Subgroup temperaments: Difference between revisions

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.

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

== 2.3.25 subgroup ==

This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Comma list: [[2048/2025]]

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

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

==== 2.3.23.25.41 subgroup ====

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

== 2.9.5.11 subgroup ==

{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

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

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

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

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

* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==

Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

[[Subgroup]]: 2.9.7.11

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

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

[[Badness]]: 0.00263

{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].  

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

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

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

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

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

Machine is every other step of [[supra]], most interesting for its scale patterns.  

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

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

: sval mapping generators: ~2, ~9

 

: [[gencom]]: [2 8/7; 64/63 99/98]

* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128

* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

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

=== Penta a.k.a. mechanism ===

[[Subgroup]]: 2.9.7.11

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

: sval mapping generators: ~2, ~14/9

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

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

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

[[Badness]]: 0.00439

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

== 2.9.7.13.17 subgroup ==

Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].

[[Subgroup]]: 2.9.7.13.17

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

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

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836

Badness (Dirichlet): 0.142

Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] 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 [[User:CompactStar|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 demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

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

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

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

=== Genius ===

Named 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]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

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

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

: sval mapping generators: ~2, ~14/9

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

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====

Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==

A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

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

: sval mapping generators: ~2, ~21/16

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

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====

Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

 

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

 

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

 

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

By tempering out the comma [[24576/24565]] in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.  

[[Subgroup]]: 2.75.85

[[Optimal tuning]]s:  

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }}

 

== 2.75.9/7.85 subgroup ==

A fairly natural way to extend [[#MVP archagall|MVP archagall]] is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.  

 

[[Subgroup]]: 2.75.9/7.85

[[Comma list]]: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})

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

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.  

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

: mapping generators: ~2, ~7/6

* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455

* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

=== Argentic ===

Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].  

[[Subgroup]]: 2.3.7/5

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

 

 

 

 

 

 

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

[[Comma list]]: 64/63, 91/90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Badness]] (Sintel): 0.005

This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Comma list: [[323/322]], [[392/391]]

Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}

* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}

* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}

Badness (Sintel): 0.029

==== 2.25/7.17/7.19/7.23/7 subgroup ====

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]], [[476/475]]

Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}

* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}

* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.053