User:Overthink/2.3.7 duals of 5-limit chords: Difference between revisions

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=== 5-limit Major ===

The 5-limit major chord is 1-5/4-3/2 = {1, 3, 5} with steps 5/4-6/5-4/3. Note that 4/3 is the interval between 3/2 and an octave above the root, 2/1. Replacing the 5 with a 7 gives us the chord {1, 3, 7}, which reduces to 1-3/2-7/4 with steps 3/2-7/6-8/7. This can be seen as the fundamental chord of 2.3.7 harmony, with 7/6 and 8/7 as fundamental units rather than 5/4 and 6/5.

The 5-limit major chord is 1-5/4-3/2 = {1, 3, 5} with steps 5/4-6/5-4/3. Note that 4/3 is the interval between 3/2 and an octave above the root, 2/1. Replacing the 5 with a 7 gives us the chord {1, 3, 7}, which reduces to 1-3/2-7/4 with steps 3/2-7/6-8/7. This can be seen as the fundamental chord of 2.3.7 harmony, with 7/6 and 8/7 as fundamental units for building chords rather than 5/4 and 6/5.

'''2.3.7 dual:''' Notes 1-3/2-7/4 with steps 3/2-7/6-8/7

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=== 5-limit Minor ===

The 5-limit minor chord is 1-6/5-3/2 = {3, 5, 15=3*5} with steps 6/5-5/4-4/3. The 7-limit dual is {3, 7, 21=3*7}, which reduces to the notes 21/16-3/2-7/4 with steps 8/7-7/6-3/2. This doesn't contain the root, but by shifting every note up by 8/7 (or equivalently down by 7/4), we get a chord with notes 1-3/2-12/7 and steps 3/2-8/7-7/6. This is similar to the 2.3.7 major chord, but with the 7/6 and 8/7 swapped. The note between the fifth and the octave is lowered compared to the 2.3.7 major by [[49/48]], which can be considered the 2.3.7 equivalent of the 5-limit interval [[25/24]], the difference between 5/4 and 6/5.

The 5-limit minor chord is 1-6/5-3/2 = {3, 5, 15=3*5} with steps 6/5-5/4-4/3. The 7-limit dual is {3, 7, 21=3*7}, which reduces to the notes 21/16-3/2-7/4 with steps 8/7-7/6-3/2. This doesn't contain the root, but by shifting every note up by 8/7 (or equivalently down by 7/4), we get a chord with notes 1-3/2-12/7 and steps 3/2-8/7-7/6. This is similar to the 2.3.7 major chord, but with the 7/6 and 8/7 swapped. It therefore functions as the minor variant of the 2.3.7 major chord. The note between the fifth and the octave is lowered compared to the 2.3.7 major by [[49/48]], which can be considered the 2.3.7 equivalent of the 5-limit interval [[25/24]], the difference between 5/4 and 6/5.

'''2.3.7 dual:''' Notes 1-3/2-12/7 with steps 3/2-8/7-7/6

=== 5-limit Major 7th ===

The 5-limit major 7th chord is 1-5/4-3/2-15/8 = {1, 3, 5, 15} with steps 5/4-6/5-5/4-16/15. The 7-limit dual would be {1, 3, 7, 21}, which would have notes 1-21/16-3/2-7/4 with steps 21/16-8/7-7/6-8/7. The 5-limit major 7th chord has tension from the 15/8 being close to the octave; this chord doesn't have the same tension, which is important to note.

'''2.3.7 dual:''' Notes 1-21/16-3/2-7/4 with steps 21/16-8/7-7/6-8/7

=== 5-limit Minor 7th ===

The 5-limit minor 7th chord is 1-6/5-3/2-9/5 = {3, 5, 9, 15} with steps 6/5-5/4-6/5-10/9. The 7-limit dual would be {3, 7, 9, 21}, which would have notes 9/8-21/16-3/2-7/4 with steps 7/6-8/7-7/6-9/7. This can be rearranged to 1-9/7-3/2-12/7 with steps 9/7-8/7-7/6-8/7. This chord has a quality (section incomplete)

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 === 5-limit Major ===
-The 5-limit major chord is 1-5/4-3/2 = {1, 3, 5} with steps 5/4-6/5-4/3. Note that 4/3 is the interval between 3/2 and an octave above the root, 2/1. Replacing the 5 with a 7 gives us the chord {1, 3, 7}, which reduces to 1-3/2-7/4 with steps 3/2-7/6-8/7. This can be seen as the fundamental chord of 2.3.7 harmony, with 7/6 and 8/7 as fundamental units rather than 5/4 and 6/5.
+The 5-limit major chord is 1-5/4-3/2 = {1, 3, 5} with steps 5/4-6/5-4/3. Note that 4/3 is the interval between 3/2 and an octave above the root, 2/1. Replacing the 5 with a 7 gives us the chord {1, 3, 7}, which reduces to 1-3/2-7/4 with steps 3/2-7/6-8/7. This can be seen as the fundamental chord of 2.3.7 harmony, with 7/6 and 8/7 as fundamental units for building chords rather than 5/4 and 6/5.
 '''2.3.7 dual:''' Notes 1-3/2-7/4 with steps 3/2-7/6-8/7
@@ Line 11: / Line 11: @@
 === 5-limit Minor ===
-The 5-limit minor chord is 1-6/5-3/2 = {3, 5, 15=3*5} with steps 6/5-5/4-4/3. The 7-limit dual is {3, 7, 21=3*7}, which reduces to the notes 21/16-3/2-7/4 with steps 8/7-7/6-3/2. This doesn't contain the root, but by shifting every note up by 8/7 (or equivalently down by 7/4), we get a chord with notes 1-3/2-12/7 and steps 3/2-8/7-7/6. This is similar to the 2.3.7 major chord, but with the 7/6 and 8/7 swapped. The note between the fifth and the octave is lowered compared to the 2.3.7 major by [[49/48]], which can be considered the 2.3.7 equivalent of the 5-limit interval [[25/24]], the difference between 5/4 and 6/5.
+The 5-limit minor chord is 1-6/5-3/2 = {3, 5, 15=3*5} with steps 6/5-5/4-4/3. The 7-limit dual is {3, 7, 21=3*7}, which reduces to the notes 21/16-3/2-7/4 with steps 8/7-7/6-3/2. This doesn't contain the root, but by shifting every note up by 8/7 (or equivalently down by 7/4), we get a chord with notes 1-3/2-12/7 and steps 3/2-8/7-7/6. This is similar to the 2.3.7 major chord, but with the 7/6 and 8/7 swapped. It therefore functions as the minor variant of the 2.3.7 major chord. The note between the fifth and the octave is lowered compared to the 2.3.7 major by [[49/48]], which can be considered the 2.3.7 equivalent of the 5-limit interval [[25/24]], the difference between 5/4 and 6/5.
 '''2.3.7 dual:''' Notes 1-3/2-12/7 with steps 3/2-8/7-7/6
+=== 5-limit Major 7th ===
+The 5-limit major 7th chord is 1-5/4-3/2-15/8 = {1, 3, 5, 15} with steps 5/4-6/5-5/4-16/15. The 7-limit dual would be {1, 3, 7, 21}, which would have notes 1-21/16-3/2-7/4 with steps 21/16-8/7-7/6-8/7. The 5-limit major 7th chord has tension from the 15/8 being close to the octave; this chord doesn't have the same tension, which is important to note.
+'''2.3.7 dual:''' Notes 1-21/16-3/2-7/4 with steps 21/16-8/7-7/6-8/7
+=== 5-limit Minor 7th ===
+The 5-limit minor 7th chord is 1-6/5-3/2-9/5 = {3, 5, 9, 15} with steps 6/5-5/4-6/5-10/9. The 7-limit dual would be {3, 7, 9, 21}, which would have notes 9/8-21/16-3/2-7/4 with steps 7/6-8/7-7/6-9/7. This can be rearranged to 1-9/7-3/2-12/7 with steps 9/7-8/7-7/6-8/7. This chord has a quality (section incomplete)