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Geometry of Chords
Thomas Noll
Department of Computer Science
Technical University of Berlin
November 4, 2001
Abstract
This article provides an introduction to basic geometric investigations
of the 12-tone sytem and its subsets, the chords. The various definitions
and results are intended to lay a theoretical basis for 12-tone-based explo-
rative and empirical research on occidental harmony. The entire approach
is motivated by the assumption that the 12 tones do not constitute an ar-
bitrary 12-element set with arbitrary paradigmatic relations, but rather
a discrete geometrical space. Transpositions obviously are instances of
paradigmatic relations, that are reflected by affine transformations. From
the mathematical p oint of view, the article presents a straightforward
elaboration of this observation. From the music theoretical point of view
it is of speculative nature and asks for experimental studies.
1 Tone PerspectivesConsider the ring Z12 of residue classes of integers modulo 12. Let T denotethe 12-tone module - i.e., Z12 understood as a module over itself. Its pointsare called tones.1 Affine endomorphisms f : T T of the tone module arecalled tone perspectives. Each one is given by a multiplication factor a Z12,together with a translation summand b T and we write ba in order to denotethe corresponding tone perspective ba : T T with ba(t) = at+b. The 144 toneperspectives form a monoid A with respect to the operation of concatenation.One has dc ba = cb+d(ca).
There are 6 submodules K T, namely the two trivial ones: T and 0T = {0}and 4 proper submodules: 2T, 3T, 4T and 6T. For each submodule K Tconsider the submonoid A(K) = {f A|f(K) K} of selfperspectives of K,
consiting of those tone perspectives mapping K into itself. Furthermore, we willInterdisciplinary Research Group KIT-MaMuTh for Mathematical Music Theory financed
by the Volkswagen-Stiftung1In a music theoretical application of this model one has to be more careful in dealing with
the meaning of the elements of T. One might formally distinguish the carrier sets Z12 and T
from one another in order to avoid confusion between scalar factors and tones.
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be concerned with the factor module T/K, whoose elements are affine subspacest + K associated with K.
The following definition and lemma focus on the proper submodules 3T and 4Tand the corresponding factor modules T/3T and T/4T.
Definition 1 (Outer decomposition of the 12-tone module)
1. The factor module T3 := T/3T is called the outer 3-cycle of the 12-tonemodule T and its elements are called inner 4-cycles or dimtones.
2. The factor module T4 := T/4T is called the outer 4-cycle of the 12-tonemodule T and its elements are called inner 3-cycles or augtones.
3. The direct sumT34 := T3 T4 is called the outer decomposition ofthe 12-tone-module.
Remark 1 The technical terms dimtone and augtone shall, on the onehand, refer to the traditional terms diminished seventh chord and augmentedtriad. On the other hand we do not intend to refer to the operations of diminu-tion and augmentation in this definition.
Lemma 1 Let ?3 and ?4 denote the projection maps fromT onto T3 and T4respectively and consider their product map ?34 onto T34, i.e.
?3 : T T3 with t3 := t + 3T,?4 : T T4 with t4 := t + 4T,?34 : T T34 with t34 := (t3, t4).
Then the following holds:1. The tone perspectives 04 and 09 induce module injections ?4 : T3 T
and ?9 : T4 T such that (?4)3 = idT3 and (?9)4 = idT4 .
2. ?34 is an isomorphism ofZ12-modules. Its inverse map is given through(?, ?)49 : T34 T with (t, s)49 := 4t + 9s.
Proof:
1. The value 04(t + 3k) = 4t does not depend on k, and, furthermore we have(4t)3 = t3. Similarily, the value
09(t + 4k) = 9t does not depend on k, and(9t)4 = t4.
2. ((t, s)49)34 = (4t + 9s)34 = (t + 3(t + 3s), s + 4(t + 2s). The last pair
represents the same element of T34 as (t, s) does. Conversely, we have(t34)49 = (t3, t4)49 = (4t + 9t) = t.
According to this lemma, one has a natural identification of the augtone-moduleT4 with the dimtone 3T = {0, 3, 6, 9} as well as of the dimtone-module T3 withthe augtone 4T = {0, 4, 8}, such that they can be viewed as retracts ofT. The
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situation is different in the case of the two-element module T2 = T/2T whencompared with the two-element tritone-chord 6T = {0, 6}. They are isomorphic
as Z12-modules, but 6T is not a retract ofT. Analogously, 2T is not a retractofT.
Definition 2 LetA3 andA4 denote the monoids of affine endomorphisms ofthe Z12-modules T3 andT4 respectively. The elements ofA3 are calleddimtoneperspectives and the elements ofA4 are called augtone perspectives.LetA34 := A3 A4 denote the direct product of the monoids A3 andA4. Itselements are called outer tone perspectives.
Lemma 2 Each tone perspective f A induces a dimtone perspective f3 A3as well as an augtone perspective f4 A4 by virtue of f3(t3) := (f(t))3 andf4(t4) := (f(t))4. The mappings ?3 : A A3 and ?4 : A A4 are surjectivemonoid morphisms. Especially, the restriction of ?3 to A(4T) (and of ?4 toA
(3T
)) yields a monoid isomorphismA
(4T
)
=A3 (and
A
(3T
)
=A4).
Proof: Take f = ba. Then
f3((t + 3k)3) = (f(t + 3k))3 = (a(t + 3k) + b)3 = (at + b)3 = (f(t))3 = f3(t3).
Hence the definition of f3 does not depend on the representatives of an ar-gument. Further, (f g)3(t3) = (f(g(t))3 = f3((g(t))3) = (f3 g3)(t3) andobviously (01)3 is the identity in A3. Finally, two tone perspectives
ba and dcinduce the same dimtone perspective, if and only if both differences b d anda c are multiples of 3. An inspection ofA(4T) = {ba | a, b 0, 4, 8} shows thatthese nine tone perspectives represent nine different dimtone perspectives, justbecause 0, 4, 8 represent different residue classes modulo 3. Hence we are donewith f3. The same line of arguments works for f4.
The following lemma shows that tone perspectives are in a natural 1-1-correspondencewith the outer tone perspectives.
Lemma 3 The mapping ?34 : A A34 with f34 := (f3, f4) is an isomor-phism of monoids.
Proof: We construct the inverse mapping ?49 : A34 A as follows: Letf = b+3k(a + 3l) and g = d+4m(c + 4n) be arbitrary representatives of anargument (f3, g4). Its image (f3, g4)49 A has to be independent of thevariables k,l,m,n. Indeed, we set
(f3, g4)49 :=4(b+3k)+9(d+4m)(4(a + 3l) + 9(c + 4n)) = 4b+9d(4a + 9c).
Hence the mapping is well-defined. Now we check that it is inverse to ? 34.
For f =b
a and (f3, g4) = ((b
a)3, (d
c)4) we compute,(f34)49 = ((
ba)3, (ba)4)49 =
4b+9b4a + 9a = ba = f,((f3, g4)49)34 = (4
b+9d4a + 9c)34 = ((ba)3, (
dc)4) = (f3, g4).
Finally, the product map ?34 = ?3?4 : A A3 A4 is a monoid morphism,because its factors ?3 : A A3 and ?4 : A A4 are monoid morphisms.
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Remark 2 Besides the monoid structure ofA,A3,A4 andA34 these sets carrythe structure of aZ12-module (e.g., addition inA is given by
ba+ dc = b+d(a+c)
and scalar multiplication by k(ba) = kb(ka). The above calculations make clearthat the monoid morphisms ?3, ?4 and?34 are, at the same time, linear modulemorphisms.However, the additive module structure and the multiplivative monoid structuredo not combine into a ring structure, because the distributivity law is not fulfilled:Left distributivity is fulfilled:
(b1a1 +b2a2) dc = (a1+a2)d+b1+b2(a1 + a2)c
= a1d+b1(a1c) +a2d+b2(a2c) =
b1a1 dc + b2a2 dc.
But right distributivity is not fulfilled:
dc (b1a1 + b2a2) = c(b1+b2)+dc(a1 + a2)d
c b1a1 +
d
c b2a2 =
cb1+
d
(ca1) +cb
2+d
(ca2) =c(b1+
b2)+2
d
c(a1 + a2).
Some contructions for rings like ideals are also meaningful in the present situ-ation. A useful generalization of right and left ideals in rings to a monoid likeA are sieves and cosieves in the categoryA (a monoid is a category consist-ing of one object, while its elements are interpreted as arrows). It appears inour situation of the pseudo-ring A, that the cosieves share the properties ofleft-ideals, while only a few out of many sieves show the additive properties ofright-ideals.
Definition 3 A set R A of tone perspectives is said to be an A-sieve, ifR A = R, i.e., if r R implies r f R for all f A. A set L A oftone perspectives is said to be anA-cosieve, if L A = L, i.e., if l L implies
f l L for all f A.
While there are thousands ofA-sieves, there are only 6 A-cosieves, namely thesubmodules T(kZ12) A (for k {0, 1, 2, 3, 4, 6}) (see Remark 2). All A-sievesare semigroups, but not vice versa (e.g., the only sieve R among the submonoidsM A is A itself, because 01 R always implies 01 f = f R for all f A).
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2 Tone Symmetries
Among the 144 tone perspectives there are 48 invertible ones, namely thosehaving multiplicative units in Z12 = {1, 5, 7, 11} as multiplication factors. Theyform a group A A and are called tone symmetries.Similarly, we write A3 A3 to denote the 3-element group of dimtone symme-tries within the 9-element monoid of dimtone perspectives and we write A4 A4for the 8-element group of augtone symmetries within the 16-element monoid ofaugtone perspectives.We consider three actions of the group A on A:
left : A A A with left(s, f) := s f.right : A A A with right(s, f) := f s.conj : A A A with conj(s, f) := s f s1.
These three actions induce actions left3,right3,conj3 ofA
3on A3 as well as
actions left4,right4,conj4 ofA4 on A4 (e.g. left3(s3, f3) := (left(s, f))3).
Proposition 1 LetA3/left3,A3/right3 andA3/conj3 denote the sets of orbitsof dimtone perspectives with respect to the actions left3,right3 and conj3, re-spectively, and letA4/left4,A4/right4 andA4/conj4 denote the sets of orbitsof dimtone perspectives with respect to the actions left4,right4 and conj4. Indetail one has the following orbits:
1. A3/left3 consists of two orbits: {(00)3, (40)3, (80)3} andA3.
2. A3/right3 consists of four orbits: {(00)3}, {(40)3}, {(80)3} andA3.
3. A3/conj3 consists of four orbits: {(00)3, (40)3, (80)3}, {(04)3}, {(44)3, (84)3}and {(08)3, (48)3, (88)3}.
4. A4/left4 consists of three orbits: {(00)4, (90)4, (60)4, (30)4},{(06)4, (96)4, (66)4, (36)4} andA4.
5. A4/right4 consists of seven orbits: {(00)4}, {(90)4}, {(60)4},{(30)4}, {(06)4, (66)4}, {(96)4, (36)4} andA4.
6. A4/conj4 consists of seven orbits: {(00)4, (90)4, (60)4, (30)4}, {(06)4, (66)4},{(96)4, (36)4}, {(09)4}, {(69)4}, {93)4, (33)4} and{(03)4, (93)4, (63)4, (33)4}.
The verification of this propositions is left to the reader.
Let A34 = A3 A
4 denote the group of outer tone symmetries within the
monoid A34 of all outer tone perspectives. The actions left3,right3, conj3
and left4,right4, conj4 induce three actionsleft34,right34, conj34 : A
34 A34 A34
(e.g. left34((s3, s4), (f3, f4)) := (left3(s3, f3),left4(s4, f4)). Obviously, thesets A34/left34,A34/right34 and A34/conj34 of orbits of these three ac-tions consist of cartesian products O3 O4 of orbits O3 and O4 with respect to
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the corresponding actions on A3 and A4. As a consequence, the above propo-sitions provide a complete picture of the three orbit structures on A34. Fur-
thermore, the natural isomorphy between (inner) tone perspectives and outertone perspectives implies, that the group actions left,right and conj yield thesame orbit structures on A as the actions left34,right34 and conj34 do onon A34. This is subsumed in the following proposition:
Proposition 2 LetA/left,A/right,A/conj denote the sets of orbits of toneperspectives with respect to the group actions left,right andcons : AA A.The monoid isomorphism ?34 : A A34 induces natural bijections
A/left = A34/left34 = {O3 O4 | O3 A3/left3, O4 A4/left4},A/right = A34/right34 = {O3 O4 | O3 A3/right3, O4 A4/right4},A/conj = A34/conj34 = {O3 O4 | O3 A3/conj3, O4 A4/conj4}.
We conclude this paragraph by a series of tables displaying these orbit struc-tures.
Remark 3 In order to display all the tone perspectives, their various orbits andother structures in a suitable and coherent way, we will be using the following9 16-table.
00
10
20
30
40
50
60
70
80
90
100
110
01
11
21
31
41
51
61
71
81
91
101
111
02
12
22
32
42
52
62
72
82
92
102
112
03
13
23
33
43
53
63
73
83
93
103
113
04
14
24
34
44
54
64
74
84
94
104
114
05
15
25
35
45
55
65
75
85
95
105
115
06
16
26
36
46
56
66
76
86
96
106
116
07
17
27
37
47
57
67
77
87
97
107
117
08
18
28
38
48
58
68
78
88
98
108
118
09
19
29
39
49
59
69
79
89
99
109
119
010
110
210
310
410
510
610
710
810
910
1010
1110
011
111
211
311
411
511
611
711
811
911
1011
1111
The displayed vertical and horizontal lines in such aTP-Table may be varied inorder to group tone perspectives. In the figure above, the lines indicate, that thetable is based on a recursive embedding of a standardized 3 4-table into itself.The outer large 3 4 - frame groups the 144 tone perspectives into 12 small3 4 - frames according to their multiplication factors, while they are displayedaccording to their translations inside of each small frame. The standardized3 4-table is the following:
0 6 9 34 10 1 78 2 5 11
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TP-Table 1 Orbits of tone perspectives under the action of tone symmetriesby concatenation from the left: A/left
00
10
20
30
40
50
60
70
80
90
100
110
01
11
21
31
41
51
61
71
81
91
101
111
02
12
22
32
42
52
62
72
82
92
102
112
03
13
23
33
43
53
63
73
83
93
103
113
04
14
24
34
44
54
64
74
84
94
104
114
05
15
25
35
45
55
65
75
85
95
105
115
06
16
26
36
46
56
66
76
86
96
106
116
07
17
27
37
47
57
67
77
87
97
107
117
08
18
28
38
48
58
68
78
88
98
108
118
09
19
29
39
49
59
69
79
89
99
109
119
010
110
210
310
410
510
610
710
810
910
1010
1110
011
111
211
311
411
511
611
711
811
911
1011
1111
These orbits are in bijection with the 6 factor spaces ofT (f f(T) T/K).TP-Table 2 Orbits of tone perspectives under the action of tone symmetriesby concatenation from the right: A/right
00
10
20
30
40
50
60
70
80
90
100
110
01
11
21
31
41
51
61
71
81
91
101
111
02
12
22
32
42
52
62
72
82
92
102
112
03
13
23
33
43
53
63
73
83
93
103
113
04
14
24
34
44
54
64
74
84
94
104
114
05
15
25
35
45
55
65
75
85
95
105
115
06
16
26
36
46
56
66
76
86
96
106
116
07
17
27
37
47
57
67
77
87
97
107
117
08
18
28
38
48
58
68
78
88
98
108
118
09
19
29
39
49
59
69
79
89
99
109
119
010
110
210
310
410
510
610
710
810
910
1010
1110
011
111
211
311
411
511
611
711
811
911
1011
1111
These orbits are in bijection with the 28 affine subspaces of T (f f(T)).
TP-Table 3 Orbits of tone perspectives under the action of tone symmetriesby conjugation: A/conj
00
10
20
30
40
50
60
70
80
90
100
110
01
11
21
31
41
51
61
71
81
91
101
111
02
12
22
32
42
52
62
72
82
92
102
112
03
13
23
33
43
53
63
73
83
93
103
113
04
14
24
34
44
54
64
74
84
94
104
114
05
15
25
35
45
55
65
75
85
95
105
115
06
16
26
36
46
56
66
76
86
96
106
116
07
17
27
37
47
57
67
77
87
97
107
117
08
18
28
38
48
58
68
78
88
98
108
118
09
19
29
39
49
59
69
79
89
99
109
119
010
110
210
310
410
510
610
710
810
910
1010
1110
011
111
211
311
411
511
611
711
811
911
1011
1111
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3 Chords and their Perspectives
Definition 4 Nonempty sets of tones are called chords. For a chord X Tthe module X X generated by all differences X X = {x y | x, y X}within a chord X is called the module of that chord and X := x + X X(for any x X) is called the affine subspace generated by that chord X. Achord X is called special if its module X X is a proper submodule of T.Non-special chords with X X = T are called general.
A typology for special chords is given by the 27 proper affine subspaces and thecorresponding 5 proper submodules ofT (see TP-Tables 2 and 1).
Definition 5 A tone perspective f A is said to be achord perspective withrespect to a given ordered pair (X, Y) of chords, if f(X) Y. The set of allchord perspectives with respect to the ordered pair (X, Y) of chords is denoted
byA(X, Y). Elements ofA(X) := A(X, X) are called selfperspectives of thechord X.
Lemma 4 The collection 2T of all chords as objects together with all chordperspectives as arrows forms a category CH. The 12 singletons {x} with x Tare terminal objects of that category.
The proof is straightforward.
Similar definitions can be given by replacing T by the modules T3,T4 and T34.Sets of dimtones are called dimchords, sets of augtones are called augchords andsets of outer tones are called outer chords. Similarly, one has generated modules,like X3 X3, generated subspaces X3, as well as the notions of special and
generic dimchords, augchords and outer chords. The resulting categories aredenoted by CH3, CH4 and CH34.
Lemma 5 The morphisms ?3 : T T3, ?4 : T T4 and ?34 : T T34induce functors ?3 : CH CH3, ?4 : CH CH4 and ?34 : CH CH34respectively. The latter is an isomorphism of categories whose inverse ?49 :CH34 C H is induced by the morphism ?49 : T34 T.
The proof is straightforward.
So far we used the symbols ?3, ?4, ?34 on two levels, namely applied totones t T and to tone perspectives f A. Without causing confusion thisnotation can be extended to chords and chord perspectives. However, there has
one detail to be mentioned:The morphism ?34 : T T34 on tones is defined as the diagonal morphism?3?4 of ?3 : T T3 and ?4 : T T4.
Consider the product category CH3CH4. Its objects are pairs (X3, X4) con-sisting of any dimchords X3 and augchords X
4. These are in 1-1-correspondence
to outer cartesian chords X3 X4, i.e., the cartesian products of dimchords
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A(T)
q
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In order to refer to the isomorphy class of the cartesian closure X of a chord
X, we will speak of its cartesian type. For the 15 possible cartesian types weuse the symbols (arrangement like above):
. .. . . ... ....
: :: : : ::: ::::...
......
......
.........
............
Any chord X is special if and only if its cartesian closure X is special. Inthe Appendix we list the 3- and 4-chords according to the their general andspecial cartesian types.
The following considerations shed some light on the notion of chord perspec-
tives. For each chord X one may study the set-valued representable functors@X and X@, i.e., the images of X under the Yoneda embeddings of the categoryCH into the functor categories SetsCH and SetsCH
op
. We recall the definitionof these functors:
Definition 7 Fix a chord X |CH|.
The covariant functor X@ : Ch Sets maps a chord Y to the setX@Y := A(X, Y) of chord perspectives from X into Y and takes chorda perspective f Y1, Y2 to the set map
X@f : X@Y1 X@Y2 withX@f(g) := f g
.
The contravariant functor @X : Ch Sets takes a chord Y to the setY@X := A(Y, X) of chord perspectives from Y into X and takes a chordperspective f Y2, Y1 to the set map
f@X : Y1@X Y2@X with f@X(g) := g f
.
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The functor X@ collects all chord perspectives starting from a fixed viewpointX with variable scope, whereas the functor @X collects all chord perspectives
with a fixed scope X and varying viewpoints. The functoriality is reflected in anatural control of the change of perspectives under scope change and viewpointchange respectively.
In order to systematically refer to the images f(X) of a chord X under var-ious tone perspectives f we define the map imgX : A |CH| with imgX(f) :=f(X).
Lemma 6 Fix a chordX |CH|. The collection of allXimg@Y := imgX(X@Y) for varying chords Y determines a covariant functor Ximg@ : CH Sets. The family of the restrictions imgX |X@Y of imgX for varying chords Y defines anatural epimorphism imgX of the functor X@ onto the functorXimg@.
Proof: For a fixed scope change f A(Y1, Y2). The functor X@ yields the set
map X@f : X@Y1 X@Y2 mapping any g X@Y1 to f g. Now, supposethat g(X) is an element of imgX |X@Y1 , and that g
(X) represents the sameelement, i.e., g(X) = g(X). Then Ximg@f(g(X)) can be defined as f(g(X)),which in turn coincides with f(g(X)), and hence is well-defined.
X@Y1 X@Y2
Ximg@Y1 Ximg@Y2
imgX |X@Y1 imgX |X@Y2
X@f
Ximg@f-
-
? ?
As the diagram illustrates, imgX is a natural transformation. Its surjectivity ateach scope Y is obvoius.
Remark 4 The collection imgY(Y@X) for a fixed scope chord X and varyingviewpoint chords Y does not give rise to a contravariant functor from CH toSets in a natural way. Consider, for example, the fixed scope X = T, thetwo viewpoint chords Y1 = T, Y2 = {0, 1, 2, 3, 4, 5}, and the tone perspective02 A(Y2, Y1) as a change of viewpoint. Suppose we are about to define thevalue F u(f) : imgY1(Y1@X) imgY2(Y2@X) of a candidate F u for such a
functor. We look, for example, at the argumentT = 01(Y1) =11(Y1). But then
there would be more than one natural value for F u(f)(T), namely 01(f(Y2)) ={0, 2, 4, 6, 8, 10} and 11(f(Y2)) = {1, 3, 5, 7, 9, 11}.
Definition 8 Isomorphisms in the category CH are called chord symme-tries, automorphisms are called selfsymmetries. We introduce the notationsA(X, Y) := {f A | f(X) Y} for the set of selfsymmetries from X to Y,
and we write A(X) := A(X, X) for the group of symmetries of a chord X.
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The category CH consists of 157 isomorphy classes. These are called chordclasses and are listed in [2], [4], [3]. Further, for each chord X one has an action
conjX : A(X) A(X) A(X) induced by the conjugation conj : A A A(cf. Paragraph 2). The monoids A(X) of selfperspectives, the groups A(X)of selfsymmetries as well as the resulting conjugation classes A(X)/conjX arestudied in [4].
4 Harmonic morphemes
The considerations of this paragraph are based on the perspectivic incidencerelation H between tone perspectives and chords.
H = {(f, X)|X |CH| , f A(X)} A |CH|.
Let us take a closer look at this relation in order to generalize the usual studyof common chord tones2 to that of common chord perspectives. Technically, webuild formal concepts on the basis of perspectivic incidence:
Definition 9 Condider the following two maps:
Ext : 2A 2|CH| with Ext(M) := {X| f M : (f, X) H},Int : 2|CH| 2A with Int(U) := {f| X U : (f, X) H}.
A (formal) harmonic morpheme is an ordered pair (M, U) 2A 2|CH|
satisfying Ext(M) = U and Int(U) = M.
We introduce the following terminology: Ext and Int are called the (formal)harmonic extension and intension maps, respectively. The two coordinatesM and U of a harmonic morpheme (M, U) are called the intension and theextensionof this morphem. Harmonic morphemes can be obtained in two ways:
1. Start with any set U |CH| of chords and construct the morpheme(Int(U),Ext(Int(U)). The concatenation Ext Int : 2|CH| 2|CH| iscalled the extensional completion
2. Start with any set M A of tone perspectives and construct the mor-pheme (Int(Ext(M)),Int(M)). The concatenation Int Ext : 2A 2A
is called intensional saturation.
Lemma 7 The intension of each harmonic morpheme is a monoid.
The proof is straightforward.
Let MON denote the set of all submonoids ofA. According to the lemma,it would be sufficient to run through all monoids M MON in order to get
2In functional harmony common chord tones indicate the possibility of functional synonymy
of two chords.
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all harmonic morphemes through M (Int(Ext(M),Ext(M)). Let furtherMONs denote the subset of all saturated submonoids of the form Int(Ext(M))
A. The set of all harmonic morphemes is parametrized by MONs.As a preliminary step of our investigation we study two maps int and ext
in opposite direction, that are related to single chords rather than to sets ofchords.
MON |CH|.int
ext-
Recall that for singleton chordsets U = {X} one has Int({X}) = A(X), i.e.,the intension of a single chord consists of its chord perspectives. We define:
int : |CH| MON with int(X) := Int({X}) = A(X).
A chord X can be fully reconstructed from its constant selfperspectives, i.e.,from the tone perspectives ba A(X) with a = 0. Obviously b0 is a chordperspective of X if and only if b X. This reconstruction can be expressed interms of the evaluation at tone t,
evt : A T with evt(f) = f(t).
The restriction of ev0 toT0 yields a bijection between the 12 constant tone
perspectives b0 and the 12 tones b = b0(0), b T. For any set M of toneperspectives we set M0 := M T0 and [M] := ev0(M0). For monoids weintroduce a separate symbol for this map:
ext : MON(A) |CH| with ext(M) := [M]
Remark 5 The set |CH| can be viewed as a retract of MON. The map ext issurjective and the composition ext int yields the identity map on |CH|.
The lemma suggests to study the fibers ext1(X) for all chords X in order toget a systematic overview over all morphemes. These fibers are partially orderedaccording to the inclusion of morphemes and have an upper and a lower limit,which are characterized through the following proposition:
Proposition 4 Let M MON be a monoid of tone perspectives. The sat-urated monoids Int(Ext(M0)),Int(Ext(M)) and Int({ext(M)}) belong to thesame fiber ext1(ext(M)) and one has the inclusions:
Int(Ext(M0)) Int(Ext(M)) Int({ext(M)})
Int(Ext(M0)) is the infimum and Int(ext(M)) the supremum of the partiallyordered set of saturated monoids MONs ext1(ext(M)).
Proof: The chord ext(M) belongs to all three chordsets Ext(M0),Ext(M)and {ext(M)}. Hence, the corresponding intensions contain no other constanttone perspectives than those in M0, i.e., the three monoids indeed belong to
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the same fiber ext1(ext(M)). Further these chord sets satisfy the inclusions{ext(M)} Ext(M) Ext(M0), since M0 M. Hence the corresponding
intensions have inclusions in the reverse order. Finally, the inclusions holdindependent of the particular choice of M ext1(ext(M)), defining the sameM0 and ext(M).
Definition 10 A chordX is calledpoor if the identity01 and the constant toneperspectives x0, x X, are its only selfperspectives, i.e., if int(X) = X0 {01}.A chord X is said to be primitive if its selfperspectives are shared by all of itssuperchords, i.e., if Ext(int(X)) = Super(X) := {Y|X Y}.
The primitive morphemes, generated by primitive chords X, have the structure(int(X),Super(X)). Poor chords are always primitive, but there are also non-poor primitive chords. To be more precise, among the 157 chord classes thereare 31 classes, whose chords are primitive, but there are only 5 classes of poor
chords. The latter are represented by
{0, 1, 3}, {0, 1, 2, 4}, {0, 1, 2, 5}, {0, 1, 2, 3, 5}, {0, 1, 2, 4, 5}.
In order to get the full picture of all harmonic morphemes one has to investigateall the saturated monoids within the partially ordered sets ext1(X), where Xruns through representatives of the 157 chord classes. For this it is sufficient tocalculate representatives for the conjugation classes MONs(X)/conjX of satu-rated monoids within MONs(X) := MONs ext1(X) under the conjugationaction of the selfsymmetries A(X) of X.
These calculations have been carried out for all morphemes
(Int(Ext(M)),Ext(M), with M 0 =
by the help of a Mathematica notebook. On the whole, there are 25364 suchmorphemes. The detailed results cannot be published within this article.3
What remains, is the calculation of those morphemes with no constant toneperspectives in their intensions. The saturated groups have first been describedas musical groups in [1]. The general case can be studied in a refined way interms of global morphemes (see next paragraph).
3Contact the author in order to get a textfile with the results or the program.
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5 Harmonic Topology
The following topological constructions are studied by Guerino Mazzola (cf. [3],chapter 24) in the much more general situation of functorial local compositionsand their endomorphisms. However, we recall some aspects within the narrowcontext of chords and chord perspectives.
For a given set M A of tone perspectives let M := {M MON|M M} denote the set of all supermonoids of M. These sets M are called basicmonoid neighbourhoods. Similarly, we call chord extensions Ext(M) basic chordneighbourhoods.
Lemma 8 The familiy of basic monoid neighbourhoods is closed with respect tointersection: For any M1, M2 A let M1, M2 denote the monoid generated
from M1 and M2. One has
(M1
M2
) = M1
M2
= M1
, M2
Similarly, the family of basic chord neighbourhoods is closed with respect tointersection: For any M1, M2 A one has
Ext(M1 M2) = Ext(M1) Ext(M2)
Definition 11 We introduce the following two topologies on chords and monoids:
The topology EX T on the set |CH| of all chords generated by the familyof all basic chord neighbourhoods {Ext(M) | M MONs} is called theharmonic extension topology.
The topology IN T on the set MON of all monoids of tone perspectivesgenerated by the family of all basic monoid neighbourhoods {M | M
MON} is called the harmonic intension topology.Proposition 5 The map int : |CH| MON is continous with respect to thetopologies EX T and IN T. Moreover, EX T is the inverse-image topology ofINTwith respect to the map int.
Proof: For any given monoid M and and any given chord X we have
X int1(M) int(X) M X Ext(M).
Hence, int1(M) = Ext(M).
Remark 6 The map ext : MON |CH| is not continous with respect tothe topologies IN T and EX T. A general open set M in the topology IN Tis characterized by the property M M = M M (M M). On the
other hand, the minimal chord neighbourhoodExt(int(X)) of any non-primitivechord X does not contain all superchords of X. If Y X is such a superchordwith Y Ext(int(X)), we have X0 ext1(Ext(int(X))) and Y0 X0, butY0 ext1(Ext(int(X))).The non-continuity of ext reflects the essential difference between chords asobjects of the category CH and chords just as sets.
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The topologies IN T and EX T are rather exotic ones. In terms of the axiomsof separation from general topology T0, T1, T2, we have the following character-
izations:
Proposition 6 Both topologies IN T and EX T satisfy the T0-axiom, but notthe T1-axiom. The closures of singletons are explicitly given as follows:
For any M MON one has M = M := {M MON | M M}.
For anyX |CH| one has X = {X |CH| | Ext(int(X)) Ext(int(X))}.
Proof: Two monoids M1, M2, are not equal, if and only if M1 M2 or M2
M1 , hence the topology IN T is T0. The same holds for any two chords andtheir minimal neighbourhoods: X1 = X2 if and only if X1 Ext(int(X2)) orX2 Ext(int(X1)). Hence the topology EX T is T0.
The closure M of a monoid M is the complement of the largest open set not
containing M, i.e.,
M = (
MM
M
)c = {M | M M}c = {M | M M} = M
Analogously, we find for chords:
X = (
XExt(int(X))
Ext(int(X)))c = {X | X Ext(int(X))}
A set M of monoids is closed, if and only if M M implies M M for allM M. Analogously, a set V of chords is closed if and only if X V impliesX V for all X satisfying X Ext(int(X)).
The closures X for the 157 representatives for all the chord classes are easilycalculated. We mention that there are 14 classes, whose chords generate closedsingletons X = {X}, namely the 1-chords, the 2-chords except {0, 6}, the 3-chords except {0, 4, 8} and the 4-chord {0, 1, 2, 6}). On the other hand, the onlychords generating all their subchords, i.e., X = Sub(X) := {Y |CH| | Y X}are the 6 classes of affine subspaces ofT, namely 0T, 6T, 4T, 3T, 2T, T.
The general chord neighbourhoods, i.e., unions of basic chord neighbour-hoods Ext(M), provide a good means to define global morphemes.
Definition 12 Let IN Ts denote the topology induced by IN T on the subsetMONs of all saturated monoids. A global harmonic morphem is given asa pair (M, {Ext(M) | M M}) consisting of a set M of saturated monoids asits local intensions and the corresponding family {Ext(M) | M M} of basic
chord neighbourhoods as its local extensions. The open setMM Ext(M) is
called the extensional carrier of M.
As an example, consider the global morpheme determined by M = {P, Q},
P = {48,0 4,0 0,4 0,0 1}, Q = {13,4 9,1 0,4 0,0 1}.
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The open set Ext(P) Ext(Q) being covered by Ext(P),Ext(Q), providesa suitable model for chords representing the same tonal function. Observe
that the generated monoid P, Q = int({0, 1, 4}) is the intension of the C-Major-triad. The extension Ext(int({0, 1, 4}))) contains only particular su-perchords of {0, 1, 4}, but not the parallel-triad {0, 1, 3} or the counter-triad {1, 4, 5}. These and other prototypical representatives of the tonic triad{0, 1, 4} in functional harmony are suitably modelled by the global morpheme({P, Q}, {Ext(P),Ext(Q)}). An even more refined filtering of all chords in theirpartially ordered roles as fuzzy representatives of a given chord X is suitablydescribed in terms of the global morphem
(int(X)s, {Ext(M) | M int(X)s
}).
See [4] and [5] for a music theoretical discussion of this proposal.Another situation where global morphemes can be considered is the study of
saturated monoids M MONs without constant tone perspectives. We giveonly a sketch of this procedure. For any monoid M of tone perspectives one hasthe relation
M = {(s, t) T T | f Mwithf(s) = t}
M0 = implies that the equivalence closure of M splits into more than onechord as equivalence classes. The stable images ImM(Xi) of these chords Xi(i = 1, , , , , n) under iterated application of the elements of M yield the minimallocal placeholders ImM(Xi)0 for the missing global constant tone perspectiveswithin M. The whole variety of candidates is given by collections of chords{Yi Ext(M) | ImM(Xi) Yi Xi}, where i runs at least over two indicesbetween 1 and n.
6 Perspectivic Interpretation of Voice Leadings
The last two paragraphs were solely dedicated to the study chord selfperspec-tives. This final paragraph now focusses on an idea concerning the possiblesyntactic roles of chord perspectives in the study of chord successions and voiceleading.
While chords are just sets of tones, one is often interested in the distributionof tones along an ensemble of k voices. The simultaneous presence of toneswithin these voices can be represented in terms of vectors, which are calledk-voicings:
v =
t1...
tk
.
In order to describe a k-voice leading, where, from an melodic point ofview, k voices make their individual tone-steps simultaneously, or where, froman harmonic point of view, one has a succession of two k-voicings, one canconveniently switch between the 3 interpretations of k 2 matrices,
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=
s1 t1......
sk tk
, melo =
(s1 t1)...(sk tk)
, harmo =
s1...sk
t1...tk
.
Explorative studies suggest to interpret voice-leadings as follows: Considerthe functions k : T
k |CH| mapping voicings to the underlying chordsk((t1,...,tk)) := {t1,...,tk} and let 1harmo and
2harmo denote the first and
the second voicing of harmo, respectively. We distinguish two interpretations:
In the causal interpretation of the voice leading one considers the chordperspectives
caus := A(k(1harmo), k(
2harmo))
and attributes to each coordinate i
melo
= (si, ti) the subset icaus of those
chord perspectives f caus satisfying f(si) = ti.
In the final interpretation of the voice leading one considers the chordperspectives
fin := A(k(2harmo), k(
1harmo))
and attributes to each coordinate imelo = (si, ti) the subset ifin of those
chord perspectives f fin satisfying f(ti) = si.
The sets icaus and ifin are never empty, because they always contain exactly
one constant tone perspective ti0 or si0. The following diagrams show threeexamples:
Example 1 Typical 4-voice leading of the succession G - C
=
5 01 12 41 0
-
-
-
-
1
2
1
5
1
2
1
5
0
4
1
0
0
4
1
0
B
T
A
S
111, 84
103, 84
103
88
11, 14
23
58
58, 59
On each arrow from left to right we list the non-constant elements oficaus, andon each arrow from right to left we list the non-constant elements of ifin .
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Example 2 Typical 4-voice leading of the succession G7 - C
=
5 011 41 12 0
-
-
-
-
2
1
11
5
2
1
11
5
0
1
4
0
0
1
4
0
B
T
A
S
88
84
88
23, 29
58
113, 116
56, 58, 59
Example 3 Typical 4-voice leading of the succession D7b - C
=
5 011 48 17 0
-
-
-
-
7
8
11
5
7
8
11
5
0
1
4
0
0
1
4
0
B
T
A
S
84
84
88
71, 74, 710
71, 53, 119
71, 74, 116, 119, 710
53, 56
The main idea behind these perspectivic interpretations is to provide a link be-tween single tone successions within the voices and chord perspectives betweenthe chords that are connected through the specific voicings. A systematic in-vestigation of all typical and rare voice leadings from or to a fixed chord X onthe background of the functor Ximg@ will be subject of a future study.
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References
[1] Mazzola, G.: Gruppen und Kategorien in der Musik, Heldermann,Berlin 1985.
[2] Mazzola, G.: Die Geometrie der Tone, Birkhauser, Basel 1990.
[3] Mazzola, G.: The Topos of Music, Birkhauser, Basel 2002 (to appear).
[4] Noll, Th: Morphologische Grundlagen der abendlandischen Harmonik,Musikometrika 7, Brockmeyer, Bochum 1997.
[5] Noll, Th: Harmonische Morpheme, Musikometrika 8, Brockmeyer,Bochum 1998.
[6] Noll, Th., Nestke, A.: Die Apperzeption von Tonen, Electronic Jour-
nal of the GMTH 2002 (http://www.gmth.de, to appear).
7 Appendix: Chordicon
Table 1: General 3- and 4-chords of cartesian type : :
Class TP-Table Representatives
10
q
qq
q
q
q
q
q
31
q
qq
q
q
q
q
q q
q
q
qq
q
q
q
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Table 2: General 3- and 4-chords of cartesian type :::
Class TP-Table Representatives
9
q
q
q
q
12
q
q q
q
q q
q
22
q
q
q
q
qq
q q
25
q
q
q
q
q
q
q
q q
q
26
q
q
qq
q
q
q
qq
q
q
q
29
q
q q q
q
q
q
q q
q
q
q
Table 3: General 3- and 4-chords of cartesian type...
...
Class TP-Table Representatives
11
q
q
q
q
q
28
q
qq
q
q
qq
q
30
q
qq
q
q
q
q
q
q
q
q
q
q
q
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Table 4: General 4-chords of cartesian type ::::
Class TP-Table Representatives
24
q
q
qq
q
q q
q
27
q
q
q
q
q
q
q
q
33
q
q
q
q
q q
q q
q
q
q
q
q q
q q
Table 5: General 4-chords of cartesian type.........
Class TP-Table Representatives
08
q
q
q
q
q
18
q
q
q
q
q
19
q
q
q q
q
20
q
q
q
q
q
q q
23
q
q
q
q
q
q
32
q
q
q
q
q
q
q q
q
q
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Table 6: General 3- and 4-chords of cartesian type
.
..
.
..
.
..
.
..Class TP-Table Representatives
17
q
q
q
q
q
q
21
q
q
q
q
q
q
q q
q
q
Table 7: Special 3-chords of cartesian type...
Class TP-Table Representatives
15
q qq
q
q q
q
q qq
q
q q
q
q qq
q
q q
q
Table 8: Special 4-chords of cartesian type ....
Class TP-Table Representatives
37
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
q q q q
Table 9: Special 3- and 4-chords of cartesian type ::
Class TP-Table Representatives
14
q
q
q
q
q
q q
q
q
q
q
q
q
q q
q
36
q
q
q
q
q q
qq
q
q
q
q
q q
qq
q
q
q
q
q q
qq
q
q
q
q
q q
qq
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Table 10: Special 3- chords of cartesian type...
Class TP-Table Representatives
16
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
Table 11: Special 3- and 4-chords of cartesian type......
Class TP-Table Representatives
13
q
q
q
q
q
q
q
q
q
q
34
q
q
q
q
q
q q
q
q
q
q
q
q
q q
q
35
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
24
