re: various MOS

>"The process of producing a scale of melodic integrity by the superposition
>of a single interval (generator)."
>
>"Those points where there are only 2 different size intervals are called
>moments of symmetry."

Joe, may I suggest: "A pythagorean-type scale (the generator need not be
the 3/2, and the interval of equivalence need not be the 2/1) is MOS iff
the generating interval occurs at only one scale degree in all modes of the
scale (i.e. 3/2 is always a 5th in the diatonic scale)."?

>"This cycle has the property that any occurrence of an interval will always
>be subtended by the same number of steps."

That's not true. I think I said it in a post once, and posted a correction
shortly after. Sorry if I have contributed to a misunderstanding. This
condition is equivalent to strict propriety.

>So, equal temperaments would all qualify?

What qualifies or not is a set of three things- generator, interval of
equivalence (IE), and number of tones. It is true that all possible
generators in an x root of n tuning have an x-tone MOS when the interval of
equivalence is n, and some generators will also have MOS's with < x tones,
if x is not prime.

>Plus a succession of fourths would qualify? How about a succession of
>variable fourths?

That depends on what you mean by "fourths"? In "variable fourths," Wilson
means intervals that subtend four scale degrees, but may be of different
acoustic sizes. To be MOS or not, the acoustic size of the generator must
be known; the scalar size of the generator is determined later, by the
number of places the chain is carried out to.

Remember that not all chains of an interval are MOS- only those chains
where the chained interval (generator) always covers the same number tones
in the IE-reduced scale are MOS.

-C.

I wrote...

>Joe, may I suggest: "A pythagorean-type scale (the generator need not be the
>3/2, and the interval of equivalence need not be the 2/1) is MOS iff the
>generating interval occurs at only one scale degree in all modes of the
>scale (i.e. 3/2 is always a 5th in the diatonic scale)."?
>
>It is true that all possible generators in an x root of n tuning have an
>x-tone MOS when the interval of equivalence is n, and some generators will
>also have MOS's with < x tones...

I should point out that this allows cases where the generator evenly
divides the IE (like the augmented triad in 12tET) to be MOS. Many authors
probably wouldn't consider them MOS, but I do.

-C.