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Subgroup temperaments: Difference between revisions

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== 2.3.… subgroups ==
== 2.3.… subgroups ==
=== Shrub ===
=== Shrub ===
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.  
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.  


==== 2.3.25 subgroup ====
==== 2.3.25 subgroup ====
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Badness (Sintel): 0.450
Badness (Sintel): 0.450


=== Darian calendar ===
=== Hypnosis ===
Darian calendar is described as 24 & 668 temperament in the 2.3.11.19 [[subgroup]] and is named after a certain calendar layout by the same name. The generator is close to the [[36/35]] quartertone, and this allows an extension to the 2.3.35.11.19 subgroup. 5 of them make [[11/8]], 8 of them make [[3/2]], and 6 of them make [[32/19]].
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].  


==== 2.3.11.19 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13
The temperament is simplest in this subgroup, although there is a tradeoff of breaking up the simplicity of the 36/35 quartertone.


[[Subgroup]]: 2.3.11.19
[[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| 4 5 13 18 | 0 8 5 -6 }}
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9


: sval mapping generators: ~6291456/5285401, ~25289/24576
[[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 tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
=== Darian calendar ===
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].


==== 2.3.35.11.19 subgroup ====
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.


Subgroup: 2.3.35.11.19
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.  


Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
[[Subgroup]]: 2.3.35.11.19


: sval mapping generators: ~2240/1881, ~36/35
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1  0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})


Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
[[Optimal tuning]]s:  
 
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
=== Hypnosis ===
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]
 
[[Subgroup]]: 2.3.7.11/5.13
 
[[Comma list]]: 169/168, 540/539, 729/728
 
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
 
[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518


{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents


=== Hydrothermal ===
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.


[[Subgroup]]: 2.3.7/5
[[Subgroup]]: 2.3.7/5


[[Comma list]]: [[50/49]]
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})


{{Mapping|legend=3| 2 3 1 | 0 1 0 }}
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}


[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}


[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}
<nowiki/>* Wart for 7/5


=== Argentic ===
=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].  


[[Subgroup]]: 2.3.7/5
[[Subgroup]]: 2.3.7/5


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


{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
<small> based on subgroup TE </small>


Badness (Sintel): 0.119
[[Badness]] (Sintel): 0.119


==== Edson ====
==== Edson ====
{{See also| Chromatic pairs #Edson }}
{{See also| Chromatic pairs #Edson }}


Edson is related to [[pele]] and [[andromeda]].  
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  


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


{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3


{{Mapping|legend=5| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


{{Optimal ET sequence|legend=1| 12, 17, 29 }}
{{Optimal ET sequence|legend=1| 12, 17, 29 }}
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{{See also| Chromatic pairs #Haumea }}
{{See also| Chromatic pairs #Haumea }}


Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].  
Related temperaments include [[#Bridgetown|bridgetown]], [[aberschismic family #Namaka|namaka]], [[schismatic family #Hemigari|hemigari]], [[#Barbados|barbados]], and [[the Archipelago #Parizekmic|parizekmic]].  


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


[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]
[[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=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


{{Mapping|legend=5| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]
: mapping generators: ~2, ~26/15


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491
[[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, 565d, 763bd, 961bd }}
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents
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=== Historical ===
=== Historical ===
{{Distinguish|Historical temperaments}}
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.
{{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 S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.
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
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Comma list]]: 364/363, 441/440, 1001/1000
[[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|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016
[[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, 72, 101, 130, 159 }}
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}


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


=== Direct breedsmic ===
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]
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]].


[[Subgroup]]: 2.3.49/5
[[Subgroup]]: 2.3.49/5


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


{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
: mapping generators: ~2, ~49/40


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966
[[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}}


{{Optimal ET sequence|legend=1|7, 10, 17}}
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}


[[Tp tuning #T2 tuning|RMS error]]: ?
<nowiki/>* Wart for 49/5


==== Bridgetown ====
=== Bridgetown ===
{{See also| Chromatic pairs #Bridgetown }}
{{See also| Chromatic pairs #Bridgetown }}


Bridgetown, the 5 &amp; 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].  


[[Subgroup]]: 2.3.11/5.13/5
[[Subgroup]]: 2.3.11/5.13/5


[[Comma list]]: [[352/351]], [[676/675]]
[[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=3| 1 0 -6 -1 | 0 2 9 3 }}


{{Mapping|legend=5| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]
: mapping generators: ~2, ~26/15


[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399
[[Optimal tuning]]s:
* [[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, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
 
<nowiki/>* Wart for 11/5


[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


=== Blackweed ===
=== 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.
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
[[Subgroup]]: 2.3.11/7


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


{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
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{{See also| Chromatic pairs #Pepperoni }}
{{See also| Chromatic pairs #Pepperoni }}


Pepperoni is generated by a fifth and can be described as the 5 &amp; 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.
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
[[Subgroup]]: 2.3.11/7.13/7


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


{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}


{{Mapping|legend=5| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
{{Mapping|legend=5| 1 0 0 -19/3 2/3 17/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*<sup>†</sup> }}
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}
: <nowiki />* wart for 11/7
: <sup>†</sup> wart for 13/7


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