Subgroup temperaments: Difference between revisions

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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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{{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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: mapping generators: ~2, ~2250/2197

~~{{Todo|inline=1| comment=Optimal tunings and error maps showed below is not yet precise enough.}}~~

Optimal tunings:

* WE: ~2 = 1200.~~000~~{{c}}, ~2250/2197 = 41.~~291~~{{c}}

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

: error map: {{val| +0.~~000~~ +0.~~001~~ -0.~~007~~}}

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

* CWE: ~2 = ~~1200.000{{c}}~~, ~2250/2197 = 41.~~292~~{{c}}

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

: error map: {{val| +0.~~000~~ +0.~~001~~ -0.~~007~~}}

: 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

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Optimal tunings:

* WE: ~2 = 1199.~~993~~{{c}}, ~169/165 = 41.~~292~~{{c}}

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

: error map: {{val| -0.~~007~~ -0.~~000~~ +0.~~157~~ -0.~~010~~}}

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

* CWE: ~2 = ~~1200.000{{c}}~~, ~169/165 = 41.~~292~~{{c}}

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

: error map: {{val| +0.~~000~~ +0.~~005~~ +0.~~163~~ -0.~~005~~}}

: 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

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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

@@ Line 88: / Line 88: @@
 === 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
@@ Line 506: / Line 506: @@
 {{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 ==
@@ Line 1,704: / Line 1,738: @@
 {{Mapping|legend=2| 1 1 1 | 0 17 11 }}
 : mapping generators: ~2, ~2250/2197
-{{Todo|inline=1| comment=Optimal tunings and error maps showed below is not yet precise enough.}}
 Optimal tunings:
-* WE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.291{{c}}
+* WE: ~2 = 1200.0001354{{c}}, ~2250/2197 = 41.2914993{{c}}
-: error map: {{val| +0.000 +0.001 -0.007}}
+: error map: {{val| +0.0001354 +0.0006234 -0.0073197}}
-* CWE: ~2 = 1200.000{{c}}, ~2250/2197 = 41.292{{c}}
+* CWE: ~2 = 2/1, ~2250/2197 = 41.2915011{{c}}
-: error map: {{val| +0.000 +0.001 -0.007}}
+: 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
@@ Line 1,727: / Line 1,759: @@
 Optimal tunings:
-* WE: ~2 = 1199.993{{c}}, ~169/165 = 41.292{{c}}
+* WE: ~2 = 1199.9934923{{c}}, ~169/165 = 41.2918271{{c}}
-: error map: {{val| -0.007 -0.000 +0.157 -0.010}}
+: error map: {{val| -0.0065077 -0.0004485 +0.1565720 -0.0103579}}
-* CWE: ~2 = 1200.000{{c}}, ~169/165 = 41.292{{c}}
+* CWE: ~2 = 2/1, ~169/165 = 41.2917463{{c}}
-: error map: {{val| +0.000 +0.005 +0.163 -0.005}}
+: 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
@@ Line 1,791: / Line 1,823: @@
 === 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