No-threes subgroup temperaments: Difference between revisions

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

* 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.

== Temperaments with a 2.5.7 gene ==

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

 

{{See also| Chromatic pairs #Augment }}

[[Comma list]]: 128/125

 

{{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 }}

==== 2.5.7.11 subgroup ====

Comma list: 56/55, 128/125

 

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

Optimal tunings:  

* 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 }}

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 }}

 

 

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

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

 

{{Optimal ET sequence|legend=1| 7, 9, 16, 25, 41, 66, 305ccdd, 371ccddd }}

 

 

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.

 

 

 

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

 

 

 

 

Subgroup: 2.5.7.11.13

 

 

 

* WE: ~2 = 1199.9194{{c}}, ~1664/1225 = 528.9681{{c}}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* [[WE]]: ~2 = 1200.3462{{c}}, ~28/25 = 204.3386{{c}}

: [[error map]]: {{val| +0.346 -2.630 +2.189 }}

: error map: {{val| 0.000 -3.125 +1.755 }}

[[Badness]] (Sintel): 0.886

==== 2.5.7.11 subgroup ====

Subgroup: 2.5.7.11

 

 

 

{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}

 

Badness (Sintel): 0.463

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=3| 1 2 3 | 0 5 -3 }}

: mapping generators: ~2, ~256/245

[[Optimal tuning]]s:

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

: [[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 }}

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]]

[[Subgroup]]: 2.5.7

[[Comma list]]: 4194304/4117715

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

: mapping generators: ~2, ~8/7

 

{{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:  

* 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 }}

 

==== 2.5.7.11.13.17 subgroup ====

Subgroup: 2.5.7.11.13.17

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 }}

Optimal tunings:  

* 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]].  

[[Subgroup]]: 2.5.7

[[Comma list]]: 823543/819200

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

{{Mapping|legend=5| 1 0 3 3 | 0 0 -7 -2 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.2169{{c}}, ~343/320 = 116.3116{{c}}

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

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

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

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 }}

Optimal tunings:  

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

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

==== 2.5.7.13.17 subgroup ====

Subgroup: 2.5.7.13.17

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

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

{{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

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 }}

Optimal tunings:  

* 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}}

* [[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

{{Mapping|legend=2| 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

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:  

* 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 }}

 

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]].

 

 

 

 

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9911{{c}}, ~{{monzo| -380 159 4 }} = 505.7532{{c}}

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

 

 

 

 

[[Comma list]]: 5021863/5000000

 

 

 

 

 

 

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

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 16, 21, 29, 37, 87, 103, 124, 161, 227, 264, 565e, 689e }}

==== 2.5.7.11.13.17 subgroup ====

=== Sephiroth ===

 

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

[[Comma list]]: 34375/32768

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Optimal tunings:  

* 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 }}

: [[error map]]: {{val| +0.366 -1.697 +1.252 }}

* [[CWE]]: ~125/88 = 600.0000{{c}}, ~5/4 = 383.9977{{c}}

{{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 }}

* [[WE]]: ~2 = 1201.1238{{c}}, ~55/32 = 928.5003{{c}}

: [[error map]]: {{val| +1.124 -0.813 -2.700 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~55/32 = 927.7384{{c}}

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

=== Movila ===

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 generators: ~2, ~11/8

* [[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 }}

{{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 }}

* [[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 }}

: mapping generators, ~2, ~20/13

* [[WE]]: ~2 = 1199.5925{{c}}, ~20/13 = 740.6286{{c}}

: [[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

== 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]].

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

: mapping generators: ~2, ~7

 

: [[error map]]: {{val| -0.015 +0.051 -0.010 }}

 

{{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]].

 

 

 

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

 

 

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

 

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.

It was named by [[CompactStar]] in 2026, referring to ''{{w|Scaevola}}'' (the genus of flowering plants), which is meant to sound reminiscent of ''score'' and ''mavila'' (the score and joan temperaments were actually named prior to this temperament).

 

 

 

 

 

 

 

==== 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:

* WE: ~2 = 1201.5484{{c}}, ~11/8 = 540.7963{{c}}

* 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: 2.7.11.13

Comma list: 1573/1568, 15488/15379

{{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 }}

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 }}

: [[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 }}

* [[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 }}

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

: mapping generators, ~2, ~64/53

[[Optimal tuning]]s:  

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

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

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

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

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 141, 167, 193p, 219p, 245p }}

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]]: 1352/1331

 

 

 

[[Optimal tuning]]s:

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

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

 

[[Badness]] (Sintel): 0.175

{{Distinguish| Bluebird }}

{{See also| Chromatic pairs #Bluebirds }}

[[Subgroup]]: 2.11.13

[[Comma list]]: 265837/262144

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

 

[[Optimal tuning]]s:

: [[error map]]: {{val| +0.880 -1.174 -2.013 }}

: error map: {{val| 0.000 -4.002 -5.405 }}

 

 

[[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 }}

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

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

{{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]].  

 

 

{{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 }}

 

Arsenic was named by [[Xenllium]] in 2026 after the 33rd chemical element.  

[[Subgroup]]: 2.11.13.17.19

{{Mapping|legend=3| 33 0 8 249 26 | 0 1 1 -1 1 }}

[[Optimal tuning]]s:  

* [[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 }}

: mapping generators: ~44/37, ~11

[[Optimal tuning]]s:

* [[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 }}