24576/24565: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = | | Name = mavka comma, archagallisma | ||
| Color name = | | 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]] that represents 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 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]]. | |||
'''24576/24565''', the ''' | |||
It can be factored into [[4096/4095]] × [[4914/4913]]. | 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 ''' | 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]])<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 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 | [[Subgroup]]: 2.3.5.7.11.13.17 | ||
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Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13 | Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13 | ||
{{ | {{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 }}. | ||
=== Archagall === | === Archagall === | ||
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Comma list: {{monzo| 13 1 -3 }} = 24576/24565 | Comma list: {{monzo| 13 1 -3 }} = 24576/24565 | ||
{{mapping|legend=1| 1 5 6 | 0 3 1 }} | |||
[[CTE]] generator: 85/64 = 491.541{{cent}} | |||
Optimal | {{Optimal ET sequence|legend=1| 5, 17, 22, 61, 83 }} | ||
==== 2.75.85.9/7 subgroup ==== | ==== 2.75.85.9/7 subgroup ==== | ||
A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(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. | A fairly natural way to extend archagall is by tempering S15/S17 which [[square superparticular|(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 | Subgroup: 2.75.85.9/7 | ||
Comma list: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo| 2 -2 0 1 }} = 2025/2023 | Comma list: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo| 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 | 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|legend=1| 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. | 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) ==== | ==== 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|legend=1| 1 11 -3 20 9 | 0 -23 13 -42 -12 }} | |||
[[CTE]] generator: 85/64 = 491.222{{cent}} | |||
Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364 | 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 === | === Archagallic === | ||
Subgroup: 2.3.5.17 | Subgroup: 2.3.5.17 | ||
{{mapping|legend=1| 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 | {{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 }} | ||
Badness: 9.335 × 10<sup>-6</sup> | Badness: 9.335 × 10<sup>-6</sup> | ||
== | == 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]] | |||