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21/16
|-4 1 0 1>
470.7809 cents
sound sample

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: Gallery of Just Intervals