The Outer Limits

So far, all of the limits I have seen discussed have been _upper_ limits.
Has anyone explored the idea of lower limits - say, a lower 3-limit, which
would not allow a power-of-two to stand alone in a numerator or denominator?
I think that would mean that every note of the scale, every interval created
by notes of the scale, would have at least a certain degree of complexity. I
drew up a few of these scales on paper but haven't yet listened to them.
Using a lower limit of 5 and an upper limit of 7 looks very strange on
paper!

BTW, I like a concept that Graham Breed introduced me to: harmonic
dimensionality using prime limits. As I understand it, using all octaves
(like P. D. Q. Bach's Don Octave) would be 1-dimensional, Pythagorean scales
(3-limit in practice) would be 2-dimensional, 5-limit would be 3-dimensional.
So a lower limit of 3 with an upper limit of 5 would insure that every
member of the scale except the tonic itself would have a 3-dimensional
relationship to the tonic; and I think it might insure that every interval
created with the scale members would also be 3-dimensional (with respect to
the tonic; 2-D standing alone, right?).

Any comments?

David J. Finnamore

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>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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