RE: [tuning] Re: Otonal/utonal pairs: 3-limit trines, 5-limit tri ads

I'd agree that, if you don't allow the transpositions implied by octave
equivalence, a fourth over a fifth is otonal, and a fifth over a fourth is
utonal. However, in no way is a fourth alone more utonal than a fifth alone,
any more than a first-inversion major triad is more utonal than a
root-position major triad. In Partch's theory, whether a triad is in root
position, first inversion, or second inversion does not affect its important
characteristics: whether it is utonal or otonal and what identities it
contains. In this sense, a fourth may be construed as either the 1 and 3
odentitites or the 1 and 3 udentities. A fifth may also be construed both
ways.

>In contrast, the utonal trine (3,4,6) or triad (10,12,15)
>has for its lowest note a partial which is not an even octave of the
>fundamental.

Since the lowest note doesn't matter for Partch's purposes, I see an
important difference here: the utonal trine (i.e., 3-limit Utonality)
contains a note which is an octave of the "fundamental", while the utonal
triad (i.e., 5-limit Utonality) does not. In fact, no higher-limit utonal
chord will contain a note octave-equivalent to the "fundamental" (which, in
Partch's terms, is really the Unity of the simplest otonal representation of
the utonality -- not very simple even in the 7-limit: (60,70,84,105)).