No-threes subgroup temperaments: Difference between revisions

== Overview by mapping of 5 ==

* Ostara, movila and vengeance have variantly expressed generators, three of which give the ~5/2.

Others have a more complex mapping of 5.

* Jubilic ([[50/49]]) → [[Jubilismic clan #Jubilic|Jubilismic clan]]

* Llywelyn a.k.a. shoe ([[4194304/4117715]]) → [[Llywelynsmic clan #Llywelyn a.k.a. shoe|Llywelynsmic clan]]

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.

[[Subgroup]]: 2.5.7

[[Comma list]]: [[2100875/2097152]]

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

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

: mapping generators: ~2, ~256/245

 

 

 

 

 

 

 

 

 

 

 

 

==== 2.5.7.11 subgroup ====

Subgroup: 2.5.7.11

 

 

 

 

 

 

 

 

 

 

 

 

 

 

==== 2.5.7.11 subgroup ====

 

 

 

 

 

 

 

 

[[Comma list]]: 1071875/1048576

 

 

 

* [[WE]]: ~2 = 1201.2543{{c}}, ~175/128 = 527.7872{{c}}

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

Optimal tunings:  

* WE: ~2 = 1200.0932{{c}}, ~13/8 = 842.7764{{c}}

* 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.

 

 

 

 

 

 

==== 2.5.7.11.13 subgroup ====

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

* [[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 }}

 

 

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:  

* 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 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

==== 2.5.7.11.13.17 subgroup ====

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Comma list]]: 33275/32768

 

: mapping generators, ~2, ~55/32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Comma list: 65/64, 170/169, 221/220

 

 

 

* WE: ~2 = 1203.6741{{c}}, ~5/4 = 373.3775{{c}}

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Amaranthine is the high-accuracy 2.7.11-subgroup strong [[restriction]] of [[Gamelismic clan #11-limit 3|undecimal mothra]].

[[Subgroup]]: 2.7.11

[[Comma list]]: 5767168/5764801

{{Mapping|legend=2| 1 0 -19 | 0 1 8 }}

: 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 }}

Badness (Sintel): 0.0309

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

[[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 }}

{{See also| Chromatic pairs #Score }}

Score is a low-accuracy [[extension]] of the unnamed 2.7.11-subgroup temperament tempering out 14641/14336.  

[[Subgroup]]: 2.7.11.13

[[Comma list]]: 343/338, 847/832

{{Mapping|legend=2| 1 1 3 1 | 0 4 1 6 }}

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

[[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 }}

{{See also| Chromatic pairs #Bossier }}

[[Subgroup]]: 2.7.11

[[Comma list]]: 214358881/210827008

: mapping generators: ~2, ~14/11

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.1886{{c}}, ~14/11 = 421.2661{{c}}

: [[error map]]: {{val| +0.189 +1.303 -2.266 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/11 = 421.2365{{c}}

: error map: {{val| 0.000 +1.066 -2.662 }}

{{Optimal ET sequence|legend=1| 17, 20, 37, 57, 94, 151 }}

[[Badness]] (Sintel): 1.73

==== 2.7.11.13 subgroup ====

 

 

 

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

 

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

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

=== Counterultrakleismic ===

[[Subgroup]]: 2.7.17

 

 

 

 

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

 

 

 

 

 

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

 

 

 

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.

[[Subgroup]]: 2.11.13

[[Comma list]]: 1352/1331

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

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

: mapping generators, ~2, ~13/11

* [[WE]]: ~2 = 1199.5223{{c}}, ~13/11 = 279.2064{{c}}

: [[error map]]: {{val| -0.478 +5.662 -4.341 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~13/11 = 278.9918{{c}}

: error map: {{val| 0.000 +6.666 -3.552 }}

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

[[Badness]] (Sintel): 0.175

[[Subgroup]]: 2.11.13

[[Comma list]]: 265837/262144

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

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

: mapping generators, ~2, ~143/128

* [[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

{{See also | Chromatic pairs #Blackbirds }}

Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.

[[Subgroup]]: 2.11.13

[[Comma list]]: 29282/28561

{{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 ===

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]].  

[[Subgroup]]: 2.11.13.17.19

{{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 }}

 

 

 

 

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

 

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

 

* [[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.

 

 

[[Comma list]]: 1874161/1874048

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

: mapping generators: ~44/37, ~11

[[Optimal tuning]]s:

 

 

 

 

 

 

[[Optimal tuning]]s:

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

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

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 0.106