View source for User:Overthink/2.3.7 duals of 5-limit chords

On this page we will look at certain 5-limit chords, and find 2.3.7 versions of them:
== First Approach ==
In this section, we will find the dual of a chord as follows: For example, take the 5-limit major, [[4:5:6]]. Assuming octave equivalence, this chord can be described as the list of odd numbers {1, 3, 5}. Note that two chords with different pitches but the same pitch classes, where notes an octave apart are considered the same pitch class, are considered different voicings of the same chord. We find the 2.3.7 dual of this chord by replacing each factor of 5 with a factor of 7, so we get {1, 3, 7}. This can be voiced as [[6:7:8]], or a wider voicing of [[4:7:12]]. Below are the duals of important 5-limit chords.

=== 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 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

=== 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. 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)