24576/24565

24576/24565, the mavka comma or archagallisma, is an unnoticeable 17-limit comma that represents the difference between 256/255 and 289/288 – two adjacent square superparticulars, making it an ultraparticular, and identifies itself as 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

Tempering out this comma in the full 17-limit results in the rank-6 mavka a.k.a. archagallismic temperament. Tempering it out in the 2.3.5.17 subgroup results in the rank-3 archagallic temperament. If we restrict it to the 2.75.85 subgroup, we get the rank-2 archagall temperament. You may find a list of good equal temperaments supporting them below. The rank-6 temperament can be thought of as being equivalent to the 17-limit with the exception that 5/4 is reached by going down by 17/16 three times, starting at 3/2. In other words, 5/4 = (3/2)/(17/16)³. Similarly, archagallic can be thought of as the 2.3.5.17 subgroup with that same equivalence (so that it is essentially being expressed through 2.3.17). Archagall has its own, more complex mapping of prime 5 at +13 archagall fourths (85/64's) (octave-reduced; specifically: minus five octaves).

Mavka a.k.a. archagallismic

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]]

Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13

Optimal ET sequence: 46, 58, 80, 103, 137, 149, 159, 171, 183, 217, 296, 320, 342f, 354, 400, 422, 525, 571, 581, 742, 764, 935, 1084, 1106, 1323, 1506, 3593g, 3947eg, 5053fgg, 6559defgg, 8065cdefggg, 10152cdeffgggg.

Archagall

2.75.85 subgroup (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

Mapping: [⟨1 5 6], ⟨0 3 1]]

CTE generator: 85/64 = 491.541 ¢

Optimal ET sequence: 5, 17, 22, 61, 83

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.

Subgroup: 2.75.85.9/7

Comma list: [13 1 -3 0⟩ = 24576/24565, [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

Mapping: [⟨1 5 6 2], ⟨0 3 1 -4]]

CTE generator: 85/64 = 491.338

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.

Subgroup: 2.3.5.7.17

Comma list: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35

Mapping: [⟨1 11 -3 20 9], ⟨0 -23 13 -42 -12]]

CTE generator: 85/64 = 491.222 ¢

Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364

Srutal archagall

This lower-accuracy temperament is an extension of srutal 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 2.3.5.17 subgroup, implying ideal tunings of 34edo, 46edo or 80edo.

See Diaschismic family #Srutal archagall.

Archagallic

Subgroup: 2.3.5.17

Mapping: [⟨1 1 2 4], ⟨0 1 1 0], ⟨0 0 -3 1]]

CTE generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201

Optimal ET sequence: 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg

Badness: 9.335 × 10^-6

Etymology

The mavka comma was named by Eliora in 2022. Its other name archagallismic comma derives from archagall, the esoteric subgroup temperament named by Scott Dakota earlier.