No-threes subgroup temperaments: Difference between revisions

== Temperaments with a 2.5.7 gene ==

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

 

{{See also| Chromatic pairs #Augment }}

 

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

 

[[Subgroup]]: 2.5.7

 

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

 

{{Mapping|legend=5| 3 0 7 0 | 0 0 0 1 }}

: mapping generators: ~5/4, ~7

 

[[Optimal tuning]]s:

* [[WE]]: ~5/4 = 399.0128{{c}}, ~7/4 = 974.7085{{c}}

: [[error map]]: {{val| -2.962 +6.776 -0.040 }}

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

 

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

 

[[Badness]] (Sintel): 0.296

 

==== 2.5.7.11 subgroup ====

Subgroup: 2.5.7.11

 

Comma list: 56/55, 128/125

 

 

{{Mapping|legend=4| 3 0 7 0 2 | 0 0 0 1 1 }}

 

* WE: ~5/4 = 398.9239{{c}}, ~7/4 = 969.1106{{c}}

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

 

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

 

Badness (Sintel): 0.196

 

Mabilic is the no-3 [[restriction]] of [[mavila family #Armodue|armodue]] (7 & 16), [[magic family #Trismegistus|trismegistus]] (16 & 25), and [[mabila family #Semabila|semabila]] (25 & 34). It is the 7 & 9 temperament in the [[2.5.7 subgroup]], and tempers out [[1071875/1048576]], the mabilisma. [[Weak extension]]s of mabilic include [[archytas clan #Sixix|sixix]] (7 & 25), [[vishnu family #Vishnean|vishnean]] (16 & 34), [[marvel temperaments #Septimin|septimin]] (9 & 41) and [[tetracot family #Monkey|monkey]] (7 & 34).  

[[Comma list]]: 1071875/1048576

 

 

 

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

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

 

 

[[Badness]] (Sintel): 1.70

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Subgroup: 2.5.7.11.13.17.19

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

{{Mapping|legend=2| 1 1 20 -49 35 42 21 | 0 3 -39 119 -71 -86 -38 }}

Optimal tunings:  

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

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

Badness (Sintel): 1.29

{{Redirect|Frostburn|the software developer|Lumi Pakkanen}}

 

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

{{Mapping|legend=3| 1 3 4 | 0 -4 -7 }}

[[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 ====

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.0939{{c}}, ~256/245 = 77.2107{{c}}

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

 

Llywelyn a.k.a. shoe is the common [[restriction]] of [[gamelismic clan #Gorgo|gorgo]] (16 & 21), [[breedsmic temperaments #Gorgik|gorgik]] (21 & 37), and [[gariboh clan #Semaja|semaja]] (16 & 37). It can be described as 16 & 21, and tempers out the [[llywelynsma]]. Its generator is a flattened [[~]][[8/7]] whole tone, seven of which give the interval class of [[5/1|5]]. Its [[ploidacot]] is alpha-heptaseph. [[Weak extension]]s to the full 7-limit include [[marvel temperaments #Misneb|misneb]] (21 & 32), which splits 8/7 into two for [[15/14]]~[[16/15]] each, [[archytas clan #Ultrapyth|ultrapyth]] (32 & 37), which splits [[16/7]] into two for a [[3/2|perfect fifth]], and [[buzzardsmic clan #Septimal buzzard|buzzard]] (53 & 58), which splits 16/7 into three, each for [[21/16]].

 

* [[Shoe5]], [[Shoe6]], [[Shoe11]], [[Shoe16]]

 

[[Comma list]]: 4194304/4117715

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

{{Mapping|legend=5| 1 0 1 3 | 0 0 7 -1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.1974{{c}}, ~8/7 = 226.7586{{c}}

: [[error map]]: {{val| -0.803 +0.194 +2.008 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.8388{{c}}

: error map: {{val| 0.000 +1.558 +4.335 }}

{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37, 164d }}

[[Badness]] (Sintel): 2.06

Comma list: 176/175, 1310720/1294139

 

{{Mapping|legend=2| 1 1 3 1 | 0 7 -1 13 }}

{{Mapping|legend=4| 1 0 1 3 1 | 0 0 7 -1 13 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0438{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 37, 206cd, 243cd, …, 354ccdd }}

 

Badness (Sintel): 0.651

Comma list: 176/175, 640/637, 847/845

{{Mapping|legend=2| 1 1 3 1 2 | 0 7 -1 13 9 }}

{{Mapping|legend=4| 1 0 1 3 1 2 | 0 0 7 -1 13 9 }}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

{{Mapping|legend=4| 1 0 -4 0 0 3 | 0 0 9 4 0 1 }}

: mapping generators: ~2, ~13/8

 

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

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

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 0.504

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=3| 1 -4 254 | 0 15 -596 }}

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

[[Optimal tuning]]s:  

* [[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}}

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

 

=== No-threes naiad (rank-3) ===

{{Todo|inline=1| review | comment = Devise the permanent name for this temperament; determine its status with respect to its extensions linked below. }}

{{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]].

[[Comma list]]: 5021863/5000000

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

[[Optimal tuning]]s:

* [[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 }}

 

Comma list: 847/845, 1001/1000

 

Optimal tunings:  

* WE: ~2 = 1200.0343{{c}}, ~5/4 = 386.6098{{c}}, ~20/13 = 745.4658{{c}}

* 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 }}

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

 

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 }}

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

 

=== Sephiroth ===

Sephiroth is the common [[restriction]] of [[magic family #Muggles|muggles]] and [[avicennmic temperaments #Submerged|submerged]].

[[Subgroup]]: 2.5.11

[[Comma list]]: 34375/32768

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

: mapping generators: ~2, ~5

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1203.3290{{c}}, ~5/4 = 373.6097{{c}}

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

{{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

{{Mapping|legend=2| 1 0 15 6 | 0 1 -5 -1 }}

{{Mapping|legend=4| 1 0 0 0 15 6 | 0 0 1 0 -5 -1 }}

Optimal tunings:  

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

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

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

Badness (Sintel): 0.410

Subgroup: 2.5.11.13.17

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

{{Mapping|legend=2| 1 0 15 6 11 | 0 1 -5 -1 -3 }}

{{Mapping|legend=4| 1 0 0 0 15 6 11 | 0 0 1 0 -5 -1 -3 }}

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

* 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

Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, tempers out [[15625/15488]], and is the common [[restriction]] of [[jubilismic clan #Astrology|astrology]] and [[stearnsmic clan #Wizard|wizard]].

[[Comma list]]: 15625/15488

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

{{Mapping|legend=5| 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=3| 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

This temperament has a structure very similar to [[mavila family #Mavila|mavila]] but is somewhat more accurate because the generator is a flat [[11/8]] rather than a sharp [[4/3]]. The major third is still ~[[5/4]], but the minor third is now ~[[64/55]] instead of ~[[6/5]].

{{Mapping|legend=3| 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 }}

 

=== Trader ===

[[Subgroup]]: 2.5.13

[[Comma list]]: 26/25

{{Mapping|legend=3| 1 0 -1 | 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=3| 1 -2 0 | 0 7 6 }}

 

 

 

 

 

 

 

 

 

 

* [[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 }}

 

 

 

 

 

 

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

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

 

 

 

Scaevola is the 2.7.11-subgroup temperament tempering out 14641/14336, with a generator of [[11/8]] and mapping +4 generators to [[7/4]]. Two strong extensions are [[#Score|score]] to 2.7.11.13 and [[subgroup temperaments #Joan|joan]] to 2.9.7.11.

 

Scaevola has a structure that is very similar to [[mavila family #Mavila|mavila]], but whereas mavila has a "fifth" of [[3/2]], "major third" of [[5/4]], and "minor third" of [[6/5]], scaevola has a "fifth" of [[16/11]], "major third" of [[14/11]], and "minor third" of [[8/7]], substituting mavila's 4:5:6 for 11:14:16. Essentially, Scaevola can be seen as a counterpart of mavila for fifths even flatter than [[9edo]], such as [[11edo]]'s fifth.

 

 

 

 

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

: mapping generators: ~2, ~11/8

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1201.2479{{c}}, ~11/8 = 542.1162{{c}}

: [[error map]]: {{val| +1.248 +0.887 -5.458 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 541.9403{{c}}

: error map: {{val| 0.000 -1.065 -9.378 }}

{{Optimal ET sequence|legend=1| 2, 7d, 9, 11, 20, 31, 51, 82e, 113e, 195eee }}

 

 

 

 

 

 

 

 

 

 

=== Bossier ===

 

 

[[Subgroup]]: 2.7.11

 

 

 

{{Mapping|legend=5| 1 0 0 0 1 | 0 0 0 8 7 }}

 

 

 

 

==== 2.7.11.13 subgroup ====

 

{{Mapping|legend=4| 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=5| 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=3| 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=3| 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 }}

 

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

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

 

 

 

 

 

 

 

 

 

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

 

 

[[Badness]] (Sintel): 0.175

 

 

 

 

 

 

 

[[Badness]] (Sintel): 0.451

=== 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

 

{{Mapping|legend=3| 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 }}

 

 

 

 

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

 

{{Mapping|legend=3| 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}}

 

 

[[Badness]] (Sintel): 2.30

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

 

 

 

[[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}}

 

[[Category:Catalogs of subgroup temperaments]]

[[Category:Catalogs of rank-2 temperaments]]