More from Brian

From: mclaren
Subject: The excellence of Paul Rapoport's
articles on microtonality
--
Having criticized Paul Rapoport's bibliographies,
let me also point out the *remarkable excellence*
of his articles on microtonality.
Among theorists of the equal-tempered scales,
four stand above all the rest, Bosanquet, Blackwood,
Rasch and Rapoport.
I've already mentioned that Paul's article "Just
Shape, Nothing Central" in MusicWorks 60 is
to date the definitive discussion of Erv Wilson's
CPS tunings in print. The importance of this
article should not be underestimated.
While some members of this tuning forum consider
Erv Wilson's CPS tunings "just another approach,"
the truth is that Wilson's CPS approach offers
a number of unique advantages found in only a
few other methods of generating and modulating
through just intonation scales.
Paul's article touches on some of these advantages
(order the back issue of MusicWorks 60 from Gayle
Young if you want more info) and uniquely offers
a number of excellent geometrical models that make
the tunings remarkably easy to visualize. Even Erv
has not drawn his CPS ratio-space structures as
well as the photographs present them in Paul's
remarkable article.
Equally impressive is Paul's MusicWorks 43 article
"Some Temperaments are More Equal than Others--
and Decidedly More Temperamental." This is the
best single article introduction to xenharmonic
equal temperaments, along with Ivor Darreg's
Xenharmonic Bulletin Number 5. However, Paul's
article covers more ground and covers it with
extraordinary lucidity. In particular, the
open-mindedness with which Paul approaches
different divisions of the octave is a splendid
change from the usual "good tuning/bad tuning"
categorization of Hall, Stoney, Yunik & Swift,
von Foerster, et al.
This article, along with the 1978 "Tempo"
article "Toward the Infinite Expansion of
Tonal Resources," is in my "beginner's packet"
of xeroxes that get sent out to everyone who
contacts me for more info about non-12
tunings.
However, Paul's most important contribution
to date is "The Notation of Thirds and Fifths
In Equal Temperaments," in J. Mus. Theory,
1993, along with its companion article "The
Notation of Equal Temperaments" in Xenharmonikon
16, 1995.
These are really two halves of a single monumental
article which treats in superb detail not only the
notation, but also the melodic and harmonic properties,
of the entire range of equal temperaments.
Paul proves a number of important results: for example,
that as long as the Pythagorean komma is positive,
the fifth in any scale can never be higher than
700 cents. He also investigates in remarkable
detail the progression of fifths for any p as
the number of tones per octave increases without
limit.
To my knowledge, no previous theorist has discussed
this issue in the detail and with the comprehensive
intelligence Paul Rapoport has shown in this article.
Paul also shows how and why the pythagorean komma
can take on values greater than 11. This is a startling
idea in 12-TET, but can easily occur for large numbers
of tones per octave.
Paul also points out that Easley Blackwood's *music*
as opposed to his theories) "also suggests accepting
or even transcending these limits under some
conditions, despite the odd Pythagorean relationships
that result." This is a superb example of Paul's prescient
recognition that music theory only goes so far. Beyond
a certain point, the ears must stand as final arbiters.
As Ludwig Wittgenstein put it in the Tractatus Logico-
Philosophicus: ""My propositions serve as elucidations
in the following way: anyone who understands me
eventually recognizes them as nonsensical, when he
has used them--as steps--to climb up beyond them.
(He must, so to speak, throw away the latter after
he has climbed it.)" [Wittgenstein, L., "Tractatus
Logico-Philosphicus," 6.54, 1921]
But this is not the end of Rapoport's contributions
to music theory: it is only the beginning.
Paul describes not just one but several methods
of determining the divisions of the octave in which
the fifth is not recognizable. This significantly
extends Blackwood's work.
Paul also points out that "Suppose we wish to
write music for the tuning having the lowest a
with a certain number of v's (thus the same
number of p's), for some basic p. We cannot
easily determine which p's there will be in the
unknown a. There are two methods for determining
the lowest a in accordance with what we have
developed so far." He derives a set of formulas
which apply to various numbers of tones per octave,
all unprecedented (to the best of my knowledge).
Paul also constructs a numebr of fascinating
t
ever, to my knowledge, compiled by
previous theorists.
Paul also defines fomulas which yield the
best t in a given division of the octave.
A related question, the problem of determining
in general how many fifths there are in
a third (ignoring octaves) Paul tackles by
using Euler's theorem. This again is an
orioginal result and extremely valuable.
To date no other articles have dealt with
the relationship between thirds and fifths
in equal temperaments to the extent and
withthe insight as Paul's two papers on the
subject.
Anyone who misguidedly imagines that my
critique of his bibliographies equates to
a criticism of Paul Rapoport's superb and
extremely insightful music theoretical
work is badly mistaken.
In fact, Paul Rapoport is one of the master
theorists of the equal temperaments. Few
xenharmonists can claim to equal, and
none surpass, his signal achivements in this
area.
--mclaren






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