Subgroup temperaments: Difference between revisions

A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  

For temperaments that omit various prime harmonics, see:

* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

This page uses an extended [[wart notation]] where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.  

== 2.3.… subgroups ==

Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  

It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.

[[Subgroup]]: 2.3.25

[[Comma list]]: 2048/2025

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

[[Optimal tuning]]s:  

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}

{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}

[[Badness]] (Sintel): 0.234

===== 2.3.25.23 subgroup =====

Subgroup: 2.3.25.23

Comma list: 576/575, 736/729

Optimal tunings:  

* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}

{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}

Badness (Sintel): 0.223

Sburb is every other step of [[pajara]], extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a [[59/1|59th harmonic]] by setting the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.

Comma list: 50/49, 64/63

{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}

Optimal tunings:  

* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}

Badness (Sintel): 0.253

Subgroup: 2.3.25.7.23

Comma list: 50/49, 64/63, 162/161

{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17 }}

Badness (Sintel): 0.419

Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})

{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}

{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}

 

 

 

 

 

 

 

 

 

 

 

 

 

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}

 

 

 

 

[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})

 

 

 

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}

 

 

 

 

[[Subgroup]]: 2.3.7/5.11/5.13/5

 

[[Comma list]]: [[196/195]] ({{monzo| 2 -1 2 0 -1 }}), [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[364/363]] ({{monzo| 2 -1 1 -2 1 }})

 

 

 

 

 

 

 

 

 

 

 

 

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}

 

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}

 

 

 

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

: mapping generators: ~2, ~26/15

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}

 

Fendo tempers out [[40/39]] in the 2.3.13/5 subgroup. This equates [[4/3]] with the tendo third [[13/10]]. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with [[father family #Father|father]], as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by [[User: VectorGraphics|Vector]] in 2025 as a contraction of ''fourth'' and ''tendo third''.  

In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.

[[5edo]] and [[13edo]] make good fendo tunings.  

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 40/39 ({{monzo| 3 -1 -1 }})

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

: mapping generators: ~2, ~3

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}}

{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}

: <nowiki/>* Wart for 13/5

=== Barbados ===

Barbados is the restriction of [[#Bridgetown|bridgetown]] to the 2.3.13/5 subgroup, and tempers out [[676/675]] alone. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 ({{monzo| 2 -3 2 }})

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

: mapping generators: ~2, ~26/15

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}}

{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}

[[Badness]] (Smith): 0.002335

* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9]

Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

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

: mapping generators: ~2, ~3

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})

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

: mapping generators: ~2, ~3

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

[[Badness]] (Sintel): 0.082

== 2.9.… subgroups ==

See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  

Baldy is every other step of [[schismatic family #Garibaldi|garibaldi]]. It is also the add-9 extension of [[no-threes subgroup temperaments #Frostburn|frostburn]]. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }}

: mapping generators: ~2, ~9

* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}}

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }}

[[Badness]] (Sintel): 0.274

Subgroup: 2.9.5.7.13

Comma list: 225/224, 325/324, 640/637

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}

Optimal tunings:  

* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}}

{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }}

RMS error: 0.5999 cents

==== Baldanders ====

Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }}

Optimal tunings:  

* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}}

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

Subgroup: 2.9.5.7.11.13

{{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}}

Badness (Sintel): 0.653

{{See also| Schismatic family #Tertiaschis }}

Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}

: mapping generators: ~2, ~33/20

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}}

[[Badness]] (Sintel): 0.439

{{See also| Dual-fifth temperaments #Dual-3 A-Team }}

A-team is every other step of [[gamelismic clan #Mothra|mothra]].

[[Subgroup]]: 2.9.5.21

[[Comma list]]: 81/80 ({{monzo| -4 2 -1 0 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})

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

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 468.9189{{c}}

 

{{Optimal ET sequence|legend=0| 18, 23, 41, 64, 87, 151, 238b′ }}

 

 

=== Mabon ===

 

[[Subgroup]]: 2.9.7

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

[[Subgroup]]: 2.9.7.11

{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}

{{Mapping|legend=5| 1 0 0 6 13 | 0 1/2 0 -1 -3 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1197.2650{{c}}, ~9/8 = 213.8957{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 214.9961{{c}}

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

Apparatus is the restriction of [[gamelismic clan #Revelation|revelation]] to the 2.9.7.11 subgroup, and every fourth step of [[breedsmic temperaments #Quasiorwell|quasiorwell]], without prime 5.  

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=3| 1 -14 1 -11 | 0 19 2 16 }}

{{Mapping|legend=5| 1 -7 0 1 -11 | 0 19/2 0 2 16 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.0088{{c}}, ~144/77 = 1084.4377{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/77 = 1084.4304{{c}}

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=5| 1 0 0 -1 6 | 0 5/2 0 6 -4 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.4843{{c}}, ~14/9 = 761.2079{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 761.4979{{c}}

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

Joan is the restriction of [[starling temperaments #Casablanca|casablanca]] to the 2.9.7.11 subgroup, as well as every other step of [[orwell]], without prime 5.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

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

{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 542.8446{{c}}

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115b′d }}

[[Badness]] (Sintel): 0.407

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

Named by {{u|Pailiaq}} in 2026, novisept is generated by a one-cent-flat [[9/7]], such that stacking 5 of them gives [[7/4]]. It can be formed by doubling both generator and period of [[gizzard]].

[[Subgroup]]: 2.9.7.13.17

[[Comma list]]: 442/441, 729/728, 833/832

{{Mapping|legend=3| 1 1 1 -1 3 | 0 6 5 13 3 }}

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 433.8360{{c}}

{{Optimal ET sequence|legend=1| 11, 25f, 36, 47, 83, 130, 213g, 296g }}

[[Badness]] (Sintel): 0.142

Genius tempers out the [[Alpharabian comma]] in the 2.9.11 subgroup; it is therefore every other step of [[no-fives subgroup temperaments #Alphaxenean|alphaxenean]]. It was named by {{u|Akselai}} in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: 131769/131072

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

[[Optimal tuning]]s:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 650.7321{{c}}

{{Optimal ET sequence|legend=1| 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee }}

[[Badness]] (Sintel): 0.198

Demon is a temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by {{u|CompactStar}} while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") [[mos]], known for its lack of representation of simple temperaments. The optimal tuning for the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament.

[[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.11

[[Comma list]]: 1331/1296

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

: mapping generators: ~2, ~18/11

* [[WE]]: ~2 = 1201.1151{{c}}, ~18/11 = 869.4724{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/11 = 869.3196{{c}}

[[Badness]] (Sintel): 0.332

== 4.3.… subgroups ==

{{Main| Tetrameantone }}

Named by [[Budjarn Lambeth]] in 2024, tetrameantone is the double-octave restriction of [[meantone family #Meantone|meantone]].

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

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

: mapping generators: ~4, ~3

* [[WE]]: ~4 = 2402.7812{{c}}, ~3 = 1898.4361{{c}}

* [[CWE]]: ~4 = 2400.0000{{c}}, ~3 = 1896.6512{{c}}

[[Optimal ET sequence]]: [[5ed4|a′5]], [[14ed4|a′14]], [[19ed4|a′19]], [[43ed4|a′43]], [[62ed4|a′62]], [[81ed4|a′81]], [[262ed4|a′262b]], [[343ed4|a′343b]]

[[Badness]] (Sintel): 0.0866

Named by {{u|CompactStar}} in 2023, tetrahanson is the double-octave restriction of [[kleismic family #Kleismic a.k.a. hanson|hanson]].

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

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

[[Optimal tuning]]s:

* [[CWE]]: ~4 = 2400.0000{{c}}, ~12/5 = 1517.0308{{c}}

[[Optimal ET sequence]]: [[8ed4|a′8]], [[11ed4|a′11]], [[19ed4|a′19]], [[68ed4|a′68]], [[87ed4|a′87]], [[106ed4|a′106]], [[511ed4|a′511]], [[617ed4|a′617]], [[723ed4|a′723]], [[829ed4|a′829]], [[935ed4|a′935c]], [[1764ed4|a′1764c]]

[[Badness]] (Sintel): 0.155

Named by [[Budjarn Lambeth]] in 2024, tetramagic is the double-octave restriction of [[magic family #Magic|magic]].

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

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

: mapping generators: ~4, ~5/4

* [[WE]]: ~4 = 2402.4898{{c}}, ~5/4 = 380.4527{{c}}

* [[CWE]]: ~4 = 2400.0000{{c}}, ~5/4 = 380.2194{{c}}

[[Optimal ET sequence]]: [[6ed4|a′6]], [[13ed4|a′13]], [[19ed4|a′19]], [[63ed4|a′63]], [[82ed4|a′82]], [[101ed4|a′101]], [[120ed4|a′120]], [[221ed4|a′221c]], [[341ed4|a′341c]]

[[Badness]] (Sintel): 0.459

Named by [[Budjarn Lambeth]] in 2024, blacktetra is the double-octave restriction of [[blackwood family #Blackwood|blackwood]].

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

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

[[Optimal tuning]]s:  

* [[CWE]]: ~4/3 = 480.0000{{c}}, ~5/4 = 395.1255{{c}} (~16/15 = 84.7745{{c}})

[[Optimal ET sequence]]: [[5ed4|a′5]], [[20ed4|a′20]], [[25ed4|a′25]], [[30ed4|a′30]]

[[Badness]] (Sintel): 0.748

== 4.6.… subgroups ==

=== Meanquad ===

Meanquad is the double-octave restriction of [[meantone family #Meantone|meantone]] to the 4.6.5 subgroup. Similarly to meantone, it has a flat [[3/2]] as generator and equates [[5/4]] and [[81/64]]. It possesses [[mos scale]]s of the forms [[3L 4s (4/1-equivalent)|3L 4s<4/1>]], [[7L 3s (4/1-equivalent)|7L 3s<4/1>]], and [[7L 10s (4/1-equivalent)|7L 10s<4/1>]], which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively.  

[[17ed4]] is one of the smallest tunings that [[support]] this temperament and features the decatonic scale. Like [[12edo]], it can be viewed as the boundary between meanquad and [[#quarchy|quarchy]], the 4.6.7-subgroup restriction of [[archy]]. Larger options include 24ed4 (12edo), [[31ed4]] and [[55ed4]].

[[Subgroup]]: 4.6.5

[[Comma list]]: 81/80 ({{monzo| -4 4 -1 }})

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

* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 696.6512{{c}}

[[Optimal ET sequence]]: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[31ed4|a′31]], [[100ed4|a′100]], [[131ed4|a′131]], [[293ed4|a′293*]]

[[Badness]] (Sintel): 0.0146

==== Tetrominant ====

Subgroup: 4.6.5.7

Comma list: 36/35 ({{monzo| 0 2 -1 -1 }}), 64/63 ({{monzo| 4 -2 0 -1 }})

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

Optimal tunings:  

* Subgroup WE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}

* Subgroup CWE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}

Optimal ET sequence: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[41ed4|a′41]], [[65ed4|a′65cd]]

Badness (Sintel): 0.0974

[[Comma list]]: 64/63 ({{monzo| 4 -2 -1 }})

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

: mapping generators: ~4, ~6

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2393.9105{{c}}, ~3/2 = 707.5215{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 709.3901{{c}}

Optimal ET sequence: [[7ed4|a′7]], [[10ed4|a′10]], [[17ed4|a′17]], [[27ed4|a′27]], [[44ed4|a′44]], [[159ed4|a′159d]], [[203ed4|a′203*dd]]

: <nowiki/>* Wart for 6

Badness (Sintel): 0.0134

Named by {{u|CompactStar}} in 2023, meansquared tempers out the double syntonic comma, [[6561/6400]], in the 4.9.25 subgroup.