21/16 - Revision history

Hotcrystal0 at 20:54, 22 May 2026

2026-05-22T20:54:18Z

← Older revision		Revision as of 20:54, 22 May 2026
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	}}		}}

	'''21/16''', the '''septimal subfourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], ~~a microtone of~~ approximately 27.3¢. It can be treated as the 21st ~~overtone~~, octave reduced. Since 21 is 3 × 7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.		'''21/16''', the '''septimal subfourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], approximately 27.3¢. It can be treated as the 21st harmonic, octave reduced. Since 21 is 3 × 7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

	21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.		21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.

FloraC: Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

2025-11-03T13:05:58Z

Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"

← Older revision		Revision as of 13:05, 3 November 2025
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	In [[septimal meantone]], this interval is represented by the augmented third.		In [[septimal meantone]], this interval is represented by the augmented third.
	== Approximation ==		== Approximation ==
	{{~~Interval_Edo_Approximation~~ \| 21/16}}		{{Interval edo approximation\|21/16}}
	== See also ==		== See also ==
	* [[32/21]] – its [[octave complement]]		* [[32/21]] – its [[octave complement]]

Pailiaq at 07:18, 3 November 2025

2025-11-03T07:18:49Z

← Older revision		Revision as of 07:18, 3 November 2025
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	In [[septimal meantone]], this interval is represented by the augmented third.		In [[septimal meantone]], this interval is represented by the augmented third.
			== Approximation ==
			{{Interval_Edo_Approximation \| 21/16}}
	== See also ==		== See also ==
	* [[32/21]] – its [[octave complement]]		* [[32/21]] – its [[octave complement]]

Plumtree: Normalising usage of Infobox Interval

2022-10-25T12:26:53Z

Normalising usage of Infobox Interval

← Older revision		Revision as of 12:26, 25 October 2022
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	{{Infobox Interval		{{Infobox Interval
	~~\| Ratio = 21/16~~		\| Name = septimal subfourth, narrow fourth, 8ve-reduced 21st harmonic
	~~\| Monzo = -4 1 0 1~~
	~~\| Cents = 470.78091~~
	\| Name = septimal subfourth, ~~<br>~~narrow fourth, ~~<br>~~8ve-reduced 21st harmonic
	\| Color name = z4, zo 4th		\| Color name = z4, zo 4th
	~~\| FJS name = P4<sup>7</sup>~~
	\| Sound = jid_21_16_pluck_adu_dr220.mp3		\| Sound = jid_21_16_pluck_adu_dr220.mp3
	}}		}}
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	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]

	~~[[Category:7-limit]]~~
	[[Category:Subfourth]]		[[Category:Subfourth]]
	[[Category:Fourth]]		[[Category:Fourth]]
	~~[[Category:Octave-reduced harmonics]]~~

Fredg999 category edits: Removing from Category:Pages with internal sound examples using Cat-a-lot

2022-02-22T23:46:43Z

Removing from Category:Pages with internal sound examples using Cat-a-lot

← Older revision		Revision as of 23:46, 22 February 2022
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	[[Category:Fourth]]		[[Category:Fourth]]
	[[Category:Octave-reduced harmonics]]		[[Category:Octave-reduced harmonics]]
	~~[[Category:Pages with internal sound examples]]~~

FloraC: Clarify the context where it works as an augmented third

2021-12-17T20:10:10Z

Clarify the context where it works as an augmented third

← Older revision		Revision as of 20:10, 17 December 2021
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	\| Monzo = -4 1 0 1		\| Monzo = -4 1 0 1
	\| Cents = 470.78091		\| Cents = 470.78091
	\| Name = septimal subfourth, <br/> narrow fourth, <br~~/> augmented third, <br/~~> 8ve-reduced 21st harmonic		\| Name = septimal subfourth, <br>narrow fourth, <br>8ve-reduced 21st harmonic
	\| Color name = z4, zo 4th		\| Color name = z4, zo 4th
	\| FJS name = P4<sup>7</sup>		\| FJS name = P4<sup>7</sup>
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	}}		}}

	'''21/16''', the '''septimal subfourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.		'''21/16''', the '''septimal subfourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3 × 7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

	21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.		21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.

	The 7-limit is known for its subminor and supermajor 2nds, 3rds, 6ths and 7ths. 21/16 is also an essential interval of the 7-limit and worth distinguishing.		The 7-limit is known for its subminor and supermajor 2nds, 3rds, 6ths and 7ths. 21/16 is also an essential interval of the 7-limit and worth distinguishing.

			In [[septimal meantone]], this interval is represented by the augmented third.

	== See also ==		== See also ==
			* [[32/21]] – its [[octave complement]]
			* [[8/7]] – its [[fifth complement]]
	* [[Gallery of just intervals]]		* [[Gallery of just intervals]]
	* [[32/21]] its [[inverse interval]]

	[[Category:7-limit]]		[[Category:7-limit]]

Fredg999: Typography

2021-12-17T19:44:12Z

Typography

← Older revision		Revision as of 19:44, 17 December 2021
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	\| Monzo = -4 1 0 1		\| Monzo = -4 1 0 1
	\| Cents = 470.78091		\| Cents = 470.78091
	\| Name = septimal ~~sub-fourth~~, <br/> narrow fourth, <br/> augmented third, <br/> 8ve-reduced 21st harmonic		\| Name = septimal subfourth, <br/> narrow fourth, <br/> augmented third, <br/> 8ve-reduced 21st harmonic
	\| Color name = z4, zo 4th		\| Color name = z4, zo 4th
	\| FJS name = P4<sup>7</sup>		\| FJS name = P4<sup>7</sup>
Line 9:		Line 9:
	}}		}}

	'''21/16''', the '''septimal ~~sub-fourth~~''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.		'''21/16''', the '''septimal subfourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

	21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.		21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.

Fredg999: FJS name, categories

2021-12-15T18:25:30Z

FJS name, categories

← Older revision		Revision as of 18:25, 15 December 2021
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	\| Cents = 470.78091		\| Cents = 470.78091
	\| Name = septimal sub-fourth, <br/> narrow fourth, <br/> augmented third, <br/> 8ve-reduced 21st harmonic		\| Name = septimal sub-fourth, <br/> narrow fourth, <br/> augmented third, <br/> 8ve-reduced 21st harmonic
			\| Color name = z4, zo 4th
			\| FJS name = P4<sup>7</sup>
	\| Sound = jid_21_16_pluck_adu_dr220.mp3		\| Sound = jid_21_16_pluck_adu_dr220.mp3
	~~\| Color name = z4, zo 4th~~
	}}		}}

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	[[Category:Fourth]]		[[Category:Fourth]]
	[[Category:Octave-reduced harmonics]]		[[Category:Octave-reduced harmonics]]
			[[Category:Pages with internal sound examples]]

Fredg999: Misc. edits, categories

2021-12-15T17:32:39Z

Misc. edits, categories

← Older revision		Revision as of 17:32, 15 December 2021
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	{{Infobox Interval		{{Infobox Interval
	~~\| JI glyph =~~
	\| Ratio = 21/16		\| Ratio = 21/16
	\| Monzo = -4 1 0 1		\| Monzo = -4 1 0 1
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	\| Color name = z4, zo 4th		\| Color name = z4, zo 4th
	}}		}}
	'''21/16''', the '''septimal sub-fourth''', is ~~an interval of the~~ [[7~~-limit\|7 prime~~-limit]] measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.
			'''21/16''', the '''septimal sub-fourth''', is a [[7-limit]] interval measuring approximately 470.8¢. It is a narrow fourth, differing from the Pythagorean perfect fourth of [[4/3]] by [[64/63]], a microtone of approximately 27.3¢. It can be treated as the 21st overtone, octave reduced. Since 21 is 3*7, 21 can be also treated as the 3rd harmonic above the 7th or the 7th harmonic above the 3rd, or both. This identity can be made clear in a chord such as 8:12:14:21, which has a just perfect fifth of [[3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4]]) in this chord, between 8 and 14 and between 12 and 21. The voicing of this chord is significant, as 3/2 sounds more consonant than its inversion 4/3 and 21/8 (an octave above 21/16) sounds more consonant than 21/16.

	21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.		21/16 is [[21/20]] away from [[5/4]]. This is an interval of about 84.5¢, a small semitone. This introduces the possibility of treating 21/16 as a dissonance to resolve down to 5/4. It can just as easily step up to 3/2 by [[8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit]] system, [[11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21]] (about 80.5¢) to 11/8. These are all movements that assume an unchanging fundamental, of course, and other movements are possible.
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	[[Category:Subfourth]]		[[Category:Subfourth]]
	[[Category:Fourth]]		[[Category:Fourth]]
	[[Category:~~Interval]]~~		[[Category:Octave-reduced harmonics]]
	~~[[Category:Just interval]]~~
	~~[[Category:Ratio]]~~
	~~[[Category:Overtone~~]]

Xenwolf: +cat

2020-07-12T15:04:26Z

+cat

← Older revision		Revision as of 15:04, 12 July 2020
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	[[Category:7-limit]]		[[Category:7-limit]]
			[[Category:Subfourth]]
	[[Category:Fourth]]		[[Category:Fourth]]
	[[Category:Interval]]		[[Category:Interval]]