24576/24565 - Revision history

FloraC: - duplicate data (moved to their respective proper places)

2026-07-22T16:08:59Z

- duplicate data (moved to their respective proper places)

@@ Line 9: / Line 9: @@
 [[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).
-If we retract it to the 2.75.85 subgroup, we get the rank-2 '''MVP archagall''' temperament, which can be extended to '''archagall''' in the 2.75.9/7.85 subgroup. 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]].
+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]].
 === Archagallic ===
@@ Line 53: / Line 55: @@
 [[Badness]] (Sintel): 15.8
 == Etymology ==

FloraC: Complete the rest. Fix RTT errors

2026-07-22T15:43:15Z

Complete the rest. Fix RTT errors

← Older revision		Revision as of 15:43, 22 July 2026
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	== 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. 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 charismic fourths ([[85/64]]'s) octave reduced.		[[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).

			If we retract it to the 2.75.85 subgroup, we get the rank-2 '''MVP archagall''' temperament, which can be extended to '''archagall''' in the 2.75.9/7.85 subgroup. 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]].

	=== Archagallic ===		=== Archagallic ===
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	[[Badness]] (Sintel): 15.8		[[Badness]] (Sintel): 15.8

	=== ~~Archagall ===~~		=== MVP archagall (2.75.85) ===
	~~====~~ 2.75.85 ~~subgroup (MVP archagall~~) ====		By tempering out the comma 24576/24565 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. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. 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 2.75.85 subgroup.
	By tempering the comma ~~S16/S17 =~~ 24576/24565 ~~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. (It is because of this combination of accuracy, efficiency and ~~simplicity (~~mapping-wise) and its corresponding explanatory power in what this comma does that the comma has been named the "archagallisma".) The "MVP" stands for ~~"Minimum Viable Product"~~, as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. 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		[[Subgroup]]: 2.75.85

	Comma list: {{monzo\| 13 1 -3 }} ~~= 24576/24565~~		[[Comma list]]: 24576/24565 ({{monzo\| 13 1 -3 }})

	{{~~mapping~~\|legend=1\| 1 5 6 \| 0 3 1 }}		{{Mapping\|legend=2\| 1 2 5 \| 0 3 1 }}
			: mapping generators: ~2, ~85/32

	[[~~CTE~~]] ~~generator~~: 85/64 = 491.~~541~~{{~~cent~~}}		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}

	{{Optimal ET sequence\|legend=1\| 5, 17, 22, 61, 83 }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118, 5939, 7760* }}

	==== 2.75~~.85~~.9/7 ~~subgroup~~ ====		==== Archagall (2.75.9/7.85) ====
	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 out [[2025/2023]] ([[S-expression\|S15/S17]]), which equates a stack of 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 out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.

	Subgroup: 2.75~~.85~~.9/7		[[Subgroup]]: 2.75.9/7.85

	Comma list: {{monzo\| 13 1 -3 0 }} = 24576/24565, {{monzo\| 2 -~~2 0 1~~ }} ~~= 2025/2023~~		[[Comma list]]: 2025/2023 ({{monzo\| 2 -2 1 0 }}), 24576/24565 ({{monzo\| 13 1 0 -3 }})

	~~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=2\| 1 2 6 5 \| 0 3 -4 1 }}
			: mapping generators: ~2, ~85/32

	{{~~mapping\|legend~~=1\| ~~1 5 6~~ 2 ~~\| 0 3 1 -4~~ }}		[[Optimal tuning]]s:
			* [[Tp tuning\|Subgroup]] [[WE]]: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
			* [[Tp tuning\|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}

	~~[[CTE]] generator: 85/64~~ = ~~491.338~~		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441* }}

	~~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~~.		=== 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.

	~~====~~ 2.3.5.7.17 ~~subgroup (prime archagall) ====~~		Subgroup: 2.3.5.7.17
	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: 1225/1224, 24576/24565, 57375/57344

	~~Comma list~~: ~~24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35~~		Subgroup-val mapping: {{mapping\| 1 -12 10 -22 -3 \| 0 23 -13 42 12 }}

	{{~~mapping\|legend~~=~~1\| 1 11 -3 20 9 \| 0 -23 13 -42 -12~~ }}		Optimal tunings:
			* WE: ~2 = 1200.0516{{c}}, ~128/85 = 708.8057{{cent}}
			* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 708.7758{{cent}}

	~~[[CTE]] generator: 85/64 = 491.222~~{{~~cent~~}}		{{Optimal ET sequence\|legend=0\| 22, …, 149, 171, 1219, 1390 }}

	~~Some good~~ (~~relative to their size~~) ~~EDOs supporting it~~: ~~22, 149, 171, 193, 215, 320, 364~~		Badness (Sintel): 0.421

	==== Srutal archagall ====		==== Srutal archagall ====

FloraC: Complete data for mavka and archagallic

2026-07-22T15:12:26Z

Complete data for mavka and archagallic

@@ Line 7: / Line 7: @@
 == Temperaments ==
-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).
+[[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. 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 charismic fourths ([[85/64]]'s) octave reduced.
 === Mavka a.k.a. archagallismic ===
 [[Subgroup]]: 2.3.5.7.11.13.17
-[[Mapping]]:<br>
+[[Comma list]]: 24576/24565
-Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13
+{| class="right-all"
 {{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 ===
@@ Line 70: / Line 98: @@
 See [[Diaschismic family #Srutal archagall]].
 == Etymology ==

FloraC: Rework intro (- the inefficient way of pumping it)

2026-07-22T15:01:33Z

Rework intro (- the inefficient way of pumping it)

← Older revision		Revision as of 15:01, 22 July 2026
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	\| 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 '''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]]).

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

	== Temperaments ==		== Temperaments ==

BudjarnLambeth at 01:59, 16 November 2024

2024-11-16T01:59:22Z

← Older revision		Revision as of 01:59, 16 November 2024
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	[[Category:Mavka]]		[[Category:Mavka]]
			[[Category:Commas with unknown etymology]]

Godtone: /* Temperaments */ added mappings and cleaned up format

2024-01-16T16:27:04Z

Temperaments: added mappings and cleaned up format

← Older revision		Revision as of 16:27, 16 January 2024
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	Comma list: {{monzo\| 13 1 -3 }} = 24576/24565		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 }}		{{Optimal ET sequence\|legend=1\| 5, 17, 22, 61, 83 }}
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	Subgroup: 2.75.85.9/7		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		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.
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	Subgroup: 2.3.5.17		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 }}		{{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 }}

Godtone: /* 2.3.5.7.17 subgroup (prime archagall) */ style and generator

2024-01-16T16:09:58Z

2.3.5.7.17 subgroup (prime archagall): style and generator

← Older revision		Revision as of 16:09, 16 January 2024
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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.		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		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		Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364

Godtone at 18:48, 15 January 2024

2024-01-15T18:48:08Z

← Older revision		Revision as of 18:48, 15 January 2024
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	==== Srutal archagall ====		==== 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]].		See [[Diaschismic family #Srutal archagall]].

Godtone at 19:20, 31 July 2023

2023-07-31T19:20:42Z

← Older revision		Revision as of 19:20, 31 July 2023
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	\| 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 [[~~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]].		It can be factored into [[4096/4095]] × [[4914/4913]].

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T17:52:54Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 17:52, 7 May 2023
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	Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13		Mapping generators: ~2, ~3, ~17/16, ~7, ~11, ~13

	{{~~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 }}.		{{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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	Sval mapping generators: ~2, ~85/32		Sval mapping generators: ~2, ~85/32

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 5, 17, 22, 61, 83 }}		{{Optimal ET sequence\|legend=1\| 5, 17, 22, 61, 83 }}

	==== 2.75.85.9/7 subgroup ====		==== 2.75.85.9/7 subgroup ====
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	Sval mapping generators: ~2, ~3, ~17/16		Sval mapping generators: ~2, ~3, ~17/16

	Optimal ~~GPV~~ sequence~~: {{Val list~~\| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }}		{{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>