MOS continued

MOS by definition are not equal divisions or scales consisting of N
exactly repeated blocks within the interval of equivalence (usually the
octave). Thus scales such as the whole tone scale, the string of pearls
octatonic, the symmetrical hexatonic mode that Blackwood likes
(313131), or the modes of limited transposition that Messiaen used are
not MOS. They lack Myhill's property as the generator (in this case the
repeating block span) comes in only one size, not two, and a cycle of
this interval does not generate all the tones. In fact, Rothenberg
computations are done on just one block of these sub-octaval repeating
scales and the block span functions as the interval of equivalence.

>Imagine this scene: you're at a party. A lovely young woman arrives,
>and somehow you find yourself in conversation with her, and somehow it
>comes around to tuning, and she asks, "But isn't equal temperament the
>perfect way to tune? Why would you want to change it?".

The appropriate response is "I love it when you talk dirty."

--John

John Chalmers wrote,

>MOS by definition are not equal divisions or scales consisting of N
>exactly repeated blocks within the interval of equivalence (usually the
>octave). Thus scales such as the whole tone scale, the string of pearls
>octatonic, the symmetrical hexatonic mode that Blackwood likes
>(313131), or the modes of limited transposition that Messiaen used are
>not MOS. They lack Myhill's property as the generator (in this case the
>repeating block span) comes in only one size, not two, and a cycle of
>this interval does not generate all the tones. In fact, Rothenberg
>computations are done on just one block of these sub-octaval repeating
>scales and the block span functions as the interval of equivalence.

This is a particularly interesting point; Rothenberg would assume that any
interval, not just the octave, can function as an interval of equivalence if
the scale is symmetrical at that interval. This is quite the opposite of
supposing an acoustical foundation for octave equivalence. However, there is
clearly an "ambiguity of tonal function" between the block-equivalent notes
in any symmetrical scale.

I wrote,

>> This is a particularly interesting point; Rothenberg would assume that
any
>> interval, not just the octave, can function as an interval of equivalence
if
>> the scale is symmetrical at that interval. This is quite the opposite of
>> supposing an acoustical foundation for octave equivalence. However, there
is
>> clearly an "ambiguity of tonal function" between the block-equivalent
notes
>> in any symmetrical scale.

Daniel Wolf wrote,

>Rothenberg's assumption is quite a musical one, if "equivalence" is here
>divorced from the issue of exact identity and associated with melodic
>transformations, for example, repetition of a figure within a given
>tetrachord in another tetrachord conjunct or disjunct with the first. This
is
>a basic procedure in much of the world's music.

And fundamental to my own understanding of the diatonic scale in Western
music, and my construction of new scales! However, the acoustic simplicity
of the fourth and fifth are in my view essential here for projecting the
similarity relation, and hence for explaining why the diatonic scale is so
remarkably effective. For Rothenberg all that matters is abstract structure
along the pitch axis, and scales with any other intervals of repetition
would be equally effective.