Interval Matrices
Added 2023-07-17 02:29:45 +0000 UTCIt took a long time to get around to this. I was preparing some of these diagrams for the book and realized that I never put them on the website.
So, several hours later I have 2048 of these tables, rendered with a handsome clarendon font... and on the website they are annotated with circles and lines showing contradictions and ambiguities.
A contradiction is when a small generic interval has a larger specific size than a larger generic. A nice example of a contradiction is found in the Double Harmonic scale (https://ianring.com/musictheory/scales/2483). Presuming it's in C, note how this scale has a generic second step between Db and E with a specific size of 3 semitones; and there is a generic third between B and Db that has a specific size of 2 semitones. The third is smaller than the second. That's the contradiction. That scale has four of them.
An ambiguity happens when two generics have the same size. It's the less heinous version of a contradiction.
Contradictions and Ambiguities relate directly to the notion of propriety, also known as coherence. Those properties tend to indicate scale candidacy, ie whether a scale is going to be "musically useful".
On the website I included the interval matrix only if the cardinality is greater than 2. I just launched them a few minutes ago.
Enjoy!
