No-threes subgroup temperaments: Difference between revisions

* Trader has a ~2/1 period and ~5/4 generator. There is a one-to-one correspondence between the 2.5 subgroup and mapped intervals.

=== Augment ===

 

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

[[Comma list]]: 128/125

 

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

: mapping generators: ~5/4, ~7

* [[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}}

: 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

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).  

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

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

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

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

=== Llywelyn a.k.a. shoe ===

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]].  

 

[[Comma list]]: 4194304/4117715

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

 

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

: mapping generators: ~2, ~8/7

* [[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 }}

* WE: ~2 = 1199.1784{{c}}, ~8/7 = 226.9582{{c}}

* 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 }}

* WE: ~2 = 1199.1643{{c}}, ~8/7 = 226.9495{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.0315{{c}}

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

Badness (Sintel): 0.319

==== 2.5.7.11.13.17 subgroup ====

Comma list: 176/175, 221/220, 640/637, 847/845

{{Mapping|legend=2| 1 1 3 1 2 2 | 0 7 -1 13 9 11 }}

{{Mapping|legend=4| 1 0 1 3 1 2 2 | 0 0 7 -1 13 9 11 }}

* WE: ~2 = 1199.3162{{c}}, ~8/7 = 227.1125{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.1774{{c}}

{{Optimal ET sequence|legend=0| 16, 21, 37, 95 }}

Named by {{u|KingHyperio}} in 2021, mercy is the common restriction of [[gamelismic clan #Miracle|miracle]] and [[porwell temperaments #Countermiracle|countermiracle]]. It tempers out the [[quince comma]]. In this temperament, two generators make [[~]][[8/7]], five make ~[[7/5]], and seven make ~[[8/5]]. The [[ploidacot]] for this temperament is omega-heptaseph. [[Weak extension]]s of this temperament include [[starling temperaments #Cassablanca|cassablanca]], [[octagar temperaments #Slendi|slendi]], [[ragismic microtemperaments #Quincy|quincy]], [[septischismic clan #Cotoneum|cotoneum]], and [[varunismic temperaments #Octowerck|octowerck]].  

[[Comma list]]: 823543/819200

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

: mapping generators: ~2, ~343/320

: [[error map]]: {{val| +0.217 +0.156 -0.798 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/320 = 116.2576{{c}}

: error map: {{val| 0.000 -0.117 -1.341 }}

{{Optimal ET sequence|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}

[[Badness]] (Sintel): 0.428

==== 2.5.7.13 subgroup ====

Presented here is a low-accuracy extension, which may be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]].

Subgroup: 2.5.7.13

Comma list: 343/338, 640/637

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

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

Optimal tunings:  

* WE: ~2 = 1199.2166{{c}}, ~14/13 = 116.0181{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 116.2099{{c}}

{{Optimal ET sequence|legend=0| 10, 21, 31 }}

Badness (Sintel): 0.546

==== 2.5.7.13.17 subgroup ====

Comma list: 170/169, 224/221, 343/338

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

* WE: ~2 = 1198.5476{{c}}, ~14/13 = 115.6294{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.9132{{c}}

{{Optimal ET sequence|legend=0| 10, 21, 31, 83fg }}

Badness (Sintel): 0.451

==== 2.5.7.13.17.19 subgroup ====

Subgroup: 2.5.7.13.17.19

Comma list: 170/169, 224/221, 343/338, 476/475

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

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

* WE: ~2 = 1198.4839{{c}}, ~14/13 = 115.5700{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 115.7299{{c}}

{{Optimal ET sequence|legend=0| 10h, 21, 31, 52f, 83fg }}

Badness (Sintel): 0.759

=== Huntington ===

{{See also| Chromatic pairs #Huntington }}

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=3| 1 -4 0 | 0 9 4 }}

{{Mapping|legend=5| 1 0 -4 0 | 0 0 9 4 }}

: mapping generators: ~2, ~80/49

* [[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}}

: 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

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

{{Mapping|legend=4| 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

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

Subgroup: 2.5.7.13.17

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

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

{{Mapping|legend=4| 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

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

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

 

 

[[Badness]] (Sintel): 7.18 × 10³

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

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

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

* [[WE]]: ~2 = 1200.0805{{c}}, ~5/4 = 386.6593{{c}}, ~77/50 = 745.4622{{c}}

: [[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

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

* 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 }}

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

* 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 }}

=== Sephiroth ===

 

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

[[Comma list]]: 34375/32768

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* 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 }}

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:

* [[WE]]: ~125/88 = 600.1831{{c}}, ~5/4 = 383.8848{{c}}

* [[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

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

[[Optimal tuning]]s:

: [[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 }}

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]].

[[Subgroup]]: 2.5.11

[[Comma list]]: 1331/1280

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

* [[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}}

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41e, 66ee }}

[[Badness]] (Sintel): 0.718

[[Comma list]]: 26/25

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

: mapping generators, ~2, ~5

[[Optimal tuning]]s:

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

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

: error map: {{val| 0.000 +22.589 -22.722 }}

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

: mapping generators, ~2, ~20/13

[[Optimal tuning]]s:

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

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

{{Mapping|legend=3| 1 0 -19 | 0 1 8 }}

 

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

 

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

 

 

Argument tempers out [[1372/1331]] in the 2.7.11 subgroup, equating [[14/11]] to a step of [[3edo]]. It is the no-3 [[restriction]] of [[#Augment|augment]].

 

 

 

 

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

 

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

 

 

 

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|legend=5| 1 0 0 1 3 | 0 0 0 4 1 }}

: mapping generators: ~2, ~11/8

* [[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 }}

 

==== Score ====

Score is an extension of scaevola that maps [[13/8]] to +6 generators.

 

Comma list: 343/338, 847/832

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

Optimal tunings:

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

{{Optimal ET sequence|legend=0| 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]].  

[[Subgroup]]: 2.7.11

[[Comma list]]: 214358881/210827008

 

 

 

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

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

 

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

 

[[Badness]] (Sintel): 1.73

 

 

Comma list: 1573/1568, 15488/15379

{{Mapping|legend=4| 1 0 0 0 1 3 | 0 0 0 8 7 2 }}

Optimal tunings:

* 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=3| 1 0 3 | 0 4 1 }}

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

: mapping generators: ~2, ~13

[[Optimal tuning]]s:  

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

: [[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=3| 1 2 3 | 0 3 4 }}

: mapping generators, ~2, ~17/14

[[Optimal tuning]]s:

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

: error map: {{val| 0.000 +4.295 -7.460 }}

{{Mapping|legend=3| 1 0 1 | 0 10 11 }}

[[Optimal tuning]]s:

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

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

: error map: {{val| 0.000 -0.205 +0.528 }}

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

[[Optimal tuning]]s:

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

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

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]]: 265837/262144

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

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

: mapping generators, ~2, ~143/128

[[Optimal tuning]]s:

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

* [[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

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

{{Mapping|legend=5| 4 0 0 0 12 13 | 0 0 0 0 1 1 }}

[[Optimal tuning]]s:

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

 

 

=== 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=3| 1 1 8 9 11 | 0 4 -7 -8 -11 }}

 

* [[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 }}

 

 

[[Subgroup]]: 2.11.13.17.19

[[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}}

{{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=3| 4 0 7 | 0 1 1 }}

[[Optimal tuning]]s:

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

* [[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