21/16: Difference between revisions

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| Monzo = -4 1 0 1

| Cents = 470.78091

| Name = septimal ~~sub-fourth~~, narrow fourth, augmented third, 8ve-reduced 21st harmonic

| Name = septimal subfourth, narrow fourth, augmented third, 8ve-reduced 21st harmonic

| Color name = z4, zo 4th

| FJS name = P47

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}}

'''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.

@@ Line 3: / Line 3: @@
 | Monzo = -4 1 0 1
 | 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
 | 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.
 /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.