21/16: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| | | Name = septimal subfourth, narrow fourth, 8ve-reduced 21st harmonic | ||
| Color name = z4, zo 4th | |||
| | |||
| Sound = jid_21_16_pluck_adu_dr220.mp3 | | Sound = jid_21_16_pluck_adu_dr220.mp3 | ||
}} | }} | ||
'''21/16''', the '''septimal | |||
'''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. | ||
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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. | |||
== Approximation == | |||
{{Interval edo approximation|21/16}} | |||
== See also == | == See also == | ||
* [[32/21]] – its [[octave complement]] | |||
* [[8/7]] – its [[fifth complement]] | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
[[Category: | [[Category:Subfourth]] | ||
[[Category:Fourth]] | [[Category:Fourth]] | ||
Latest revision as of 20:54, 22 May 2026
| 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, 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.
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.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 5 | 2\5 | 480.00 | +9.22 | +3.84 |
| 10 | 4\10 | 480.00 | +9.22 | +7.68 |
| 18 | 7\18 | 466.67 | -4.11 | -6.17 |
| 23 | 9\23 | 469.57 | -1.22 | -2.33 |
| 28 | 11\28 | 471.43 | +0.65 | +1.51 |
| 33 | 13\33 | 472.73 | +1.95 | +5.35 |
| 38 | 15\38 | 473.68 | +2.90 | +9.19 |
| 41 | 16\41 | 468.29 | -2.49 | -8.50 |
| 46 | 18\46 | 469.57 | -1.22 | -4.66 |
| 51 | 20\51 | 470.59 | -0.19 | -0.82 |
| 56 | 22\56 | 471.43 | +0.65 | +3.02 |
| 61 | 24\61 | 472.13 | +1.35 | +6.86 |
| 69 | 27\69 | 469.57 | -1.22 | -6.99 |
| 74 | 29\74 | 470.27 | -0.51 | -3.15 |
| 79 | 31\79 | 470.89 | +0.11 | +0.69 |