Re: regarding Jackys question

> From: "David J. Finnamore" <daeron@bellsouth.net>
>
> Here's another monkey that could be thrown into that wrench. What happens
> when you have two scale degrees within one of those ranges, one toward the
> middle and one toward the outside? Say you have a scale with degrees at both
> 694_ and 736_. Or you stretch your octave to 1242_ and end up with a
> "leading tone" of 1195_. [In the voice of the Godfather:] What then? Huh?

This scale tool I've been playing with about ran into exactly the this
problem. Although one of the charming things about microtonality is the
ability to have 'bruised unisons' and the like, I see them as fundamentally
different than a useful melodic interval.

I see the problem as very similar to consistency, and am working
on a classification system for the diatonic systems I've been looking
at. [I'm looking for systems which hopefully will have 1) useful
melodic properties and 2) many consonances 3) be transposable as
means of exploring new sounds through counterpoint].

Here 'scale' refers to the set of all rotations of an ordered set
of pitches and 'mode' refers to a specific rotation.

C0 : No tones in a mode are different approximations for the same
rational interval. This eliminates exactly the case you refer
to above.

C1 : No tones in a scale use different approximations for the same
rational interval. In this case, if two modes have a 9/5, then
they also use the same approximation, even if the scale is
improper to the point that the 9/5 is on different degrees. This
may not be so meaningful since historically, composers have
sometimes seemed to prefer this attribute. As one used to the
surety of an EDO, I at least want to identify when this occurs.

C2 : Composite scale with one cycle of 'sharps' added has no tones
using different approximations to the same rational interval.

C3 : Composite scale with two cycles of 'sharps' has...etc.

Of course, if adding a cycle of sharps got you back to where you
started then its a closed system. If it is unclosed, then eventually
you will crash at one of these "C" dimensions.

>
> Sticky hypothetical situations aside, I would prefer to define the octave
> in terms of Western music theory. I would use a different term for
> approximations of the 2:1 in other musics.

In the recent thread about this, I was looking for ways of approaching
the 2/1 with the same approach towards consistency as any other interval.
If across the range of the tuning, going up N steps is always going up
by the best 2/1, then that is "2-limit consistent" and whatever the
actual ratio, that is the functional "octave" in the tuning,

Bob Valentine