Reinhard/Monzo/Schoenberg/Partch

}Partch's point is that the assignment of partials to 12-eq
}representations by Schoenberg has a margin of error which nearly equals
}the smallest interval in the 12-equal scale, so that the partials could
}just as easily be represented by the next nearest 12-eq note as by the
}one assigned by Schoenberg.

That's right. Essentially, Schoenberg's idea falls flat on its face.
Reinhard will argue otherwise. "Tomfoolery" or not, I only consider a
theory to have content if it can conceivably be falsified by a
contradictory body of evidence. The longer a theory goes without such
contradictory evidence actually occuring, the more believable the theory
becomes. Partch either succesfully falsified the theory, or the theory
is so vague and "above falsifiablity" that it is entirely lacking in
content.

This, of course, says nothing about the validity of Schoenberg's
intuitions and the
quality of the music that resulted. Good music is not an excuse for bad
theory. (Neither is bad music.)

}Unfortunately (for Riemann) acoustical science knows nothing of
}undertones. These hallucinatory tones would have to be mechanically
}_multiplex_ tones in the same sense in which overtones are mechanically
}_partial_ tones. But this would require the imagination of a sort of
}fourth dimension for space. A volume of mass under tension can indeed
}vibrate in parts of itself. This is plain mechanics. But how could it
}vibrate in multiples of itself? The multiples would have to lie in the
}fourth, the invisible dimension.

I don't know how a fourth or invisible dimension of space would help,
but again, Monzo (or is it Meyer) is correct. Meyer is of course subject
to a similar criticism as Schoenberg, as the former used quartertones to
realize septimal harmony, despite the fact that 24ET is not consistent
within the 7-limit, much as the latter used 12ET to realize 13-limit
harmony despite the inconsistency. As for "intellectual machinations," I
seem to be accused of that every time I make a decisive theoretical
point against Reinhard. The importance and beauty of utonal formations
in Partch's music and elsewhere is undeniable (I heard Catler play some
beautiful 7-limit utonalities on Sunday). The fact that undertone series
as scales can be more easily constructed by man than overtone series as
scales is also undeniable. The fact that utonal chords have a lower
first common overtone, and a greater rate of occurence of higher common
overtones, and hence are in a sense easier to tune with beats, that any
comparable chords including otonal ones, is also undeniable. That is not
the issue. In fact, anyone who has bothered to read my paper on 22tet
will note a complete equality between the way I treat otonal and utonal
formations. The point is that the debate on the nature of minor chords
was already quite old in Partch's day. Partch took a certain position in
this debate (although not a wholly self-consistent one) and moved on, to
extend minor-type formations to tetrads, pentads, and hexads.

Tomfoolery and intellectual machinations aside, I hope we can all agree
that the music is the most important thing, our attempts to understand
music are extremely limited, and arguing about famous musicians'
attempts to do so can get a bit too far from the business of making
music.