From: mclaren
Subject: group theoretic approaches to microtonality
--
In topic 4 of tuning digest 650, John Chalmers commented
cogently on G. J. Balzano's papers on microtonality:
"Re Balzano: B seems to have independently discovered the principle
of propriety, though he called it 'coherence.' Basically it means that
the scale has no overlapping interval classes, i.e. that the largest
2nd is less than or equal to the smallest third for all melodic seconds
(intervals between adjacent tones) and melodic thirds (intervals between
every other tones), irrespective of their acoustical size and similarly
for all scale interval classes." -- John Chalmers
David Rothenberg independently came up with this idea, as did
Carlton Gamer. Rothenberg hypothesized that proper scales would
be heard as coherent wholes, while improper scales would tend
to be heard as modes of smaller numbers of tones.
There are several problems with the idea of "propriety" or
"coherence." For one thing, real music in the real world doesn't
appear to back up this hypothesis. A classic improper scale,
the Greek Enharmonic, appears to be heard as 7 entirely
seamless and sensible notes. No listener in my experience
has ever heard the Greek Enharmonic as a 5-note mode surrounded
by microtones. This may be due to the fact that in real music
in the real world, leading tones are often considerably sharpened
on variable-pitch instruments, so we listeners are already
used to hearing a seamless 7-note diatonic scale with a much smaller
microtone in it just as part of what is laughingly called
western diatonic "12-TET" performance practice.
The other more serious problem is that I see no way to
set up an A/B controlled experiment to test Rothenberg's
hypothesis with hard cold numbers. The reason is that I
can't think of any way to devise a control composition whose
musical style won't influence the way in which the tuning
is heard. Alas, all compositions have their own style which
in turn changes the way the tuning is heard. So how can you
be sure that the style of the composition isn't responsible
for how the listener hears the scale, rather than the tuning?
Those of you who don't believe this strong influence of
musical style will want to listen to some of my
23-TET compositions. Depending on style I can make 23-TET
sound either near-just and diatonic and smoothly tonal,
or radically dissonant and chromatic and anti-tonal.
This bodes ill for the idea of a "neutral" musical style
in a control composition. And without a control of some
kind, psychoacoustic tests on Rothenberg's propriety
hypothesis would be meaningless...and what use is
a hypothesis you can't test?
John C. goes on to say:
"Balzano's other contribution to scale theory is his application
of Group Theory to scale generation. In analogy with the major mode
in 12, Balzano created a class of scales partitionable into chains
of triads of the form 0 k 2k+1 and 0 k+1 2k+1, where k and k+1 are
the number of scale degrees in the mediant and conjugate-mediant
('Thirds') and 2k+1 is the dominant of the triads, following Riemann's
and Lewin's terminology. The "chromatic" sets C thus have k(k+1) or
k^2 +k tones, the generators (G) of the diatonic sets have 2k+1
tones and the diatonic sets (D) have 2k+1 tones.
The K and k+1 are also the generators of groups of cardinality C.
Hence for 12-tet, k = 3, k+1 = 4, 2k+1 = 7, k^2+k = 12. Balzano studied
the case where k = 4, 5, and 6 and mentioned k=8 briefly. These correspond
to equal temperaments of 20, 30, 42, and 72 tones (56 is presumably
included). The diatonic sets and their generators have 9, 11, 13, and 17
tones respectively (15 for k = 7)." -- John Chalmers
Balzano went on to derive group theoretic analogs of diatonic scales
from these chromatic fields. (Alternatively, you could think of him
as having built the chromatic fields from the diatonic analogs.)
However Balzano does not appear to have heard any music in the scales
he discusses. Back when Balzano was writing his papers, circa 1978-1985,
no affordable retunable digital synthesizers existed.
One big criticism I would venture of the Riemann/Lewin group theoretic
idea about generators of the diatonic sets is that by definition this
stuff ONLY works for ETs whose cardinality is a composite number
rather than a prime. But this is a big problem, because it tells us
that the Riemann/Lewin/Forte set theory stuff arose specifically from
an ET whose cardinality was a composite number--in short, the
Riemann/Lewin/Forte set theory stuff is extremely 12-centric.
Alas, the explosion of xenharmonic compositions since 1986 has
proven that many worthwhile and interesting-sounding ETs have
prime numbers of pitches: 17, 19, 29, 31, 37, 43, etc. Thus
a set theory viewpoint of diatonic scale generation which is
12-centric and which cannot handle prime numbers of tones is
de facto suspicious (as are all theories derived from 12-centric
concepts once we move out into the infinite realm of microtonal
equal temperaments).
Moreover, the 0 k 2k+1 definition of the triad is VERY 12-centric.
In some ETs, the fifth will NOT be 2k+1, but 2k. Example:
17/oct, whose most euphonious triad has the form 0 k 2k (0 cents,
352.9 cents, 705.9 cents) and whose next most euphonious triad
doesn't exist, since there is no major and minor but merely
a single consonant neutral triad. Abolishing major and minor
leads not to the Balzano series of tunings k(k+1) but to a
different series k(k) + 1 or k^2 + 1, since we must form chains
on a series of neutral triads. This series of scales gives
5 10 17 26 37 50 65 82 101, a really interesting set of equal
temperaments, every one of which is particularly worth
investigating musically and all of which are unjustly
overlooked.
John Chalmers goes on to say:
"There are several problems with this theory. The set of predicted scales
misses several harmonically much better tunings by 1 degree with the
exceptions of 12 and 72-tet (19, 31, 41, also 29 and 43 are better by most
criteria than 20, 30 and 42). Secondly, the triads are continually
shrinking as k gets larger, and the number of tones in the scale compared
to the number in the chromatic set also decreases. For example, in 72-tet,
only 17 of the 72 tones are in the scale, leaving the remaining 55 tones
as auxiliaries, alternates, or ornaments. The triads 0 8 17 and 0 9 17 are
essentially tone clusters of 0 133.3 283.3 and 0 150 283.3 cents. The
scale itself, consisting of 4 repeated blocks of 5 4 4 4 degrees and a
final interval of 4 degrees, is coherent (actually, strictly proper),
however. It is an MOS (as are all of Balzano's scales), though not a
'deep scale' (Winograd, Gamer, L= C/2 or C/2+1)." -- John Chalmers
Other problems which John doesn't mention include: [1] the fact that
as the number of tones/oct grows large, the number of pitches in
the diatonic analog also grows large. But this conflicts with
Irwin Pollack's and George Miller's findings about the human
ability to process musical information. Pollack and Miller found
that there is a limit of roughly 2.5 bits on the number of musical
pitches we can keep in short-term memory, plus or minus 2. This
means that as a practical matter it is unrealistic to ask an
audience to distinguish more than 9 modal pitches. Beyond
that number, some of the pitches tend to be confused with one
another. This explains the prevalence of 7-note modes throughout
the world's music: that modal number of pitches is optimal for
human short-term memory. However, 17 pitches is far too many--
the resulting music would doubtless sound even more incoherent
and indistinguishable than the music of the post-Webern serialists,
who tried to use all 12 chromatic pitches of 12-TET as a single
12-note mode. The post-Webern tykes failed because of the
hardwired limitations of the human brain, and Balzano's proposed
modes for ETs > 12/oct would also fail for the same reason.
What's needed is a set of between 5 and 9 pitches in ETs > 12/oct.
[2] John's criticism of the Balzano series of scales K(K+1)
2, 6, 12, 20, 30, 42, 56, 72, 90, 110 tones per octave &c.
is faulty because Balzano explicitly abandons harmonic-series
considerations in evaluating these tunings. It is therefore
not fair to point out that the Balzano series of ETs missing
"harmonically much better tunings by 1 degree," since B was
not after harmonically better tunings.
A valid variant of John's criticism is based not on members
of the harmonic series, but on simple musical experience.
Having composed in every equal temperament from 5 through 53
tones per octave, it seems to me that every one of 'em has
something valuable to offer. I have not yet encountered
any equal temperament in which interesting and emotionally
compelling music can't be composed. In particular, while
12/oct, 20/oct, 30/oct and 42/oct are all fine scales with
wonderful possibilties, 21/oct, 22/oct, 33/oct, 27/oct, 29/oct,
and particularly 37/oct and 39/oct are at least as splendid
sounding and at least as interesting musically. Thus Balzano's
series simply fails the test of experience. He picks out as
unusually noteworthy those equal temperaments which are worthwhile,
but no more so musically than the ETs around them.
Thus the Balzano series of scales must be classed with the
Yasser/Kornerup Fibonacci series of scales K-1 + K (2,5,7,
12,19,31,50,81,131 &c.) as numerology that doesn't have
an awful lot to do with the real world. Yes, some of Yasser's
and Kornerup's Fibonacci ETs are excellent and produce fine
music (particularly 7, 12, 19 and 31) -- but as Paul Erlich
will readily attest, non-Yasser ETs produce music
that sounds just as good: 22, 17, 26, 15.
[3] Because Balzano's series of ETs is essentially
a recursion relation, it depends crucially on the starting
number. This is a bit of black-magic hocus-pocus that
Balzano never bothers to justify, since it cannot be
justified. The series K(K+1) yields very different
results when you start with 6/oct. The 6 tone scale
doesn't impress anyone who's ever heard it as being
particularly suited to tonal or diatonic music, and B
seems to brush this aside--he starts at K = 3, which
gives 12 pitches to the octave. 2 or 6 equal pitches to
the octave gives lousy results, as does any number that
pushes you above 72 equal pitches. Thus Balzano's
series is only useful for a tiny number of ETs, yet
he never gives us any hard mathematical justification
for focussing on that tiny number of ETs.
The same problem, incidentally, afflicts the Yasser/Kornerup
series: if you start with 2 and 5, everything's peachy.
But what if you start with 1 and 4? 1, 4, 5, 9, 14, 23,
37, 60, 97 &c. This is a set of ETs that would give
the Yasser the willies. He inveighed at length against
14-equal because of its purportedly unacceptably symmetry,
yet his Fibonacci series of scales produces 14 equal if
you start with a different seed. Ditto if you start with
3 and 6, giving 3, 6, 9, 15, 24, 39, 63, 100 &c. You
can even get 22 equal if you start with 2 and 6: 2, 6,
8, 14, 22, 36, 58, 104 &c. These kinds of results
would make Yasser yip. But since it's just a
recursive series, there's no mathematical reason at
all why it has to start with 2 and 5.
[4] The final criticism with Balzano's group theory
approach really harks back to the Riemann/Lewin
idea of chains of triads. Why is the triad sacred
or necessary once we exit 12/oct? Many 12/oct tunings
do not have useful or recognizable fifths, so triads
are sometimes not even available or worthwhile in
some ETs...this raises the question: why ought we
to buy into a group theoretic construct based on
triads? Of course, this criticism allows us to
extend and/or alter Balzano's theory by using
chains of something other than triads--perhaps
dyads? Or pentads? And how about 7/6 constructs?
John Chalmers goes on to point out:
"One might modify the theory and increase the number of 'thirds' in the
basic chords and harmonize with 7th chords, whole-tone scales, etc.
(as did Yasser in 19-tet). Another possibility is to lengthen the chain
of generators to produce larger MOS's. Eleven tones out of 20-tet is a
better analog of the major scale than are 9, which really belong in
16-tet, assuming a fifth-like generator. By 11 out of 30, one is already
into an essentially non-diatonic realm of scales." -- John Chalmers
Since Pollack's and Miller's findings on the human auditory
information channel capacity limit the number of useful modal
pithces to 9 or less, this means that by the time we arrive at
30/oct the mode is not only non-diatonic but will be heard as
an indistinguishable mishmash of chromatic notes (just as
post-Webern serialist compositions are heard by John
Q. Public as an indistinguishable mishmash of chromatic notes).
Diatonicism need not be the be-all and end-all of music--entirely
anti-diatonic sets such as all 9 of 9-equal or all 7 of 7-equal
or 5 out of 8 equal can and have been used musically to produce
wonderfully vivid and emotionally powerful music. The real
problem with using more than 9 pitches is that it exceeds the
human short-term memory capacity and results in a rapid and
progressive loss of ability by the listener to distinguish
pitches, with a concomittant rapid and progressive loss of
interest in the composition.
The Pollack/Miller "Magic Number 7 Plus or Minus 2" limit
suggests an entirely different direction: by using 2 overlapping
sets of 9 or less pitches out of the 11 out of 30, a composer
should be able to keep the pitches separate. But this would
require substantial modification of Balzano's, Riemann's
and Lewin's theory since we are now far outside of 12/oct
paradigms.
John concludes:
"All in all, Balzano's is an extremely interesting theory, which
should be tried compositionally (I believe a visiting composer
at UCSD did write a piece using larger chords in 20-tet, but
I've not heard it)." -- John Chalmers
The composer was Charles Wuorinen, one of the diehard post-Webern
serialists. Alas, since these guys wildly exceeded the human
information channel capacity in 12, it goes without saying
that applying those kinds of musical procedures to 20/oct
would exceed human short-term memory even more flagrantly.
Thus Wuorinen was probably not the guy to test Balzano's
theory.
--mclaren
