Friday, January 6, 2012

Missing Intervals?

I tried inverting the ranking function to get an estimate of the missing items on the hypothetical list. The estimates turn out to be about 13/12 for the "second" and 1 for the "first".

Checking the intervals a, b and c found for the subdivision of the octave one finds that the first two are seconds and the third squeeks in as a first according to the empirical ranking function. Following the same system of nomenclature used for the other intervals the smaller second is a minor and the larger one a major. Adding these items to the hypothetical list appears to complete it.

The constant is somewhat arbitrary in the ranking function some other value may work better. What seems to be clear is that ranking proceeds from smaller intervals to larger ones.

Supplemental (Jan 7): The ranking mechanism is really important. A simple one would be to rank the intervals by size using decimal or hexagesimal division of the ratios. Two digit accuracy is enough to get the correct ranking. The tone and semitone have separate names and aren't really missing.

Supplemental (Jan 9): The list of intervals is the correct one for the just diatonic scale.

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