Mike Sheiman's Alternative Interval Categorizations
People often say xenharmonic intervals like 16/11 are "sour" and mathematically similar intervals (e.g. octave inverses like
1/(16/11) or 11/8) are "sweet". Doesn't that seem a bit counter intuitive?
We've been told via standard music theory to accept everything, even xenharmonic/microtonal intervals, be pigeon-holed into some sort of diatonic category.
In 12EDO C is the tonic/"first".
C# (apx. 17/16) is a minor second
D (apx. 9/8) is a major second
D# (apx. 6/5) is a minor third
E (apx. 5/4) is a major third
F (apx 4/3) is a perfect fourth (Why not a major or minor? Inconsistency...)
F# (apx. 7/5) is on the borderline between a fourth and fifth
G (apx. 3/2) is a perfect fifth (Again, no major or minor. Inconsistency...)
G# (apx. 8/5)is a minor sixth
A (apx. 5/3) is a major sixth
A# (apx. 9/5) is a minor seventh
B (apx 15/8) is a major seventh
Notice how...even in 12EDO, interval categories seem a bit shaky and inconsistent.
So how, then, to you categorize something like an 11/8 or 16/11 between a fourth and a fifth? Or an interval like 14/9, between a fifth and a sixth? Furthermore, how do explain when, for example, a 16/11 feels "sour" while an 11/8 slightly below it feels upbeat/sweet?
Usually we simply add additional names as necessary and further complicate the system. 16/11? That's sour because it's a diminished fifth. Around 14/9? That's upbeat because it's an augmented fifth. Why not just stick with major (more upbeat) and minor (more downbeat) and neutral (in-between upbeat and downbeat and a bit sour)...equally distributed among 4ths, 5ths, 6ths...?
Here's a proposal for a major/minor/neutral-only system
C is the tonic/"first".
(15/14 and less) is a minor second
(13/12 to 11/10) is a neutral second
(10/9 to 9/8) is a major second
(8/7) is a minor second-half
(15/13) is a neutral second-half
(7/6) is a major second-half
(19/16 to 6/5) is a minor third
(11/9) is a neutral third
(5/4-9/7) is a major third
(4/3) is a minor fourth (not a perfect fourth)
(15/11) is a neutral fourth
(11/8) is a major fourth (a more upbeat fourth)
(7/5) is a minor fourth-half (not the usual tritone)
(10/7) is a neutral fourth-half (not the usual tritone)
(13/9) is a major fourth-half (a "more upbeat tritone")
(16/11) is a minor fifth
(22/15) is a neutral fifth
(3/2) is a major fifth (not a perfect fifth)
(17/11) is a minor fifth-half
(14/9-11/7) is a major fifth-half
(8/5) is a minor sixth
(13/8-18/11) is a neutral sixth
(5/3) is a major sixth
(12/7) is a minor sixth-half
(26/15) is a neutral sixth-half
(7/4) is a major sixth-half
(16/9-9/5) is a minor seventh
(11/6) is a neutral seventh
(15/8) is a major seventh
Note there is only one gap where there isn't an equal minor/neutral/major sub-type categorization for every interval number/type! Only the fifth-half isn't perfectly even with two parts instead of 3.
At a quick glance...the point is with the latter system, you can hopefully quickly/easily tell which intervals to use to get upbeat (major), downbeat and a tad tense (minor), somewhat tense and mixed-mooded (neutral), or relatively sour (fourth-half) intervals.