Mike Sheiman's Alternative Interval Categorizations

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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.