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Lowest-badness vs lowest-complexity transversal

🔗Mike Battaglia <battaglia01@gmail.com>

12/9/2011 3:31:23 PM

I worked out a bunch of stuff related to mavila modal harmony here:

http://xenharmonic.wikispaces.com/Mavila+Temperament+Modal+Harmony

For each scale I listed the "diatonic" names, and a "transversal." I'm not
sure if the Armodue people have "superdiatonic" based interval names, and
if so I'll add those too.

I did something interesting with the transversal, rather than just picking
the lowest-complexity JI match for each tempered dyad. For example, I used
12/11 to describe the size of the "major second" in mavila[9], which in
16-EDO is 150 cents, instead of 9/8.

Although I did it by intuition and not by using any mathematical algorithm,
what I'm tending towards here is some kind of transversal method which
factors in both the complexity and the error of the underlying JI dyads in
picking a winner. Such an algorithm would have to be defined with respect
to a reference tuning, such as TOP, POTE, etc - something reasonable.

So say we're in 11-limit mavila. For example, the algorithm would look at
the "major second" in mavila[9], find the tuning in POTE mavila for that
interval, see that the intervals 9/8, 11/10, 12/11, 16/15, etc, are all
mapped to it, and then pick from that the interval which is lowest in JI
complexity * tempered error, or something similar to that. This will
probably produce JI scales for high error temperaments that sound really
good, still resemble the original scale, and fix high-error simple dyads
like 675 cents that have no competing simple JI dyads which better
approximate it.

-Mike