24576/24565 - Revision history

BudjarnLambeth at 01:59, 16 November 2024

2024-11-16T01:59:22Z

← Older revision		Revision as of 01:59, 16 November 2024
Line 88:		Line 88:

	[[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
Line 34:		Line 34:
	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 }}
Line 44:		Line 44:

	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.
Line 74:		Line 76:
	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
Line 56:		Line 56:
	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
Line 65:		Line 65:

	==== 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
Line 4:		Line 4:
	\| 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
Line 24:		Line 24:
	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 ===
Line 38:		Line 38:
	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 ====
Line 76:		Line 76:
	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>

FloraC: +naming info

2023-04-22T10:34:01Z

+naming info

← Older revision		Revision as of 10:34, 22 April 2023
Line 79:		Line 79:

	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]]

FloraC: The ratio difference has been mentioned above

2023-04-22T10:24:13Z

The ratio difference has been mentioned above

← Older revision		Revision as of 10:24, 22 April 2023
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma\|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between ~~two adjacent~~ [[~~square superparticular~~]]~~s –~~ [[289/288]] ~~and~~ [[~~256/255~~]], 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~~]]. What's more, it is the interval separating [[256/255]] and [[289/288~~]].		'''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]].

	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 '''mavka ~~temperament~~''' a.k.a. '''archagallismic'''. 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'''. 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 ===		=== Mavka a.k.a. archagallismic ===

Aura: Deleted excess links to 171edo's page.

2023-04-22T00:54:38Z

Deleted excess links to 171edo's page.

← Older revision		Revision as of 00:54, 22 April 2023
Line 54:		Line 54:

	==== 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.		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

Aura at 00:12, 22 April 2023

2023-04-22T00:12:09Z

← Older revision		Revision as of 00:12, 22 April 2023
Line 4:		Line 4:
	\| Comma = yes		\| Comma = yes
	}}		}}
	'''24576/24565''', the '''mavka comma''' or '''archagallisma''', is an [[Unnoticeable comma\|unnoticeable]] [[17-limit]] [[comma]] that represents the difference between two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], 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 two adjacent [[square superparticular]]s – [[289/288]] and [[256/255]], 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]]. What's more, it is the interval separating [[256/255]] and [[289/288]].

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