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

 

 

[[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=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}

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

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

{{See also| Chromatic pairs #Haumea }}

Haumea is the add-7/5 extension of [[#Bridgetown|bridgetown]], which is in turn the add-11/5 extension of [[#Barbados|barbados]]. Other related temperamnets include [[schismatic family #Hemigari|hemigari]], [[the Archipelago #Parizekmic|parizekmic]], and [[aberschismic family #Namaka|namaka]].

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

[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})

{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}

: mapping generators: ~2, ~26/15

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}

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

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

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

{{Distinguish|Historical temperaments}}

{{Distinguish|History (temperament)}}

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.

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

[[Comma list]]: 364/363 ({{monzo| 2 -1 1 -2 1 }}), 441/440 ({{monzo| -3 2 2 -1 0 }}, 1001/1000 ({{monzo| -3 0 1 1 1 }})

: mapping generators: ~2, ~21/20

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}

{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].  

[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})

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

: mapping generators: ~2, ~49/40

[[Optimal tuning]]:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}

: <nowiki/>* Wart for 49/5

{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})

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

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

: mapping generators: ~55/39, ~26/15

* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}

{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

[[Subgroup]]: 2.3.11.13/5.17  

[[Comma list]]: 221/220 ({{monzo| -2 0 -1 1 1 }}), 243/242 ({{monzo| -1 5 -2 0 0 }}), 289/288 ({{monzo| -5 -2 0 0 2 }})

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

* [[Tp tuning|subgroup WE]]: ~17/12 = 600.1781{{c}}, ~26/15 = 950.7656{{c}} (~15/13 = 249.2344{{c}})

* [[Tp tuning|subgroup CWE]]: ~17/12 = 600.0000{{c}}, ~26/15 = 950.6011{{c}} (~15/13 = 249.3989{{c}})

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}

: <nowiki/>* Wart for 13/5

=== Bridgetown ===

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of [[#Haumea|haumea]] and the add-11/5 [[extension]] of [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})

{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}

: mapping generators: ~2, ~26/15

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}

{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}

: <nowiki/>* Wart for 11/5

=== Seventeen-cot ===

Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026.

[[Subgroup]]: 2.3.11/5.13/5

[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})

: mapping generators: ~2, ~169/165

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}}

: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}}

: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }}

{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161… }}

[[Badness]] (Sintel): 0.080

=== Blackweed ===

Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out [[256/243]] alike but in the 2.3.11/7 subgroup. [[20edo]] is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.

[[Subgroup]]: 2.3.11/7

[[Comma list]]: 256/243 ({{monzo| 8 -5 0 }})

{{Mapping|legend=3| 5 8 0 | 0 0 1 }}

: mapping generators: ~9/8, ~11/7

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}

{{Main| Parapyth }}

{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain [[retraction]] of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

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

 

 

 

 

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

 

 

 

 

 

 

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

 

 

 

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.

 

 

 

 

 

 

* ''[[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 ===

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 generators: ~2, ~3

[[Optimal tuning]]s:

* [[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]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}

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

: <nowiki/>* Wart for 19/7

[[Badness]] (Sintel): 0.082

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

{{See also| Chromatic pairs #Baldy }}

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

[[Optimal tuning]]s:  

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

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

[[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 }}

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

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

Badness (Sintel): 0.386

RMS error: 0.5999 cents

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:  

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

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

Badness (Sintel): 0.389

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{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}}

{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }}

Badness (Sintel): 0.653

=== Daemotertiaschis ===

{{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 }}

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}}

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

{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459… }}

{{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 }}

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.9303{{c}}, ~21/16 = 464.7465{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 464.5029{{c}}

{{Optimal ET sequence|legend=1| 13, 18, 31 }}

[[Badness]] (Sintel): 0.177

===== 2.9.5.21.11 subgroup =====

Subgroup: 2.9.5.21.11

Comma list: 81/80 ({{monzo| -4 2 -1 0 }}), 99/98 ({{monzo| -1 2 0 -2 1 }}), 385/384 ({{monzo| -7 -1 1 1 1 }})

{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

Optimal tunings:  

* Subgroup WE: ~2 = 1201.3979{{c}}, ~21/16 = 464.6020{{c}}

* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 464.1242{{c}}

{{Optimal ET sequence|legend=0| 13, 31, 44, 75b′ }}

Badness (Sintel): 0.213

{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

: mapping generators: ~2, ~9

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1202.6916{{c}}, ~9/8 = 186.1446{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 186.1740{{c}}

{{Optimal ET sequence|legend=1| 6, 13, 45b′f }}

[[Badness]] (Sintel): 0.281

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[Magic family #Magic|magic]].  

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224 ({{monzo| -5 0 2 -1 }}), 245/243 ({{monzo| 0 -3 1 2 }})

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

: mapping generators: ~2, ~14/9

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1201.0786{{c}}, ~14/9 = 761.3878{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 760.9151{{c}}

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 71, 112* }}

[[Badness]] (Sintel): 0.0884

 

 

Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 }}), 225/224 ({{monzo| -5 0 2 -1 0 }}), 245/243 ({{monzo| 0 -3 1 2 0 }})

 

 

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

Optimal tunings:  

* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 761.4017{{c}}

 

 

 

==== 2.9.15.7.11.13 ====

Subgroup: 2.9.15.7.11.13

 

Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 0 }}), 105/104 ({{monzo| -3 0 1 1 0 -1 }}), 144/143 ({{monzo| 4 1 0 0 -1 -1 }}), 196/195 ({{monzo| 2 0 -1 2 0 -1 }})

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Subgroup]]: 2.9.7.11

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 0.199

 

 

[[Subgroup]]: 2.9.7.11

 

 

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

 

 

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

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

 

 

 

 

 

 

 

 

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

 

 

 

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

[[Comma list]]: 131769/131072

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

: mapping generators: ~2, ~16/11

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.3181{{c}}, ~16/11 = 650.8508{{c}}

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

[[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.

[[Comma list]]: 1331/1296

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

: mapping generators: ~2, ~18/11

[[Optimal tuning]]s:  

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

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

{{Optimal ET sequence|legend=1| 4, 7, 11, 18, 29, 76e }}

[[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]].  

[[Comma list]]: 81/80

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

: mapping generators: ~4, ~3

[[Optimal tuning]]s:  

* [[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

{{Main| Tetrahanson }}

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

[[Comma list]]: 15625/15552

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

: mapping generators: ~4, ~12/5

[[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

[[Optimal tuning]]s:

* [[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 }}

: mapping generators: ~4, ~5

* [[WE]]: ~4/3 = 477.7018{{c}}, ~5/4 = 397.6806{{c}} (~16/15 = 80.0212{{c}})

* [[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

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]].

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

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

: mapping generators: ~4, ~6