24576/24565: Difference between revisions

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{{Infobox Interval

| Name = mavka comma, archagallisma

| Color name = ~~trisu~~-agu ~~negative 2nd~~

| Color name = 17u<sup>3</sup>g-2, Trisu-agu comma

| 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 ==

~~=== Rank 6 temperament ===~~

[[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).

~~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

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]].

[[~~Mapping~~]]:

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

[ ~~⟨ 1 0 1 0 0 0 4~~ ]

=== Archagallic ===

[[Subgroup]]: 2.3.5.17

~~⟨ 0 1 1 0 0 0 0~~ ]

[[Comma list]]: 24576/24565

⟨ 0 0 ~~3 0~~ 0 0 -1 ]

: mapping generators: ~2, ~3, ~17/16

~~⟨ 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}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}

~~⟨ 0 0 0 0~~ 1 ~~0 0 ]~~

{{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 }}

~~⟨ 0 0 0 0 0 1~~ 0 ~~] ⟩~~

[[Badness]] (Sintel): 0.127

~~{{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 }}.

=== Mavka a.k.a. archagallismic ===

[[Subgroup]]: 2.3.5.7.11.13.17

==~~= 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

~~Mapping: [ ⟨ 1~~ 5 ~~6 ] ⟨ 0 3 1 ] ⟩~~

~~Some good (relative to their size) EDOs supporting it~~: ~~5, 17, 22, 39, 61, 83~~

[[Comma list]]: 24576/24565

=~~== Srutal archagall ===~~

{| class="right-all"

~~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:

|-

| [⟨ || 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

~~Subgroup~~: 2.3.5.17

[[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}}

~~Comma list: 256/255~~, ~~289/288~~

{{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 }}.

~~Mapping:~~ [ ~~⟨ 2 3 5 8~~ ] ~~⟨ 0 1 -2 1~~ ] ⟩

[[Badness]] (Sintel): 15.8

~~Some good (relative to their size) EDOs supporting it: 12~~, ~~22, 34, 46, 56, 58, 80~~

== 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:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Name = mavka comma, archagallisma
-| Color name = trisu-agu negative 2nd
+| Color name = 17u<sup>3</sup>g-2, Trisu-agu comma
 | 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 ==
-=== Rank 6 temperament ===
+[[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).
-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
+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]].
-[[Mapping]]:
+Finally, tempering out 256/255 and/or 289/288 results in a lower-accuracy temperament, [[srutal archagall]], of the [[diaschismic family]].
-[ ⟨	1	0	1	0	0	0	4	]
+=== Archagallic ===
+[[Subgroup]]: 2.3.5.17
-⟨	0	1	1	0	0	0	0	]
+[[Comma list]]: 24576/24565
-⟨	0	0	3	0	0	0	-1	]
+{{Mapping|legend=2| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}
+: mapping generators: ~2, ~3, ~17/16
-⟨	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}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9799{{c}}, ~17/16 = 105.2003{{c}}
-⟨	0	0	0	0	1	0	0	]
+{{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 }}
-⟨	0	0	0	0	0	1	0	] ⟩
+[[Badness]] (Sintel): 0.127
-{{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 }}.
+=== Mavka a.k.a. archagallismic ===
+[[Subgroup]]: 2.3.5.7.11.13.17
-=== 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
-Mapping:  [ ⟨	1	5	6	]      ⟨	0	3	1	] ⟩
-Some good (relative to their size) EDOs supporting it: 5, 17, 22, 39, 61, 83
+[[Comma list]]: 24576/24565
-=== Srutal archagall ===
+{| class="right-all"
-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:
+|-
+| [⟨ || 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
-Subgroup: 2.3.5.17
+[[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}}
-Comma list: 256/255, 289/288
+{{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 }}.
-Mapping:  [ ⟨	2	3	5	8	]      ⟨	0	1	-2	1	] ⟩
+[[Badness]] (Sintel): 15.8
-Some good (relative to their size) EDOs supporting it: 12, 22, 34, 46, 56, 58, 80
+== 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:Commas with unknown etymology]]