24576/24565: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Name = mavka comma, archagall comma
| Name = mavka comma, archagallisma
| Color name = trisu-agu negative 2nd
| Color name = 17u<sup>3</sup>g-2, Trisu-agu comma
| Comma = yes
| Comma = yes
}}
}}
 
'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] measuring about 0.775 [[cents]]. It is the difference between [[256/255]] and [[289/288]] – two adjacent [[square superparticular]]s, 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]]).  
'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an unnoticeable 17-limit comma that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], [[Square superparticular#Sk/S(k + 1) (ultraparticulars)|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 ==
== 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]])<sup>3</sup>. 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 ===
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]].  
By tempering it (S16/S17) 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
 
Mapping:  [ ⟨ 1 5 6 ]      ⟨ 0 3 1 ] ⟩
 
Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83


=== Srutal archagall ===
Finally, tempering out 256/255 and/or 289/288 results in a lower-accuracy temperament, [[srutal archagall]], of the [[diaschismic family]].  
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 hence the list of EDOs mentioned) on the following prior-discussed subgroup:


Subgroup: 2.3.5.17
=== Archagallic ===
[[Subgroup]]: 2.3.5.17


Mapping:  [ ⟨ 2 3 5 8 ]     ⟨ 0 1 -2 1 ] ⟩
[[Comma list]]: 24576/24565


Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
{{Mapping|legend=2| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}
: mapping generators: ~2, ~3, ~17/16


=== Rank 6 temperament ===
[[Optimal tuning]]s:
By tempering it out in the full [[17-limit]], the rank-6 '''mavka temperament''' is defined, to which extensions are also rank 6.
* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}


Subgroup: 2.3.5.7.11.13.17
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}


Mapping:
[[Badness]] (Sintel): 0.127


[ ⟨ 1 0 1 0 0 0 4 ]
=== Mavka a.k.a. archagallismic ===
[[Subgroup]]: 2.3.5.7.11.13.17


⟨ 0 1 1 0 0 0 0 ]
[[Comma list]]: 24576/24565


⟨ 0 0 3 0 0 0 -1 ]
{| class="right-all"
|-
| [⟨ || 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


⟨ 0 0 0 1 0 0 0 ]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9692{{c}}, ~3/2 = 701.9798{{c}}, ~17/16 = 105.1973{{c}}, ~7/4 = 968.8875{{c}}, ~11/8 = 551.4103{{c}}, ~13/8 = 840.6200{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}, ~7/4 = 968.8842{{c}}, ~11/8 = 551.3897{{c}}, ~13/8 = 840.6045{{c}}


⟨ 0 0 0 0 1 0 0 ]
{{Optimal ET sequence|legend=1| 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 }}.


⟨ 0 0 0 0 0 1 0 ] ⟩
[[Badness]] (Sintel): 15.8


{{Val list|legend=1| 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 }}.
== 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.  


[[Category:Mavka]]
[[Category:Mavka]]
[[Category:Commas with unknown etymology]]

Latest revision as of 16:08, 22 July 2026

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 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

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.

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