24576/24565: Difference between revisions

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'''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 [[~~Square superparticular #Sk/S(k + 1) (ultraparticulars)|''~~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 '''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]].

It can be factored into [[4096/4095]] × [[4914/4913]].

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Comma list: {{monzo| 13 1 -3 }} = 24576/24565

~~Sval mapping: [~~{{~~val~~| 1 2 5 ~~}}], {{val~~| 0 3 1 }}]

~~Sval mapping generators~~: ~~~2, ~~~85/32

[[CTE]] generator: 85/64 = 491.541{{cent}}

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Subgroup: 2.75.85.9/7

~~Sval mapping: [{{val| 1 2 5 6 }}, {{val| 0 -4 3 1 }}]~~

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

[[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.

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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

[[Subgroup]]: 2.3.5.7.17

Comma list: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35

~~Sval mapping: [~~{{~~val~~| 1 11 -3 20 9 ~~}}, {{val~~| 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

==== 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.

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]].

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Subgroup: 2.3.5.17

~~Sval mapping: [~~{{~~val~~| 1 0 1 ~~0 0 0~~ 4 ~~}}, {{val~~| 0 1 1 0 ~~0 0 0 }}, {{val~~| 0 0 -3 ~~0 0 0~~ 1 }}]

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

[[CTE]] generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201

{{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 }}

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[[Category:Mavka]]

[[Category:Commas with unknown etymology]]

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 | 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 [[Square superparticular #Sk/S(k + 1) (ultraparticulars)|''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 '''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]].
 It can be factored into [[4096/4095]] × [[4914/4913]].
@@ Line 34: / Line 34: @@
 Comma list: {{monzo| 13 1 -3 }} = 24576/24565
-Sval mapping: [{{val| 1 2 5 }}], {{val| 0 3 1 }}]
+{{mapping|legend=1| 1 5 6 | 0 3 1 }}
-Sval mapping generators: ~2, ~85/32
+[[CTE]] generator: 85/64 = 491.541{{cent}}
 {{Optimal ET sequence|legend=1| 5, 17, 22, 61, 83 }}
@@ Line 44: / Line 44: @@
 Subgroup: 2.75.85.9/7
-Sval mapping: [{{val| 1 2 5 6 }}, {{val| 0 -4 3 1 }}]
 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
+{{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.
@@ Line 56: / Line 58: @@
 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
+[[Subgroup]]: 2.3.5.7.17
 Comma list: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35
-Sval mapping: [{{val| 1 11 -3 20 9 }}, {{val| 0 -23 13 -42 -12 }}]
+{{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
 ==== 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.
+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]].
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 Subgroup: 2.3.5.17
-Sval mapping: [{{val| 1 0 1 0 0 0 4 }}, {{val| 0 1 1 0 0 0 0 }}, {{val| 0 0 -3 0 0 0 1 }}]
+{{mapping|legend=1| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }}
-Sval mapping generators: ~2, ~3, ~17/16
+[[CTE]] generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201
 {{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 }}
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 [[Category:Mavka]]
+[[Category:Commas with unknown etymology]]