= iff m=p and n=q.
Thus our notation is both necessary and sufficient to identify an ME
string. But, although all ME strings are necklaces while all
necklaces are not ME, any ME string can be taken as a token for an
infinite "class" of necklaces. For instance, s = <1121112> is
maximally even, but all strings of the form are necklaces,
whether they are ME or not (e.g., <2212221> is ME, <1161116> is not
ME, <5525552> is not ME, <5565556> is ME, etc.) Thus we will use the
notation <(m;n)> to denote the class of all necklaces that can be
derived from by substituting other integers for 's
elements. <(7;12)> then denotes all strings of the form ,
and we will write <(7;12)> DEF <9222922> if we need a specific string
example from the class.(2) The need for specification is due in part
to the fact that WARP is sensitive to the rotation of its generic and
scale strings, as will now be made evident.
=============
2. There is some redundancy in this notation since any necklace class
has any number of ME tokens; for example, <(7;12)> represents the same
necklace class as, say, <(7;9)> and <(7;16)>. While it would be more
cumbersome and less intuitive, this notation could be improved by
partitioning all ME strings with M(m,x,k) defined as all strings of
the form where m,x,k are integers with m >= 1; x = 1,2,...; k
<= [[m/2]]; xm%y >= m; and % stands for "plus or minus." N(m,x,k)
would then represent all necklaces which can be derived from any
member of M(m,x,k) by substitution. Or, using the "generating" case
and the above notation, we might simply use N(m,k) defined as
<(m;m+k)>. Thus <7;12>, <7;9>, <7;16>, as well as <1171117> and so
on, all belong to the necklace class N(7,2) = <(7;9)> = .
=============
[5.4] MAXIMALLY EVEN WARPS. If q = __ and r = are ME
strings, they automatically fulfill the inequality conditions in the
definition for the r-WARP of q: (u = #(q)) <= (v = SUM(q) = #(r)) <=
(w = SUM(r)). ____ serves as a generic string and as a scale
string. But generally, two necklaces q = <(u;v)> and r = <(v;w)> are
not WARP-compatible unless SUM(q) = #(r).
[5.5] Let q = <3;11> and r = <11;13> be two ME strings. If we
stipulate permutations q = <3;11> DEF <344> and r = <11;13> DEF
<11211111211>, then WARP(q,r) = <445>, and the resultant string is a
permutation of the ME string <3;13>. But if we choose a different
permutation of q, q' = <3;11> DEF <434>, WARP(q',r) = <535> which,
although it is a necklace in the <(3;13)> class, is not ME. Now let x
= <11;18> DEF <12212212212>. WARP(q,x) = <567> and WARP(q',x) =
<657>, both of which are asymmetric (and thus, of course, neither ME
nor necklaces). If we generalize without caution, the WARP may be
undefined. For example, if p = <(3;11)> DEF <177>, then WARP(p,r) is
undefined since SUM(p) = 15 does not equal #(r) = 11; but if p' =
<(3;11)> DEF <155> we have WARP(p',r) = <166>.
[5.6] Again let q = <3;11> DEF <344>. If we now stipulate the 0-
permutation of the scale string as r(0) = <11;13> DEF <11211111211>,
then WARPSET(q,r) = {<445>, <445>, <454>, <355>, <355>, <355>, <445>,
<454>, <454>, <355>, <355>}. Let a = <445>, a' = <454>, and b =
<355>; then CHORDSET(q,r) (based on the scale
{0,1,2,4,5,6,7,8,9,11,12}) is {0a, 1a, 2a', 4b, 5b, 6b, 7a, 8a', 9a',
11b, 12b}.(3)
=============
3. This particular WARPSET presents a situation that occasionally
arises when resultant strings are symmetric. Note that a' is a
circular permutation of a, i.e., a and a' represent the same shape, so
for any p in C13, (p,a') = (p-4,a) (mod 13) (see [1.13] in Part I).
If we "canonize" a = <445> as the "correct" string form and require
that all a-based structures be put in that "root position" form, the
CHORDSET given in [5.6] must be abandoned for
{0a,1a,11a,4b,5b,6b,7a,4a,5a,11b,12b). This in turn makes scale
elements 4, 5, and 11 into roots which can be associated with both a
and b strings, whereas scale elements 2, 8, and 9 can't be associated
as roots with any chords.
=============
[5.7] While much of what we are about to discuss can be said about
necklaces generally, it will be most profitable to concentrate on
maximal evenness, but we will still generalize when doing so points to
interesting relations and structures. We will now adopt the following
notation for WARPSETs whose generic and scale strings are both
maximally even:
WARPSET(____,) = ||u;v;w||
Thus the designation "maximally even" applied to a WARP refers only to
the variable strings (i.e., the generic and scale strings) since, as
noted, resultant strings may not be maximally even.
[5.8] I = is a left identity (generic) string, and J = is
a right identity (scale) string for ME WARPs. Thus WARP(I,) =
WARP(,J) = .
[5.9] DIATONIC STRINGS. Using Clough and Douthett's Theorem 2.2 (4),
for any odd x > 1, a diatonic string is any ME string of the form d =
= , where j is a substring of (x-3)/2 2's and k is
a substring of (x-1)/2 2's. For any pitch class p, pd is a
(hyper)diatonic set based on p. For p=0,
x 2(x-1) d (p,d)
3 4 <121> {0,1,3}
5 8 <21221> {0,2,3,5,7}
7 12 <2212221> {0,2,4,5,7,9,11}
9 16 <222122221> {0,2,4,6,7,9,11,13,15}
... ... ... ...
We will now focus closely on WARPSETs of the form ||w;x;2(x-1)|| and
their related CHORDSETs as we generalize the usual diatonic.
=============
4. Clough and Douthett, op. cit., pp.170-71.
=============
6. THE USUAL DIATONIC AS A MODEL FOR AN ABSTRACT DIATONIC SYSTEM.
[6.1] RD(1). To model the usual diatonic in the notation presented in
Part I, we will make the following assignments:
In C7,
J = <7;7> = <1111111>
g = <3;7> DEF <223>
p = 0
X = WARPSET(g,J) = ||3;7;7||
Y = pJ = {0,1,2,3,4,5,6}
Z = CHORDSET(X)
and in C12,
h = <7;12> DEF <2212221>
q = 0
W = WARPSET(g,h) = ||3;7;12||
H = qh = {0,2,4,5,7,9,11}
K = CHORDSET(W)
The usual diatonic system(5), in our present terminology (see
[2.25]), is then the ordered triple
RD(1) = (W,H,K).
Since (X,Y,Z) (=) (W,H,K), the entire model can be displayed as a set
of correspondences:
___________C7_____________ ___________C12______________
/ \ / \
X Y Z W H K
<223> = g 0 {0,2,4} = 0g <435> = t 0 {0,4,7} = 0t
<223> = g 1 {1,3,5} = 1g <345> = t' 2 {2,5,9} = 2t'
<223> = g 2 {2,4,6} = 2g <345> = t' 4 {4,7,11} = 4t'
<223> = g 3 {3,5,0} = 3g <435> = t 5 {5,9,0} = 5t
<223> = g 4 {4,6,1} = 4g <435> = t 7 {7,11,2} = 7t
<223> = g 5 {5,0,2} = 5g <345> = t' 9 {9,0,4} = 9t'
<223> = g 6 {6,1,3} = 6g <336> = u 11 {11,2,5} = 11u
Names for the strings <435> = t, <345> = t', and <336> = u have been
assigned to identify the usual major, minor, and diminished triads
respectively.(6)
=============
5. "System" is used here in the sense of the ordered triple of [2.25]
in Part I. "Diatonic system" may be confused with that term as used
by C&D (---) and Agmon (---), and for that reason we will mostly use
the term "Riemann Diatonic System" introduced below.
6. Diatonic tetrachords (e.g., the dominant 7th-chord) are not
included here since they would unduly complicate the present study.
But the reader may want to keep these important structures in mind as
we procede since they present an interesting dilemma. In the usual
diatonic system, clearly one could view a 7th-chord as either a triad
with an added third *or* the complement of a triad with respect to the
diatonic scale (at the end of the day they produce the same
structure). However, as the reader may note after reading about the
next RD system, these two possible construction principles, when
lifted into RD(2), produce different results. In RD(2) the basic
structure is a 5-note chord. If we use the additive principle, the
"hyper7th-chord" will have six notes. But if we use the complement
principle, the "hyper7th-chord" will have eight notes.
=============
[6.2] The interval matrix for the diatonic scale string, which is not
partitioned, is
/ 2 5 0 0 0 0 0 0 0 0 0 0 \
| 0 0 4 3 0 0 0 0 0 0 0 0 |
| 0 0 0 0 6 1 0 0 0 0 0 0 |
MINT(h) = | 0 0 0 0 0 1 6 0 0 0 0 0 |
| 0 0 0 0 0 0 0 3 4 0 0 0 |
| 0 0 0 0 0 0 0 0 0 5 2 0 |
\ 0 0 0 0 0 0 0 0 0 0 0 7 /
and this yields the ic-mapping
in C7 in C12
1 --> 1,2
2 --> 3,4
3 --> 5,6
4 --> 6,7
5 --> 8,9
6 --> 10,11
7 --> 12 (=0)
As noted above, dealing with non-partitioned interval matrices can be
somewhat complex, and we purposely have chosen examples which avoid
them. However, in this case, the ambiguity presented by the tritone
does not present any unusual difficulties. If we wish to write the
vector associated with MINT(h), it is helpful to display it thus:
U(h) = [254361
1634527]
instead of [254362634527]. The vector can then be "partitioned" as
[2,5;4,3;6,1
1,6;3,4;5,2;7]
so that
U'(h) = [2+5;4+3;6+1;
1+6;3+4;5+2;7] = [7777777] = U(J)
Keeping the usual caution in mind while dealing with the tritone in
vector applications, we can then write the corresponding interval-
class vector relationships on which generic covariance is based:
V(h) = [254361]
V'(h)= [2+5;4+3;6+1] = [777] = V(J)
[6.3] Thus by G-COV, the partition sum ic-vector of any string in W
will equal V(g). W contains only two unique strings, t (t') and u,
for which V(t) = V(t') = [001110] and V(u) = [002001], so V'(t) =
V'(t') = [0+0;1+1;1+0] = V'(u) = [0+0;2+0;0+1] = [021] = V(g). If we
had chosen a different string for g we would have created a different
white note system, but by G-COV the same partition-vector pattern
would still hold due to the diatonic scale string being held constant;
however, we would then be headed in a direction which would soon
diverge radically from our chosen diatonic model.
[6.4] The important thing to take from G-COV, with respect to RD
systems, is the characteristic partition pattern imposed by the scale
string which will always be of the int-multiplicity form
[a,b; c,d; ...; p,q;
q,p; ...; d,c; b,a; z]
and which can always be reduced to the ic-multiplicity form
[a,b; c,d; ...; p,q].
The sum of the elements in any comma-separated pair in both forms will
always equal the sum of the elements in the scale string (i.e., z).
[6.5] To investigate scale covariance in RD(1) we now concentrate on
ic-vectors spanning sets in the generic system (X,Y,Z). From this
point on, when dealing with ic-vectors, we will adopt the more
intuitive and widely used notation which places the ic0 multiplicity
as the first vector component. Thus V(pg,(p+1)g) = [0513] for g =
<223> indicates that between the sets, say, {1,3,5} and {2,4,6}, there
are 0 ic0's (no common tones), 5 ic1's, 1 ic2, and 3 ic3's.
[6.6] Given the generic string g = <3;7> DEF <223> as before, if p is
any pitch class in C7, then pg is a triad in Z and (p+t)g is pg's t-
transpose. The following list exhausts the possible ic spanning
vectors for Z (% indicates plus-or-minus)
V(pg,pg) = [3042]
V(pg,(p%1)g) = [0513]
V(pg,(p%2)g) = [2142]
V(pg,(p%3)g) = [1323]
The distribution of components in this list isn't as random as it
might at first appear since it derives from the list of interval
spanning vectors U. The reader will note that the ic-vector is simply
the int-vector with the non-zero components "folded back" on their
inversions. For U(pg,(p+1)g) we have:
[ 0 3 0 2 1 1 2 ]
| | | | | | |
int: 7=0 1 2 3 4 5 6
ic: 0 1 2 3 3 2 1
\ \ \___/ / /
\ \_________/ /
\_______________/
and [0302112] --> [0513]. The pattern behind the ic vectors listed is
then made clear by listing the int vectors for the same sets:
U(pg,(p+1)g) = [3021120]
U(pg,(p+2)g) = [0302112]
U(pg,(p+3)g) = [2030211]
U(pg,(p+4)g) = [1203021]
U(pg,(p+5)g) = [1120302]
U(pg,(p+6)g) = [2112030]
U(pg,(p+0)g) = [0211203] (= U(g))
(note that we retain the ic0 component in the last position here since
this display is taken directly from MINT(g) -- see [3.11-14] in Part
I).
[6.7] We can now relate any ic vector spanning two sets in K to one of
the spanning vectors just listed for Z since, by S-COV, the partition-
sum ic vector spanning any pair of sets in K will equal the ic vector
spanning a corresponding pair of sets in Z. Some examples, referring
back to the system correspondences in [6.1], are:
V(9t',2t') = [1121130]
V'(9t',2t') = [1;1+2;1+1;3+0]
= [1323]
= V(5g,1g)
V(4t',5t) = [0230121]
V'(4t',5t) = [0;2+3;0+1;2+1]
= [0513]
= V(2g,3g)
V(5t,7t) = [0141021]
V'(5t,7t) = [0;1+4;1+0;2+1]
= [0513]
= V(3g,4g)
[6.8] By S-COV, the list of ic spanning vectors in [6.6] summarizes
all the voice-leading possibilities for any pair of triads in Z and
thus in K as well. If we display this list as an "available voice-
leading" matrix for interval classes,
/3042\
AVLIC(Z) = |0513|
|2142|
\1323/
it can be read in two ways. First, any given row indicates the
multiplicity of generic interval classes spanning two triads in Z or
the multiplicity of diatonic interval classes spanning two
corresponding triads in K. For example, row 3 indicates the voice-
leading multiplicities found in V(pg,(p%2)g), so all seven of the
corresponding chord pairs in K (whose "roots are a major or minor
third apart") have available connections of 2 ic0, 1 ic1/ic2, 4
ic3/ic4, and 2 ic5/ic6. Second, any column will list all possible
chord pairs spanned by a given interval class. If we wish to conduct
an "interval class search" -- say we wish to know how all possible
chord progressions in K are spanned by a perfect or diminished fifth -
- we simply go to the fourth column (representing ic3 in C7 which
corresponds to ic5/ic6 in C12) and read down: there are 2 fifths
available between correspondents of pg and pg; 3 fifths between
correspondents of pg and (p%1)g; 2 between corespondents of pg and
(p%2)g; and 3 between correspondents of pg and (p%3)g.
[6.9] We can also form the more complete "available voice-leading"
matrix for intervals,
/3021120\
|0302112|
|2030211|
AVLINT(Z) = |1203021|
|1120302|
|2112030|
\0211203/
and make interval-spanning statements about corresponding chords of
the same sort as we did for interval-class spans. Due to AVLINT's
symmetry ((row m, column n) = (row n, column m)), we can state a more
general theorem relating chord pairs which can be extended via S-COV
to correspondence-related white note systems. We state it here
without proof and using int rather than ic notation.
[6.10] For any integers p,x,y in the chromatic white note system
(A,B,C) which has been generated by a generic string g,
#intx(pg,(p+y)g) = #inty(pg,(p+x)g).
For example, returning to g = <2,2,3> and (X,Y,Z) in the usual
diatonic world, this theorem tells us, among many other things, that
#int5(pg,(p+1)g) = #int1(pg,(p+5)g) = 1. To give us some idea of the
amount of information packed into an AVLIC or AVLINT matrix, we can
translate this example (by correspondence) into more familiar diatonic
language, remembering that int5 and int1 in C7 "translate" to int(8-
or-9) and int(1-or-2), respectively, in C12. What the example says,
loosely, is that there is always one (and only one) 6th within a
diatonic scale that can be found to span two triads within that scale
whose roots are a 2nd apart and furthermore this implies that there is
always one (and only one) 2nd within that same scale that can be found
to span two triads within that scale whose roots are a 6th apart.
[6.11] The theorem in [6.10] will, of course, hold for any simple
white note system, not only those generated by ME strings. Its
application is intimately related to the principle of scale
covariance, since changing the scale string while keeping the generic
string will consistently yield the same values for corresponding sets,
but changing the generic string will normally yield different
AVLINTs.(7)
=============
7. The reader is invited to generate and compare the AVLINTs for the
three other possible triadic white note systems which can be
associated with the usual diatonic string using g = <115>, <133>
(necklaces) or <124> (asymmetric). It is then an interesting exercise
to search for non-diatonic scale strings for each of these generic
strings which might be profitably employed as a basis for
compositional praxis (both inside and outside C12).
=============
[6.12] Finally, cover covariance (C-COV) in RD(1) is particularly
interesting. The C-7 chromatic scale Y = {0,1,2,3,4,5,6} can be
covered by the g-triads in Z in a number of ways, but we know that any
minimal cover of Y must consist of three triads. Perhaps the most
important cover of Y is the one which, by correspondence, underlies
the "dominant" related triples in K:
COVER1(Z) = {pg, (p+3)g, (p-3)g}.
For istance, setting p = 5, R = {5g,1g,2g} = {{5,0,2},{1,3,5},{2,4,6}}
and R//Y (see [4.16-19] in Part I). Also note the common-tone
relationships among these triads: #int0(pg,(p+3)g) = #int0(pg,(p-3)g)
= 1 but #int0((p+3)g,(p-3)g) = 0. The only other minimal cover from Z
is of the form
COVER2(Z) = {pg, (p+1)g, (p-1)g}.
with the common-tone relationships: #int0(pg,(p+1)g) = #int0(pg,(p-
1)g) = 0 and #int0((p+1)g,(p-1)g) = 2. Any triple in Z which cannot
be expressed in one of these two forms (COVER1 or COVER2) will not
cover Y (e.g., {0g,2g,3g} will not cover Y).
[6.13] Now, bracketing possible psycho-acoustical, semiological,
sociological, and historical "reasons" (which we've at least been
trying to avoid up until now at any rate -- saving them for later to
sit in judgement on any synthetic structures we might come to enjoy),
is there any reason that we might expect a privileged status for
either COVER1 or COVER2? This is tantamount to asking: Is there
really any (non-bracketed) reason to "prefer" progressions whose roots
are a diatonic 4th-or-5th apart to those whose roots are a 2nd-or-7th
apart, or vice versa? Well, if we examine the possible (diatonicized)
triads in K corresponding to the sets in Z (refer back to the
correspondence chart in [6.1]), we note that, for whatever choice we
make for a correspondence to COVER2, the resulting cover of H will be
"mixed" with respect to chord "quality." For example, the cover
{2t',4t',5t} contains two minor and one major triad. On the other
hand, the choices corresponding to COVER1 yield two possibilities:
"mixed" and "pure." With respect to symmetry, COVER1 can retain the
kinds of (mirror) "quality" symmetries found in COVER2 while adding a
kind not found there: two unique covers which each contain three
chords of the same quality. Furthermore, COVER1 seems to most
directly point to the overall inversional symmetry at the heart of
RD(1) through the partition of K by major, minor, and diminished
chords:
R1(Z) = {5t,0t,7t}
R1'(Z) = {2t',9t',4t'}
R2(Z) = {11u}
where R2 is a sort of "residue" set. So, fully aware that this is a
choice made, in this particular context, by a preference for maximal
symmetry coupled with a desire to discover as much variety as
possible, we will privilege the sets R1 and R1' and call them "Riemann
covers."(8)
=============
8. Clearly the converse, a preference for maximal variety coupled with
a secondary desire to discover as much symmetry as possible, is most
attainable by choosing COVER2 -- or by choosing neither or both. In
the larger sense, if we were to adopt a preference for variety
throughout, we could have stopped with Part I since, when we find
ourselves in any (structured) musical universe, what we can say about
any string can be said about symmetric strings as well, and cover
covariance is simply an objective fact associated with whatever system
we happen to find ourselves in at the moment. But in beginning Part
II by focusing on necklaces and then on maximal evenness and
(hyper)diatony we already (perhaps covertly) stated a preference for
symetric structures which a fortiori guides our present and subsequent
choices.
=============
[6.14] A Riemann cover R will always have the following
characteristics ($(A) indicates the interval string associated with
set A):
(1) The set R contains three chords which
will be labelled S, T, D.
(2) #int0(S,T) = #int0(T,D) = 1
(3) #int0(S,D) = 0
(4) $(S) = $(T) = $(D)
(5) V(T,S) = V(T,D)
(6) S.&.T.&.D = H
(7) There is an inversion of R, R' = {S',T',D'}, such that R' is
a Riemann cover.
[6.15] Any CHORDSET whose scale string is (hyper)diatonic and that
generates a Riemann cover will be called a Riemann Diatonic (RD)
system. As we shall soon see, the usual diatonic based on ||3;7;12||,
which we have been calling RD(1), represents the "smallest" or "first"
such system available. This represents a somewhat different approach
to what David Lewin has called "Riemann Systems"(9). The two
approaches will be reconciled shortly, but a more complete congruence
will not be demonstrated at this time. Our purpose here is more
toextend systems rather than to simply find an alternate path to those
that are known (and have been extensively used) in C12.
=============
9. David Lewin, "A Formalized Theory of Generalized Tonal Functions,"
Jounal of Music Theory 26.1 (1982)
=============
[6.16] The triple {S,T,D} and its inversion display a characteristic
pattern when projected onto a C12 circle. We plot the "roots" of each
chord triple {5t,0t,7t} and {2t',9t',4t'} as vertices for the two
triangles (5,0,7) and (2,9,4) on C12 (see Figure 1). Note that the
two triangles overlap one another. A more complex version of this
configuration will be met in RD(2)'s Riemann covers where we will find
four overlapping triangles. In other words, where RD(1) has two
"natural tonics" (the apexes of the triangles--one (right) "major" and
one (left) "minor"), RD(2) will have four "natural" tonics, each
supported by its own "dominant" and "subdominant." In general, any
system RD(n) (to be defined) will have 2n, or n pairs of, Riemann
covers.
****FIGURE 1****
[6.17] Note that nothing has yet been said about the cardinality of
the basic RD(1) chords, so in higher RD systems the basic chords need
not be triads. Nor has any stipulation been made concerning the
specific shape (string) of the chords, so "dominant," "mediant," and
any other intervallic constituents are temporarily left undefined.
7. AN ABSTRACT DIATONIC SYSTEM.
[7.1] Let us assume that the set of strings W = ||3;7;12|| in RD(1) is
the first member of a series of similar RD WARPSETs ||x;y;z||. We
first note, given that the scale string must be diatonic (an
assumption that can be stretched within limits -- see [] below), there
can't be any other WARPSET of the form ||3;y;z|| since (from [6.14])
chord T must have one (and only one) tone in common with both S and D,
and only y=7 satisfies that condition, concomitantly fixing z=12. So
we begin by varying x. x=4 won't work since, to satisfy the common-
tone condition, this would imply a WARPSET of the form ||4;10;z||, and
<10;z> can't be diatonic since 10 is even; and the same applies for
any even x. x=5 is the next possibility. Here the common-tone
condition demands a WARPSET of the form ||5;13;z||, and the diatonic
string from y=13 is <13;24>. In fact, as we will now see, the system
(W',H',K') based on ||5;13;24||, which will be labelled RD(2), is the
next RD system. After confirmation of this fact, we will return to
find a general expression for any RD system.
[7.2] RD(2). We begin by making the following assignments:
In C13,
J' = <13;13> = <1111111111111>
g' = <5;13> DEF <23233>
p = 0
X' = WARPSET(g',J') = ||5;13;13||
Y' = pJ' = {0,1,2,3,4,5,6,7,8,9,10,11,12}
Z' = CHORDSET(X')
and in C24,
h' = <13;24> DEF <2222212222221>
q = 0
W' = WARPSET(g',h') = ||5;13;24||
H' = qh' = {0,2,4,6,8,10,11,13,15,17,19,21,23}
K' = CHORDSET(W')
[7.3] Since (X',Y',Z') (=) (W',H',K'), we can display RD(2) fully by
the following table of correspondences:
____________C13____________ _______________C24______________
/ \ / \
X' Y' Z' W' H' K'
<23233>=g' 0 {0,2,5,7,10} = 0g' <46365> = r 0 {0,4,10,13,19} = 0r
<23233>=g' 1 {1,3,6,8,11} = 1g' <45465> = s' 2 {2,6,11,15,21} = 2s'
<23233>=g' 2 {2,4,7,9,12} = 2g' <45465> = s' 4 {4,8,13,17,23} = 4s'
<23233>=g' 3 {3,5,8,10,0} = 3g' <45456> = s 6 {6,10,15,19,0} = 6s
<23233>=g' 4 {4,6,9,11,1} = 4g' <36456> = r' 8 {8,11,17,21,2} = 8r'
<23233>=g' 5 {5,7,10,12,2}= 5g' <36456> = r' 10 {10,13,19,23,4}= 10r'
<23233>=g' 6 {6,8,11,0,3} = 6g' <46365> = r 11 {11,15,21,0,6} = 11r
<23233>=g' 7 {7,9,12,1,4} = 7g' <46365> = r 13 {13,17,23,2,8} = 13r
<23233>=g' 8 {8,10,0,2,5} = 8g' <45465> = s' 15 {15,19,0,4,10} = 15s'
<23233>=g' 9 {9,11,1,3,6} = 9g' <45456> = s 17 {17,21,2,6,11} = 17s
<23233>=g' 10 {10,12,2,4,7}=10g' <45456> = s 19 {19,23,4,8,13} = 19s
<23233>=g' 11 {11,0,3,5,8} =11g' <36456> = r' 21 {21,0,6,10,15} = 21r'
<23233>=g' 12 {12,1,4,6,9} =12g' <36366> = t 23 {23,2,8,11,17} = 23t
[7.4] We will not give the 13X24 interval matrix MINT(h') here, but we
will note the ic-mapping it generates for RD(2):
j-in-C13 k-in-C24
1 --> 1,2
2 --> 3,4
3 --> 5,6
4 --> 7,8
5 --> 9,10
6 --> 11,12
7 --> 12,13
8 --> 14,15
9 --> 16,17
10 --> 18,19
11 --> 20,21
12 --> 22,23
13(=0)--> 24(=0)
[7.5] Illustrating the situation for G-COV in RD(2), the partition
vectors produce the following (where a=10, b=11, c=12, d=13):
U(h') = [2b496785a3c1
1c3a587694b2d]
U'(h') = [2+b;4+9;6+7;8+5;a+3;c+1;
1+c;3+a;5+8;7+6;9+4;b+2;d]
= [ddddddddddddd] = U(J')
V(h') = [2b496785a3c1]
V'(h') = [2+b;4+9;6+7;8+5;a+3;c+1] = [dddddd] = V'(J')
W' contains three unique strings, r (r'), s (s'), and t, for which
V(r) = V(r') = [001112003110],
V(s) = V(s') = [000221003110],
V(t) = [002003004001].
Thus
V'(r) = V'(r') = [0+0;1+1;1+2;0+0;3+1;1+0] =
V'(s) = V'(s') = [0+0;0+2;2+1;0+0;3+1;1+0] =
V'(t) = [0+0;2+0;0+3;0+0;4+0;0+1] =
= [023041] = V(g').
[7.6] For ic "voice leading" in RD(2) we have
/5046082\
|0732715|
|2354344|
AVLIC(Z') = |3246163|
|0731905|
|4146082|
\1543525/
By S-COV we can then determine that, for example, when we move from
the chord 8r' to the chord 17s, with a span of 5 generic ic steps
between "roots," that V(8r',17s) corresponds to V(pg',(p%5)g') =
[4146082], which means that between 8r' and 17s there are available 4
ic0 connections, 1 ic1/2, 4 ic3/4, ..., 2 ic11/12. Leaving out the
trivial case of the progression
pg'-->pg', there are a total of 6X13=78 possible triad progressions in
RD(2). This is in contrast to 3X7=21 possible triad progressions in
RD(1). In RD(1) there are 10 distinct spanning vectors; in RD(2)
there are 30. All of this is summarized in detail in Table 1 for
RD(1) and Table 2 for RD(2).
*** Table 1 ***
*** Table 2 ***
[7.7] That RD(2) is indeed a Riemann diatonic system can now be
demonstrated by investigating its C-COV relationships. The C13
chromatic scale Y' can be minimally covered by three pentachords from
Z' in two ways. Following RD(1), and our preference for maximal
symmetry, we will select
COVER(Z') = {pg', (p+6)g', (p-6)g'}.
This in turn discloses the following partition of the chords in K':
R1(Z') = {0r,11r,13r}
R1'(Z') = {21r',8r',10r'}
R2(Z') = {6s,17s,19s}
R2'(Z') = {15s',2s',4s'}
R3(Z') = {23t}.
It is easily verified from [6.14] that R1, R1', R2, and R2' are
Riemann covers.
[7.8] We may now see the characteristic triangular pattern for RD(2)
alluded to in [6.16]. If we plot the roots of each of the four
Riemann covers as sets of vertices on the C24 circle, (0,11,13),
(21,8,6), (6,17,19), and (15,2,4), we note again that these four
(congruent) triangles all cross one another in a characteristic "RD
pattern" first suggested in [6.16] (see Figure 2).
*** FIGURE 2 ***
[7.9] We now return to finding a general expression for any Riemann
Diatonic system RD(n). First note that both generic strings <223> =
<3;7> for RD(1) and <23233> = <5;13> for RD(2), which are not
themselves diatonic, can be derived from the diatonic strings <112> =
<3;4> and <12122> = <5;8>, respectively, by simply adding 1 to each
entry in the strings. Thus <223> = <1+1,1+1,2+1> = <3;4+3> = <3;7>
and <23233> = <1+1,2+1,1+1,2+1,2+1> = <5;8+5> = <5;13>. Generally,
*any* ME string can be derived by writing out the ME string
and then adding k to each entry in the string. But apparently
only k=1 will produce a generic string which can be used as a basis
for the type of RD string we have been looking at.
[7.10] In any RD WARPSET ||x;y;z|| there is a "hidden" diatonic string
such that the generic = , and since (by [5.9]) y'
= 2(x-1), any RD WARPSET can be expressed in the form ||x;3x-2;6(x-
1)|| for odd x > 1.
[7.11] = <2,j,2,k> (where j is a substring of (x-3)/2 3's and
k is a substring of (x-1)/2 3's) will be called a Riemann generic
string. Thus RD(3)'s generic string is <2332333> (whose "hidden"
diatonic string is, incidentally, also the usual diatonic scale string
<1221222>).
[7.12] RD(n). We may also express any higher RD system in terms of
RD(1) as its generator. Thus any RD WARPSET can also be expressed as
||3+2n;7+6n;12+12n|| where n >= 0. By simple algebraic manipulation,
we may then define a generalized RD system as RD(n) = (A,B,C), where A
= ||2n+1;6n+1;12n|| for n>0.
8. RIEMANN NON-DIATONIC SYSTEMS.
[8.1] Riemann covers, which we have seen are a significant
characteristic of RD systems, may also be associated with non-diatonic
systems. The following general string set formulae assume that x is
odd and greater than 1, y=3x-2, and k >= 0. Other variations beyond
those given here are possible and finding them is a fruitful exercise.
[8.2] The simplest case is ||x;y;(k+1)y|| which represents a series of
"smoothly" expanding string sets. CHORDSET(,J), where J= is
a string of y 1's, will trivially generate y Riemann covers of the
form {pq,(p+z)q,(p-z)q}, where p is a pitch class in Cy, q is some
stipulated string in , and z = [y/2]. We already know that this
triple will be a cover, but it is also a Riemann cover since all three
chords share the string q. then simply enlarges all
intervals in the scale string by the same amount which
concomitantly enlarges all intervals in q by the same amount, keeping
the Riemann cover relationship for any k.
[8.3] ||x;y;6(x-1)+ky|| represents an expansion of RD(n)'s scale
string by adding multiples of y. ||x;y;6(x-1)|| is the basic formula
for RD(n) given above in [7-10]. Adding k to each entry in a diatonic
string expands the string into another ME necklace form, keeping the
original ME pattern. Thus <7;12> = <2212221> and <7;12+7> = <7;19> =
<3323332> retains the basic ME pattern. When this happens, the
initial composite WARPSET's resultant strings are uniformly increased
by successively adding the (constant) generic string to each of those
resultant strings. To continue the example, stipulating <3;7> =
<223>, we have ||3;7;12|| = {<435>,<345>,...,<336>}, so (by adding
<223> to each of these strings) ||3;7;12+7|| = ||3;7;19|| =
{<4+2,3+2,5+3>,<3+2,4+2,5+3>,...,<3+2,3+2,6+3>} =
{<658>,<568>,...,<559>}. Since the relative positions of identical
strings are retained, the Riemann cover is likewise retained.
[8.4] Perhaps the most interesting Riemann non-diatonic system is
based on what might be termed the "conjugate-diatonic" WARPSET,
||x;y;3x||. While the generic string is again retained, the
values of the elements of the diatonic scale string are reversed; so
the diatonic <2212221> becomes the "conjugate-diatonic" <1121112>.
[8.5] Two ME strings s = and t = are conjugates of
one another if b+b' = x. Thus <5;4X5+3> = <5;23> = <45455> and
<5;4X5+2> = <5;22> = <54544> are conjugates. To find the conjugate of
the general scale string in RD(n) we first note that, through simple
manipulation, <3x-2;6(x-1)> = <3x-2;(3x-2)+(3x-4)>. Since (3x-2)-(3x-
4) = 2, the conjugate-diatonic scale string will be of the form <3x-
2;(3x-2)+2> = <3x-2;3x>.
[8.6] Swapping the diatonic scale string for its conjugate has the
effect of changing the string's values while retaining its basic ME
pattern, thereby retaining the Riemann cover as well. For example,
the string set ||5;13;15|| is the basis for the conjugate of RD(2).
Its basic scale string is <1111121111112> and, using <5;13> = <23233>
as the generic string, the members of the WARPSET can easily be
calculated: {<23334>, <24234>, ..., <33333>}. Presence and location
of the two pairs of Riemann covers can now easily be determined.
9. EXTENSION OF DAVID LEWIN'S RIEMANN SYSTEMS.
[9.1] Returning to Riemann diatonic systems, we now procede to
reconcile our "topological" approach with David Lewin's "tonal
function" approach. Consider first Lewin's definition which we have
been studiously avoiding until now: "By a Riemann System (RS) we shall
understand an ordered triple (T,d,m), where T is a pitch class and d
and m are intervals, subject to the restrictions that [neither d nor m
is 0 and m does not equal d]."(10) He then procedes to define the
tonic triad as the unordered set {T,T+m,T+d}. This, and the rest of
the relationships investigated by Lewin, can be transported to RD(1)
since, for example, his RS (T,7,4) produces the triad {0,0+4,0+7} =
{0,4,7} which coincides with 0t in RD(1). But can the Lewin-Riemann
concept of chord construction (and concomitant relationships) be
modified to fit *any* RD system? Using 0r = {0,4,10,13,19} as a test
chord in RD(2) we guess that (T,13,m) might be an irredundant RS since
int13 seems to fulfill a "dominant" function. But what do we make of
the quality-determining "mediant interval" m? And how do we generate
the other three chord members? The answers to these questions will
lead us, at the end of our study, to some bizarre (but nevertheless
plausible) conclusions about diatonic systems.
=============
10. Lewin, op. cit., p.26.
=============
[9.2] Interestingly, any of the four (apex) pentachords (privileged as
"tonic" by the definition of a Riemann cover) in RD(2) can be
constructed from the RS triple (T,d=13,m=9) as follows:
red: {T, T+d,T+d-m,T+d-2m,T+d-3m}
green: {T,T+m, T+d,T+d-m,T+d-2m }
yellow:{T,T+m,T+2m, T+d,T+d-m }
blue: {T,T+m,T+2m,T+3m,T+d }
The spacing in each set is used to draw attention to the grouping of
elements. The colors on the left are randomly assigned "shape" or
"chord quality" or "mode" designations analogous to "major" and
"minor." Note that each set contains both T and T+d, as we would
expect. But whereas in the simpler triadic system the only other
element is T+m, here there is a combination of three additional
elements, each with the form T+km or T+d-km (k=0,1,2,3); that is, T
and T+d are "anchor points" to which positive and negative integral
multiples of m are added.
[9.3] Setting T=0,15,6,21, we may now calculate the four basic Lewin-
Riemann tonic pentachords in the scale
{0,2,4,6,8,10,11,13,15,17,19,21,23}.
red: set T=0
{0,0+13,0+13-9,0+13-18,0+13-27} mod 24
= {0,13,4,19,10}
(= 0r)
green: set T=15
{15,15+9,15+13,15+13-9,15+13-18} mod 24
= {15,0,4,19,10}
(= 15s')
yellow: set T=6
{6,6+9,6+18,6+13,6+13-9} mod 24
= {6,15,0,19,10}
(= 6s)
blue: set T=21
{21,21+9,21+18,21+27,21+13} mod 24
= {21,6,15,0,10}
(= 21r')
[9.4] "Dominant" and "subdominant" pentachords related to each of the
four basic pentachords can be found by adding 13 and -13 (=11),
respectively, to each T-value in [9.3]. Thus there are four sets of
primary pentachords in this extended RS. Lewin's secondary chords can
similarly be transported, but we will concentrate here on the primary
chords.
[9.5] A Lewin-Riemann System (LRS) may be found in any RD system by
generalizing [9.2] in the following way. If ||x;y;z|| is an RD
WARPSET (where x is odd and > 1, y = 3x-2, and z = 6(x-1)), then an
embedded LRS can be identified as the ordered triple (T,d,m) where T
is a pitch class (mod z), d=y, and m=y-4. A tonic x-ad of the RS
(T,d,m) is the unordered set TON = {T+fm,T+d-gm}, a dominant x-ad is
DOM = {T+d+fm,T+2d-gm}, and a subdominant x-ad is SUB = {T-d+fm,T-gm},
where f = 0,...,a and g = 0,...,b are related by a+b = x-2.
[9.6] The triple (TON,DOM,SUB) generated by the RS (T,7,4) can be
traditionally interpreted as defining a chord-generated "key" in
relationship to the pitch class T. Thus the RS (T=3,d=7,m=4) implies
the "key"
KEY(E ,+) DEF {(E ,+).&.(B ,+).&.(A ,+)}
with (E ,+) stipulated as "the harmonic goal".
This in turn implies the "relative key"
KEY(C,-) DEF {(C,-),&.(G,-).&.(F,-)}
with (C,-) stipulated as "the harmonic goal".
Looking back at the four basic (T,13,9) chords listed in [9.3] and
generating their respective dominants and subdominants, using string
notation for clarity we can create the following modal key
relationships ("/" here denotes "with").
KEY(0,red) DEF [{0r.&.11r.&.13r} / 0r DEF "goal" KEY0]
KEY(15,green) DEF [{15s'.&.2s'.&.4s'} / 15s' DEF "goal" KEY1]
KEY(6,yellow) DEF [{6s.&.17s.&.19s} / 6s DEF "goal" KEY2]
KEY(21,blue) DEF [{21r',.&.8r'.&.10r'} / 21r' DEF "goal" KEY3]
Note that each tonic ("goal" KEYk) is generated by our original choice
for the pitch class T = 0 by running through the values of a (see
[9.5])
a = 0, KEY0 = T - (0 X m) = T - 0 = 0
a = 1, KEY1 = T - (1 X m) = T - 9 = 15
a = 2, KEY2 = T - (2 X m) = T - 18 = 6
a = 3, KEY3 = T - (3 X m) = T - 27 = -3 (mod24) = 21
This may all be generalized as follows.
[9.7] HYPERKEYS. Given the variables stipulated in [9.5],
(1) a "key stipulation" generated by the LRS (T,d,m) is defined
as KEYa = T-am (mod z)
(2) a hyperkey structure KEY(KEYa,name) is defined as
[{TON.&.DOM.&.SUB} / TON DEF "goal" KEYa], where name is purely
referential and can be chosen at random.
(3) the "relative" of (KEYa,name1) is (KEYb,name2), where b = x-
2-a from [9.5].
[9.8] NB: [9.5] makes a significant modification to the original Lewin
definition quoted in [9.1]. Whereas RS (tri)chords are always
oriented toward T as a "monopole," in order to extend this system into
higher order RDs we have been forced to define the generalized LRS
chord as bipolar, i.e., the x-2 mediants are added to and/or
subtracted from either T or T+d. The effect of this is to destroy the
uniformly one-directional nature of Riemann's original concept where
the entire chord is described as "up" or "down" from a root/tonic.
However it is possible to "save the appearances" with respect to up-
ness and down-ness in higher RD systems.
[9.9] OPERATIONS IN LRS. In this section we have thus far been
concentrating on definitions in order to transport RS structures into
higher RD systems as LRS structures. But the heart of Lewin's Riemann
System is the family of operations
IDENT (identity)
CONJ (conjugate)
TDINV (tonic-dominant inversion)
RET (retrograde)
collectively called the "serial group" of operations (GSER).(11)
These operations on (T,d,m) consist essentially of alternations of
three elements:
location (whether the root is T or T+d)
polarity (whether d and m are positive or negative)
mode (whether the mediant element is m or m'=d-m).
So GSER consists of
IDENT: change nothing,
CONJ: keep T and change mode,
TDINV: add d to T and change polarity
(of d and m),
RET: add d to T and change both polarity
(of d and m) and mode.
Thus for RS (0,7,4), if the original triad is X = (0,4,7) = C major,
IDENT(X) = (0,4,7) = C major
CONJ(X) = (0,3,7) = C minor
TDINV(X) = (7,3,0) = dual G minor
RET(X) = (7,4,0) = dual G major.
But our revised definition for an LRS keeps the mode constant to
accomodate multiple mediant elements, centering instead on the
separate polarity alterations of d and m, a "+" value being measured
from T and a "-" value being measured from T+d. Thus the revised
family of operations, GSER*, can be described
IDENT: change nothing,
CONJ: change polarity of m,
TDINV: add d to T and change polarity
of both d and m,
RET: add d to T and change polarity of d.
=============
11. Lewin, op. cit., p.37-39.
=============
[9.10] The reader may verify that GSER* produces the same set of
chords as GSER in the previous example. But to illustrate that GSER*
generalizes to higher RD systems, consider the LRS (T=0,d=13,m=9) and
generate a "green" set as defined in [9.3]: X = (T,T+m; T+d-2m,T+d-
m,T+d) = (0,9; 19,4,13). Here we have reversed the order of elements
on the right to illustrate the pentachord's bipolar structure,
(T)-->(T+m) (T+d-2m)<--(T+d-m)<--(T+d),
\ /
\ /
\--------------->---------------/
a short right arrow indicating +9, a short left arrow indicating -9,
and the long right arrow indicating +13 (all mod 24). GSER* then
yields:
IDENT(X) = (0,9; 19,4,13) = 0-green
CONJ(X) = (0,9,18; 4,13) = 0-yellow
TDINV(X) = (13,4; 18,9,0) = dual 13-yellow
RET(X) = (13,4,19; 9,0) = dual 13-green.
[9.11] The arrangement which places T on the far left and T+d on the
far right with the set of mediant elements in between will be called
the canonical arrangement of an LRS chord. In the above example, the
intervals between the elements of the canonically ordered X, read left
to right, form a substring /9A99/, where A=10 (just why this is called
a substring here will be evident shortly). Since the "subdominant"
and "dominant" pentachords which join X to form the primary chords of
the system each share a common (polar) tone with X, the substring of
the canonical (sub)string form of the entire structure
(11,20,6,15,0,9,19,4,13,22,8,17,2) is /9A999A999A99/.
[9.12] THE SHIFT OPERATION. If we now clone the canonical substring
/9t99/ an indefinite number of times we have the infinite repeating
string Q = <...9A999A999A999A999A99...>. We may then directly apply
Lewin's definition of the SHIFT operation: "Given an integer N ... we
will define a formal operation SHIFT(N) which operates on any given
[L]RS to produce a transformed [L]RS whose canonical listing is
'shifted N places' from that of the given system."(12)
=============
12. Lewin, op. cit., p.48.
=============
[9.13] For a triadic system such as (T,7,4) in C12, K is an
alternation of m and d-m, <...343434...>. When paired with
appropriate pitch classes, e.g., ( ...g Bb d F a C e G b D f# A ...),
SHIFT in effect produces modulations. But even more, it describes a
functional relationship between virtually any pair of diatonic chords.
For instance, SHIFT(+2) applied to segment (B d F a C e G) modulates
to (F a C e G b D), i.e., from F major to C major (or from a B -13th
chord to an F-13th chord). In RD(2), SHIFT produces analogous pairing
relationships.
[9.14] Returning to numerical notation for chord elements (mod 24 for
clarity) and assigning 0 to reflect the example in [9.3], the string Q
= <...9A999A999A999A999...> produces the canonically ordered structure
(...11,20,6,15,0,9,19,4,13,22,8,17,2,11,21,6,15,0,...).
Again using red, green, yellow, and blue from [9.3] as mode
designations, 0-green indicates the canonic structure
(11,20,6,15,0,9,19,4,13,22,8,17,2) with the substring /9A999A999A99/
which can be rearranged to disclose the "0-green scale"
{0,2,4,6,8,9,11,13,15,17,19,20,22} with the C24 string
<2222122222122>.
[9.15] SHIFT(+1) applied to 0-green produces SHIFT(+1)(0-green) =
(20,6,15,0,9,19,4,13,22,8,17,2,11) = 9-yellow with the substring
/A999A999A999/. The "9-yellow scale"
{9,11,13,15,17,19,20,22,0,2,4,6,8} with the string <2222212222221>
shows that SHIFT(+1) here has kept all of 0-green's pcs but shifted
the "tonic" from pc0 to pc9 and rotated the string, i.e., produced a
"modal modulation."
[9.16] On the other hand, SHIFT(+2)(0-green) = SHIFT(+1)(9-yellow) =
19-blue = (6,15,0,9,19,4,13,22,8,17,2,11,21) has the substring
/999A999A999A/ and represents a "chromatic-modal modulation" with
respect to both 0-green and 9-yellow. The "19-blue scale" is
{19,21,22,0,2,4,6,8,9,11,13,15,17} with string <2122222122222>; its
string is not only a different rotation of <13;24> than either 0-green
or 9-yellow, but it also adds pc21 which neither of the other two
systems (keys) contains.
[9.17] As we have seen, by modifying Lewin's original definitions for
Riemann Systems, both the group of operations GSER (as GSER*) and the
operation SHIFT(N) are preserved in the hyperdiatonic system RD(2).
Extending to larger hyperdiatonic systems, the following progression
is easily verified:
RD(1) <--> (T,7,3) in C12
RD(2) <--> (T,13,9) in C24
RD(3) <--> (T,19,15) in C36
... ...
RD(n) <--> (T,7+6(n-1),3+6(n-1)) in C12n
It is left as an exercise to determine the LRS canonical form of the
substring supporting any basic chord in RD(n), i.e., to complete the
following: (T,7,3) --> /43/; (T,13,9) --> /A999/; (T,19,15) --> /?/;
(T,?,?) --> /?/.
***************
INSTEAD OF A CONCLUSION.
There is no truth beyond magic....
One, when you've discovered the truth ...
it does have the most extraordinary
magical quality about it. It's the
payoff, to recognize the deep order ...,
you feel you are in touch with something
fundamental. But there's also a poetic
sense in it: reality is strange. Many
people think reality is prosaic. I don't.
We don't explain things away....
We get closer to the mystery.
Brian Goodwin, theoretical biologist,
quoted in Roger Lewin, Complexity:
life at the edge of chaos
(Macmillan, 1992)
Despite the promise made at the end of Part I, I must break off the
development at this point solely for reasons of space, leaving the
promise only partially fulfilled. This study grew over the past two
years from a simple premise, the development of which I foolishly
believed would be at least short if not sweet. (I have now learned a
valuable lesson about simple premises and promises.) Though I had not
planned it this way, I will have to publish the conclusion to this
tonal fantasy in the future as Part III. But I will take my time with
it, perhaps to savor it.
Part III begins as a study in hyperdiatonic voice leading based on
Richard Cohn's parsimonious chord pairs and Eytan Agmon's efficient
linear transformations. These voice-leading considerations then force
some startling, even bizarre, conclusions about the nature of higher
order hyperdiatonic systems. While maintaining much of the usual
diatonic's salient features, a fourth-order hyperdiatonic system
contains "hypertones," sets which behave much like single tones in the
usual diatonic (this was first hinted at in [9.1] with the "mediant
problem"). This behavior then implies a nascent theory of
"counterset" which might arise through compositional praxis. But,
even without an existing body of hyperdiatonic works, by re-examining
Schenker's "axioms" it is possible to devise a "hyper-Schenkerian
conjecture" which asserts that counterpoints generally arise as more
or less natural products of white note systems, rather than magically
as the result of a fortuitous history. Examination of this idea will,
finally, conclude Part III of this particular fantasy.
But there is more; much more, I think. While examining the "simple
premise" in relation to tonal theory, it has become increasingly
obvious to me that the WARP function, which effectively transforms any
musical structure from one "space" to another, can be useful in the
study of (hyper)atonal structures. This idea is now a work-in-
progress and has already provided some startling connections and
pointed to unsolved problems.
But for now I am convinced enough of WARP's power to flush out
invariants, that I am willing to make the following statement, not
because I think it will necessarily turn out to be "true," but because
I think it is the right goal for speculative theory.
Just as there are abstract geometries which
can only be reached by studying their
various models, there are abstract musics which
can only be reached by studying their various
models.
It may be that we will ultimately learn that
there is only one abstract music;
and, at least in this sense if no other,
distinctions melt away:
There is only ONE music.
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