Subgroup temperaments: Difference between revisions
m →MVP archagall: comma |
m + link |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 508: | Line 508: | ||
== 2.75.85 subgroup == | == 2.75.85 subgroup == | ||
=== | === Archagall === | ||
{{See also| Fifthplus }} | |||
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. | 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. | ||
| Line 526: | Line 528: | ||
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760* }} | {{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 = | ==== 2.75.9/7.85 subgroup ==== | ||
A fairly natural way to extend 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. | |||
A fairly natural way to extend | |||
Subgroup: 2.75.9/7.85 | |||
Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }}) | |||
{{ | Subgroup-val mapping: {{mapping| 1 2 6 5 | 0 3 -4 1 }} | ||
Optimal tunings: | |||
* | * Subgroup WE: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}} | ||
* | * Subgroup CWE: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }} | ||
== 4.3.5 subgroup == | == 4.3.5 subgroup == | ||
| Line 2,026: | Line 2,026: | ||
= Related temperament collections = | = Related temperament collections = | ||
* [[Dual-fifth temperaments]] | * [[Dual-fifth temperaments]] | ||
* [[Irrational-subgroup temperaments]] | |||
* [[Equalizer subgroup]] temperaments | * [[Equalizer subgroup]] temperaments | ||
* [[Substitute harmonic]] temperaments | * [[Substitute harmonic]] temperaments | ||