3072edo - Revision history

ArrowHead294 at 22:42, 20 February 2025

2025-02-20T22:42:09Z

← Older revision		Revision as of 22:42, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|3072~~}}		{{ED intro}}

	== Theory ==		== Theory ==
Line 18:		Line 18:
	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
			\|-
	! rowspan="2" \| [[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \| [[Comma list]]		! rowspan="2" \| [[Comma list]]
Line 30:		Line 31:
	\| {{monzo\| 37 25 -33 }}, {{monzo\| 161 -84 -12 }}		\| {{monzo\| 37 25 -33 }}, {{monzo\| 161 -84 -12 }}
	\| {{mapping\| 3072 4869 7133 }}		\| {{mapping\| 3072 4869 7133 }}
	\| -0.002		\| −0.002
	\| 0.003		\| 0.003
	\|		\|
Line 37:		Line 38:
	\| 250047/250000, {{monzo\| -2 -25 1 14 }}, {{monzo\| -53 -1 9 12 }}		\| 250047/250000, {{monzo\| -2 -25 1 14 }}, {{monzo\| -53 -1 9 12 }}
	\| {{mapping\| 3072 4869 7133 8624 }}		\| {{mapping\| 3072 4869 7133 8624 }}
	\| 0.006		\| +0.006
	\| 0.013		\| 0.013
	\|		\|
Line 44:		Line 45:
	\| 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840		\| 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840
	\| {{mapping\| 3072 4869 7133 8624 10627 }}		\| {{mapping\| 3072 4869 7133 8624 10627 }}
	\| 0.013		\| +0.013
	\| 0.019		\| 0.019
	\|		\|
Line 51:		Line 52:
	\| 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427		\| 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427
	\| {{mapping\| 3072 4869 7133 8624 10627 11638 }}		\| {{mapping\| 3072 4869 7133 8624 10627 11638 }}
	\| 0.006		\| +0.006
	\| 0.022		\| 0.022
	\|		\|

Fredg999: /* External links */ Misc. edits

2024-07-06T05:32:55Z

External links: Misc. edits

← Older revision		Revision as of 05:32, 6 July 2024
Line 77:		Line 77:

	== External links ==		== External links ==
	* [http://tonalsoft.com/enc/number/8mu.aspx ~~Tonalsoft Encyclopedia \| ''~~8mu / oktamu'']		* [http://tonalsoft.com/enc/number/8mu.aspx 8mu / oktamu] on [[Tonalsoft Encyclopedia]]

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

FloraC: /* Theory */ fix grammar

2023-11-07T08:22:03Z

Theory: fix grammar

← Older revision		Revision as of 08:22, 7 November 2023
Line 8:		Line 8:

	=== As a tuning standard ===		=== As a tuning standard ===
	3072edo~~'s step~~ is known as '''octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.		A step of 3072edo is known as an '''octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Cleanup; +external link

2023-11-07T08:01:09Z

Cleanup; +external link

← Older revision		Revision as of 08:01, 7 November 2023
Line 7:		Line 7:
	Although consistent to the 11-odd-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 [[subgroup]] tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the [[patent val]] still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.		Although consistent to the 11-odd-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 [[subgroup]] tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the [[patent val]] still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.

	=== ~~Significance in digital audio software~~ ===		=== As a tuning standard ===
	3072edo's step is known as '''~~Octamu~~''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.		3072edo's step is known as '''octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 55:		Line 55:
	\|		\|
	\|}		\|}

			== Music ==
			; [[Eliora]]
			* [https://www.youtube.com/watch?v=sksIgNTJ-XY ''Etude for Celtic Harp in Whoosh''] (2023)

	== See also ==		== See also ==
Line 72:		Line 76:
	** [[196608edo]] (14mu tuning)		** [[196608edo]] (14mu tuning)

	== ~~Music~~ ==		== External links ==
	~~; [[Eliora]]~~		* [http://tonalsoft.com/enc/number/8mu.aspx Tonalsoft Encyclopedia \| ''8mu / oktamu'']
	* [~~https~~://~~www.youtube~~.com/~~watch?v=sksIgNTJ-XY~~ ''~~Etude for Celtic Harp in Whoosh~~''] ~~(2023)~~

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

FloraC: Adopt template: Factorization and cleanup

2023-10-30T09:58:06Z

Adopt template: Factorization and cleanup

← Older revision		Revision as of 09:58, 30 October 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|3072}}		{{EDO intro\|3072}}
	~~==Theory==~~
	3072edo is consistent to the [[11-limit]] and it is an extremely accurate 5-limit tuning, tempering out {{monzo\|37 25 -33}} (whoosh) and {{monzo\|161 -84 -12}} ([[Kirnberger's atom\|atom]]) in the 5-limit; 250047/250000, {{monzo\|-2 -25 1 14}}, and {{monzo\|-53 -1 9 12}}; in the 7-limit; 9801/9800, 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit.

	Although consistent to the 11-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the patent val still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.		== Theory ==
			3072edo is [[consistent]] to the [[11-odd-limit]] and it is an extremely accurate 5-limit tuning, tempering out {{monzo\| 37 25 -33 }} (whoosh) and {{monzo\| 161 -84 -12 }} ([[Kirnberger's atom\|atom]]) in the 5-limit; 250047/250000 ([[landscape comma]]), {{monzo\| -2 -25 1 14 }}, and {{monzo\| -53 -1 9 12 }}; in the 7-limit; [[9801/9800]], 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit.

			Although consistent to the 11-odd-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 [[subgroup]] tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the [[patent val]] still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.

	=== Significance in digital audio software ===		=== Significance in digital audio software ===
	3072edo's step is known as '''Octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.		3072edo's step is known as '''Octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{~~harmonics~~ in equal\|3072}}		{{Harmonics in equal\|3072}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	3072 factors as ~~2<sup>10</sup> x 3~~, with subset edos {{EDOs\|1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536}}.		3072 factors as {{factorization\|3072}}, with subset edos {{EDOs\| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, and 1536 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \|[[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \|[[Comma list]]		! rowspan="2" \| [[Comma list]]
	! rowspan="2" \|[[Mapping]]		! rowspan="2" \| [[Mapping]]
	! rowspan="2" \|Optimal		! rowspan="2" \| Optimal<br>8ve Stretch (¢)
	8ve Stretch (¢)		! colspan="2" \| Tuning Error
	! colspan="2" \|Tuning Error
	\|-		\|-
	![[TE error\|Absolute]] (¢)		! [[TE error\|Absolute]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|37 25 -33}}, {{monzo\|161 -84 -12}}		\| {{monzo\| 37 25 -33 }}, {{monzo\| 161 -84 -12 }}
	\|[{{~~val~~\|3072 4869 7133}}]		\| {{mapping\| 3072 4869 7133 }}
	\| -0.002		\| -0.002
	\|0.003		\| 0.003
	\|		\|
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|250047/250000, {{monzo\|-2 -25 1 14}}, {{monzo\|-53 -1 9 12}}		\| 250047/250000, {{monzo\| -2 -25 1 14 }}, {{monzo\| -53 -1 9 12 }}
	\|[{{~~val~~\|3072 4869 7133 8624}}]		\| {{mapping\| 3072 4869 7133 8624 }}
	\|0.006		\| 0.006
	\|0.013		\| 0.013
	\|		\|
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840		\| 9801/9800, 151263/151250, 184549376/184528125, 73525096183/73466403840
	\|[{{~~val~~\|3072 4869 7133 8624 10627}}]		\| {{mapping\| 3072 4869 7133 8624 10627 }}
	\|0.013		\| 0.013
	\|0.019		\| 0.019
	\|		\|
	\|-		\|-
	\|2.3.5.7.11.13		\| 2.3.5.7.11.13
	\|9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427		\| 9801/9800, 140625/140608, 151263/151250, 196625/196608, 3327500/3326427
	\|[{{~~val~~\|3072 4869 7133 8624 10627 11638}}]		\| {{mapping\| 3072 4869 7133 8624 10627 11638 }}
	\|0.006		\| 0.006
	\|0.022		\| 0.022
	\|		\|
	\|}		\|}
	==See also==
	*[[Equal-step tuning\|Equal multiplications]] of MIDI-resolution units		== See also ==
	**[[24edo]] (1mu tuning)		* [[Equal-step tuning\|Equal multiplications]] of MIDI-resolution units
	**[[48edo]] (2mu tuning)		** [[24edo]] (1mu tuning)
	**[[96edo]] (3mu tuning)		** [[48edo]] (2mu tuning)
	**[[192edo]] (4mu tuning)		** [[96edo]] (3mu tuning)
	**[[384edo]] (5mu tuning)		** [[192edo]] (4mu tuning)
	**[[768edo]] (6mu tuning)		** [[384edo]] (5mu tuning)
	**[[1536edo]] (7mu tuning)		** [[768edo]] (6mu tuning)
	**[[6144edo]] (9mu tuning)		** [[1536edo]] (7mu tuning)
	**[[12288edo]] (10mu tuning)		** [[6144edo]] (9mu tuning)
	**[[24576edo]] (11mu tuning)		** [[12288edo]] (10mu tuning)
	**[[49152edo]] (12mu tuning)		** [[24576edo]] (11mu tuning)
	**[[98304edo]] (13mu tuning)		** [[49152edo]] (12mu tuning)
	**[[196608edo]] (14mu tuning)		** [[98304edo]] (13mu tuning)
			** [[196608edo]] (14mu tuning)

	== Music ==		== Music ==
	; [[Eliora]]		; [[Eliora]]
	* ''[https://www.youtube.com/watch?v=sksIgNTJ-XY Etude for Celtic Harp in Whoosh ~~(Op. 3, No. 5)]~~'' (2023)		* [https://www.youtube.com/watch?v=sksIgNTJ-XY ''Etude for Celtic Harp in Whoosh''] (2023)

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

Eliora: +music, -redundant categories

2023-10-24T10:46:29Z

+music, -redundant categories

← Older revision		Revision as of 10:46, 24 October 2023
Line 69:		Line 69:
	**[[196608edo]] (14mu tuning)		**[[196608edo]] (14mu tuning)

	[[Category:~~Equal divisions of the octave\|####~~]] ~~<!-- 4-digit number -->~~		== Music ==
			; [[Eliora]]
			* ''[https://www.youtube.com/watch?v=sksIgNTJ-XY Etude for Celtic Harp in Whoosh (Op. 3, No. 5)]'' (2023)

			[[Category:Listen]]

Eliora: rtp table, i'll fill in the relative error a bit later

2023-07-12T21:40:15Z

rtp table, i'll fill in the relative error a bit later

@@ Line 4: / Line 4: @@
 edo is consistent to the [[11-limit]] and it is an extremely accurate 5-limit tuning, tempering out {{monzo|37 25 -33}} (whoosh) and {{monzo|161 -84 -12}} ([[Kirnberger's atom|atom]]) in the 5-limit; 250047/250000, {{monzo|-2 -25 1 14}}, and {{monzo|-53 -1 9 12}}; in the 7-limit; 9801/9800, 151263/151250, 184549376/184528125, and 73525096183/73466403840 in the 11-limit.
-Although consistent to the 11-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.
+Although consistent to the 11-limit, it makes more sense to actually see 3072edo as a 2.3.5.7.13 subgroup tuning, due to lower relative error. There it tempers out 140625/140608 and 1990656/1990625. Overall in the 13-limit, the patent val still has smaller errors than any other val despite incosistency. In higher limits, it is not as impressive, with only [[53/32]] being 17% off and 2.3.5.67.71 subgroup having less than 4% error.
 === Significance in digital audio software ===
 edo's step is known as '''Octamu''' (eighth MIDI-resolution unit, 8mu, 2<sup>8</sup> = 256 equal divisions of the [[12edo]] semitone). The internal data structure of the 8mu requires two bytes, with the first bits of each byte reserved as a flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down, and 5 other bits which are not used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
@@ Line 13: / Line 13: @@
 factors as 2<sup>10</sup> x 3, with subset edos {{EDOs|1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536}}.
 ==See also==
 *[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

BudjarnLambeth at 07:03, 9 July 2023

2023-07-09T07:03:59Z

← Older revision		Revision as of 07:03, 9 July 2023
Line 1:		Line 1:
	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|3072}}		{{EDO intro\|3072}}
	==Theory==		==Theory==

BudjarnLambeth at 04:32, 9 July 2023

2023-07-09T04:32:02Z

← Older revision		Revision as of 04:32, 9 July 2023
Line 1:		Line 1:
	{{Infobox ET}}		{{novelty}}{{Infobox ET}}
	{{EDO intro\|3072}}		{{EDO intro\|3072}}
	==Theory==		==Theory==

Eliora: /* Subsets and supersets */

2023-07-07T00:47:01Z

Subsets and supersets

← Older revision		Revision as of 00:47, 7 July 2023
Line 11:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	3072 factors as 2<sup>10</sup> x 3, with subset edos 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536.		3072 factors as 2<sup>10</sup> x 3, with subset edos {{EDOs\|1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536}}.

	==See also==		==See also==