24576/24565
| Interval information |
archagallisma
24576/24565, the mavka comma or archagallisma, is an unnoticeable 17-limit comma that represents the difference between two adjacent square superparticulars – 289/288 and 256/255, therefore, it is the amount by which a stack of three 17/16's fall short of a 6/5 minor third. It is also the amount by which a stack of two 128/85's octave-reduced exceeds 17/15 and the amount by which a stack of three 85//64's octave-reduced falls short of 75/64.
It can be factored into 4096/4095 × 4914/4913.
Temperaments
Mavka
By tempering out this comma in the full 17-limit, the rank-6 mavka temperament is defined. You may find a list of good equal temperaments supporting it below.
Subgroup: 2.3.5.7.11.13.17
Mapping:
[⟨1 0 1 0 0 0 4]
⟨0 1 1 0 0 0 0]
⟨0 0 3 0 0 0 -1]
⟨0 0 0 1 0 0 0]
⟨0 0 0 0 1 0 0]
⟨0 0 0 0 0 1 0]]
Archagall
2.75.85 (MVP archagall)
By tempering the comma S16/S17 = 24576/24565 out 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 simplicity (mapping-wise) 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 following subgroup:
Subgroup: 2.75.85
Comma list: [13 1 -3⟩ = 24576/24565
Sval mapping: [⟨1 2 5]], ⟨0 3 1]]
Sval mapping generators: ~2, ~85/32
Optimal GPV sequence: Template:Val list
2.75.85.9/7 subgroup
A fairly natural way to extend archagall is by tempering S15/S17 which (because of how semiparticulars work) equates 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 S15/S16 instead (equating three 16/15's with 17/14), however it is unclear whether 16/15 can even be reached so it is preferred to think of it as adding S15/S17 = 2025/2023. If you do want to reach 16/15 look to the next extension listed here that includes prime 5.
Mapping:
[⟨1 2 5 6]
⟨0 -4 3 1]]
Commas: [13 1 -3 0⟩ = 24576/24565, Template:Monzo 2 -2 0 1 = 2025/2023
Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171
It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension.
2.3.5.7.17 subgroup (prime archagall)
We may observe that in a good tuning of archagall there is an accurate 5/4 at +13 fourths (85/64's) minus five octaves (2/1's). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to 171edo for which 171edo is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = 1225/1224 and (S18/S20)/S49 = 5832/5831 while not tempering any of {S49, S50, S18/S20} individually. Note that 171edo is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it.
Commas: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35
Mapping:
[⟨1 11 -3 20 9]
⟨0 -23 13 -42 -12]]
Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364
Srutal archagall
2.3.5.17 subgroup (MVP srutal archagall)
Named because this lower-accuracy temperament is also an extension of (the 5-limit) srutal temperament that adds prime 17 (and which thereby is able to express the harmonics 75 and 85 in their appropriate prime subgroup). It achieves this by equating 85/64 with 4/3 by tempering their difference of S16 = 256/255. Therefore it also tempers S17 = 289/288 and thus equates 17/15 with 9/8 due to tempering S16 × S17. It could be described as the 10 & 12 temperament (with strong emphasis on 12edo being the better tuning) on the following prior-discussed subgroup:
Subgroup: 2.3.5.17
Comma list: 256/255, 289/288
Sval mapping: [⟨2 0 11 5]], ⟨0 1 -2 1]]
Sval mapping generators: ~17/12, ~3
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
Optimal GPV sequence: Template:Val list
Badness: 0.00575