

* CWE: ~2 = 1200.0000{{c}}, ~19/14 = 529.0045{{c}}

 

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

 

 

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

[[Optimal tuning]]s:  

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

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

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

[[Badness]] (Sintel): 148

==== 2.5.7.11 subgroup ====

Subgroup: 2.5.7.11

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

Optimal tunings:  

* WE: ~2 = 1200.0130{{c}}, ~5/4 = 386.3653{{c}}

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

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

Badness (Sintel): 52.2

==== 2.5.7.11.13 subgroup ====

Subgroup: 2.5.7.11.13

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

Subgroup-val mapping: {{mapping| 1 0 372 1255 -398 | 0 1 -159 -539 173 }}

* WE: ~2 = 1200.0137{{c}}, ~5/4 = 386.3655{{c}}

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

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

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|legend=2| 1 -4 254 | 0 15 -596 }}

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

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

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

{{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]].

[[Comma list]]: 5021863/5000000

 

 

 

 

 

 

 

 

 

 

 

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

Optimal tunings:

* WE: ~2 = 1200.4068{{c}}, ~5/4 = 386.6701{{c}}, ~17/11 = 745.3706{{c}}

* 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

{{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 ===

 

 

[[Comma list]]: 34375/32768

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{{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 }}

 

: [[error map]]: {{val| -0.408 -1.098 +3.244 }}

: error map: {{val| 0.000 -0.673 +4.307 }}

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

[[Badness]] (Sintel): 1.93

* 31-17/13-commatic → [[31st-octave temperaments #31-17/13-commatic|31st-octave temperaments]]

* 37-11-commatic (rank-1) → [[37th-octave temperaments #37-11-commatic (rank-1)|37th-octave temperaments]]

=== Amaranthine ===

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Subgroup]]: 2.7.11

 

 

 

 

 

 

 

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 }}

=== Voltage ===

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.

[[Subgroup]]: 2.7.13

[[Comma list]]: 28672/28561

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7827{{c}}, ~13/8 = 842.1707{{c}}

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

: error map: {{val| 0.000 +0.201 +1.729 }}

{{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 }}

 

[[Optimal tuning]]s:  

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

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

 

 

[[Badness]] (Sintel): 0.224

{{See also | Chromatic pairs #Lovecraft }}

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.  

 

 

 

 

 

{{Optimal ET sequence|legend=1| 4, 9, 13, 30, 43, 73, 116e }}

 

 

 

[[Subgroup]]: 2.11.13

 

 

 

 

 

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

 

 

 

 

 

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

: mapping generators, ~13/11, ~11

[[Optimal tuning]]s:

* [[WE]]: ~13/11 = 299.9728{{c}}, ~11/8 = 546.6107{{c}}

: [[error map]]: {{val| -0.109 -5.033 +5.730 }}

* [[CWE]]: ~13/11 = 300.0000{{c}}, ~11/8 = 546.4664{{c}}

: error map: {{val| 0.000 -4.852 +5.939 }}

{{Optimal ET sequence|legend=1| 4, 12e, 16, 20, 24, 44, 68 }}

Yamablu, with a generator of ~26/17, is named for its tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). It extends the 2.11.13-subgroup temperament tempering out 556573090931/549755813888. The [[Kite's Genchain mode numbering|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].  

 

[[Optimal tuning]]s:

* [[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 }}

 

[[Badness]] (Sintel): 0.386

 

 

[[Subgroup]]: 2.11.13.17.19

 

[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432

: mapping generators: ~247/242, ~11

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

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

 

 

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.

 

 

 

 

 

[[Optimal tuning]]s:

: [[error map]]: {{val| +0.001 +0.007 -0.017 }}

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

 

 

[[Badness]] (Sintel): 0.00188

 

[[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}}

: error map: {{val| 0.000 -0.960 +2.127 }}

 

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

 

[[Badness]] (Sintel): 0.559

 

 

 

[[Optimal tuning]]s:

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