source file: mills2.txt
Date: Sun, 6 Jul 1997 08:12:33 +0200

Subject: Re: Going to extremes

From: mr88cet@texas.net (Gary Morrison)

>This is the chord progression I've used, in 15TET. For your information, it
>is the tuning that has the largest wandering tonic, 80 cents.

   Any equal-temperament in the group that I call "fibonacci+3" will do that.

   By fibonacci, what I'm referring to is J. Yasser's fibonacci-like
sequence of tunings (5 7 12 19 31 50 81...).  (By fibonacci-like I mean
that the next number in the sequence is the sum of the previous two.)  12
and up in this sequence have meaningful representations of major scales,
with a single size of whole-step and a single size of half-step in the
usual W W H W W W H setup.

   Those sizes of whole- and half-steps follow the REAL fibonacci sequence
(1 1 2 3 5 8...).  So, 7TET has a whole-step size of 1 and a half-step size
also of 1, thus it takes the first two positions in the sequence.  12TET
takes the second two positions:  2 steps for whole, and 1 for half.  19TET
takes the 2 and 3 positions, 31 takes the 5 and 3 positions, and so forth.


   Now, there is a parallel series that mimics the syntonic comma
difference in the Ptolemaic major-scale tuning, by increasing the size of
three of the whole tones by one chromatic degree.  That takes on the
pattern lW sW H lW sW lW H (lW large whole tone, sW small whole tone),
that pattern being dictated by where Ptolemaic tuning has 9:8 vs 10:9 whole
tones.

   That parallel "fibonacci+3" series then is (10), 15, 22, 34, 53...
These tunings, since they all have two different sizes of whole-step, can
simulate the syntonic comma error that gives rise to this particular
wandering-tonic effect.  With 22 the effect is very striking, since the
pitch change is big, whereas with 53, the effect is more of a hauting one.

   With 15 it ought to prove even more striking to say the least, because,
as Kami pointed out, the amount by which it wanders is HEEEEYUUUUGE!

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