An algebra of chords for a non-degenerate Tonnetz

Abstract

For an n-TET tuning system, we propose a formalism to study the transformations of k-chords over a generalized non-degenerate Tonnetz generated by a given interval structure. Root and mode are the two components of a directed chord on which the algebra operates, so that chord transformations within one chord cell or towards other cells, and paths or simple circuits over the chord network can be determined without resorting to computational algorithms or geometrical representations. The one-step transformations over the edges of the chord network associated with the $k-1$ drift operators generalize the basic operators P, R and L of the Neo-Riemmanian triadic progressions and the maximally smooth cycles of the 12-TET system to any higher dimensional space.

Keywords:

• Chord spaces
• generalized Tonnetze
• orbifold
• voice leading
• modes
• symmetry group

2010 Mathematics Subject Classifications:

• 05C12
• 11A05
• 54E35

Acknowledgements

I wish to thank an anonymous referee for very useful and constructive comments that helped to improve the manuscript and reducing its length significantly.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We use the term trichord as similar to triad, meaning a set of three notes, not necessarily in a particular segment of a scale. This meaning is extended to tetrachords, hexachords, etc.

2 Mode shifts mean rotations of the mode intervals. Shifts will refer to modes and rotations to directed chords.

3 Full rank modes are also called successive interval-arrays (Chrisman 1971) or circular interval series (Chrisman 1977), while partial modes are called interval-arrays or interval series.

4 Greek characters will be used to represent modes to easily distinguish them from chords and notes.

5 The expression $r=p+qmodn$, although written without parentheses, will mean that r is the reminder in the Euclidean division of p + q by n, otherwise we would write $r=p+(qmodn)$.

6 In general, directed chords will be notated with lowercase Latin letters, without subindices. Subindices will refer to their notes.

7 While dealing with directed chords, chords, and modes of dimension k we assume that the indices for notes $a_i$, interval modes $A_i$, and chord rotations $a^i$ are defined in the group $({\mathbb{Z}}_k,+)$.

8 In general, if ${\mu }_1$ and ${\mu }_2$ are generalized modes, ${\mu }_1\cdot {\mu }_2\ne R\left({\mu }_1\right)\cdot R\left({\mu }_2\right)$, $R\left({\mu }_1\cdot {\mu }_2\right)\ne R\left(R\right({\mu }_1\left)\cdot R\right({\mu }_2\left)\right)$, ${\mu }_1+{\mu }_2\ne R\left({\mu }_1\right)+R\left({\mu }_2\right)$, $R({\mu }_1+{\mu }_2)\ne R\left(R\right({\mu }_1)+R({\mu }_2\left)\right)$; but full modes with the operations ${\mu }_1\odot {\mu }_2\equiv R\left({\mu }_1\cdot {\mu }_2\right)$, ${\mu }_1\oplus {\mu }_2\equiv R({\mu }_1+{\mu }_2)$ are a commutative group.

9 A single shift is obtained by the permutation $\sigma =(k,1,\dots ,k-1)$ of the symmetric group ${\mathcal{S}}_k$ with $k!$ elements. Since ${\sigma }^k=1$, a shift generates a cyclic subgroup $〈\sigma 〉$ of ${\mathcal{S}}_k$ of order k, which provides a number $\left(k-1\right)!$ of cosets in the quotient group ${\mathcal{S}}_k/〈\sigma 〉$. The group ${\mathcal{S}}_k$ may be obtained from generators in several ways, e.g. from the product of transpositions in the form $(1,i)$, $1<i\le k$, or from products of σ and the transposition $(1,2)$.

10 In a general, particular families of music objects can be defined according to their behaviour with regard to some transformations. For example, a k-chord is defined as a subset of k notes, regardless their order. Therefore, a k-chord is any family of k-tuples of notes which is invariant or closed with regard to permutations. If a k-chord is defined from directed chords, i.e. a k-tuple of notes arranged clockwise direction on the octave, then a chord is a family invariant under rotations, either direct or inverse. Similarly, a mode class is defined as a family of modes which is invariant with regard to shifts. Then there is a group G that acts on a set C by transforming their elements, such as symmetry transformations, and particular families of elements in C are invariant with regard to specific transformations of G. Redfield-Pólya's theorem allows to enumerate in a general way the classes resulting from the action of the group G on C. In Fripertinger and Lackner (2015) these techniques are thoroughly studied.

11 To simplify the notation, the chord and mode components of the cases shown in graphs and figures will be without subindices.

12 Musicans call translations transpositions, however, this may be confusing since in mathematics a transposition is a permutation which exchanges two elements and keeps all others fixed.

13 According to Tymoczko (2012) the Tonnetz may be organized in other ways than a generalized torus, depending on the properties to study.

14 We use the term note for a vertex of the tonal network, while the term vertex alone refers to the chord network.

15 In an n-TET system the translations form a group isomorphic to ${\mathbb{Z}}_n$ and the inversions are a group isomorphic to ${\mathbb{Z}}_2$. Then, the operations on the root generated by ${\tau }_1$ and ι have the structure of a semidirect product ${\mathbb{Z}}_n⋊{\mathbb{Z}}_2$ with $2n$ elements $\{{\tau }_0,{\tau }_1,{\tau }_1^2,\dots ,{\tau }_1^{n-1},\iota ,{\tau }_1\iota ,{\tau }_1^2\iota ,\dots ,{\tau }_1^{n-1}\iota \}$, where ${\mathbb{Z}}_2$ acts on ${\mathbb{Z}}_n$ by inversion, since the generators of the factor groups satisfy $\iota {\tau }_1\iota ={{\tau }_1}^{-1}$. Therefore, the operations on the root of directed chords are isomorphic to the dihedral group $D_n$ (e.g. Dummit and Foote 2004), so called because it is the group of symmetries of a regular polygon.

16 Instead of referring to the components of a full mode $\mu \in M(n,k)$ from 1 to k, we refer to the mode intervals from 0 to $k-1$ in order to use their indices according to the cyclic structure of ${\mathbb{Z}}_k$.

17 The set of mode operations $\{\mathfrak{e},\mathfrak{n},\mathfrak{p},\mathfrak{r}\}$ is an abelian Klein four-group, which is the direct product $〈\mathfrak{n}〉\times 〈\mathfrak{p}〉$ of cyclic groups of order 2. However, if the modes are used strictly clockwise definite, we only have to deal with $〈\mathfrak{p}〉$.

18 Originally, the concept of parsimonious voice leading was applied when two notes of a trichord were maintained during a transformation and the third note moved by a minor second or a major second. Douthett and Steinbach (1998) gave a more flexible definition by requiring that just one note were maintained, with the other two moving by minor or major second.

19 Equivalently, ${\mathfrak{t}}_i^{-1}={\mathfrak{t}}_i$, ${\mathfrak{t}}_j{\mathfrak{t}}_i={\mathfrak{t}}_i{\mathfrak{t}}_j\iff |i-j|\ne 1$, and ${\mathfrak{t}}_{i+1}{\mathfrak{t}}_i{\mathfrak{t}}_{i+1}={\mathfrak{t}}_i{\mathfrak{t}}_{i+1}{\mathfrak{t}}_i$. The second condition only applies for k>3.

20 The transposition ${\mathfrak{t}}_{k-1}$, since $\mathfrak{s}={\mathfrak{t}}_{k-2}\cdots {\mathfrak{t}}_0$, can be expressed as ${\mathfrak{t}}_{k-1}={\mathfrak{t}}_{k-2}\cdots {\mathfrak{t}}_1{\mathfrak{t}}_0{\mathfrak{t}}_1\cdots {\mathfrak{t}}_{k-2}$.

21 In an n-TET system, the translations and inversions are made by intervals $x\in {\mathbb{Z}}_n$, being ${\tau }_x={\tau }_1^x$. Since translations are generated by ${\tau }_1$, they satisfy ${\tau }_1^n={\tau }_0$ and are isomorphic to ${\mathbb{Z}}_n$. On the other hand, an inversion by x is obtained as ${\mathfrak{I}}_x={\tau }_x{\mathfrak{I}}_0$. Since ${\mathfrak{I}}_0^2=ϵ$, then ${\mathfrak{I}}_0$ generates a group isomorphic to ${\mathbb{Z}}_2$. Therefore, such a group of operations on a chord is generated by ${\tau }_1$ and ${\mathfrak{I}}_0$. It is a semidirect product ${\mathbb{Z}}_n⋊{\mathbb{Z}}_2$ with $2n$ elements $\{{\tau }_0,{\tau }_1,{\tau }_1^2,\dots ,{\tau }_1^{n-1},{\mathfrak{I}}_0,{\tau }_1{\mathfrak{I}}_0,{\tau }_1^2{\mathfrak{I}}_0,\dots ,{\tau }_1^{n-1}{\mathfrak{I}}_0\}$ satisfying ${\mathfrak{I}}_0{\tau }_1{\mathfrak{I}}_0={{\tau }_1}^{-1}$. Similarly to the operations on the root of a directed chord, the operations on a chord are isomorphic to the dihedral group $D_n$.

22 The translation may be rewritten according to Equation (39(39) $\tau [{\mu }^j{]}_{-1}^{-i}=\tau {\left[\mathfrak{p}{\mu }^j\right]}_0^i(=\tau [A_{j+k-1}+\dots +A_{j+k-i}])$(39) ), so that it is carried out by the same mode ν.

23 Here, the translation may be also rewritten according to Equation (39(39) $\tau [{\mu }^j{]}_{-1}^{-i}=\tau {\left[\mathfrak{p}{\mu }^j\right]}_0^i(=\tau [A_{j+k-1}+\dots +A_{j+k-i}])$(39) ), so that it is carried out by the same mode $\mathfrak{p}\nu$.

24 In a non-degenerate Tonnetz, these shifts are always different, otherwise, for $p\ne q$, the shifts ${\mu }^p$ and ${\mu }^q$ may coincide.

25 This is a way to test whether two chords $\left\{x|\mu \right\}$ and $\left\{x^{\mathrm{\prime }}|{\mu }^{\mathrm{\prime }}\right\}$ belong to the same co-cycle. Since it is not sure that both chords may be expressed as directed chords with the same root (in which case we should only to compare whether both modes belong to the same class), we write both chords as generated by the same mode $\mu =[A_0,\dots ,A_{k-1}]$, e.g. $\left\{x^{\mathrm{\prime }}|{\mu }^{\mathrm{\prime }}\right\}=\left\{y|\mu \right\}$. Then, both chords are in the same co-cycle if, and only if, there exist two indices i, j so that the interval difference among the roots x and y can be expressed as a series $A_{k-j}+A_{k-j+1}+\cdots +A_{k-1}+A_0+\cdots +A_{i-2}+A_{i-1}$, i.e. as a fraction of any shift of the mode μ.

26 Let us remember that this equality has a geometric meaning. According to Equation (16(16) $m=a_0+p_0{A}_0+\cdots +p_{k-2}{A}_{k-2}modn$(16) ), the series of intervals $A_0,\dots ,A_{i-1}$ (i<k) are associated with independent directions. Similarly, the series $A_{k-j},\dots ,A_{k-1}$ (j<k) are independent intervals. A number of k consecutive terms are dependent and form a cycle, hence they can be suppressed. In that case, the remaining terms on the left-hand side are at most k−2 in number and are independent. In addition, the intervals $A_p$ and $A_q$ must be included among those terms.

27 However, if the Tonnetz is degenerate and the interval vectors $A_p$ and $A_q$ have the same length, the cells $x_0+A_p$ and $x_0+A_q$, and the cells $x_0-A_p$ and $x_0-A_q$ have respective roots with a similar value.

28 In this case we obtain the mode $-{\mu }^2$, which belongs to the co-cycle of the mode $-\mu$, but in a general case with k>3, we may reach other co-cycles which are not in such a relation with the mode μ, since there will be more than two mode classes.

29 In this case, $A_0=A$, $A_1=B$, $A_2=C$, $x_0=a$, $x_1=a+A$, and $x_2=a+A+B=a-C$.