21/16: Difference between revisions
m Typography |
Clarify the context where it works as an augmented third |
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| Monzo = -4 1 0 1 | | Monzo = -4 1 0 1 | ||
| Cents = 470.78091 | | Cents = 470.78091 | ||
| Name = septimal subfourth, <br | | 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 | '''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]] | ||
[[Category:7-limit]] | [[Category:7-limit]] | ||
Revision as of 20:10, 17 December 2021
| Interval information |
narrow fourth,
8ve-reduced 21st harmonic
reduced harmonic
[sound info]
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.
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.