Radian - Revision history

Lériendil: added reference to lucy tuning

2025-03-26T13:29:46Z

added reference to lucy tuning

← Older revision		Revision as of 13:29, 26 March 2025
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	Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo\|25]], 7\[[44edo\|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.		Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo\|25]], 7\[[44edo\|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.

	Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents. ~~EDOs which temper out~~ the ~~difference between the radian and exactly 1/6~~ of ~~a circle are said to temper out the~~ [[~~engineer's comma~~]], ~~as it equates pi with 3~~.		Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents. It is definitionally the [[meantone]] whole tone of [[Lucy tuning]], which is well approximated by [[88edo]].

	Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.		Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.

Secretcoffee: Added novelty tag

2023-07-25T17:37:43Z

Added novelty tag

← Older revision		Revision as of 17:37, 25 July 2023
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			{{Novelty}}
	{{Infobox Interval		{{Infobox Interval
	\| Ratio = 2^{\frac{1}{2\pi} }		\| Ratio = 2^{\frac{1}{2\pi} }

Plumtree: Infobox Interval added

2022-10-27T18:54:51Z

Infobox Interval added

← Older revision		Revision as of 18:54, 27 October 2022
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			{{Infobox Interval
			\| Ratio = 2^{\frac{1}{2\pi} }
			\| Cents = 190.98593171027437
			\| Name = radian, radial major second, radial whole tone
			}}
	'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math>, or 190.98593 cents.		'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math>, or 190.98593 cents.

Eliora at 10:45, 17 June 2022

2022-06-17T10:45:57Z

← Older revision		Revision as of 10:45, 17 June 2022
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	Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.		Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.
			[[Category:Transcendental]]

Eliora: /* Relative Radian */

2022-06-17T10:45:32Z

Relative Radian

← Older revision		Revision as of 10:45, 17 June 2022
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	The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.		The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

	~~Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents.~~		== Approximations ==

	~~== Approximations ==~~		Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo\|25]], 7\[[44edo\|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.

	~~Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\~~[[~~25edo\|25~~]]~~, 7\~~[[~~44edo\|44~~]], ~~53\333, and 113\710~~. ~~7\44~~ and ~~113\710~~ are ~~complementary~~ to the ~~historically notable 22/7 and 355/113 approximations of~~ pi.		Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents. EDOs which temper out the difference between the radian and exactly 1/6 of a circle are said to temper out the [[engineer's comma]], as it equates pi with 3.

	Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.		Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.

	== ~~Relative~~ Radian ==		== Radian in other intervals of equivalence ==
	The '''Relative Radian''' is ~~an interval based on~~ a ~~given interval~~. ~~In being~~ the ~~interval of~~ <math>\frac{1}{2\pi}</math> ~~of this interval, pitches, it is the generalization~~ of the ~~common radian.~~		The '''Relative Radian''' is a generalization of the common radian to nonoctave intervals of equivalence. Just as the octave radian, it is defined as <math>\frac{1}{2\pi}</math> of the original interval on the logarithmic scale.

	~~The~~ interval ~~has an interpretation that relates to equal divisions of all intervals. Since pitch classes in equal divisions of all intervals form in~~ the ~~shape of a circle, radian interval therefore occurs as the radius of this perceptional circle~~.

	Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller.		Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.

Moremajorthanmajor at 21:58, 18 January 2022

2022-01-18T21:58:18Z

← Older revision		Revision as of 21:58, 18 January 2022
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	Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.		Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.

			== Relative Radian ==
			The '''Relative Radian''' is an interval based on a given interval. In being the interval of <math>\frac{1}{2\pi}</math> of this interval, pitches, it is the generalization of the common radian.

			The interval has an interpretation that relates to equal divisions of all intervals. Since pitch classes in equal divisions of all intervals form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

			Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller.

Eliora: Had to take out the template because it automatically puts the category oof

2022-01-18T12:26:35Z

Had to take out the template because it automatically puts the category oof

← Older revision		Revision as of 12:26, 18 January 2022
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	~~{{Infobox Interval\|Ratio=2^(1/(2pi))\|Monzo=undefined\|Cents=190.98593\|Name=radian, radial major second, radial whole tone}}~~		'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math>, or 190.98593 cents.

	'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math> cents.

	The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.		The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

Xenwolf: fixed math

2021-09-25T21:31:20Z

fixed math

← Older revision		Revision as of 21:31, 25 September 2021
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	{{Infobox Interval\|Ratio=2^(1/(2pi))\|Monzo=undefined\|Cents=190.98593\|Name=radian, radial major second, radial whole tone}}		{{Infobox Interval\|Ratio=2^(1/(2pi))\|Monzo=undefined\|Cents=190.98593\|Name=radian, radial major second, radial whole tone}}

	'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}/{2\pi}</math cents.		'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math> cents.

	The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.		The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

Eliora at 22:19, 24 September 2021

2021-09-24T22:19:25Z

← Older revision		Revision as of 22:19, 24 September 2021
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	Radian, radial major second, or radial whole tone is an interval of 1200/~~2pi~~ cents.		{{Infobox Interval\|Ratio=2^(1/(2pi))\|Monzo=undefined\|Cents=190.98593\|Name=radian, radial major second, radial whole tone}}

			'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}/{2\pi}</math cents.

			The interval has an interpretation that relates to all [[EDO\|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

	~~{{Infobox Interval\|Ratio~~=~~2^(~~1/(2pi))\|~~Monzo=undefined~~\|~~Cents=190~~.~~98593~~\|~~Name=radian~~, ~~radial major second~~, ~~radial whole tone}}~~		Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents.

			== Approximations ==

			Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo\|25]], 7\[[44edo\|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.

			Other approximations include 5\[[31edo\|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.

Eliora: page, edits to continue

2021-09-24T22:02:40Z

page, edits to continue

New page

Radian, radial major second, or radial whole tone is an interval of 1200/2pi cents.

{{Infobox Interval|Ratio=2^(1/(2pi))|Monzo=undefined|Cents=190.98593|Name=radian, radial major second, radial whole tone}}