2016edo - Revision history

Hotcrystal0: fix

2026-04-24T16:46:14Z

fix

← Older revision		Revision as of 16:46, 24 April 2026
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	== Theory ==		== Theory ==
	2016edo shares the mapping for 3 with [[224edo]], albeit with a 28 relative ~~cent~~ error. First 7 prime harmonics with less than 25% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47.		2016edo shares the mapping for 3 with [[224edo]], albeit with a 28% relative error. First 7 prime harmonics with less than 25% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47.

	2016edo has two reasonable mappings for 7. The 2016d val, {{val\| 2016 3195 4681 5659 }}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo\| 14 11 -22 7 }}. As such, its circle of the interval 7/6 is the same as in [[9edo]]. The patent val, {{val\| 2016 3195 4681 5658 }} tempers out [[250047/250000]], along with {{monzo\| 7 18 -2 -11 }} and {{monzo\| 43 -1 -13 -4 }}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo's patent val corresponds to [[63/50]].		2016edo has two reasonable mappings for 7. The 2016d val, {{val\| 2016 3195 4681 5659 }}, tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and {{monzo\| 14 11 -22 7 }}. As such, its circle of the interval 7/6 is the same as in [[9edo]]. The patent val, {{val\| 2016 3195 4681 5658 }} tempers out [[250047/250000]], along with {{monzo\| 7 18 -2 -11 }} and {{monzo\| 43 -1 -13 -4 }}. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo's patent val corresponds to [[63/50]].

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:31:44Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:31, 13 March 2026
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	\| [[Jamala]]		\| [[Jamala]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==

Lucius Chiaraviglio: /* Subsets and supersets */ Last edit left awkward phrasing

2025-09-07T08:55:30Z

Subsets and supersets: Last edit left awkward phrasing

← Older revision		Revision as of 08:55, 7 September 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	2016 is a significantly composite number, with its subset edos being {{EDOs\| 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008 }}. Its abundancy index is 2.25. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music\|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába\|Alois Haba]] and used by [[Ivan Wyschnegradsky]], and ~~used by~~ jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]], and 224edo is a member of [[The Riemann zeta function and tuning\|zeta]] edos.		2016 is a significantly composite number, with its subset edos being {{EDOs\| 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008 }}. Its abundancy index is 2.25. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music\|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába\|Alois Haba]] and used by [[Ivan Wyschnegradsky]] and jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]], and 224edo is a member of [[The Riemann zeta function and tuning\|zeta]] edos.

	2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160).		2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160).

Lucius Chiaraviglio: /* Subsets and supersets */ The use of 72edo by Ivan Wyschnegradsky was not merely theoretical — see Arc en ciel (also one other composition that I have yet to find a recording of)

2025-09-07T08:54:28Z

Subsets and supersets: The use of 72edo by Ivan Wyschnegradsky was not merely theoretical — see Arc en ciel (also one other composition that I have yet to find a recording of)

← Older revision		Revision as of 08:54, 7 September 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	2016 is a significantly composite number, with its subset edos being {{EDOs\| 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008 }}. Its abundancy index is 2.25. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music\|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába\|Alois Haba]] and [[Ivan Wyschnegradsky]], and used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]], and 224edo is a member of [[The Riemann zeta function and tuning\|zeta]] edos.		2016 is a significantly composite number, with its subset edos being {{EDOs\| 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008 }}. Its abundancy index is 2.25. Some of its divisors have found applied use. 72edo has been used in [[Wikipedia:Byzantine music\|Byzantine chanting]], has been theoreticized by [[Wikipedia:Alois Hába\|Alois Haba]] and used by [[Ivan Wyschnegradsky]], and used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]], and 224edo is a member of [[The Riemann zeta function and tuning\|zeta]] edos.

	2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160).		2016 is a divisor of some [[highly composite edo]]s, such as [[10080edo]], [[20160edo]], etc. As a subset of 20160edo, one step of 2016edo is exactly 10 pians (10\20160).

Neutraldown: /* Fractional-octave temperaments */

2025-04-10T06:20:26Z

Fractional-octave temperaments

← Older revision		Revision as of 06:20, 10 April 2025
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	In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments\|dike temperament]], defined as {{nowrap\|1600 & 2016dijk}} in the 37-limit with period 32.		In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments\|dike temperament]], defined as {{nowrap\|1600 & 2016dijk}} in the 37-limit with period 32.

	In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as {{nowrap\|1944 & 2016}} and named after an eponymous song. It has a comma basis of {47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000{.		In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as {{nowrap\|1944 & 2016}} and named after an eponymous song. It has a comma basis of {47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000}.

	=== Odd harmonics ===		=== Odd harmonics ===

ArrowHead294 at 23:00, 20 February 2025

2025-02-20T23:00:49Z

← Older revision		Revision as of 23:00, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 15:27, 16 January 2025

2025-01-16T15:27:31Z

← Older revision		Revision as of 15:27, 16 January 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro~~\|2016~~}}		{{EDO intro}}

	== Theory ==		== Theory ==
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	In the 11-limit, 2016edo tempers out the {{monzo\| 0 0 -22 0 3 11 }} comma, which equates a stack of eleven [[25/13]]'s with three [[11/1]]'s. However, it does ''not'' temper out the [[jacobin comma]].		In the 11-limit, 2016edo tempers out the {{monzo\| 0 0 -22 0 3 11 }} comma, which equates a stack of eleven [[25/13]]'s with three [[11/1]]'s. However, it does ''not'' temper out the [[jacobin comma]].

	2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked. One such temperament is 311 & 2016, produced by stacking 1465\2016, and defined for the 2.5.11.13.19.41 subgroup with the comma basis 16777475/16777216, 1171280/1171001, 615288025/615120896, 1180029296875/1179517976576.		2016 has a total of 576 numbers coprime to it, which means this is how many generators can reach any point in the octave by being stacked. One such temperament is {{nowrap\|311 & 2016}}, produced by stacking 1465\2016, and defined for the 2.5.11.13.19.41 subgroup with the comma basis 16777475/16777216, 1171280/1171001, 615288025/615120896, 1180029296875/1179517976576.

	=== Fractional-octave temperaments ===		=== Fractional-octave temperaments ===
	The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of [[chromium]] with period 24 and the [[akjayland]], period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18.		The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of [[chromium]] with period 24 and the [[akjayland]], period 21. The 2016d val gives rise to {{nowrap\|171 & 306}}, period 9 and {{nowrap\|270 & 936bd}}, period 18.

	In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments\|dike temperament]], defined as 1600 & 2016dijk in the 37-limit with period 32.		In the 2016dijk val, which is tuned better than the patent val, it supports the [[32nd-octave temperaments\|dike temperament]], defined as {{nowrap\|1600 & 2016dijk}} in the 37-limit with period 32.

	In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as 1944 & 2016 and named after an eponymous song. It has a comma basis 47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000.		In the 2.5.11.13.19.41.47, 2016edo supports the period 72 Jamala temperament, defined as {{nowrap\|1944 & 2016}} and named after an eponymous song. It has a comma basis of {47012251/47000000, 2502280/2501369, 2680291328/2679296875, 410041489/410000000, 52448351813/52428800000{.

	=== Odd harmonics ===		=== Odd harmonics ===
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	\| {{monzo\| -83 26 18 }}, {{monzo\| 30 47 -45 }}		\| {{monzo\| -83 26 18 }}, {{monzo\| 30 47 -45 }}
	\| {{mapping\| 2016 3195 4681 }}		\| {{mapping\| 2016 3195 4681 }}
	\| 0.036		\| +0.036
	\| 0.050		\| 0.050
	\| 8.4		\| 8.4
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	\| 250047/250000, {{monzo\| 7 18 -2 -11 }}, {{monzo\| 43 -1 -13 -4 }}		\| 250047/250000, {{monzo\| 7 18 -2 -11 }}, {{monzo\| 43 -1 -13 -4 }}
	\| {{mapping\| 2016 3195 4681 5660 }}		\| {{mapping\| 2016 3195 4681 5660 }}
	\| 0.007		\| +0.007
	\| 0.066		\| 0.066
	\| 11.1		\| 11.1
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	\| 5250987/5242880, 40353607/40310784, {{monzo\| 14 11 -22 7 }}		\| 5250987/5242880, 40353607/40310784, {{monzo\| 14 11 -22 7 }}
	\| {{mapping\| 2016 3195 4681 5659 }} (2016d)		\| {{mapping\| 2016 3195 4681 5659 }} (2016d)
	\| 0.060		\| +0.060
	\| 0.060		\| 0.060
	\| 10.1		\| 10.1
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	\| {{monzo\| 14 8 -10 -1 }}, {{monzo\| -26 15 -5 4 }}, {{monzo\| -29 27 3 -6 }}		\| {{monzo\| 14 8 -10 -1 }}, {{monzo\| -26 15 -5 4 }}, {{monzo\| -29 27 3 -6 }}
	\| {{mapping\| 2016 3195 4681 6974}}		\| {{mapping\| 2016 3195 4681 6974}}
	\| 0.036		\| +0.036
	\| 0.043		\| 0.043
	\| 7.3		\| 7.3
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	\| 196625/196608, 53144100/53094899, {{monzo\| 14 8 -10 -1 0 }}, {{monzo\| -13 9 5 -8 4 }}		\| 196625/196608, 53144100/53094899, {{monzo\| 14 8 -10 -1 0 }}, {{monzo\| -13 9 5 -8 4 }}
	\| {{mapping\| 2016 3195 4681 6974 7460 }}		\| {{mapping\| 2016 3195 4681 6974 7460 }}
	\| 0.032		\| +0.032
	\| 0.040		\| 0.040
	\| 6.7		\| 6.7
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	\| 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608		\| 2601/2600, 120285/120224, 140625/140608, 161109/161051, 196625/196608
	\| {{mapping\| 2016 3195 4681 6974 7460 8240}}]		\| {{mapping\| 2016 3195 4681 6974 7460 8240}}]
	\| 0.034		\| +0.034
	\| 0.036		\| 0.036
	\| 6.2		\| 6.2
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	\| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865		\| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865
	\| {{mapping\| 2016 4681 6974 7460 8564 10801 11198 }}		\| {{mapping\| 2016 4681 6974 7460 8564 10801 11198 }}
	\| 0.002		\| +0.002
	\| 0.019		\| 0.019
	\| 3.2		\| 3.2
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	\| [[Jamala]]		\| [[Jamala]]
	\|}		\|}
	<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if ~~it is~~ distinct		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if distinct

	== Music ==		== Music ==
	; [[Mercury Amalgam]]		; [[Mercury Amalgam]]
	* [~~https~~://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022)		* [http://www.youtube.com/watch?v=ILMS8XT1bPs ''Hemoclysm Totem''] (2022)

	[[Category:Akjayland]]		[[Category:Akjayland]]
	[[Category:Listen]]		[[Category:Listen]]

ArrowHead294 at 12:05, 16 November 2024

2024-11-16T12:05:44Z

← Older revision		Revision as of 12:05, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[Comma list]]
			! rowspan="2" \| [[Mapping]]
			! rowspan="2" \| Optimal<br />8ve stretch (¢)
			! colspan="2" \| Tuning error
			\|-
			! [[TE error\|Absolute]] (¢)
			! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.019		\| 0.019
	\| 3.2		\| 3.2
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 21		\| 21
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	\| 73205/53248<br />(?)		\| 73205/53248<br />(?)
	\| [[Jamala]]		\| [[Jamala]]
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

	== Music ==		== Music ==

ArrowHead294 at 05:33, 16 November 2024

2024-11-16T05:33:34Z

← Older revision		Revision as of 05:33, 16 November 2024
Line 27:		Line 27:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3.5		\| 2.3.5
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	\| 0.066		\| 0.066
	\| 11.1		\| 11.1
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.3.5.7		\| 2.3.5.7
	~~\| style="border-top: double;"~~ \| 5250987/5242880, 40353607/40310784, {{monzo\| 14 11 -22 7 }}		\| 5250987/5242880, 40353607/40310784, {{monzo\| 14 11 -22 7 }}
	~~\| style="border-top: double;"~~ \| {{mapping\| 2016 3195 4681 5659 }} (2016d)		\| {{mapping\| 2016 3195 4681 5659 }} (2016d)
	~~\| style="border-top: double;"~~ \| 0.060		\| 0.060
	~~\| style="border-top: double;"~~ \| 0.060		\| 0.060
	~~\| style="border-top: double;"~~ \| 10.1		\| 10.1
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.3.5.11		\| 2.3.5.11
	~~\| style="border-top: double;"~~ \| {{monzo\| 14 8 -10 -1 }}, {{monzo\| -26 15 -5 4 }}, {{monzo\| -29 27 3 -6 }}		\| {{monzo\| 14 8 -10 -1 }}, {{monzo\| -26 15 -5 4 }}, {{monzo\| -29 27 3 -6 }}
	~~\| style="border-top: double;"~~ \| {{mapping\| 2016 3195 4681 6974}}		\| {{mapping\| 2016 3195 4681 6974}}
	~~\| style="border-top: double;"~~ \| 0.036		\| 0.036
	~~\| style="border-top: double;"~~ \| 0.043		\| 0.043
	~~\| style="border-top: double;"~~ \| 7.3		\| 7.3
	\|-		\|-
	\| 2.3.5.11.13		\| 2.3.5.11.13
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	\| 0.036		\| 0.036
	\| 6.2		\| 6.2
	\|-		\|- style="border-top: double;"
	\| style="border-top: double;" \| 2.5.11.13.19.41.47		\| 2.5.11.13.19.41.47
	~~\| style="border-top: double;"~~ \| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865		\| 7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865
	~~\| style="border-top: double;"~~ \| {{mapping\| 2016 4681 6974 7460 8564 10801 11198 }}		\| {{mapping\| 2016 4681 6974 7460 8564 10801 11198 }}
	~~\| style="border-top: double;"~~ \| 0.002		\| 0.002
	~~\| style="border-top: double;"~~ \| 0.019		\| 0.019
	~~\| style="border-top: double;"~~ \| 3.2		\| 3.2
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center~~-~~all left-5"~~		{{rank-2 begin}}
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	~~! Associated<br>Ratio~~
	~~! Temperaments~~
	\|-		\|-
	\| 21		\| 21
	\| 983\2016<br>(23\2016)		\| 983\2016<br />(23\2016)
	\| 585.119<br>(13.690)		\| 585.119<br />(13.690)
	\| 91875/65536<br>(126/125)		\| 91875/65536<br />(126/125)
	\| [[Akjayland]]		\| [[Akjayland]]
	\|-		\|-
	\| 24		\| 24
	\| 979\2016<br>(55\2016)		\| 979\2016<br />(55\2016)
	\| 582.738<br>(32.738)		\| 582.738<br />(32.738)
	\| 7/5<br>(?)		\| 7/5<br />(?)
	\| [[Chromium]]		\| [[Chromium]]
	\|-		\|-
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	\|-		\|-
	\| 72		\| 72
	\| 925\2016<br>(1\2016)		\| 925\2016<br />(1\2016)
	\| 550.595<br>(0.595)		\| 550.595<br />(0.595)
	\| 73205/53248<br>(?)		\| 73205/53248<br />(?)
	\| [[Jamala]]		\| [[Jamala]]
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	== Music ==		== Music ==

Francium: +categories

2024-02-13T18:31:01Z

+categories

← Older revision		Revision as of 18:31, 13 February 2024
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	[[Category:Akjayland]]		[[Category:Akjayland]]
			[[Category:Listen]]