24576/24565

Interval information
Ratio	24576/24565
Factorization	213 × 3 × 5-1 × 17-3
Monzo	[13 1 -1 0 0 0 -3⟩
Size in cents	0.7750585¢
Names	mavka comma,; archagallisma
Color name	17u3g-2, Trisu-agu comma
FJS name	[math]\displaystyle{ \text{d}{-2}_{5,17,17,17} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	29.1693
Weil norm (log2 max(n, d))	29.1699
Wilson norm (sopfr(nd))	85
Comma size	unnoticeable
S-expression	S16/S17
	Open this interval in xen-calc

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).

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 the 2.75.9/7.85 subgroup as archagall. 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.

Finally, tempering out 256/255 and/or 289/288 results in a lower-accuracy temperament, srutal archagall, of the diaschismic family.

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

Optimal tunings:

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	4	],
⟨	0	1	1	0	0	0	],
⟨	0	0	-3	0	0	1	],
⟨	0	0	0	1	0	0	],
⟨	0	0	0	0	1	0	]]

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

Optimal tunings:

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

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.