McLaren on Group Theory, etc.

From: mclaren
Subject: visions of the future
--
I have a vision.
In the darkness, a performer floats
weightless. She lifts her hand: in a
holographic display she grabs points
in ratio space. Sounds collide.
With her other hand she reaches out,
stretching and compressing the overtones
of the timbre.
Microlasers which detect the focal point
of her eyes allow her to control by the
direction of her gaze the path of
the tuning through N-dimensional space--
a superset of ratio space. The harmonies
and melodies of this unheard future
music issue from a DIN jack plugged
into the back of her neck. But for the
virtues of proprioceptive feedback, the
entire composition could issue from her brain.
Yet even here, in the microtonal future,
there is great virtue in using the body
in performance. This performer knows (as
David Gelerntner has pointed out) that we
think as much with our muscles as
with our minds: musical composition is a
matter of tactile feedback as well as
cerebration. The story goes that
when Stravinsky used to take a break from
rehearsing his orchestral pieces he
would sit and outstretch his hands,
fingering imaginary patterns on a
nonexistant piano. Every so often,
he'd smile--discovering a new sound/
touch pattern.
This imagined future does not yet exist.
Someday, though, someone will create it.
Therefore it remains to us to prepare
ourselves by understanding as clearly
as possible the nature of the xenharmonic
landscape which we have entered, having
exited the 12-tone equal tempered
scale.
In the digital domain, discreteness (as they
say) is the better part of valor. But
discreteness applies not only to the bitstream
which nowadays makes up all the music we
hear on recordings and ever more of the
music we hear live: tunings are also
discrete, composed of dissociated atoms
of pitch which nonetheless cohere into
larger wholes.
How do the individual intervals of a tuning
form larger patterns? What is the nature
of those patterns, and what are their
similarities and differences to the
wider continuum of tunings of which
each intonation forms one specimen?
--
Group theory is a subject which has been
greatly underrated in music theory. Recently
some efforts have been made by Balzano (and
others) to relate concepts from group theory
to microtonal tunings, but these efforts have
been clouded by excessively complex language
and unduly obscurantist presentation.
In fact the basic ideas of group theory are
extremely simple--almost in proportion to
their power. Thus the ideas which I propose
to filch from group theory and apply to
the equal temperaments are practically
ludicrous in their extreme simplicity...yet
my experience with composing in the equal
temperaments from 5/oct through 53/oct now
convinces me that group theoretic concepts
are profoundly central to the musical behaviour
of the equal temperaments.
--
Balzano has pointed out in a convoluted way
that 12-tet exhibits unusual group theoretic
properties. For one thing, only 7 and 11
units (a word which will be used instead of
"scale steps" for the sake of brevity) of
12-tet are both non-composite and relatively
prime to the total number of pitches in the
temperament.
You may object that 5 and 1 units in 12-tet
are also non-composite and relatively prime
to the the total number of scale-steps
per octave. In raising this objection,
you have forgotten that in dealing with
modulo 12 arithmetic, 7 = 5 and 11 = 1.
7 modulo 12 = 5, and 11 modulo 12 = 1.
Thus from a group theoretic standpoint
the 12-tet scale contains only 6 unique
intervals; 1, 2, 3, 4, 5, 6. Or alternatively
11, 10, 9, 8, 7, 6. From the viewpoint of
modulo 12 arithmetic these two interval
sets are identical (albeit reversed in order).
Also from a group theoretic standpoint,
these two interval sets (once re-ordered)
exhibit identical transformation properties.
Moving through the 12 pitches by 1 scale
step at a time produces a resultant set
of scale-steps identical to moving through
the 1 pitches 11 scale steps at a time,
albeit in reversed order.
Let's see an example to convince the
doubters: 1 2 3 4 5 6 7 8 9 10 11 12=1
= (C) C# D D# E F F# G G# A A# B C
Now 11 scale steps at a time:
1 11 10 9 8 7 6 5 4 3 2 1=12
(C) B A# A G# G F# F E D# D C# C
--
It's obvious from examination of the
above lists that one is merely the
reflection of the other. More importantly,
the essential relationships between successive
members of these resultant sets is identical
in both cases: sucessive pitches of both sets differ
by 1 scale-step. The only distinction is the
sign of the difference. +1 for 12 modulo 1,
-1 for 12 modulo 11.
This essential relational similarity indicates
that one resultant set is merely a trivial
linear transformation of the other set.
In effect, the two sets are identical from
the point of view of group theory.
--
The same is equally true of fifths/fourths.
In more formal terms, 12 modulo 5 =
12 modulo 7 with a sign change in the
relation between successive members of
the resultant set. The sign change is
irrelevant because it means only
that we ascend as opposed to descend
the circle of perfect fifths. From the
point of view of group theory, the direction
of travel around the circle of fifths is
inconsequential because it is the relationship
between successive members of the circle of
fifths which unqiuely defines this substructure.
--
Let me use that word again: substructure.
The whole point here is to reveal the substructures
hidden underneath the surface of each equal
temperaments like bones beneath the skin of
an artist's model. A good painter or sculptor or
draughtsman or photographer knows that it is
*not* sufficient to understand the play of light
and shadow on the surface of the human skin;
to accurately represent a human face, one must
understand and elocidate in a photo or with
a pencil or with a chisel on marble the
brathtakingly intricate framework of bone
which underlies the cheek and brow and chin.
So with intonations.
To understand a tuning deeply enough to use
it effectively, it is not sufficient to
grasp the mere surface of the tuning--
what kind of thirds it offers, what kinds
of triads, what the tuning's approximation
is to this or that member of the harmonic series,
whether harmonic 5, or 7, or 11, or 13, etc.
These are important issues...but they are
ultimately superficial. A tuning's "sound" and
function is crucially dependent on the
invisible substructure which lies beneath
(but controls in every way) the audible
exterior of the intonation.
This is where group theoretic concepts
come into play.
Returning to 12, we note that it boasts
several unique properties: for one thing,
the minor third = 3 units is prime,
yet a factor of the total (non-prime) number
of pitches. The major third = 4 units is
non-prime, a power of 2, and also a factor
of the total number of pitches. Moreover,
the perfect fifth = 7 units is prime
and not a multiplicative composite of either
the min 3rd or maj 3rd or the total number of
pitches in an octave...but 3 + 4 units =
7 units, so the minor plus major third
yields an interval which is relatively prime
to the total number of units per octave.
Lastly, the only other number of units
relatively prime to 12 is 1 (= 11).
What do these facts mean?
[1] Because 7 units ( = p5) is both
relatively prime to 12 and absolutely
prime, it means that there is only a
single circle of fifths in this
temperament.
As a result, we can visit any pitch
in 12-tet by travelling around a
single circle of fifths.
[2] Because only 7 (=5) and 11 (=1)
units are relatively prime to the
total number of units per octave
AND non-composite, these are the
only two intervals which can
generate unqiue structural modes
in 12. Triads moving up and down
by 7 units generate the major
scale in 12, while single pitches
moving up and down by 11 (=1)
units generates the chromatic
scale in 12. These are the two
modal patterns which form the basis
of western 12-tet music.
[3] Because neither 3 nor 4 units
are relatively prime to 12,
moving by 3 or by 4 melodic
units generates a repeating
pattern which never changes.
Moving by 3 units up or down
yields only 4 pitches of 12,
while moving by 4 units up or
down yields only 3 pitches of
12. Note that the union of
these two sets (C E G# and
C Eb GB Bbb= A)=
(C D# E F# G# A) does NOT
produce a structural melodic
mode. This means that progressions
by successions of minor thirds
or by successions of major thirds
(A) will not allow us to travel
to any desired pitch in 12 and
(B) will prove highly repetitive.
[4] Progressions by alternating
sets of 3 and 4 units *will*
allow us to visit any desired
pitch in the scale, and also
sketch out the well-known
Alberti Bass pattern--e.g.,
arpeggiated major or minor
triads.
--
These structural features
buried deep inside the 12-tet
scale are not necessarily
reproduced in other equal
temperaments. 12-tet is a peculiar
combination of extreme internal
symmetries (because all but
11 and 7 units of 12 produce
endlessly repeating cyclic
patterns which reiterate only
a few pitches out of 12) and broken
symmetries (because 3 + 4 = 7,
and 7 generates a constantly
changing pattern of 12 full
pitches, along with 1 = 11).
--
Thus 12 is an equal division
of the octave which stands
midway between extreme internal
symmetry and complete internal
asymmetry.
To get a feeling for this, let's
consider next an equal temperament
which exhibits very high internal
symmetry. 16, being a power of 2,
represents almost the uttermost
extreme in internal symmetry.
16 has a unit size of 75 cents
and as a power of 2, a lot of
factors: 2, 4, 8, 16. Only 1, 3, 5,
are non-composite *and* relatively
prime to 16. Notice that 7 = 9
modulo 16 and 9 is a composite number.
11 = 5 modulo 16, 13 = 3 modulo 16,
15 = 1 modulo 16.
Progression by 1 unit yields
the chromatic scale in 16-tet.
3 units, being also relatively
prime to 16, generates a chain of
16 pitches separated successively
by 225 cents. This is the 16-tet
equivalent of a whole-tone scale--
a thoroughly anti-tonal construct.
5 units = 375 cents and being
also relatively prime to 16,
progression by 5 units yields
16 successive pitches separated
from one another by 375 cents.
This structure has no analog in
western music. It is highly anti-tonal.
Notice several things:
[1] in 16-tet, 3/16 of the
intervals are relatively prime and
non-composite, while in 12-tet 2/12
of the interals were relatively prime
and non-composite.
[2] In 16-tet there is no such thing
as a "perfect fifth." So 16 lacks
the Alberti Bass coset pattern, as
well as modulation by perfect fifths
(the closest thing to a perfect
fifth in 16 is 9 units = 675 cents).
But the minor third in 16 = 4 units,
300 cents, while the major third =
5 units 375 cents. 4 + 5 = 9, which
alas does not sound *anything*
like a p5. So in 16 it is NOT true that
the major plus minor third = p5!
Moreover (unlike 12), the closest
interval to a perfect fifth is
composite and non-relatively-prime
to the total number of pitches in
the octave, while the major third
is non-composite and is relatively
prime to the total number of pitches.
This tells us that the major third
is very un-12-like, the "perfect fifth"
in 16 is very un-12-like, and only the
minor third IS 12-like. Moreover,
chord progressions are impossible
because the Alberti Bass pattern in
16 does not "fill the space" of
16-tet. Imagine 16-tet chromatic
pitches laid out in columns and the
3-unit pattern laid out in rows
and each of these planes transposed
up and down by 5 units. This is a
spatial representation of the 3
fundamental modal substructures
in 16-tet. A moment's visualization
will suffice to convince yourself
that a 2-D tiling of 4 x 5 units
repeated vertcally up and down endlessly
*cannot* fill all of our 16-tet
group theoretic space.
A crystallographer would say that
our 16-tet modal 3-D space exhibits
no regular unit cell.
Notice the difference from 12-tet:
12 was a 2-D plane and it *did*
have a regular unit cell. Triangles
formed along the 1, 3, 4 coordinates
will completely tile 12-tet modal
space. Moreover, in 12-tet the maj 3rd
+ min 3rd = p5, whereas this is
*not* true in 16-tet.
So 16 is very different from 12
in group theoretic terms. My conjecture
is that the lack of a unit cell in 16's
modal 3-space indicates a strong conflict
between the melodic and harmonic functions
in 16.
--
Let's move on from 16's extreme internal
symmetries to a scale which lies at the
other end of the spectrum from 12...
Namely, 19. This division of the octave is
highly asymmetric because all of the intervals
in 19 form unrepeating non-interlocking
patterns. In 19, unlike 12, there are
19 different diminished 7th chords
(in 12 there are only 3 distinct dim 7th
chords). Moreover, in 12 (as Paul
Rapoport has pointed out) the conventional
notation of the dim 7th tetrad does
violence to the acoustic reality of the
construct because any conventional
notation (C-E-Gb-Bbb, or E-Gb-Bbb-Dbb
or Gb-Bbb-Dbb-Fb or A-C-E-Gb, etc.)
implies the existence of a root, which
the dim 7th chord simply doesn't have
in 12-tet.
It floats free of any root...or, if you
prefer, all of its 4 pitches serves
equally as a root. But in 19, a quartet
of stacked minor thirds *can* be
differentiated by root because transposing
to the next pitch upward inside a 19-tet
diminished 7th chord generates both a
different set of pitches *and* a different
set of pitch-names. In 19, Eb is not
just notationally but audibly different
from D#, and so dim 7th chords *do* have
entirely functional roots. In 19, dim 7th
chords *never* "float free" of the scale--
they are far more tonal--and far more
audibly consonant, from the standpoint
of both musical and sensory consonance.
Try playing a 19-tet dim 7th if your
fingers can stretch that far on a 19-tet
guitar. You'll be pleasantly surprised.
--
In 19, the p5 = 11 units, while the
minor third = 5 units and the major
3rd = 6 units. So in 19, the p5 is
both non-composite *and* relatively
prime to the total number of pitches
in the scale, while m3 + M3 = p5
(5 units + 6 units = 11 units).
This tells us that 19 is from a group
theory viewpoint somewhat similar in its
internal substructure to 12. Audible
results confirm this similarity:
because 19 matches 12 in many basic
respects and because 19 has a fine
approximation of the 3rd harmonic,
it's bound to sound somewhat 12-like.
However, 19 differs sharply from
12 in the sense that both the min 3
(5 units) and the Maj 3 (6 units) in
19 are relatively prime to the total
number of units in the octave (19).
(In 12, the min 3rd and maj 3rd are
both factors of the total number of
units in the octave = 12, not relatively
prime to 12 at all.)
This means that a cycling Alberti Bass
which moves up & down by major or minor
3rds in 19 will probably sound more
musically complex than a similar
pattern in 12 cycling by major or
minor thirds: the 12-tet version merely
moves up and down the same pitch subset,
while the 19-tet version constantly
moves to new pitches.
In 19, the fact that modulation by root
movement of maj 3rds *or* min 3rds allows
us to visit any possible pitch also raises
the possibility that triadic root
movement by 3rds (which sounds weak
and anemic in 12) might sound robust
and vivid in 19.
So 19 is somewhat 12-like, but not
entirely, from the group theoretic
viewpoint.
--
Moving on to 22, we note that the minor
3rd = 6 units while the maj 3rd =
7 units. However the p5 = 13 units.
Here, we see some marked similarities
to 19 because the p4 is composite:
p4 = 13 modulo 22 = 9. The p5 = m3 + M3
(13 units = 6 + 7) and is prime, but the
major 3rd in 22 is both non-composite and
relatively prime to the total number
of pitches in the octave, whereas in
12 the maj 3rd (with 4 units) was
composite and a factor of the total
number of pitches in the scale.
Also in 12, both the p4 and p5 are
non-composite and relatively prime to
the total number of pitches in the scale--
but in 22 the p4 is composite yet relatively
prime to 22, while the p5 is both prime
and relatively prime to 22, suggesting
that in 22 the p4 is a much more slippery
and amibigous interval than in 12-tet.
--
Let's compare 12, 19 and 22 from
a group theoretic viewpoint:
19 contains major thirds with
symmetry properties more like
12's than 22, while 22's minor
thirds are more like 12's from
this highly abstract perspective.
19's major thirds are composite,
while 22's minor thirds are
composite. Note, however, that
19 is more like 12 than 22
from this point of view--
because in 22 the number of units
in a minor third is composite
and the total number of pitches
in the scale is also composite--
and the 2 intervals have a
factor in common, 2. But 19
is a prime number and thus
non-composite, with no factor
in common with the number of
units in the 19-tet major
third of 6 units.
This raises the important
issue of symmetry-breaking,
which many subscribers didn't
seem to understand in a my
post touching on the subject
long ago & far away.
The idea is simple: movement
by a chain of major thirds (or
indeed cyclic successive motion
by *any* melodic interval) in
19 will always break melodic
symmetry because it will force
the composer to visit all 19
pitches in the scale. In 22-tet,
however, movement by minor thirds
will overlap with itself in
a self-limited entirely symmetrical
pattern. If we imagine an equal
temperament as a circle of seats
around a dinner table, moving by certain
subsets clockwise around the table
(e.g., minor thirds in 22-tet) will
land us constantly in the same set of
seats over and over again. If
one diner leaves gets up & we shift
everyone clockwise by the amount
of the gap and reinsert the diner
who left at a new point, we merely
re-order a fixed subset of dinner
guests. The roster of people in
that subset never changes.
If we try the same trick in
19-tet, we find ourselves having
to drag everyone at the dinner
table into the act. Our subset of
guests inevitably expands to
include all the people at the
dinner table.
Let's see the proof:
Moving up or down by 6 units
in 22 gives
1 7 13 19 25 (=3) 31 (=9)
37 (=15) 43 (=21) 49 (=5)
55 (=11) 61 (=17) 67 (=1)
73 (=7) & so on, since we're
now repeating the subset
enumerated at the start ad
inifinitum.
As you can see, moving by
6 units (minor 3rd) in 22
never touches a subset of the
22-tet pitches: 2,4,6,8,10,12,
14,16,18,20,22. This generates
a gapped pattern which can
be rotated through the scale
by starting on another pitch
but whose internal substructure
*cannot* be changed (i.e., the
successive intervals twixt
members of the set cannot be
altered as long as we move
by a chain of minor thirds in
22).
By contrast, 19 is a prime number...
and consequently moving by a chain
of major thirds in 19 inevitably
yields not a subset of 19, but the
entire set of pitches in the
octave.
--
These group theoretic and abstract
algebraic considerations are especially
valuable becuase [1] they reveal musical
similarities between equal temperaments
which might on the surface seem different--
and these similarties in musicalpractice
turn out to be vividly and clearly audible;
for instance, recently these considerations
led me to turn a 9-tet composition into
a 15-tet compositions by expanding the
intervals by a fixed factor and calculating
the closest pitches in 15-tet. The result
was extremely successful from a musical
standpoint: both compositions sounded
nearly identical musically, yet distinguishable
in interesting ways (from an intonational
standpoint).
Why transfer a composition in 15-tet into
9-tet, rather than into 12-tet or 18-tet?
9 is more similar to 15 than to 12 or 18
from a group theoretic viewpoint because
9 and 15 are both multiples of 3 and
thus share identical major thirds of
400 cents. 12 and 18 also boast the
same 400-cent major thirds, but in 12
the 7-unit p5 is relatively prime to 12
as well as absolutely prime.
However, in both 9 and 15 the p5 (or
its closest approximation) is composite.
In 9-tet, the closest to the p5 is
5 units (remember that 5 modulo 9 =
4, composite) while the closest to the
p5 in 15-tet is 9 units (also composite).
The 9-tet min 3rd + maj 3rd = 2 units +
3 units = 5 units, while in 15-tet the minor
3rd + major 3rd = 4 units + 5 units = 9 units.
This means that in 15-tet, the major
3rd is prime but a factor of the
total number of pitches in the octave, while
in 9 the major third is also prime and
a factor of the total number of pitches
in the octave.
Conversely, in both 9 and 15 tet
the minor third is relatively prime
to the total number of pitches. This
is very different from 12, in which
both major 3rd and minor 3rd are
factors of the total number of pitches,
while the major third is prime
but the minor third is composite.
Likewise, 9 and 15 exhibit radically
different internal substructure from
18: in that temperament, the minor 3rd
+ major 3rd = 5 units + 6 units =
11 units, the closest approach to
the p5 (733.333 cents).
But notice that in 18, the major
third is a factor of the total number
of pitches as well as being composite,
while the minor 3rd is both prime
and relatively prime to the total
number of pitches. Thus both 18
and 12 are very unlike 9 and 15
in group theoretic terms: the
symmetry groups formed by chains
of major or minor thirds in 9 will
be much more like those formed in
15 than in either 12 or 18.
--
In my experience, the most important
audible properties of equal tempered
scales are: [1] the presence or
absence of recognizable fifths
according to Blackwood's criterion;
[2] the melodic substructure (whole-
tone scale, semitone scale, 1/3-tone
scale, 1/5-tone scale, etc.) of the ET;
[3] the group theoretic/abstract
algebraic substructure of the equal
temperament.
As controversial as it might sound,
in my experience the presence or
absence of precise approximations
of members of the harmonic series
is relatively unimportant in determing
how the equal temperaments function
musically--except when an equal
temperament has a "p5" so far away
from the 3rd harmonic as to sound
grossly and unremittingly dissonant.
(In general, the limit seems to be
685.4 < p5 < 720 cents. Again, this
is the Blackwood/Rasch criterion,
and it explains what one hears
in the ETs beautifully.)
Whether a perfect 5th is 720 cents
or 685.4 cents or 700 cents is
in my musical experience a trivial
consideration--regardless of the
exact degree of approximation of
the p5, all thse types of perfect
fifths sound entirely convincing
and functional in completely
effective musical ways when used
with the equal temperaments from
which they derive.
Thus, most of the work which has
been done theoretically about the
equal temperaments seems to be
musically irrelevant to and
almost completely disconnected from
what the ear actually *hears* in
musical terms in these equal
temperaments. In particular, theoretically
fascinating structures like Clarence
Barlow's harmonicity and David Rothenberg's
propriety appear to have no audible connection
whatever to any any audible functional property of
the xenharmonic equal temperaments
which I have been able to ascertain.
Needless to say, all of these
statements will be greeted with the
utmost delight by the members of this
tuning forum.
--mclaren


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