24576/24565
24576/24565, the mavka comma or archagallisma, is an unnoticeable 17-limit comma measuring about 0.775 cents. It is 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 can be factored into (4096/4095)⋅(4914/4913).
| Interval information |
archagallisma
Temperaments
Tempering out this comma in the full 17-limit results in the rank-6 mavka a.k.a. archagallismic temperament, or in the 2.3.5.17 subgroup, the rank-3 archagallic 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).
If we retract it to the 2.75.85 subgroup, we get the rank-2 MVP archagall temperament, which can be extended to archagall in the 2.75.9/7.85 subgroup. This has its own, more complex mapping of prime 5 at +13 charismic fourths (85/64's) octave reduced, resulting in prime archagall, an extension of fifthplus.
Archagallic
Subgroup: 2.3.5.17
Comma list: 24576/24565
Subgroup-val mapping: [⟨1 1 2 4], ⟨0 1 1 0], ⟨0 0 -3 1]]
- mapping generators: ~2, ~3, ~17/16
- WE: ~2 = 1199.9692 ¢, ~3/2 = 701.9798 ¢, ~17/16 = 105.1973 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9799 ¢, ~17/16 = 105.2003 ¢
Optimal ET sequence: 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg
Badness (Sintel): 0.127
Mavka a.k.a. archagallismic
Subgroup: 2.3.5.7.11.13.17
Comma list: 24576/24565
| [⟨ | 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 | 0 | 1 | 0 | ]] |
- mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13
- WE: ~2 = 1199.9692 ¢, ~3/2 = 701.9798 ¢, ~17/16 = 105.1973 ¢, ~7/4 = 968.8875 ¢, ~11/8 = 551.4103 ¢, ~13/8 = 840.6200 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9799 ¢, ~17/16 = 105.2003 ¢, ~7/4 = 968.8842 ¢, ~11/8 = 551.3897 ¢, ~13/8 = 840.6045 ¢
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.
Badness (Sintel): 15.8
MVP archagall (2.75.85)
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 ([13 1 -3⟩)
Subgroup-val mapping: [⟨1 2 5], ⟨0 3 1]]
- mapping generators: ~2, ~85/32
- Subgroup WE: ~2 = 1199.9692 ¢, ~85/64 = 491.5853 ¢
- Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.5794 ¢
Optimal ET sequence: 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118*, 5939*, 7760*
Archagall (2.75.9/7.85)
A fairly natural way to extend archagall is by tempering out 2025/2023 (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 ([2 -2 1 0⟩), 24576/24565 ([13 1 0 -3⟩)
Subgroup-val mapping: [⟨1 2 6 5], ⟨0 3 -4 1]]
- mapping generators: ~2, ~85/32
- Subgroup WE: ~2 = 1200.0241 ¢, ~85/64 = 491.3358 ¢
- Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.3290 ¢
Optimal ET sequence: 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441*
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: 1225/1224, 24576/24565, 57375/57344
Subgroup-val mapping: [⟨1 -12 10 -22 -3], ⟨0 23 -13 42 12]]
Optimal tunings:
Optimal ET sequence: 22, …, 149, 171, 1219, 1390
Badness (Sintel): 0.421
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.
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.