NOLL, Thomas - The Topos Of Triads

8/14/2019 NOLL, Thomas - The Topos Of Triads

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COLLOQUIUM ON MATHEMATHICAL MUSIC THEORYH. Fripertinger, L. Reich (Eds.)Grazer Math. Ber., ISSN 10167692Bericht Nr. 347 (2005), 1-26

The Topos of Triads

Thomas Noll

Abstract

The article studies the topos Sets T of actions of an 8-element monoid T onsets. It is called the triadic topos as T is isomorphic to the monoid of affinetransformations of the twelve tone system Z 12 , leaving a given major or minortriad invariant. The subobject classier of this topos and its Lawvere-Tierney-Topologies j are calculated. We explore the characteristic functions of the re-stricted T -action on triads as subobjects of the corresponding full affine T -actionon Z 12 . The upgrade operations on such triad-actions which are induced by theLawvere-Tierney-Topologies relate the triads to actions on larger tone sets, suchas the modal mixture, hexatonic, and octatonic systems. Further we calculate theminimal j -dense chords for three interesting cases: j P , j R , j L , which are relatedto Riemannian and Neo-Riemannian concepts.

1 Introduction

There are three mathemusical motivations for the present study, which we recapitulatein the subsections of this introduction. In the main part of this paper (section 2) we arefocussed on mathematical facts to which we offer music-theoretical interpretations anda music-analytical application in section 3. With respect to music-theoretical aspectswe also refer to [10], [11] as well as to ongoing investigations. Despite from its music-theoretical motivation the paper may also be studied as an exercise in basic topos theory.

1.1 Affine Self-perspectives of the Triad

Our investigation is motivated by the assumption that the homogeneous structure of the 12 tone systemas exemplied by the module Z 12 and its affine transformationsisof music-theoretical relevance. The action of 24-element dihedral group of translationsand inversions on subsets of Z 12 has been studied to great extend and more rarely alsothe action of the full 48-element automorphism group (i.e. including multiplication by

The present investigation is based on my work within the interdisciplinary research group KIT-MaMuTh (1998-2003) which was nanced by the Volkswagen-Stiftung.

Mathematics Subject Classication 2000: Primary 05E20, Secondary 33C80.Keywords and phrases: Music Theory, Triads, Dissonance, Topoi, Lawvere-Tierney-Topologies.


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2 Th. Noll

5 and 7). Non-invertible transformations (with multiplication by 0, 2, 3, 4, 6, 8, 9, 10)have seldom been interpreted in music-theoretical terms. However, Guerino Mazzola(c.f. [7]) observes that the major and minor triads are among the so called circle chords

X = {t, f (t), f (f (t)) ,...} which are generated from a given tone (pitch class) t Z 12and an affine transformation f : Z 12 Z 12 iteratively applied to it. With respect tousual circle-of-semitones encoding where C = 0 , C = 1 ,...,B = 11 we can generate theset {C,E,G } = {0, 4, 7} by applying f (t) = 3 t + 7 mod 12 iteratively starting from 0 -and thereby algebraically simulating the overtone-derivation of this chord:

0 7 4 7 4

Inspired by this observation I tried to algebraically simulate other aspects of chordmorphology (see [9], [10]). The present investigations further develops selected aspectsof this approach. While focussing on the triads we recapitulate just very few basicdenitions and facts. Throughout the paper we use the a circle-of-fths encoding of the12 tones:

note name C G D A E . . . E B F t Z 12 0 1 2 3 4 . . . 9 10 11

In order to denote affine transformations of Z 12 we write

t s : Z 12 Z 12 with t s(z) := s z + tmod 12.

In the music theoretical context of the 12-tone system we call them tone perspectives .

We denote the monoid of all 144 tone perspectives byA

. The concatenation of toneperspectives t 1 s1 and t 2 s2 is given by the formula

t 1 s1 t 2 s2 = t 1 + s 1 t 2 s1 s2

We study chords in terms of their self-perspectives , i.e.with each (non-empty) subsetX Z 12 we associate a submonoid of A , namely A (X ) = {h A | h(X ) X }. Withrespect to circle-of-fths notation the C-major-triad is represented by the set X ={0, 1, 4}. We get

A (X ) = {00, 10, 40, 01, 13, 49, 48, 04}.

This concrete order of the 8 elements does not matter here, but is in accordance withthe multiplication table in 2.1.3. Isomorphic chords have conjugated monoids of self-perspectives: A ( (X )) = A (X ) 1. Hence, it is useful to embed the study of severaltriads into the more exible language of actions of an abstract monoid.

1.2 The Triad as a Concord of Consonances

Tone perspectives t s are uniquely determined by the values t s(0) = t and t s(1) = s + t.The stretching factor s can be associated with the distance s = ( s + t) t between thesetwo values. We may say that t s turns a fth (distance = 1) based in 0 into an interval


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The Topos of Triads 3

of size s based in t. Figure 1 visualizes this idea of extrapolating interval relations intotone perspectives. After choosing a xed fth interval we obtain a bijection between the144 intervals and the 144 tone perspectives with many interesting properties (c.f. [9],

[8], [10]). Especially we may shift the consonance/dissonance-dichotomy from one levelof description to the other.

Figure 1: Interval relations induce affine self-perspectives of the C -major triad {0, 1, 4}.We have 13(0) = 3 0 + 1 = 1 and 13(1) = 3 1 + 1 = 4, i.e. the tone perspective 13 isinduced by the relation between a fth with base 0 and a major sixth with base 1 (leftgraph). Analogously, we have 48(0) = 8 0 + 4 = 4 and 48(1) = 8 1 + 4 = 0, i.e. thetone perspective 48 is induces by the relation between a fth with base 0 and a minorsixth with base 4 (right graph).

As a practical side effect of the interpretation of tone perspectives as extrapolatedinterval relations we use this association in order to load the nonpersonal repertoire of the 144 tone perspectives with intuitive names:

Name FormulasPrime Perspectives {t 0 | t Z 12}Fifth Perspectives (Transpositions) {t 1 | t Z 12}Major Second Perspectives {t 2 | t Z 12}. . . . . .Minor Seventh Perspectives {t 10 | t Z 12}Fourth Perspectives (Inversions) {t 11 | t Z 12}

We mention that there is a nice music-theoretical interpretation to the extrapolationidea. While Hugo Riemann (c.f. [12]) tends to consider triads as elementary and atomicbuilding stones of harmony, Carl Stumpf (c.f. [13]) disagrees with this viewpoint froma psychological point of view. According to Stumpf a triad is a concord of consonantintervals, i.e. a kind of wholeness beyond a mere collection of 9 intervals:


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Figure 2: 9 Internal intervals of the C -major triad

But each of the 9 intervals - considered in relation to the fth based in C = 0generates a tone perspective. In each case C = 0 and G = 1 are mapped to tones of the C -major triad, by denition. But in each case we may inspect the image of thetone E = 4. Surprisingly there is only one violation, namely the fourth perspective orinversion 111, mapping E = 4 to Eb = 9. The surprise is twofold: Firstly, most 3-elementtone sets have more such violations, and secondly, this only violation corresponds to aninterval which in music theory is considered to be dissonant: the fourth based in G.

A transformational reformulation of Stumpfs characterization may run as follows: Thetriad is a concord of its consonant internal intervals in the sense that their perspectivalextrapolations constitute its self-perspectives. The dissonant fourth is also discordantin this sense. Concordance is thus geometrically interpreted as stability, even thoughwe admittedly deal with a monoidal version of the idea of a stabilizer here, rather thanwith a group.

An interesting aspect of Hugo Riemanns point of view is his characterization of non-triadic tones as dissonances. Some of these dissonances are even characteristic forcertain chords in their harmonic meaning, such as the added sixth to the subdominantand the seventh to the dominant. The idea of this article is to study triads not as sets

but as monoid actions, i.e. as concords in the above sense. As such they have richercharacteristic functions and allow a qualitative differentiation of the 9 non-triadic tones.

1.3 The Category of Monoid Actions as a Topos

The present study has also been inspired by Guerino Mazzolas efforts to develop ageneral topos-theoretic metalanguage for music theory (c.f. [8]). While Mazzolas mo-tivation for his approach is mainly rooted in arguments which themselves belong tothe meta-level and/or to general epistemological and semiological lines of thought, ourconcrete investigation is driven by the desire to directly interpret topos-theory in music-theoretical terms in connection with the two concrete issues mentioned above. The Toposof Triads is a very small topos of monoid actions and we can calculate its structure sim-ply by hand. Our interest is especially directed to the music-theoretical interpretationof the subobject classier, of characteristic arrows and of Lawvere-Tierney topologies.To position the present study within Mazzolas theoretical framework suggests a studyon its own.


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2 Topos Structure of Sets T

This section is dedicated to the detailed calculation of the topos Sets T of monoid actions

: T X X on arbitrary sets X , where the monoid T is now an abstract version of the 8-element triadic monoid A ({0, 1, 4}) introduced in subsection 1.1.

2.1 The Triadic Monoid T and its Actions on a Set X

Observe that the monoid A ({0, 1, 4}) is generated by two tone perspectives: 13 and 48.Suppose we are given any two maps f, g : X X mapping a set X into itself. Theinterplay of such two maps can be arbitrarily wild. Their mutual concatenations

f f, g f, f g, g g, f f f, g f f,...

yield a monoid M whose identity is the identity map of X (understood as the emptyconcatenation of f and g). In the special case that all these concatenations differ from oneanother, M turns out to be isomorphic to the monoid {f , g}of all words (written fromright to left) generated from the two-element alphabet {f , g} with word-concatenationbeing its binary operation and the empty word being its neutral element. Generally,however, when sending words in letters f and g to corresponding concatenations of themaps f and g several words may represent the same resulting map on X , i.e. in thegeneral case M will be isomorphic to a factor monoid {f , g}/ for some equivalencerelation {f , g} { f , g}. The triadic monoid T the central object of our interest is an 8-element factor monoid of this kind. It expresses a rather simple interplay of two maps. We re-introduce this monoid by dening an equivalence relation betweenwords from {f , g}. We do so by introducing some basic relations between small wordsand extend these relations to larger words by applying them to their subwords in allcontexts, i.e. v v is extended to uvw uv w for all prexes u and suffixes w of v.Note that we hereby assume reexivity of the relation . We furthermore extend it suchthat symmetry and transitivity are fullled as well.

Before going into the detailed calculations we anticipate a general picture of a monoidaction : T X X in Figure 3. We call the subsets F = f(X ) X and G = g(X ) X the primary images of f and g, respectively. The secondary images A = g(f(X )) = g(F )and B = f(g(X )) = f(G) are subsets of the primary images G = g(X ) and F = f(X ),

respectively. The intersection C = F G turns out to be in bijection to both sets Aand B. These three sets A, B and C exemplify the generalized triad . In our concretemusic-theoretical application they form singleton sets, each consisting of one tone of thetriad.

2.1.1 Amalgamation of the Generators

We proceed in three steps in order to construct the triadic monoid T from the freemonoid {f , g}. The two maps f and g shall in general not commute, which means


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Figure 3: Action of the Triadic Monoid T on a Set X

that we do not identify the words gf and fg in our construction. The two generatorsshall nevertheless be closely related in a way which slightly differs from commutativity.We introduce the following relation between words:

ggf ffg

In the sequel we call this the amalgamation relation. In terms of set maps f, g : X X this relation is expressed by the commutativity of the following diagram:

X F

G

A

B C

E

E

d d

d d

c

c

f

g

f

g

f

g

In our concrete music-theoretical application we have the additional nice property thatthe product map f g : X F G is injective which means that X is actually thelimit of the lower right sub-diagram in the above diagram (with X and the two arrowsstarting in X being removed). We say in that case that X is the amalgam of F and Galong C .


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2.1.2 The Hypertriadic Monoid T

Now we put a restriction on both individual maps f and g which we call semi-idempotenceand which is expressed by the relations:

fff f and ggg g.

The maps f f and g g are idempotent because ( f f) (f f) = f (f f f) = f f. Inother words, we have projections f f : X F , g g : X G mapping F and G identi-cally onto themselves, respectively. The restrictions f|F and g|G are therefore involutionsand the maps f = f (f f) and g = g (g g) decompose into an idempotent projectionfollowed by an involution of their images, respectively. By putting the amalgamationand the semi-idempotence relations together we obtain a nite factor monoid

T = {e,f , f

2

, g ,g2

,a,af,b,bg,c,c2

}consisting of 11 elements. The choice of these symbols becomes evident in the Cayleygraph below. The elements of T correspond to the nodes of this graph and wordsw {f, g }representing an element of T correspond to directed pathways along the edges(labeled f and g) leading from the node e to the node of the element being representedby a word. The words have to be parsed from right to left (observe, that from each nodeof the Cayley graph departs exactly one edge labeled with f and one edge labeled withg).

eg f E'

b aE ' d

d

c' E

c2

T T

ag bf E ' e

e e

e e u

!

f 2g2

i

I

f f f

f f f f f f f f f x

f f

f f

f f

f f

f f f

f w

e e e e e

f g

g g f f

f g

f gf g

f g

g f

g f

f g gf

The reader is invited to check the correctness of the Cayley graph. We give a samplecalculation for the correctness of the g-edge from node bf leading back to a and of thef -edge from node ag leading back to b:

gfgf gggfgf gffggf gffffg gffg gggf gf fgfg fffgfg fggffg fggggf fggf fffg fg

Another example is the equivalence gfg gff of two pathways from e to the node ag:

gfg gggfffg gffgggf gffgf gggff gff .


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We also show that the node labeled c2 and hence suggesting the pathway (ggf)(ggf)from e to c2 can alternatively be reached by the shorter path of the word ffgg:

(ggf)(ggf) ggfffg ggfgffgg.

Another interesting fact is the idempotence of c2:

(ffgg)(ffgg)ffffgfggffgfggffgg.

In terms of an action of T on a set X observe that the involution f|F bijectively exchangesthe two subsets B and C of F and that analogously g|G yields a bijection between thetwo subsets A and C of G. This is because C = f(f(g(X ))) = f(f(G)) = f(B) andf(C ) = f(f(B)) = B (analogously for g with C and A).

2.1.3 The Triadic Monoid T

In order to nally reconstruct our original 8-element monoid A ({0, 1, 4}) we have toidentify the elements a and af with one another as well as b with bg as well as c withc2. This means in terms of actions of T on a set X that we dispense from the involutionaf |A = ag |A of the set A, the involution bf |B = bg |B of the set B and the involution c|cof the set C and identify them with the identities a |A , b|B and c|C of these sets. We doso by introducing the following relations on words

gff gf and fgg fg.

The resulting factor monoid is called the triadic monoid T = {e,f , f 2, g ,g2,a ,b ,c}. The

reader may inspect and check the multiplication table as well as the associated Cayleygraph:

a b c e f f 2 g g2a a a a a a a a ab b b b b b b b bc c c c c c c c ce a b c e f f 2 g g2f b c b f f 2 f b bf 2 c b c f 2 f f 2 c cg c a a g a a g2 gg2 a c c g2 c c g g2

e

c

g f

c c

T T

b a

d

d d

d

d d d

d d s

d d

d s

g2 f 2

E'

E' gf

f g

f g f g

gf

gf g

f gf

One may immediately check that the concrete monoid A ({0, 1, 4}) of self-perspectivesof the C-major triad is actually an action of T on X = Z 12 and the restriction of theseself-perspectives to Y = {0, 1, 4} dene an action on the triad Y itself. The monoid-isomorphy between T and A ({0, 1, 4}) is given in the table below:

element of T a b c e f f 2 g g2selfperspective a = 00 b = 10 c = 40 e = 01 f = 13 ff = 49 g = 48 gg = 04


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2.2 The Subobject Classier The relation between the T -action : T X X on the 12-element set X = Z 12given by A ({0, 1, 4}) and the restricted T -action : T Y Y on the three-element setY = {0, 1, 4} is a topos-theoretical generalization of the set-theoretic relation betweenthe set X = Z 12 and its subset Y X . However the set-theoretical characteristicfunctions Y : X {false, true } with

Y (x) =true if x Y false if x /Y

have to be modied for the reason that the actions of T on the 2-element truth-value set{false, true } are not rich enough in order to classify the sub-actions in terms of equivari-ant characteristic functions: Consider any action : T { false, true } {false, true }and let

f and

f denote set maps corresponding to the element f T in the two

actions and , respectively. Now consider the two elements x1 = 9 , x2 = 2 X \ Y in the complement of Y . One of them, namely x1 = 9, is mapped by f inside of Y ,because f (x1) = 13(9) = 3 9 + 1 mod 12 = 4 Y , while the other, x2 = 2, remainsoutside of Y , because f (x2) = 13(2) = 3 2 + 1 mod 12 = 7 /Y . We easily see thatthe above characteristic function Y : X {false, true } of the subset Y of X is not anequivariant map between the actions und , as we get the following contradiction:

f (false ) =f ( Y (x1)) = Y (f (x1)) = truef ( Y (x2)) = Y (f (x2)) = false

The appropriate topos-theoretic denition of an equivariant characteristic function of the relation between T -actions requires a larger codomain , known as thesubobject classier of the topos Sets T . The following considerations are dedicated tothe study of a particular T -action : T , which generalizes the role of the set{false, true }. Instead of just the 2-element set {false, true } we have to consider the set of cosieves onT . A cosieve (or left ideal) of an arbitrary monoid M is a subset B M such that M B = {m b | m M, b B} is contained in B . For trivial reasons this is thecase for the empty set F = and for the full monoid M , which is T in our case. Thesetwo cosieves F and T replace the two values false and true . With our Cayley graphwe can easily nd the principal cosieves of T by collecting all nodes along all pathwaysstarting from a given node. The elements a, b and c generate the same principal cosieveC = {a,b,c} which furthermore is contained every nonempty cosieve simply becausefrom each node there is a pathway into that 3-element set. More precisely, we havea m = a, b m = b and c m = c for any m T . If a cosieve contains f it must alsocontain f 2 and vice versa, because of f 3 = f (similarly for g and g2). Hence, includingthe full cosieve, there are four principal cosieves: C = T a, L = T f , R = T g, T = T e.Any nonempty cosieve is the union of the principal cosieves generated by its elements.In our case there is only one additional non-principal cosieve, namely P = L R . Intotal we have a six-element subobject classier = {F , C , L , R , P , T } whose elementsform the following diagram with respect to set inclusion:


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F =

C = {a,b,c }

L = {a,b,c,f ,f 2} R = {a,b,c,g,g 2 }

P = T \{ e}

T

T

Q

k

Q

k

T

The subobject classier understood as an object of the category Sets T of T -actions onsets needs to be considered together with the following action : T where

(m, B ) = m (B ) := {n T | n m B }.

The maps m always coincide for those m T generating the same principal cosieve-Therefore we have

a (B ) = b(B ) = c(B ) =T if B = F F if B = F

as well as

f (B ) = f 2 (B ) =T if B {L , P , T }R if B {C , R }F if B = F

g(B ) = g2 (B ) =T if B {R , P , T }L if B {C , L }F if B = F

Finally, we have e = 1 : The only elements m whose concatenations m e belong to Bare the elements of B themselves. Hence e is the identical map on .

2.3 Subobjects of and their Characteristic Arrows

For each equivariant injection of T

-actions we have the characteristic arrow : || with (x) = {h T | h (x) | |} .

In this subsection we study the subactions of within in terms of their characteristicarrows. In the next section we are also concerned with the characteristic arrows of triads(as actions) within the 12-tone system.The four maps a , f , g, e form a commutative monoid. This implies, that they arealso equivariant, i.e. they are arrows in the category of T -actions. They arethe characteristic functions {C ,L ,R ,P ,T }, {L ,P ,T }, {R ,P ,T } and {T } of the four subobjects


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{C , L , R , P , T }, {L , P , T }, {R , P , T } and {T }, respectively. In preparation of the followingsubsection we introduce a third way of denoting them:

jC

= a , jL

= f , jR

= g , jT

= e.Any given map : is an equivariant map, and hence classies a subobject of if it commutes with these four maps. Since j L jR = j C and j T = 1 it is sufficient tocheck the two equations

jL = j L and jR = j R .

From this we conclude

1. If B j 1L (B ) then (B ) j 1L ((B )), because (B ) = ( jL (B )) = j L ((B ))

2. Analogously, if Bj

1R (

B) then (

B) j

1R ((

B))

3. If B is xed under j L then so is (B ) because j L (k(B )) = ( jL (B )) = k(B ), i.e. if B {F , R , T } then also k(B ) {F , R , T }

4. Analogously, if B is xed under j R then so is (R ), i.e. if B {F , L , T } then also(B ) {F , L , T }

Conditions 1 and 2 say that maps bers of j L and j R again into bers. Condition3 and 4 say that maps the images j L () and j R () into themselves. We use theseconditions in order to calculate the subobjects of in four steps.

Step 1: Considering 3. und 4. together we conclude that also maps {F , T } into itself.The value (F ) {F , T } can be freely chosen without any restriction, because {F } isa ber on its own with respect to both, j L and j R . In other words, if we have anyequivariant map : with (F ) = F then the variation

(B ) = (B ) if B = F

T if B = F

is equivariant too. Written in terms of characteristic functions this means:

K = K {F } for F /K

Step 2: Our next observation is, that if maps any nonempty B to F it must auto-matically send all nonempty cosieves B to F . This follows from the fact that the fourbers {L , P , T }, {C , R } and {R , P , T }, {C , L } are connected through common elements.If one of these bers is mapped to the one-element ber {F } (this is what happens if any nonempty B is mapped to F ) than all of them are mapped to it. Thus, we obtainthe constant arrow {} with {}(B ) = F for all B and we get {} = {F } = with

B =F if B = F T if B = F


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All other arrows : have (T ) = T which means, that {} and {F } are the onlysubobjects of not containing T .Step 3: If (T ) = T then both of the bers {L , P , T } and {R , P , T } have to be mapped

into themselves. Consequently P can only be mapped to P and to T and converselythere is not further restriction to that. In other words, if K is a subobject having T andP as elements then K \{ P } is a subobject as well.Step 4: Again we suppose that : is an arrow with (T ) = T . From({L , P , T }) {L , P , T } and ({R , P , T }) {R , P , T } and conditions 3 and 4 we con-clude that L and R can only be mapped to themselves or to T . The ber {C , R } hasto be mapped into itself or into {L , P , T }, similarly {C , L } is mapped into itself or intoand {R , P , T }. We discuss these cases below:

1. ({C , R }) {C , R } and ({C , L }) {C , L }. In this case we necessarily have(C) = C , (R ) = R , (L ) = L ,

2. ({C , R }) {C , R } and ({C , L }) {R , P , T }. In this case we necessarily have(C) = R , (R ) = R , (L ) = T ,

3. ({C , R }) {L , P , T } and ({C , L }) {C , L }. In this case we necessarily have(C) = L , (L ) = L , (R ) = T ,

4. ({C , R }) {L , P , T } and ({C , L }) {R , P , T }. In this case we have two choicesfor (C ), namely P and T and we necessarily get (L ) = T and (R ) = T .

We summarize: There are exactly 5 arrows : with (T ) = (P ) = T and(F ) = F . They are parametrized by their values (C ) {C , L , R , P , T } (c.f. step 4). Toeach of them there is an associated arrow differing from only in the value (P ) = P(c.f. step 3). There is one further arrow with (F ) = F , namely the constant arrow {} classifying the empty set as a subobject. To each of these 11 arrows there is anassociated arrow , sending F to T . The resulting 22 arrows are listed in the tablesbelow. Those with special names (see also section 2.4) are labeled accordingly.

jT j P j R j L j CF F F F F F F F F F F F

C F C C L L R R P P T T

L F L L L L T T T T T T R F R R T T R R T T T T

P F P T P T P T P T P T T F T T T T T T T T T T

j F F T T T T T T T T T T T C F C C L L R R P P T T L F L L L L T T T T T T

R F R R T T R R T T T T P F P T P T P T P T P T

T F T T T T T T T T T T


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2.4 Lawvere-Tierney Topologies

The intersection B 1 B 2 of two cosieves B 1 and B 2 is also a cosieve. Furthermore, theintersection map: : is an arrow in the category Set T of T -actions. Hence,if we are given an arrow : we may ask wether the commutation property(B 1 B 2) = (B 1) (B 2) is fullled. It expresses the commutativity of the followingdiagram in Set T :

E

E

c c

We explore two properties which are implied by the satisfaction of this property.

1. If we have an inclusion of cosieves B 1 B 2 with (B 1) = T then (B 2) = T ,because (B 2) = (B 2) T = (B 2) (B 1) = (B 2 B 1) = (B 1) = T .

2. If (B 1) = (B 2) = T , then (B 1 B 2) = T , because (B 1 B 2) = (B 1)(B 2) =T T = T .

According to these two properties it turns out that if a non-empty subobject K of a(nite) subobject classier is classied by an arrow = K commuting with intersec-tion, then K has to be a principal subset of being generated from a smallest elementB :

K = B = {B | B B }.We write j B := B . and call j B the upgrade of the cosieve B . This convention corre-sponds with our previous notation for j C = a , j L = f , j R = g, j T = e. In additionto these four upgrades which are given by the action we have two others, namely jP = {P ,T } and j F = F = {F ,C ,L ,R ,P ,T }. The six upgrades are instances of so calledLawvere-Tierney Topologies j : , which by denition satisfy the followingthree properties:

(1) j (T ) = T , (2) j j = j, (3) ( j j ) = j .

We used conditions (1) and (3) in order to choose these six upgrade arrows j F , j C , j L ,

jR , j P , j T out of the 22 possible ones. Condition (2) is satised by all six of them.

3 Triads, Characteristic Functions, and Upgrades

We are now able to apply the results of the previous section to the study of triads. Recallthat we obtained the monoid T as an abstract copy of the Monoid A ({0, 1, 4}) of the8 self-perspectives of the C-major triad {0, 1, 4}. Now we consider a small family of 144 such affine actions in order to analyze their restrictions to triads as their minimalnonempty subactions.


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Figure 4:Sub([1, 4])

Figure 5:Sub([10, 4])

3.1 Affine T -Actions on Z 12 and their Subactions

As generating tone perspectives for a family of 144 affineT -actions we consider the twelvemajor sixth perspectives m 3, m Z 12 in the role of f and twelve minor sixth perspectivesn 8, n Z 12 in the role of g. For given m, n Z 12 let [m, n ] : T Z 12 Z 12 denote theT -action which is dened for the generators f and g by virtue of

[m, n ]f = m 3 and [m, n ]g = n 8.

Our initial example A ({0, 1, 4}) now turns out to be the action [1, 4]. To each T -action we have the Heyting algebra Sub() of its subactions. In particular we have

Sub([m, n ]) = {[m, n ]|X |X Z 12 with m 3(X ) X and n 8(X ) X }

This collection consists of all restrictions of [m, n ] to chords X Z 12 , where theserestrictions are well-dened. This is the case, whenever m 3 and n 8 are self-perspectivesof X , i.e. when [m, n ] A (X ). Figures 4 and 5 display all the subobjects of the actions[1, 4] and [10, 4]. These examples represent two interesting types of triadic actions,whose qualitative difference becomes clear as our investigation proceeds.

The gures are to be interpreted as follows: The 4096 subsets of Z 12 are identiedwith pairs of whole tone sets. These pairs are identied by the points in a square of size 64 64 The horizontal lower strip represents subsets of the even whole tone set{0, 2, 4, 6, 8, 10}, while the left vertical strip represents subsets of the odd whole toneset {1, 3, 5, 7, 9, 11}. Black dots denote subobjects of [1, 4] (Figure 4) and of [10, 4](Figure 5), while light dots denote supersets of {0, 1, 4} not being subobjects.


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Observe that aside from a black dot in the very left lower corner representing theempty action there are also black dots in lower left corners of the remaining areaswith black or light dots. This means that among these subactions of [m, n ] there is

a minimal non-empty one. We denote it by [m, n ]. For the action [1, 4] we alreadyknow, that this minimal subobject [1, 4] has the C-major triad X = {0, 1, 4} asits carrier set. Generally, the elements of these minimal carrier sets correspond to thetranslation parts of the projections [m, n ]a , [m, n ]b, and [m, n ]c. However, not all of the 144 actions [m, n ] are faithfully triadic, in the sense that these three projections(or prime perspectives ) are pairwise different from one another. This depends on theconcrete choice of m and n. It turns out that the action [m, n ] is faithfully triadic if and only if 5m + 2 n {1, 11, 5, 7, 2, 10}.

We explore this more closely: The 144 actions [m, n ] with m, n Z 12 can be suitablyclassied with the help of the following expression m 3 n 8 n 8 m 3 = 5m +2 n 0, yielding

the difference between [m, n ]b, [m, n ]a . If the commutation characteristic 5m + 2 nis either a unit modulo 12, (i.e. = 1 or = 5) or if it is equal to 2 then the carrier setof [m, n ] has 3 elements. Furthermore, if 5 m + 2 n is a unit 1, 5, 7 or 11 then this carrierset | [m, n ]| belongs to the 48-element isomorphy class of the generic triads representedby the C-major-triad {0, 1, 4}. If 5m + 2 n = 2 then the chord | [m, n ]| belongs tothe 24-element isomorphy class of the stretched triads , represented by {0, 4, 10} (i.e. adominant-seventh-chord with omitted fth). In all other cases | [m, n ]| has only twotones or just one tone. The table lists commutation characteristics and representativesof the corresponding transposition classes of the 3-element chords | [m, n ]|

5m + 2 n 1 -1 5 7 2 -2

| [m, n ]| {0, 1, 4} {0, 3, 4} {0, 3, 7} {0, 4, 7} {0, 4, 6} {0, 4, 10}We therefore calculate and compare the characteristic functions for our two representa-tive examples, namely for the generic triad [1, 4] as a subaction of [1, 4] and of thestretched triad [10, 4] as a subaction of [10, 4] (see Figure 6). The calculation can bedone by looking at the effect of the two generators on a given tone t:

{0,1,4}(t) =

T if t {0, 1, 4} i.e for t = 0 , 1, 4,P if t / {0, 1, 4}, but 3t + 1 , 8t + 4 {0, 1, 4} i.e for t = 9 ,L if 3t + 1 {0, 1, 4}, but 8t + 4 / {0, 1, 4} i.e for t = 5 , 8,R if 8t + 4 {0, 1, 4}, but 3t + 1 / {0, 1, 4} i.e for t = 3 , 6, 7, 10,C if 3t + 1 , 8t + 4 / {0, 1, 4} i.e for t = 2 , 11

{0,10,4}(t) =

T if t {0, 10, 4} i.e for t = 0 , 10, 4,P if t / {0, 10, 4}, but 3t + 10 , 8t + 4 {0, 10, 4} i.e for t = 6 ,L if 3t + 10 {0, 10, 4}, but 8t + 4 / {0, 10, 4} i.e for t = 2 , 8,R if 8t + 4 {0, 10, 4}, but 3t + 10 / {0, 10, 4} i.e for t = 1 , 3, 7, 9,C if 3t + 10 , 8t + 4 / {0, 10, 4} i.e for t = 5 , 11


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Figure 6: Characteristic functions {0,1,4}, {0,10,4} of the generic and the stretched triads.

With respect to the C -major triad (left) we observe that higher truth values (i.e. P , Rand L ) represent harmonically more characteristic tones, (or characteristic dissonancesas Hugo Riemann calls them), while the typical melodic dissonances, i.e. passing tonesand suspensions F and D correspond to the least truth value C . The characteristicharmonic dissonances of the C -major triad with respect to the Riemannian operationsP (Parallel), R (Relative) and L (Leading tone) should be 9 = E , 3 = A and 5 = B .We come back to this correspondence in Subsection 3.4. We interpret the truth valuesfor the stretched triad (right) in the context of a musical example (see Subsection 3.3).

3.2 Octatonic, Hexatonic and Wholetone Systems as Upgrades

Each of the six Lawvere-Tierney topologies j : (as discussed in subsection 2.4)induces a local operator J : Sub() Sub() on the subobjects of any object in Set T .

If is such a subobject and : its classifying arrow then J ( ) is thesubobject classied by j . Three of them appear to be interesting, namely J P , J Land , J R , as the other three have no interesting effects on tone sets. We calculate theseupgrades of the generic triad [1, 4] as a subobject of [1, 4] (see also Figure 7) and forthe stretched triad [10, 4] as a subobject of [10, 4] (see also Figure 8):

|J P ( [1, 4])| = {0, 1, 4, 9} major-minor mixture|J L ( [1, 4])| = {0, 1, 4, 5, 8, 9} hexatonic system|J R ( [1, 4])| = {0, 1, 3, 4, 6, 7, 9, 10} octatonic system

|J P ( [10, 4])| = {0, 4, 6, 10} french augmented sixth chord|J L ( [10, 4])| = {0, 2, 4, 6, 8, 10} wholetone system|J P ( [10, 4])| = {0, 1, 3, 4, 6, 7, 9, 10} octatonic system

These results offer a theoretical bridge to Richard Cohns study of the solidarity betweenvoiceleading parsimony and transformations in hyper-hexatonic systems of triads andthereby motivate our usage of the symbols P , L and R , which are reminiscent of theNeo-Riemannian operators P , R and L (c.f [4]). We recall the following two facts:


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Figure 8: Concatenation of the characteristic function {0,4,10} of the stretched triadwith the P -, R - and L -upgrades j P , j R , j L .

T -action if and only if k 1 mod 3, i.e. exactly in the situation described above. Wealso nd, that hexatonic and 2 k-tonic systems conside with the upgrades of the Cohn-triad {0, k, 2k 1} with respect to the Lawvere-Tierney topologies j L and j R respectively.

In other words, the same enrichment of the rigid Riemannian triads which we derivedfrom Stumpfs idea of concordance can be extended to the Neo-Riemannian study of parsimoneous triadic systems. The typical meaning of these upgrade operations in topostheory is that of a modal operator with the interpretation It is locally the case that....Music-theoretically speaking, the associated larger tone sets can be seen as modalities,where the original triad is still locally present. We therefore suggest to interpret musical

there is a more general case, which has also been discussed by David Clampitt (c.f. [3]): To replace thesemitone by another generator of the additive group Z 3 k in the role of being a parsimoneous connection,corresponds to a switch from a P-cycle (or a Cohn cycle) to what David Clampitt calls a Q-cycle .


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passages, which exemplify a parsimoneous hexatonic cycle as instances of a situation,where the triad of departure remains locally true with respect to the j L -topology. Twotypical examples which are cited in the context of Neo-Riemannian studies are

1. the closing section of the rst movement of Schuberts piano trio in E -major, op.100 with a complete P-cycle

E + ; E ; C + ; C ; G+ ; G ; Eb+ ,

2. Franz Liszts piano composition Consolations , No. 3 with an almost complete cycle(in the opposite direction)

D + ; F ; F + ; A ; A+ ; (D ) ; D +

The identication of these cycles is based on a larger scale analysis, where applied chordsare ignored. It is therefore important to mention, that our argument does not apply toapplied dominants. Their tones are not(!) locally true with respect to their triad of reference. In Subsection 3.3 we discuss an application, where each single note of a pieceis taken into consideration. It differs from the above examples also in another point. Theconsidered triads are stretched, rather than generic ones. In Subsection 3.4 we return tothe generic case in order to interpret the parsimoneous connections in terms of density .

We close this subsection by mentioning another music-theoretical cross link. GerardBalzano ([1]) investigates the diatonic scale {0, 2, 4, 5, 7, 9, 11} (semitone encoding) andthe diatonic major and minor triads {0, 4, 7}, {7, 11, 2}, {5, 9, 0}, {9, 0, 4}, {2, 5, 9},{4, 7, 11} within the 12-tone system Z 12 . Among other properties 2 he emphasizes thedecomposition Z 12 = Z 3 Z 4 and suggests the musical exploration of generalized tonesystems sharing this property. He comes up with a family of tone systems Z k(k+1) =Z k Z k+1 with a generalized diatonic subsystem of 2 k + 1 tones. The major- and minortriads within these systems are transpositions and inversions of {0, k, 2k 1}. Observe,that they are associated with affine triadic actions on Z k (k+1) with

f = 2k 1k mod k(k + 1) and g = k ( k)mod k(k + 1) .

We list the three examples for which the sizes of the minor thirds are k = 3 , 4, 5

k k(k+1) f g triad || L -upgrade R -upgrade3 12 73 48 {0, 4, 7} {0,3,4,7,8,11} {0,1,3,4,6,7,9,10}4 20 94 515 {0, 5, 9} {0,4,5,9,10,14,15,19} {0,1,4,5,8,9,12,13,16,17}5 30 115 624 {0, 6, 11} {0,5,6,11,...,24,29} {0,1,5,6,10,11,...,25,26}

The P -upgrade is always the 4-element intersection of the L - and R -upgrade and containsthe minor third as an additional tone.

2 He pays much attention to fth-transpositions of the diatonics in solidarity parsimoneous chromaticalterations.


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3.3 Triadic Analysis of Scriabins Etude Op. 65 Nr. 3

Figure 9 displays the opening bars of the Piano Etude Op. 65 Nr. 3 by AlexanderScriabin. 3

Figure 9: Opening four bars of Scriabins Etude Op. 65 Nr. 3. The left hand chords{G,F,B }, {D , C , F }, {A,G,C }, {E , D , G} are stretched triads. The rst three basstones G, D , A constitute on the large scale a stretched triad as well.

The 102 bars of the piece can be suitably segmented into harmonic cells, each of which is based on a stretched triad played in the left hand. In the appendix we listall the segments (the rst column locates the segments [ begin, end) with respect to barnumbers, the last column lists lexicographic smallest respresentatives for the chords X constituted by the corresponding segments tones). The lengths of the segments varyaccording to the texture between half bars and three bars. Inspired by Hugo Riemannsview on harmony we consider the stretched triads as generalized instances of consonantchords and study the added right hand tones as local dissonances. The values of thesetones with respect to the corresponding characteristic functions z,z +4 ,z +10 are listed inthe second column for each individual segment of the piece. Surprisingly, the two toneswith the lowest truth value C are missing throughout the piece. The highest non-truetruth-value P corresponds to the tritone (relative to the local fundament considered), i.e.,the paradigmatic tritone pendulum between the opening halfbars can also be interpretedas a characteristic fullbar prolongation. The ner local right hand structure seems tobe governed by the truth values R and L .

A closer look at the entire bass progression shows that Scriabin avoids exactly two

tones throughout the bass: A and D. In other words, there is an analogy betweenthe common local avoid-tones (but each time relative to the local bass) and theglobal avoid-tones in the bass with respect to the large scale opening stretched triad{G, D , A}. Figure 10 displays the presence of the upgrades of this triad with respectto the topologies j T (= identity), j P , j R , and j L . The avoidance of the two tones Aand D implies that their is always a modality such that the opening (large scale) triadis locally present. The distribution of these modalities (french augmented sixth chord,octatonic and whole tone systems) is coherent with the formal analysis of the piece.

3 Skrjabin, A., Klavierwerke Band I: Et uden , Edition Peters, Leipzig 1965.


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79 95

39 63

1 17

Figure 10: Analysis of the bass tones with respect to the central chord {1, 7, 3} ={G, C , A}. The three systems display the passages bars 1 - 38, bars 39 - 78 and bars 79-102, respectively. Within each system there are four horizontal layers, corresponding

to the upgrades j T (top layer), j P (second layer), j R (third layer), j L (bottom layer).

It is promising to compare our results with Cliff Calenders characterization of Scriabinslate harmonic vocabulary which is based on arguments of voice-leading parsimony. Cal-lender comes to a similar observation, namely that these chords are located between theoctatonic and wholetone system (c.f. [2]). But an investigation of voice-leading in atopos-theoretical framework has to be subject of a separate study.

3.4 j -Dense Subobjects in [m, n ]A subobject is called j -dense with respect to a Lawvere-Tierney topology j if j is the constant map sending all of || to T . This means in the case of affinetriadic T -actions on Z 12 that j must send all tones t Z 12 to T . Consequently,the carrier chord | | of must fully contain the set complement Z 12\ J ( ). Furthermore,since is an action it must contain all the transformed chords m (Z 12 \| J ( )|) for m T .Suppose that these chords are not empty, which is the case for the topologies j B withB = T , P , R , L . Then the minimal j B -dense subobject B has the carrier chord|B | = mT m (Z 12 \| J B ( )|). In particular we have | | |B |, because for any t Z 12the three images a (t), b(t), c(t) constitute the triad | | . Now we further investigateour standard examples [1, 4] and [10, 4] by calculating the minimal j B -dense subobjectsfor varying cosieves B :

jT -Density: In this case the minimal dense subobject is Z 12 itself, because [1, 4] =|J T ( [1, 4])| and therefore |[1, 4]T | contains both, Z 12 \ [1, 4] and [1, 4]. Analogously,|[10, 4]T | = Z 12 .

jP -Density: The only one tone which is neither in | [1, 4]| = {0, 1, 4} nor inZ 12\| J P ( [1, 4])| = {2, 3, 5, 6, 7, 8, 10, 11} is t = 9. But as 9 is neither in the image13(Z 12) = {1, 4, 7, 10} of [1, 4]f = 13 nor in the image 48(Z 12) = {0, 4, 8} of [1, 4]g = 48it is also not in [1, 4]P . In other words, [1, 4]P = Z 12\{ 9}. A similar argument showsthat [10, 4]P = Z 12\{ 6}.


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jR -Density: Five tones are neither in | [1, 4]| nor in Z 12\| J R ( [1, 4])| = {2, 5, 8, 11},namely t = 3 , 6, 7, 9, 10. But since 13(2) = 7 and 13(11) = 10, we see that 7 and 10 haveto be in [1, 4]R . The remaining tones t = 3 , 6, 9 are neither in the image of 13 nor in

the image of 48 and thus we conclude [1, 4]R = Z 12\{ 3, 6, 9}. A similar argument for[10, 4] discards 1 and 7 from the list t = 1 , 3, 6, 7, 9 of possible missing tones since

103(5) = 1 and 103(11) = 7 and hence [10, 4]R = Z 12 \{ 3, 6, 9}. jL -Density: We depart from three possible missing tones tones in [1, 4]L , namely

t = 5 , 8, 9. They are neither in | [1, 4]| nor in Z 12\| J L ( [1, 4])| = {2, 3, 6, 7, 10, 11}.But since 48(2) = 8 we have to discard 8 from that list. The remaining tones t = 5 , 9are neither in the image 13 nor in the image of 48 and thus we conclude [1, 4]L =Z 12 \{ 5, 9}. In the case of [10, 4] we have Z 12\| J R ( [10, 4])| = {1, 3, 5, 7, 9, 11} andobtain the candidates t = 2 , 6, 8 from which we must discard 8 since 48(5) = 8 andhence we get [10, 4]L = Z 12\{ 2, 6}.

jC -Density: The chordZ

12\| J C ( [1, 4])| is empty. However, its characteristic func-tion {} maps every tone to the empty cosieve F , and hence the concatenation j C {}does so too. In other words, the carrier chord |[1, 4]C | must not be empty. Therefore[1, 4]C is the smallest non-empty subobject of [1, 4], i.e. the triad [1, 4]. Similarly weobtain [10, 4]C = [10, 4].

jF -Density: As in the case of j C we observe Z 12\| J F ( [1, 4])| = {}. But now theconcatenation j F {} sends every tone to the maximal cosieve, i.e. the empty chord is jF -dense.

The notion of j L -density qualitatively distinguishes the two tones with (t) = L ,i.e. t = 5 , 8 (B and A ) in the case of the C-major triad. Both are mapped into | |under f (13(5) = 4, 13(8) = 1), but one of them is also an g-image (8 = 4 8(5)), whilethe other is not.

It turns out, that the tones E and B have two competing characterizations in thelight of Neo-Riemannian theory and in the light of j L -density. In the context of RichardCohns hexatonic cycle they yield the parsimoneous voice-leading connections of the C-major triad C + to its neighbors C and E . In the sense of density the two tones areimplicit with respect to the j L -topology, i.e. they are not elements of the minimal j Ldense set {0, 1, 2, 3, 4, 6, 7, 8, 10, 11} containing the C-major triad C + = {0, 1, 4}.

Further investigations are necessary in order to understand the connection betweenthese two competing characterizations. The following observation is a possible point of departure: Among the triadic actions on the hexatonic chord {0, 1, 4, 5, 8, 9} there are sixactions corresponding to the triads in the hexatonic cycle {C + , C , A + , A , E + , E } .The identity 01, the major third transpositions 41, 81 and the inversions 111, 511, 911 areequivariant automorphisms of Z 12 and of {0, 1, 4, 5, 8, 9} and induce equivariant mapsbetween conjugated subactions. 4

4 Note ([6]) that the Neo-Riemannian transformations P and L are not inversions of Z 12 . The uniform triadic transformation P exchanges C + with C , E + with E and A + with A . The inversion 1 11exchanges C-major triad C + with the C minor triad and coincides with P for these two chords.This is not the case for the other four triads, because 1 11(E +) = 1 11({4, 5, 8}) = {9, 8, 5} = A and


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A systematic study of equivariant maps, i.e. arrows in the topos Sets T is one possiblestrategy to enrich the investigations of the present paper. In the light of [10] (Section6) another strategy is interesting, namely an enlargement of the base category T . A

good candidate is a a category V with two triadic objects (i.e. copies of T ) and abstractvoice-leading arrows between them. The topos of triads Set T will be replaced then bythe topos of functors Set V .

References

[1] Balzano, Gerard: The group-theoretic description of 12-fold and microtonal pitch systems, in: CMJ 4: 66-84, 1980.

[2] Callender, C., Voice-Leading Parsimony in the Music of Alexander Scriabin , in:

Journal of Music Theory 42 : 219-233, 1998.[3] Clampitt, David: Pairwise Well-Formed Scales: Structural and Transformational

Properties , PhD-Thesis. Buffalo. 1997.

[4] Cohn, R., Neo-Riemannian Operations, Parsimonious Trichords, and their Tonnetz Representations , in: Journal of Music Theory 41 , 1997: 1 - 66.

[5] Goldblatt, R., Topoi - the Categorial Analysis of Logic , North-Holland, New York1984.

[6] Hook, J., Uniform Triadic Transformations , in: Journal of Music Theory 46 , 2003.

[7] Mazzola, G., Gruppen und Kategorien in der Musik , Heldermann, Berlin 1985.

[8] Mazzola, G., The Topos of Music , Birkhauser, Basel 2002.

[9] Noll, Th, Morphologische Grundlagen der abendl andischen Harmonik , in: Boroda,M. (ed): Musikometrika 7, Brockmeyer, Bochum 1997.

[10] Noll, Th. and M. Brand, Morphology of Chords, in: Mazzola, G.. et al. (eds): Per-spectives in Mathematical and Computational Music Theory epOs-music, Osnabr uck2004.

[11] Noll, Th. and A. Volk, Transformationelle Logik der Dissonanzen , in: Enders, B.(ed): Musik, Mathematik und Technik . Pfau, Saarbr ucken 2004.

[12] Riemann, H., Handbuch der Harmonielehre , Breitkopf und H artel, Leipzig 1887.

[13] Stumpf, C., Konsonanz und Konkordanz . In: Zeitschrift fr Psychologie 58 , 1911:321 303.

1 11(E ) = A +. In other words, the equivariant maps 1 11, 5 11, 9 11 do not represent proper Neo-Riemannian transformations. But the same problem occurs within Richard Cohns argumentation andneeds a longer and ramied discussion.


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