No-threes subgroup temperaments: Difference between revisions

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

== Temperaments with a 2.5.7 gene ==

* Sidewalk ([[823543/800000]]) → [[2023/2000 #Sidewalk]]

 

[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.3479{{c}}, ~28/25 = 204.3389{{c}}

==== 2.5.7.11 subgroup ====

Optimal tuning (TE): ~2 = 1200.6817{{c}}, ~28/25 = 205.0745{{c}}

{{See also| Chromatic pairs #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.

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~175/128 = 527.236{{c}}

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.

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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~256/245 = 77.205{{c}}

[[Subgroup]]: 2.5.7

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

{{Mapping|legend=2| 1 2 54 | 0 1 -159 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 386.360{{c}}

{{Optimal ET sequence|legend=1| 205, 264, 469, 733, 997, 1261, 1525, 1789 }}

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

Subgroup-val mapping: {{mapping| 1 2 54 -177 | 0 1 -159 -539 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 386.361{{c}}

{{Optimal ET sequence|legend=0| 264, 733, … }}

Subgroup-val mapping: {{mapping| 1 2 54 -177 52 | 0 1 -159 -539 173 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 386.361{{c}}

{{Optimal ET sequence|legend=0| 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 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~{{monzo| 381 0 -159 -4 }} = 694.243{{c}}

{{Optimal ET sequence|legend=1| 382, 1025, 1407, 1789, 3196, … }}

{{Mapping|legend=3| 3 0 7 9 11 | 0 0 0 -1 -1 }}

[[Optimal tuning]] ([[POTE]]): ~5/4 = 400.000{{c}}, ~8/7 = 228.275{{c}}

Ostara is a temperament that is derived from 93 & 524 solar calendar leap rule scale. It was initially defined by taking the 2.7.13.17.19 subgroup, but it can also be intepreted in general no-threes 19-limit.  

Ostara can also refer to a collection of temperaments which temper out 16807/16796.{{clarify}}

{{Mapping|legend=2| 1 1 20 -49 | 0 3 -39 119 }}

* [[CTE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.003{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~5120/3773 = 529.004{{c}}

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

* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}

* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.003{{c}}

Comma list: 1001/1000, 32768/32725, 147968/147875, 537824/537251

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

* CTE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}

* CWE: ~2 = 1200.000{{c}}, ~1664/1225 = 529.005{{c}}

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 }}

* CTE: ~2 = 1200.000{{c}}, ~19/14 = 529.006{{c}}

* CWE: ~2 = 1200.000{{c}}, ~19/14 = 529.005{{c}}

{{See also| Tricesimoprimal miracloid/Eliora's approach }}

 

Tricesimoprimal miracloid is described as the 52 & 1789 temperament in the 2.5.7.11.19.29.31 subgroup, with harmonics specifically selected for 52edo and 1789edo. Its generator is [[31/29]], which is also close to the secor. Since it is conceived as the temperament in the above specific subgroup, it makes no sense to name it for smaller subgroups. In terms of microtempering, a circle of 52 generators is essentially a barely noticeable [[well temperament]] for 52edo.

Subgroup-val mapping: {{mapping| 1 419 48 177 157 624 625 | 0 -461 -50 -192 -169 -685 -686 }}

Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~58/31 = 1084.628{{c}}

{{Optimal ET sequence|legend=1| 52, 1737, 1789, … }}

Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.  

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~16/13 = 357.002{{c}}

Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.  

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/8 = 842.711{{c}}

[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.6544{{c}}, ~5/4 = 380.3004{{c}}, ~11/8 = 551.9653{{c}}

* [[WE]]: ~2 = 1200.080{{c}}, ~5 = 2786.820{{c}}, ~100/77 = 454.618{{c}}

* [[CWE]]: ~2 = 1200.000{{c}}, ~5 = 2786.740{{c}}, ~100/77 = 454.590{{c}}

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

* WE: ~2 = 1200.034{{c}}, ~5 = 2786.678{{c}}, ~13/10 = 454.569{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5 = 2786.646{{c}}, ~13/10 = 454.557{{c}}

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

* WE: ~2 = 1200.407{{c}}, ~5 = 2787.484{{c}}, ~13/10 = 455.036{{c}}

* CWE: ~2 = 1200.000{{c}}, ~5 = 2787.107{{c}}, ~13/10 = 454.906{{c}}

== Temperaments with a higher 2.5.''p'' gene ==

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

[[Optimal tuning]] (CTE): ~2 = 1200.000{{c}}, ~11/8 = 529.846{{c}}

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to [[wizard]].  

[[Optimal tuning]] ([[POTE]]): ~125/88 = 600.000{{c}}, ~5/4 = 383.768{{c}}

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

: mapping generators, ~2, ~55/32

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~55/32 = 928.032{{c}}

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

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~5/4 = 372.236{{c}}

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~5/4 = 407.079{{c}}

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

{{Mapping|legend=2| 1 5 6 | 0 -7 -6 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~13/10 = 459.281{{c}}

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

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~7/4 = 968.913{{c}}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~11/8 = 540.099{{c}}

Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.  

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~14/11 = 421.309{{c}}

Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.  

: mapping generators: ~2, ~16/13

* [[POTE]]: ~2 = 1200.000{{c}}, ~16/13 = 357.677{{c}}

* [[TOP tuning|POTT]]: ~2 = 1200.000{{c}}, ~16/13 = 357.794{{c}} (1200 - 300 log₂(7))

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 324.446{{c}}

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~17/14 = 336.858{{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 ([[POTE]]): ~2 = 1200.000{{c}}, ~64/53 = 323.034{{c}}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~13/11 = 279.318{{c}}

[[Optimal tuning]] ([[POTE]]): ~13/11 = 300.000{{c}}, ~11/8 = 546.660{{c}}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~143/128 = 182.368{{c}}

{{Mapping|legend=2| 1 5 1 1 0 | 0 -4 7 8 11 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~17/13 = 462.9606{{c}}

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~26/19 = 539.886{{c}}

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

[[Optimal tuning]] ([[TE tuning|TE]]): ~2 = 1200.4457{{c}}, ~11/8 = 548.4934{{c}}, ~16/13 = 358.638{{c}}