Subgroup temperaments: Difference between revisions

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For temperaments that omit various prime harmonics, see:

* [[No-thirteens subgroup temperaments]]

* [[No-elevens subgroup temperaments]]

* [[No-sevens subgroup temperaments]]

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= Composite subgroup temperaments =

== 2.3.35 subgroup ==

=== Darian calendar ===

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]].

==== 2.3.11.19 subgroup ====

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

: sval mapping generators: ~6291456/5285401, ~25289/24576

[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257

[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...

==== 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.

Subgroup: 2.3.35.11.19

Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}

: sval mapping generators: ~2240/1881, ~36/35

Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288

[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...

== 2.9.5.7 subgroup ==

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

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=== Daemotertiaschis ===

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, which is notoriously difficult to approximate with simple JI or RTT methods.

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

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[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== ~~Baldi~~ ===

=== Baldy ===

~~Baldi~~ 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.

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.

[[Subgroup]]: 2.9.5.7

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===== Sburb =====

This temperament sets the 413th harmonic (~~octave-reduced~~) to the diminished seventh.

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

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=== Novisept ===

Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4.

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

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

Badness (Dirichlet): 0.142

== 2.9.11 subgroup ==

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

== 2.75.85 subgroup ==

=== MVP archagall ===

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

[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})

: mapping generators: ~2, ~85/32

[[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 ==

=== Archagall ===

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 generators: ~2, ~85/32

[[Optimal tuning]]s:

* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}

* [[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* }}

== 4.3.5 subgroup ==

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== 2.….11/7.… subgroups ==

=== 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]]: {{monzo| 8 -5 }} (256/243)

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

[[Optimal tuning]]s:

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

: [[error map]]: {{val| -5.746 +8.852 -0.035 }}

* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}

: error map: {{val| 0.000 +18.045 +2.475 }}

=== Pepperoni ===

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Scales: [[Oceanfront scales]]

=== Seventeen-cot ===

Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.

====2.3.13/5 subgroup====

Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)

edo join: 29 & 146

: mapping generators: ~2, ~2250/2197

Optimal tunings:

* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}

: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}

* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}

: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}

edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262

Badness (Sintel): 0.064

====2.3.11/5.13/5 subgroup====

Comma basis: 225000/224939, 43940/43923

edo join: 29 & 146

: mapping generators: ~2, ~169/165

Optimal tunings:

* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}

: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}

* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}

: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}

edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233

Badness (Sintel): 0.080

== 2.….49/5.… subgroups ==

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=== Supramin ===

This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].

This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[cohemimabila]].

[[Subgroup]]: 2.17/7.19/7

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[[Category:Subgroup temperaments| ]]

[[Category:Temperament collections]]

[[Category:Rank 2]]

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 For temperaments that omit various prime harmonics, see:
-* [[No-thirteens subgroup temperaments]]
 * [[No-elevens subgroup temperaments]]
 * [[No-sevens subgroup temperaments]]
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 = Composite subgroup temperaments =
+== 2.3.35 subgroup ==
+=== Darian calendar ===
+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]].
+==== 2.3.11.19 subgroup ====
+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
+{{Mapping|legend=2| 4 5 13 18 | 0 8 5 -6 }}
+: sval mapping generators: ~6291456/5285401, ~25289/24576
+[[Optimal tuning]] ([[CTE]]): ~6291456/5285401 = 1\4, ~25289/24576 = 50.257
+[[Support]]ing [[ET]]s: {{EDOs|24, 596, 620, 644, 668, 692, 716}}, ...
+==== 2.3.35.11.19 subgroup ====
+edo 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
+Sval mapping: {{mapping| 4 0 5 13 18 | 0 1 8 5 -6 }}
+: sval mapping generators: ~2240/1881, ~36/35
+Optimal tuning (CTE): ~2240/1881 = 1\4, ~36/35 = 50.288
+[[Support]]ing [[ET]]s: {{EDOs|24, 668}}, ...
 == 2.9.5.7 subgroup ==
 See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].
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 === 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, which is notoriously difficult to approximate with simple JI or RTT methods.
+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
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 [[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}
-=== Baldi ===
+=== Baldy ===
 {{See also|Schismatic family #Garibaldi}}
 {{See also|No-threes subgroup temperaments #Frostburn}}
-Baldi 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.
+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.
 [[Subgroup]]: 2.9.5.7
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 ===== Sburb =====
-This temperament sets the 413th harmonic (octave-reduced) to the diminished seventh.
+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
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 === Novisept ===
-Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4.
+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
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 [[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836
+Badness (Dirichlet): 0.142
 == 2.9.11 subgroup ==
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 {{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}
+== 2.75.85 subgroup ==
+=== MVP archagall ===
+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
+[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})
+{{Mapping|legend=2| 1 2 5 | 0 3 1 }}
+: mapping generators: ~2, ~85/32
+[[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 ==
+=== Archagall ===
+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 }}
+: mapping generators: ~2, ~85/32
+[[Optimal tuning]]s:
+* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
+* [[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* }}
 == 4.3.5 subgroup ==
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 == 2.….11/7.… subgroups ==
+=== 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]]: {{monzo| 8 -5 }} (256/243)
+{{Mapping|legend=2| 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}}
+: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
+* [[Tp tuning|subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
+: error map: {{val| 0.000 +18.045 +2.475 }}
+{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}
 === Pepperoni ===
 {{Main| Parapyth }}
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 Scales: [[Oceanfront scales]]
+=== Seventeen-cot ===
+Seventeen-cot is a rank-2 temperament in the 2.3.13/5 and 2.3.11/5.13/5 subgroups. It tempers out the [[Tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~2250/2197 or ~169/165 generator which is a 17th of a ~3/2 perfect fifth. It can be described as the 29 & 146 temperament in these subgroups.
+====2.3.13/5 subgroup====
+Comma basis: {{monzo| -6 -11 17 }} (2.3.13/5)
+edo join: 29 & 146
+{{Mapping|legend=2| 1 1 1 | 0 17 11 }}
+: mapping generators: ~2, ~2250/2197
+Optimal tunings:
+* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
+: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
+* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
+: error map: {{val| 0.0000000 +0.0005177 -0.0074359}}
+edos: 29, 465, 494, 436, 523, 407, 378, 349, 30[-3], 28[+3], 320, 291, 59[-3], 262
+Badness (Sintel): 0.064
+====2.3.11/5.13/5 subgroup====
+Comma basis: 225000/224939, 43940/43923
+edo join: 29 & 146
+{{Mapping|legend=2| 1 1 1 1 | 0 17 4 11 }}
+: mapping generators: ~2, ~169/165
+Optimal tunings:
+* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
+: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
+* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
+: error map: {{val| 0.0000000 +0.0046870 +0.1627569 -0.0043781}}
+edos: 29, 465, 494, 436, 523, 407, 378, 349, 320, 291, 30[-3], 262, 28[+3], 233
+Badness (Sintel): 0.080
 == 2.….49/5.… subgroups ==
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 === Supramin ===
-This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].
+This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[cohemimabila]].
 [[Subgroup]]: 2.17/7.19/7
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 [[Category:Subgroup temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
+[[Category:Rank 2]]
 {{Todo| review | cleanup }}