Generalized Tonnetz and discrete Abel-Jacobi map
aa r X i v : . [ m a t h . M G ] M a y Generalized Tonnetz and discrete Abel-Jacobi map
Filip D. Jevtić
Mathematical InstituteSASA, Belgrade
Rade T. Živaljević
Mathematical InstituteSASA, Belgrade
May 1, 2020
Abstract
Motivated by classical Euler’s
Tonnetz , we introduce and study the combinatorics andtopology of more general simplicial complexes
T onn n,k ( L ) of Tonnetz type . Out mainresult is that for a sufficiently generic choice of parameters the generalized tonnetz
T onn n,k ( L ) is a triangulation of a ( k − -dimensional torus T k − . In the proof weconstruct and use the properties of a discrete Abel-Jacobi map , which takes values inthe torus T k − ∼ = R k − / Λ where Λ ∼ = A ∗ k − is the permutohedral lattice. Keywords: generalized Tonnetz, discrete Abel-Jacobi map, permutohedral lattice, simpli-cial complexes, polyhedral combinatorics, triangulated manifolds.MSC2010: 14H40, 52B05, 52B20, 52B70, 52C07, 57Q15
In his seminal work on music theory “Tentamen novae theoriae musicae ex certissismisharmoniae principiis dilucide expositae” (1739), Leonhard Euler introduced a lattice dia-gram –
Tonnetz – representing the classical tonal space. In more recent interpretationsthis diagram is identified as a triangulation of a torus with 24 triangles representing all themajor and minor chords. If the equal tempered scale is identified with Z , the Tonnetzcan be described as T onnetz = {{ x, x + 3 , x + 7 } | x ∈ Z } ∪ {{ x, x + 4 , x + 7 } | x ∈ Z } . Notice that if { x, y, z } ∈ T onnetz then { x − y, y − z, z − x } = ±{ , , } . This serves asan inspiration to introduce and study more general complexes of “Tonnetz type.” This work was supported by the Serbian Ministry of Education, Science and Technological Developmentthrough Mathematical Institute of the Serbian Academy of Sciences and Arts. .1 Generalized Tonnetz Suppose that L = { l i } ki =1 is a collection of k positive integers which add up to nl + l + · · · + l k = n . We say that a collection L is generic if for each pair I, J ∈ [ n ] of subsets of [ n ] X i ∈ I l i = X j ∈ J l j ⇒ I = J . (1.1)A collection L is reduced if the largest common divisor of all l i is , h l , l , . . . , l k i = 1 . (1.2) Caveat:
From here on we identify elements of [ n ] with the corresponding elements (con-gruence classes) in the additive group Z n = { , , . . . , n − } of integers, modulo n . Astandard geometric model for this set is V n = { ǫ | ǫ n = 1 } , the set of vertices of a regular n -gon. For this reason we in principle assume a counterclockwise orientation on the unitcircle S where the n -gon is inscribed. However for some readers it may be more natural touse (occasionally) more traditional presentation of the (classical) Tonnetz, with clockwiseorientation and n = 12 occupying its “usual” place.Each ordered pair ( x, y ) of elements in Z n (respectively [ n ] or V n ) defines an interval I x,y = { x, x + 1 , . . . , y } ⊂ Z n . The length of this interval is L ( I x,y ) = | I x,y | − y − x ∈ Z n .If a set τ = { v , v , . . . , v t } is a subset of Z n we always assume the cyclic order v ≺ v ≺ · · · ≺ v t of its elements, i.e. v j − v i ∈ { , , . . . , n − } for each pair i < j of indices. Definition 1.3.
A generalized tonnetz
T onn n,k ( L ) ⊆ [ n ] is a ( k − -dimensional simplicialcomplex whose maximal simplices are ∆( x ; σ ) = { x, x + l σ (1) , x + l σ (1) + l σ (2) , . . . , x + l σ (1) + . . . + l σ ( k − } (1.4) where x ∈ Z n and σ ∈ Σ k is a permutation. Our main result is the following theorem which claims that a (sufficiently generic)generalized Tonnetz is also a triangulation of a torus, as its classical counterpart.
Theorem 1.5.
Suppose that L = { l i } ki =1 is generic and reduced in the sense of (1.1) and(1.2). Then the generalized tonnetz T onn n,k ( L ) is a triangulation of a ( k − -dimensionaltorus T k − := ( S ) k − . The central idea in the proof of Theorem 1.5 is to identify the (triangulated) torus T k − ∼ = R k − / Λ as a combinatorial Jacobian , i.e. as the target of a (discrete) Abel-Jacobimap J : T onn n,k ( L ) −→ R k − / Λ where Λ ∼ = A ∗ k − is a permutohedral lattice.2 .2 Discrete Abel-Jacobi map The classical Abel-Jacobi map is a map from an algebraic curve S (Riemann surface of genus g ) into the torus C g / Λ where Λ ⊂ C g is the lattice of periods. More explicitly there exist g linearly independent holomorphic differentials ω , . . . , ω g on S and if { c j } gj =1 ⊂ H ( S ; Z ) is a collection of basic cycles then the vectors v j = h c j , ω i = ( Z c j ω , . . . , Z c j ω g ) form a basis of a lattice Λ . Then the Abel-Jacobi map J : S → C g / Λ is defined by J ( p ) = ( Z pp ω , . . . , Z pp ω g ) mod Λ . (1.6)In the special case when the curve S is elliptic the Jacobi map is an isomorphism.In analogy with this construction, we describe explicit simplicial cocycles ω i (1 i k ) on T onn n,k ( L ) , which play the role of holomorphic differentials and allow us to constructthe corresponding “discrete Abel-Jacobi map.” This can be compared to the use of discreteAbel-Jacobi maps in the construction of standard realizations of maximal abelian covers ofgraphs in topological crystallography, see [8, 9, 10] and [2]. T onn n,k ( L ) is a manifold We begin our analysis of complexes of Tonnetz type by showing that the irreducibilitycondition (1.2) can always be assumed, without an essential loss of generality.
Proposition 2.1.
The (geometric realization of the) complex
T onn pn,k ( pL ) ⊆ [ pn ] , where pL := { pl , pl , . . . , pl k } , is homeomorphic to the disjoint union of p copies of T onn n,k ( L ) . Proof:
As a consequence of (1.4) if σ = { v , v , . . . , v k } ∈ T onn pn,k ( pL ) then v i ≡ v j ( mod p ) for each i, j ∈ [ k ] . It follows that T onn pn,k ( pL ) is a disjoint union of its p subcomplexes T j ∼ = T onn n,k ( L ) where T j is spanned by vertices in the same Z n -coset ofthe group Z pn . (cid:3) Unlike the irreducibility condition (1.2), the genericity condition (1.1) is essential forthe proof of the following proposition.
Proposition 2.2.
T onn n,k ( L ) is a connected, combinatorial manifold if the “length vector” L = ( l , . . . , l k ) is both generic and reduced, in the sense of (1.1) and (1.2). Moreover,the links of vertices are isomorphic to boundaries of simplicial polytopes dual to ( k − -dimensional permutohedra. Proof:
By definition τ = { v , v , . . . , v t } ∈ T onn n,k ( L ) if and only there is a partition [ k ] = I ⊔ I ⊔ · · · ⊔ I t such that for each i the length of the interval I v i ,v i +1 ( v t +1 := v ) is L ( I v i ,v i +1 ) = X j ∈ I i l j . v = v the face poset of the star Star ( v ) ⊆ T onn n,k ( L ) is isomor-phic to the poset of all ordered partitions of [ k ] (here we use the genericity of L ), wherethe top dimensional simplices in Star ( v ) correspond to the finest ordered partitions of [ k ] .Note that the finest ordered partitions of [ k ] are in - correspondence with permutationsof [ k ] .Recall [13, Example 0.10] that the face poset of a ( k − -dimensional permutohedron P erm k − is also the poset of all ordered partitions of [ k ] , but with the reversed ordering.(The finest partitions/permutations correspond to the vertices of the permutohedron.) Itimmediately follows that Link ( v ) ∼ = ∂Q k − where Q k − = P erm ◦ k − is the (simplicial)polytope polar to the permutohedron.More generally, the link Link ( τ ) of τ = { v , v , . . . , v t } is isomorphic to the join Link ( τ ) = ∂Q s − ∗ · · · ∗ ∂Q s t − where s j = | I j | is the cardinality of the set I j . Consequently, T onn n,k ( L ) is indeed amanifold. To show that it is connected, it is sufficient to show that consecutive vertices x and x + 1 are connected. Indeed, since L is reduced we obtain the relation a l + . . . + a k l k = 1 for some a , . . . , a k ∈ Z , which describes a sequence of edges connecting x and x + 1 . (cid:3) Remark 2.3.
The following geometric model for the complex
Star ( v ) can be used foran alternative proof of Proposition 2.2. Let c i ∈ R k − ( i = 1 , . . . , k ) be a spanning setof vectors such that c + c + · · · + c k = 0 . Let Z = [0 , c ] + · · · + [0 , c k ] ⊂ R k − be theMinkowski sum of line segments I j = [0 , c j ] . Then the zonotope Z admits a triangulationwhere the maximal simplices Σ π ( π ∈ S k ) , indexed by permutations, are the following Σ π = Conv { c π (1) , c π (1) + c π (2) , . . . , c π (1) + · · · + c π ( k ) } . (2.4)This triangulation of Z is isomorphic to Star ( v ) which can be proved by comparing(1.4) and (2.4).A very special case of Theorem 1.5 can be established by an elementary, direct argu-ment. Proposition 2.5.
Let L = ( l , l , l ) be a reduced, generic length vector. Then the associ-ated, -dimensional generalized Tonnetz T onn n, ( L ) is a triangulation of the -dimensionaltorus T = ( S ) . Proof:
In light of Propositions 2.1 and 2.2
T onn n, ( L ) is a connected -manifold. Assume l < l < l . It is not difficult to see that the f -vector of T = T onn n, ( L ) is f ( T ) =( n, n, n ) , hence χ ( T ) = 0 .The complex T onn n, ( L ) is orientable. Indeed, all triangles in T onn n, ( L ) fall intotwo classes. Generalized “major triads” are the triangles τ = { v , v , v } where v − v = l , v − v = l and v − v = l (for some circular order of vertices of τ ). These simplices are4ositively oriented. Negatively oriented are generalized “minor triads”, i.e. the triangles τ = { v , v , v } where v − v = l , v − v = l and v − v = l .Summarizing, T onn n, ( L ) is a connected, orientable -manifold with vanishing Eulercharacteristic, hence it must be the torus T . (cid:3) f -vector of T onn n,k ( L ) Proposition 2.6.
Suppose that L is generic and reduced and let f ( T ) = ( f , f , . . . , f k − ) be the f -vector of the generalized Tonnetz T = T onn n,k ( L ) . Then f m − = n P ( k, m ) m = n m ! m S ( k, m ) (2.7) where P ( k, m ) is number of ordered partitions partition I ⊔ . . . ⊔ I m = [ k ] and S ( k, m ) isa Stirling number of the second kind. Proof: If S is a ( m − -dimensional face of T onn n,k ( L ) then there exists x ∈ Z n and apartition [ n ] = I ⊔ . . . ⊔ I m such that S = { x, x + µ L ( I ) , x + µ L ( I ⊔ I ) , . . . , x + µ L ( I ⊔ . . . ⊔ I m − ) } , where µ L ( I ) = P i ∈ I l i . The first equality in the formula (2.7) is an immediate consequence.(Since L is generic if µ L ( I ) = µ L ( J ) then I = J .) The second follows from the equality P ( k, m ) = m ! S ( k, m ) , where S ( k, m ) is a Stirling number of the second kind. (cid:3) Proposition 2.8.
If the vector L is generic and reduced then the Euler characteristic of T onn n,k ( L ) is 0. Proof:
Let T = T onn n,k ( L ) . Its Euler characteristic is χ ( T ) = n k X m =1 ( − m +1 m ! m S ( k, m ) . Using the well-known recurrence for Stirling numbers S ( k, m ) = mS ( k − , m ) + S ( k − , m − we obtain χ ( T ) = n k X m =1 ( − m +1 m ! m mS ( k − , m ) + k X m =1 ( − m +1 m ! m S ( k − , m − ! = n k X m =1 ( − m +1 m ! S ( k − , m ) + k − X q =0 ( − q q ! S ( k − , q ) ! = n (cid:0) ( − k +1 k ! S ( k − , k ) + S ( k − , (cid:1) = 0 (cid:3) .2 Fundamental group of T onn n,k ( L ) A consequence of Theorem 1.5 is that the fundamental group of a generalized Tonnetz
T onn n,k ( L ) is free abelian of rank k − , provided the vector L is generic and reduced.Proposition 2.12 is a key step in the direction of this result. Before we commence theproof, let us make some general observation about the edge-path groupoid of the Tonnetz T onn n,k ( L ) .Each edge-path connecting vertices a = v and b = v m is of the form α = X X · · · X m where X i = −−−→ v i − v i is an oriented edge ( -simplex) in T onn n,k ( L ) .Recall (Section 1.1) that I X = I u,v ⊂ Z n is the (oriented) interval, corresponding to X = −→ uv . (With a slight abuse of language we use the same notation for the correspondingarc in S .)Let the L -type X L of X be defined as the unique non-empty subset I ⊂ [ k ] such that L ( I u,v ) = P j ∈ I l j .We say that X = −→ uv is atomic if either v = u + l i or u = v + l i for some i ∈ [ k ] . If X is positively oriented, i.e. if v = u + l i , then we call it ⊕ -atomic (similarly ⊖ -atomic if u = v + l i ).Note that the L -type of an ⊕ -atomic oriented -simplex X = −→ uv is a singleton X L = { i } (we say that X is of type i ), while the L -type of the associated ⊖ -atomic -simplex X − = −→ vu is [ k ] \ { i } .The following lemma is an immediate consequence of Definition 1.3. Lemma 2.9.
Each oriented -simplex X = −→ uv is homotopic X ∼ = Y Y · · · Y t (relative tothe end-points u and v ) to a product of ⊕ -atomic -simplices Y j . Moreover, one can readoff the L -type of X from this representation as, X L = { Y L , Y L , . . . , Y Lt } . (2.10)The following lemma (see Figure 2 for a visual proof) shows that we can rearrange andgroup ⊕ -atomic -simplices according to their type. Lemma 2.11.
An edge-path which is a product XY of two ⊕ -atomic -simplices X and Y , respectively of type i and j (where i = j ) is homotopic (rel the end points) to a product Y ′ X ′ of two ⊕ -atomic -simplices, where type ( Y ′ ) = j and type ( X ′ ) = i . Proposition 2.12.
If the length vector L is generic and reduced then the fundamentalgroup π ( T onn n,k ( L )) of the generalized Tonnetz is abelian. Proof:
Suppose that v is the chosen base point and assume that α and β are two edge-paths (loops) based at v . We are supposed to show that the edge paths αβ and βα arehomotopic (rel v ).By Lemma 2.9 we are allowed to assume that both α = Y Y · · · Y s and β = Z Z · · · Z t are products of ⊕ -atomic -simplices. 6 Y . . . Y t Figure 1: X = Y . . . Y t X YY X X XYY
Figure 2: XY = Y X
Use Lemma 2.11 to rearrange atoms in the product αβ (similarly βα ) and write it asa product αβ = A A . . . A k , where A i is the product of ⊕ -atoms of type i . (Some A i maybe empty words.)Observe that (as a consequence of Lemma 2.11) the length of the word A i is equal tothe number of type i ⊕ -simplices in the product αβ .If βα = A ′ A ′ . . . A ′ k is the corresponding regrouped presentation of βα we observe that A j = A ′ j for each j . This completes the proof of the proposition. (cid:3) T onn n,k ( L ) We already know (Proposition 2.2) that a generalized Tonnetz T = T onn n,k ( L ) is a con-nected complex. Since, according to Proposition 2.12, the fundamental group π ( T ) is7belian, it is isomorphic to the first homology group H ( T ; Z ) := Z /B , where Z and B are the corresponding groups of cycles and boundaries.When working with the homology group it is more customary to use additive notation.For example the ⊕ -atom decomposition X ∼ = Y Y · · · Y t from Lemma 2.11 can be rewrittenas the following equality (in homology), X = P ti =1 Y i .In this section the emphasis is on (co)homology so here we follow the additive notation. Definition 3.1.
The cochains θ i,j ∈ C = Hom ( C ; Z ) , where i = j k , are definedon ⊕ -atomic -simplices as follows: θ i,j ( Y ) = +1 if Y is of L -type i − if Y is of L -type j if the L -type of Y is neither i nor j .If X = −→ uv is an oriented -simplex and X ∼ = Y Y · · · Y t its ⊕ -atom decomposition fromLemma 2.9, then by definition θ i,j ( X ) = t X m =1 θ i,j ( Y m ) . (3.2) On other oriented -chains they are extended by linearity. Proposition 3.3.
The cochain θ i,j is well defined. Moreover, it is a cocycle which definesan element of H ( T ; Z ) . These classes (cocycles) are referred to as “elementary classes”defined on T = T onn n,k ( L ) . Proof:
We check that θ i,j is well defined by showing that possibly different ways to extend θ i,j lead to the same result. Essentially the only case when this happens is when we evaluate θ i,j ( − X ) = θ i,j ( X − ) , where X = −→ uv and X − = −→ vu (by formula (3.2)) expecting to obtainthe result − θ i,j ( X ) .This is indeed the case since P km =1 θ i,j ( Y m ) = 0 , where Y m is a ⊕ -atomic -simplex oftype m , for each m ∈ [ k ] . Similarly we obtain that the coboundary δθ i,j ( τ ) = θ i,j ( −−→ u u ) + θ i,j ( −−→ u u ) + θ i,j ( −−→ u u ) = 0 is zero for each (oriented) -simplex τ = { u , u , u } . (cid:3) Definition 3.4.
For each i ∈ [ n ] let ω i be the cocycle defined by ω i = P j = i θ i,j . Moreexplicitly, ω i is the unique -cocycle defined on T onn n,k ( L ) such that for each ⊕ -atomic -simplex Y ω i ( Y ) = (cid:26) k − if Y is of type i − if Y is of type j = i .These cocycles are referred to as the “canonical” cocycles defined on T onn n,k ( L ) .
8s an immediate consequence of the definition we obtain the following relation ω + ω + · · · + ω k = 0 . (3.5)The complex T onn n,k ( L ) also has naturally defined -cycles. Definition 3.6.
For i ∈ [ k ] let c i be the -cycle defined by c i := P x ∈ [ n ] E ix where E ix = −→ xy (for x ∈ [ n ] and i ∈ [ k ] ) is the ⊕ -atomic -simplex of type i with end-points x and y = x + l i . Proposition 3.7. If [ c i ] ∈ H ( T onn n,k ( L ); Z ) is the homology class of the cycle c i then [ c ] + [ c ] + · · · + [ c k ] = 0 . (3.8) Proof:
Informally, the cycle c i is the sum of all ⊕ -atomic -simplices of type i . They canbe concatenated to form d irreducible cycles of length q , where n = qd and d = ( n, l i ) . Foreach x ∈ Z n the cycle E x = E x + E x + l + · · · + E kl + ··· + l k − is trivial by Definition 1.3. Since P ki =1 c k = P x ∈ Z n E x the equality (3.8) is an immediate consequence. (cid:3) The following proposition implies that aside from (3.5) and (3.8) there are essentiallyno other relations among { ω i } i ∈ [ k ] and { [ c i ] } i ∈ [ k ] . Proposition 3.9.
Let h· , ·i be the pairing between the cohomology and homology classesand let M = [ m i,j ] k − i,j =1 be a ( k − × ( k − -matrix where m i,j = h ω i , c j i . Then det( M ) = n ( nk ) k − . Proof:
By direct calculation we have h ω i , c j i = X ν = i h θ i,ν , c j i = (cid:26) n ( k − if i = j − n if i = j (3.10)It follows that M is a circulant matrix with the associated polynomial equal to f ( x ) = n ( k − − n ( x + x + · · · + x k − ) . Recall that the determinant of the circulant matrixwith the associated polynomial f ( x ) is equal to Q k − j =1 f ( ǫ j ) , where ǫ j are solutions of theequation x k − − . From here it immediately follows that det ( M ) = n ( nk ) k − . (cid:3) H ( T onn n,k ( L )) In this section we complete the analysis and summarize our knowledge about the homologygroup H ( T ; Z ) of the generalized Tonnetz T = T onn n,k ( L ) .We already know (Section 2) that each homological -cycle has a (multiplicative) rep-resentation X = Y Y · · · Y t where Y i are ⊕ -atomic -simplices. We also write X = v Y Y · · · Y t when we want to emphasize that the initial vertex of Y (playing the roleof the base point of the loop X ) is v . 9f Y j is of L -type i j then we can also (symbolically) record this information as the word X = E i E i . . . E i t = v E i E i . . . E i t , (4.1)where E j denotes a step of length l j in the positive direction. Most of the time we cansafely remove the base point v from the notation. For example the representation X = E E E E E E = E E E describes an edge-path which begins at v , makes one step oftype , then three steps of type and finally two steps of type .Lemma 2.11 can be interpreted as a symbolic (edge-path) relation E i E j = E j E i whichsays that one can interchange two consecutive steps (as in Fig. 2) without changing thehomotopy type of the edge-path (rel end-points). From here we easily deduce the followingproposition. Proposition 4.2.
Each homological cycle X has a representation X = E p E p · · · E p k k where p , . . . , p k are non-negative integers such that p l + p l + · · · + p k l k = p n for some p > . Moreover, if Y = E p ′ E p ′ · · · E p ′ k k has a similar representation, where p ′ l + p ′ l + · · · + p ′ k l k = p ′ n , then the cycles X and Y are homologous if and only if ( p , . . . , p k ) − ( p ′ , . . . , p ′ k ) ∈ Z . (4.3) where = (1 , . . . , ∈ Z k and Z = { m | m ∈ Z } . Proof:
If the relation (4.3) is satisfied then X and Y are clearly homologous since E = E E . . . E k is a trivial cycle.Conversely, suppose that X and Y are homologous. Then, p i − p j = θ i,j ( X ) = θ i,j ( Y ) = p ′ i − p ′ j for each i < j and the relation (4.3) follows. (cid:3) As an immediate consequence we obtain the following representation of the first homol-ogy group of the generalized Tonnetz
T onn n,k ( L ) as a lattice of rank ( k − in a hyperplane H L ⊂ R k . Theorem 4.4.
Let H L = { x ∈ R k | h x, L i = x l + · · · + x k l k = 0 } be the central hyperplanein R k , orthogonal to L . Then there is an isomorphism H ( T onn n,k ( L ); Z ) −→ H L ∩ Z k (4.5) where H L ∩ Z k is a free abelian group (lattice) of rank ( k − . Proof:
Let P ⊆ R k be the closed, convex cone P = { ( x , . . . , x k ) ∈ R k | ( ∀ i ) x i > and ( ∃ x > x n = x l + · · · + x k l k } . (4.6)Let W = P ∩ Z k be the abelian semigroup of all lattice points in P . Obviously = (1 , . . . , is in W . Let Z > = { m | m ∈ Z > } ⊂ W be the subsemigroup of W generated by the10ector . Then, as a consequence of Proposition 4.2, there is an isomorphism of abeliangroups H ( T onn n,k ( L ); Z ) ∼ = W/ ( Z > ) . It is not difficult to see that the map p : P → H L , which sends x ∈ P to p ( x ) = x − x (see (4.6)), induces an isomorphism W/ ( Z > ) −→ H L ∩ Z k . For example it induces anepimorphism since for each lattice point y ∈ H L ∩ Z k the vector y + m is in W for asufficiently large positive integer m .Let λ := l l · · · l k and λ i := λ/l i . The vectors z ( j ) = ( z ( j )1 , . . . , z ( j ) k ) , where z ( j ) j = nλ j and z ( j ) i = 0 for i = j , clearly belong to W . The corresponding vectors { ˆ z ( j ) := z ( j ) − λ } ∈ H L ∩ Z k span the hyperplane H L . From here we deduce that rank( H L ∩ Z k ) = k − . (cid:3) We conclude this section by some observations about the cycles c i , introduced in Defi-nition 3.6.Suppose that l i is not relatively prime to n , say n = qd and l i = pd , where d > and p and q are relatively prime. In this case the cycle c i can be decomposed as a sum of d (irreducible) cycles, each of length q . The following proposition claims that all these cyclesdetermine the same homology class. Proposition 4.7.
All cycles of the form v E qi are homologous. E E E v uE E E E Figure 3: u E E E − = u E Proof:
Let u be another base point such that v = u + l j . Then the cycle u E j E qi E − j isclearly homologous to the cycle v E qi . On the other hand u E j E qi E − j = u E qi E j E − j = u E qi .
11y iterating this argument we see that x E qi and x + z E qi are homologous for any integer z which can be written in the form z = p n + p l + · · · + p k l k . Since the vector L is reducedwe see that z = 1 for some choice of parameters p , p , . . . , p k , which completes the proofof the proposition. (cid:3) We already know that the fundamental group of the generalized tonnetz T = T onn n,k ( L ) is free abelian of rank k − (Theorem 4.4). For the continuation of the proof of Theorem1.5, a natural step would be to show that T onn n,k ( L ) is an aspherical manifold, in thesense that its all higher homotopy groups are trivial. Note however that asphericity aloneis not sufficient to guarantee that such a manifold is covered by an euclidean space, see [6]for examples.We offer a direct proof that the universal covering e T of a generalized Tonnetz T = T onn n,k ( L ) is homeomorphic to R k − (with a lattice Λ L ⊂ R k − of rank ( k − as a groupof deck transformations) which implies that T is homeomorphic to a ( k − -dimensionaltorus. To this end we construct a discrete Abel-Jacobi map Ω : e T −→ D Λ where H = { x ∈ R k | x + · · · + x k = 0 } , Λ := H ∩ Z k is a lattice isomorphic to the permutohedral lattice A ∗ k − and D Λ is the associated Delone triangulation . It is well known that each finite simplicial complex K admits an universal covering p : e K → K in the simplicial category.By a classical construction, see Seifert-Threlfall [7], the vertices of e K are (combinatorial)homotopy classes of (simplicial) edge-paths α = α x = v α x in K , connecting the base-point v with a (variable) vertex x ∈ K .By definition a simplex in e K is a collection of edge-paths { α i x i } di =0 (or rather theirhomotopy classes) such that the end-points form a simplex τ = { x , . . . , x d } in K andfor each i = j the edge-paths α i x i and α j x j are neighbours in the sense of the followingdefinition. Definition 5.1.
Two edge paths α x and β y are neighbors if { x, y } is an edge e ∈ K andthe edge-path α x eβ − y is a homotopically trivial loop based at v . We emphasize that the homotopy always refers to combinatorial homotopy. In par-ticular two edge-paths a α b and a β b are homotopic means that one can be obtained fromthe other by a sequence of elementary modifications (moves) which replace one side of atriangle by the remaining two sides (or vice versa).12 .2 Discrete Abel-Jacobi map Following the notation from Section 4, each edge-path α = v α x , which emanates from thebase point v and ends at a vertex x , is homotopic (rel the end-points) to an edge-path ofthe form α = v E p E p · · · E p k k x = E p E p · · · E p k k (5.2)where p i > for each i ∈ [ k ] .The canonical cocycles ω i , introduced in Section 3, together define a vector valued -cocycle ω = ( ω , ω , . . . , ω k ) on the generalized tonnetz T onn n,k ( L ) which in light of (3.5)takes values in the subspace H = { y ∈ R k | y + · · · + y k = 0 } ⊂ R k . More precisely, thecocycle ω takes values in the lattice Λ = { x ∈ Z k | k X i =1 x i = 0 and all x i are in the same (mod k ) congruence class } . This is one of incarnations of the lattice of type A ∗ k − (the dual of the root lattice A k − ),which can be also described as the projection on H of the k -fold dilatation k Z k of thecubical lattice Z k , along the main diagonal D = { ( t, t, . . . , t ) } t ∈ R ⊂ R k .In the sequel we will need a more precise description of Delone cells (simplices) of thislattice. Following [4], for i = 0 , , . . . , k let [ i ] = ( j i , ( − i ) j ) = ( j, . . . , j, − i, . . . , − i ) ∈ R k ,where i + j = k and in the vector [ i ] there are i occurrences of j (respectively j occurrencesof − i ). Similarly if π ∈ S k is a permutation then [ i ] π = π ([ i ]) is obtained from [ i ] bypermuting the coordinates. Remark 5.3.
Denote a i := ω ( E i ) . Then a + a + · · · + a k = 0 and { a i } i = j is a basis of thelattice Λ for each j ∈ [ k ] . For I = { i , . . . , i r } ⊆ [ k ] let a I := P j ∈ I a i and E I = E i · · · E i r .It is easily checked that the vector [ i ] (in the traditional notation [4]) is the same as thevector a [ i ] = ω ( E · · · E i ) = ω ( E [ i ] ) . Proposition 5.4. ([1, Theorem 4.5], [4, Chapters 4 and 21])
The Delone cells of thelattice Λ ∼ = A ∗ k − are ( k − -simplices, which are related via permutations of coordinatesand translation to the canonical simplex whose vertices are [0] = [ k ] , [1] , . . . , [ k − . Remark 5.5.
In the notation of Remark 5.3 the vertices of a Delone cell are the vectors a [ j ] = [ j ] . Each element of the lattice A ∗ k − has a representation z = p a + p a + · · · + p k a k where p j ∈ Z . (This representation is unique if p + · · · + p k = 0 .) Let Star D Λ ( z ) be theunion of all Delone cells which have z as a vertex (the star of z in the Delone complex D Λ ).Then the vertices of simplices in Star D Λ ( z ) are the vectors z + a I for all subsets I ⊆ [ k ] and each simplex σ ∈ Star D Λ ( z ) is of the form σ = Conv { z + a I , z + a I , . . . , z + a I s } where I ⊂ I ⊂ · · · ⊂ I s . 13he vector valued cocycle ω can be extended to edge-paths (5.2) by the formula ω ( α ) = ω ( E p E p · · · E p k k ) = k X i =1 p i ω ( E i ) = k X i =1 p i a i ∈ Λ . (5.6)Since ω is a cocycle, ω ( a α b ) = ω ( a β b ) for each two homotopic edge-paths (with thesame end-points). The following proposition says that ω takes different values on non-homologous cycles. Proposition 5.7.
The map ω , described by the formula (5.6), induces a monomorphism b ω : H ( T onn n,k ( L ); Z ) −→ Λ . Proof:
It is sufficient to show that if α = E p E p · · · E p k k is a loop such that b ω ( α ) = 0 then α is trivial in the homology group. However, if ω i ( α ) = 0 for each i ∈ [ k ] then kθ i,j ( α ) = ω i ( α ) − ω j ( α ) = 0 for each i = j . In turn θ i,j ( α ) = 0 for each i = j which implies p = p = · · · = p k and, inlight of Proposition 4.2, α is a trivial cycle. (cid:3) Proposition 5.8.
Suppose that two edge paths α = a α x and β = a β y , which share thesame initial point a , satisfy the equality ω ( α ) = ω ( β ) . Then x = y , i.e. they have the sameend-point as well. Proof:
Suppose that α = a E p E p · · · E p k k x and β = a E q E q · · · E q k k y . By assumption ω ( α ) = k X i =1 p i a i = k X i =1 q i a i = ω ( β ) . It follows that p i − q i does not depend on i and the result follows. (cid:3) The following proposition refines Proposition 5.7. It says that a vector b ∈ Image( b ω ) cannot be “very short” (unless it is zero). Proposition 5.9.
Suppose that ξ = E p E p · · · E p k k is cycle representing a non-trivialhomology class in H ( T onn n,k ( L )) , where p i > for each i ∈ [ k ] . Then p i > for some i .Moreover, its image ω ( ξ ) ∈ Λ in the lattice Λ cannot be expressed as a difference a I − a J of vectors described in Remark 5.3, where I and J are subsets of [ k ] . Proof:
Since ξ = E E · · · E k is a trivial cycle, by factoring out from ξ the power ( ξ ) ν ,where ν := min { p j } kj =1 , we can assume that p j = 0 for some j .Since ξ is a cycle we know that p l + · · · + p k l k = p n is divisible by n . If p > then p i > for some i ∈ [ k ] and we are done. Otherwise p = 1 and p i for each i . Thisis not possible since the equality ( p l + · · · + p k l k ) − ( l + · · · + l k ) = n − n = 0 L . As an immediate consequence we see thatthe equality ω ( ξ ) = a I − a J = a I − ( a [ k ] − a J c ) = a I + a J c is not possible. (cid:3) Formula (5.6) can be used for the definition of a simplicial map
Ω : e T −→ D Λ where D Λ is the Delone triangulation of the ( k − -dimensional, affine space H ⊂ Z k , associatedto the lattice Λ .More explicitly, if e τ = { α i x i } di =0 is a simplex in e T , then Ω( e τ ) = { ω ( α x ) , . . . , ω ( α d x d ) } . Proposition 5.10.
The map
Ω : e T −→ D Λ is an isomorphism of simplicial complexes. Proof:
The map Ω is clearly an epimorphism on vertices. Indeed, if z = p a + · · · + p k a k is a vertex of D Λ then z = Ω( E p · · · E p k k ) .We continue by showing that Ω is a local isomorphism of simplicial complexes. (Inparticular Ω is a simplicial map.) Recall that [ i ] = ω ( E E · · · E i ) = a [ i ] for each i (includingthe case i = 0 when we have the empty word). Similarly [ i ] π = ω ( E π (1) E π (2) · · · E π ( i ) ) foreach permutation π ∈ S k .Let α = v α x be an edge-path describing a vertex in e T (connecting the base point v with a vertex x in T ). Then in light of Proposition 2.2 (see also Remark 2.3) the star Star e T ( α ) of this vertex is the union of k ! simplices (one for each π ∈ S k ) e τ π = { α, αE π (1) , αE π (1) E π (2) , . . . , αE π (1) E π (2) E π ( k − } . In light of (5.6) the Ω -image of this simplex is Ω( e τ π ) = { ω ( α ) , ω ( α ) + [1] π , ω ( α ) + [2] π , . . . , ω ( α ) + [ k − π } . It follows from Propositions 2.2 (Remark 2.3) and Proposition 5.4 that Ω maps bijectivelythe star Star e T ( α ) of α in e T to the star Star D λ ( ω ( α )) of ω ( α ) in the Delone triangulationof H .The map Ω is actually a covering projection. For this it is sufficient to show that foreach z ∈ Λ the open star OpStar D Λ ( z ) = Int( Star D Λ ( z )) is evenly covered by open starsin e T . More precisely we demonstrate that the inverse image Ω − ( OpStar D Λ ( z )) = [ ω ( α )= z OpStar e T ( α ) is a disjoint union of open stars in e T . Let α = a α x and β = a β y be two edge-pathsrepresenting two vertices in e T . We want to show that if ω ( v α x ) = z = ω ( v β y ) and Star e T ( α ) ∩ Star e T ( β ) = ∅ then α and β represent the same vertex in e T .Assume the opposite. As a consequence of Proposition 5.8 we know that x = y , i.e. α and β share the same end-point. It follows that ξ = α − β is a cycle in T which defines anon-trivial homology class (otherwise α and β would represent the same vertex in e T ).The intersection K = Star e T ( α ) ∩ Star e T ( β ) is a subcomplex of both stars. If thisintersection is non-empty then it contains a vertex e of both stars, hence ω ( e ) = ω ( α )+ a I = ω ( β ) + a J for some subsets I and J of [ k ] . This implies that Ω( ξ ) = ω ( ξ ) = ω ( α ) − ω ( β ) = a I − a J = a I + a J c . ξ has toosmall image ω ( ξ ) for a non-trivial homology class. Hence the cycle α − β is trivial and theedge paths α and β represent the same vertex in the universal cover e T .Finally, since e T is connected and D Λ is simply connected, we conclude that the coveringmap Ω must be an isomorphism of simplicial complexes. (cid:3) Completion of the proof of Theorem 1.5:
The isomorphism Ω is clearly Γ -equivariant,where Γ = H ( T onn n,k ( L ); Z ) acts on D Λ via the monomorphism b ω from Proposition 5.7(see also the formula (5.6)). It immediately follows that T onn n,k ( L ) isomorphic to thesimplicial complex D Λ / Λ L where Λ L := b ω (Γ) ⊂ Λ is a free abelian group of rank k − . (cid:3) The isomorphism Ω , described in Proposition 5.10, can be used for comparison of combi-natorial types of different complexes of Tonnetz type.Note that each automorphism of the Delone simplicial complex D Λ induces an isom-etry on the ambient euclidean space H ⊂ R k . If two simplicial complexes T and T ofTonnetz type are combinatorially isomorphic then their universal covers e T and e T are alsocombinatorially isomorphic.From these two observation we conclude that each Tonnetz inherits a canonical metricfrom the euclidean space H which is an invariant of its combinatorial type.1 4 7 10 19 0 3 6 95 8 11 2 51 4 7 10 1 (a) T onn , (3 , ,
105 710 0 2 47 9 11 14 6 8 101 3 510 0 (b)
T onn , (2 , , Figure 4: Combinatorially non-isomorphic complexes of Tonnetz type.For illustration the classical Tonnetz, exhibited in Figure 4, is non-isometric (and there-fore combinatorially non-isomorphic) to the “Tonnetz” shown in the same figure on theright. 1605 710 0 2 47 9 11 14 6 8 101 3 510 0 (a)
T onn , (2 , , (b) T onn , (1 , , Figure 5: Combinatorially isomorphic complexes of Tonnetz type.This can be proved by comparing the shortest closed, non-contractible geodesics (sys-toles) of both complexes. For example the systole on the left has the length , while onthe right the length is √ .The complex T onn , (2 , , and the complex T onn , (1 , , (exhibited in Figure 5)are isometric, in particular have systoles of the same length. Moreover, they are combinato-rially isomorphic. Indeed, if we cut out the parallelogram - - - from Figure 5 (b) and glueit on the opposite side, we obtain a fundamental domain of the “Tonnetz” T onn , (1 , , which, by an automorphism of the planar Delone complex D Λ , can be mapped to the Figure5 (a).Figures 4 and 5 were originally generated by lifting the triangulations from a Tonnetz T to its universal cover e T . Informally speaking, they are obtained by gradually unfoldingthe complex T in the plane until the picture becomes periodic.Results from Section 4 and 5, as summarized in the following proposition, allow us togenerate these and related pictures (for an arbitrary T onn n,k ( L ) ) directly from the inputlength vector L = ( l , l , . . . , l k ) . Proposition 6.1.
The lattice Λ L := b ω (Γ) ⊂ Λ , which appears in the isomorphism T onn n,k ( L ) ∼ = D Λ / Λ L , has the following explicit description Λ L = { p a + · · · + p k a k | ( ∀ i ) p i ∈ Z and ( ∃ p ∈ Z ) p l + · · · + p k l k = p n } where a i = ω ( E i ) ∈ Λ are the vectors introduced in Remark 5.3. Example 6.2.
Let us explicitly describe the group Λ L for T onn , (2 , , . The lattice Γ L = { ( x, y, z ) ∈ Z | x + 3 y + 7 z = 0 } as a parametric presentation Γ L = { ( x, y, z ) ∈ Z | ( ∃ r, s ∈ Z ) | x = − r − s, y = 2 r + s, z = s } . By choosing ( r, s ) = (1 , and ( r, s ) = (0 , we obtain that { ( − , , , ( − , , } is a basisfor Γ L . It follows that the corresponding generators of the lattice Λ L are b = − a + 2 a and b = − a + a + a = − a . By interpreting (in Figure 4) a and a as the vectorsconnecting the vertex (labeled by) by the neighbouring vertices and , we easily checkthat vectors b and b preserve the labeling of this lattice. They actually generate this latticesince h b , b i is a sublattice of Λ of index . It is natural to extend the definition of the tonnetz to the case n = + ∞ , interpreted asthe limit case when k is fixed and n approaches infinity. Informally, vertices are pointson a circle C with circumference while simplices are finite subsets I ⊂ C which are L -admissible in the sense of the following definition. Definition 6.3.
Let C = R / Z = [0 , / h ≃ i be a circle with induced group structureand the corresponding (circular) order. Suppose that L = ( l , l , . . . , l k ) is a collection ofpositive real numbers such that: (1) The numbers l i add up to one, l + l + · · · + l k = 1 . (2) L is irrational in the sense that p l + · · · + p k l k = 0 for each p ∈ Z k \ { } .A subset I ⊂ C is L -admissible if there exists x ∈ C and a permutation π ∈ S k such that ∆( x ; σ ) = { x, x + l σ (1) , x + l σ (1) + l σ (2) , . . . , x + l σ (1) + . . . + l σ ( k − } . (6.4) Full (irrational) tonnetz F - T onn ∞ ,k ( L ) is the ( k − -dimensional simplicial complex of all L -admissible subsets of C . The irrational tonnetz T onn ∞ ,k ( L ) is a connected componentof the full Tonnetz F - T onn ∞ ,k ( L ) . Remark 6.5.
A rotation of the circle C induces an automorphism of the full tonnetz F - T onn ∞ ,k ( L ) . Moreover, the group C acts transitively on its connected components. Itfollows that all connected components of the irrational tonnetz are isomorphic.Note that the condition (2) in Definition 6.3 guarantees that the length vector L isgeneric in the sense that numbers l i satisfy the condition (1.1). Theorem 6.6.
Irrational tonnetz
T onn ∞ ,k ( L ) is isomorphic to the Delone triangulationof the vector space R k − associated to the permutohedral lattice A ∗ k − , T onn ∞ ,k ( L ) ∼ = D Λ . roof: Many concepts introduced in Section 3, such as atomic -simplices E i , cocycles θ i,j , canonical cocycles ω i etc., preserve they meaning in the case of the infinite tonnetz T onn ∞ ,k ( L ) . An exception are canonical cycles c i whose existence is ruled out by thecondition (2) from Definition 6.3. Proposition 2.12 still holds with essentially the sameproof so the fundamental group of the infinite tonnetz is always abelian. Let us show thatit is actually a trivial group.As before each -chain has a representation X = E p E p · · · E p k k . If this is a cycle (withwinding number p ) then p l + · · · + p k l k = p . In light of the condition (2) (Definition6.3) this is possible only if p = · · · = p k = p in which case X is a boundary.The end of the proof follows closely the idea of the proof of Proposition 5.10. Theisomorphism Ω :
T onn ∞ ,k ( L ) → D Λ is again defined with the aid of formula (5.6). (cid:3) The genericity condition (1.1) plays a central role in many arguments. For illustrationa generalized tonnetz may not be a manifold without this condition, as visible from theclassification of all -dimensional (not necessarily generic) complexes of Tonnetz type, see[3, Section 6].The smallest examples ( k > of length vectors which are generic are: (1 , , for ( n, k ) = (7 , , , for ( n, k ) = (8 , , , , (2 , , for ( n, k ) = (9 , , , , (1 , , for ( n, k ) = (10 , , , , (1 , , , (1 , , , (2 , , , (2 , , for ( n, k ) = (11 , , , , (1 , , , (1 , , , (2 , , , (3 , , for ( n, k ) = (12 , , etc.It is not difficult to construct examples of families of generic vectors as illustrated by (1 , q, q , . . . , q k − ) for q > .When k is fixed, asymptotically (when n → ∞ ) almost all vectors are generic. This canbe deduced by observing that generic vectors are positive integer vectors in a simplex withvertices ne i ( i = 1 , . . . , k ) outside the union of the hyperplane arrangement H k = { H I,J } where for two disjoint, non-empty subspaces I, J ⊂ [ k ] H I,J = { x ∈ R k | X i ∈ I x i = X j ∈ J x j } . There are other generalizations of the classical Tonnetz, see for example [3, 5, 11] or [12].The authors of these papers usually put more emphasis on combinatorial and geometricaspects of the musical theory and see mathematics primarily as a useful tool. These papersdo not overlap with our exposition with an exception of [3] where the author introduced andstudied the complexes of Tonnetz type in the case k = 3 (without the genericity condition(1.1)). In particular our Proposition 2.5 is included in [3, Theorem 23]. Moreover the19uthor provides the list of all -dimensional complexes which in the non-generic case canarise as Tonnetz-type complexes. Acknowledgements:
We would like to acknowledge valuable remarks and kind sugges-tions of the anonymous referee which helped us improve the presentation of results in thepaper.
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