24576/24565

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.

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	trisu-agu negative 2nd
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

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]]

Template:Val list.

Archagall

By tempering it 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. 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

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