21/16: Difference between revisions

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| Ratio = 21/16
| Ratio = 21/16
| Monzo = -4 1 0 1
| Monzo = -4 1 0 1
| Cents = 470.7809
| Cents = 470.78091
| Name = septimal sub-fourth
| Name = septimal sub-fourth
| Sound = jid_21_16_pluck_adu_dr220.mp3
| Sound = jid_21_16_pluck_adu_dr220.mp3
| 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|4/3]] by [[64/63|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|3/2]] between 8 and 12 as well as between 14 and 21. There are also two harmonic sevenths ([[7/4|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 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 is [[21/20|21/20]] away from [[5/4|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|8/7]], the septimal supermajor 2nd of about 231.2¢, a consonance in its own right. In an [[11-limit|11-limit]] system, [[11/8|11/8]] is also nearby, so that 21/16 can step up by the small semitone of [[22/21|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.


See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]     [[Category:7-limit]]
:''See also [[Gallery of Just Intervals]]''
[[Category:fourth]]
 
[[Category:interval]]
[[Category:7-limit]]
[[Category:just_interval]]
[[Category:Fourth]]
[[Category:ratio]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Ratio]]

Revision as of 21:32, 23 October 2018

Interval information
Ratio 21/16
Factorization 2-4 × 3 × 7
Monzo [-4 1 0 1⟩
Size in cents 470.7809¢
Name septimal sub-fourth
Color name z4, zo 4th
FJS name [math]\displaystyle{ \text{P4}^{7} }[/math]
Special properties reduced,
reduced harmonic
Tenney height (log2 nd) 8.39232
Weil height (log2 max(n, d)) 8.78463
Wilson height (sopfr(nd)) 18

[sound info]
Open this interval in xen-calc

21/16, the septimal sub-fourth, is an interval of the 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 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.

See also Gallery of Just Intervals
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