An interesting series of tunings

I just happened to notice that in many of the tunings (equal temperaments,
to be precise) that I work with, seven steps gives you an interval where the
lower note sounds like the root. Let's see what happens if we formalize this
idea -- we want the lower note of the interval to correspond to a harmonic
that is a power of 2 (so it sounds like the root), and the upper note to be
the next higher harmonic:

Interesting tuning #n = round(7/(log((2^n+1)/2^n)/log(2)))

According to this formula, interesting tuning #0 is 7-tET, which is the
standard tuning of Thailand. Siamese compositions modulate, usually by
fifths, much like those in the West. As the only other culture that relies
on modulation, Thailand is the only other culture that uses equal
temperament (though I have heard that this practice extends into Cambodia).

Interesting tuning #1 is 12-tET. Apparently, interesting enough to satisfy
the West for the last few centuries.

Interesting tuning #2 is 22-tET. This is my main avenue of research; contact
me for more info. This is also the first equal tuning with consistent
representations of all harmonies through the 12th harmonic. See previous
digests for discussions of consistency.

Interesting tuning #3 is 41-tET. Recommended by Paul van Jenko around the
turn of the century, this tuning is barely distinguishable from 9-limit just
intonation. Try a "ninth chord" tuned (in cents) 0 380.5 702.4 965.9 1404.9.
It also offers consistent representations through the 16th harmonic.

Interesting tuning #4 is 80-tET. This is the first equal tuning with
consistent representations through the 20th harmonic. That pretty much
covers any harmonies I would consider musically useful, and then some.

Interesting tuning #5 is 158-tET. Or it would be if there were anything
interesing about it. Like most numerological speculations, such as
Kornerup's and Yasser's, this one's luck runs out after a while. Then again,
who am I to say that some great composer won't come along and demonstrate
the wonderful properties of 158-tET?


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