I-Shaped Tiles in the Tonnetz

An I-shaped tile proposed in this paper is used to construct an enharmonically conformed Tonnetz space that preserves more music-theoretic information related to tonality, arguably the original purpose of a Tonnetz, than the standard parallelogram tiles currently used in the literature. The resulting tiling is monohedral, monomorphic and enantiomorphic, giving music-theoretic meaning in particular to the non-horizontal tile boundaries that cross fifths-related regions.

1. 1.

   Tilings in the \(\textit{Tonnetz}\) are considered in detail by Brower [1]. Hook [14] also considers the topic in the context of exploring musical spaces. For histories of the \(\textit{Tonnetz}\), see Cohn [5, 6]. The specific version used in this paper, a \(\textit{Tonnetz}\) with tilted rather than rectangular axes, follows Tymoczko [17] and originates with Otakar Hostinský (1879).

2. 2.

   Starting in the upper right corner moving clockwise, the interior angles are: 60\(^{\circ }\), 120\(^{\circ }\), 240\(^{\circ }\), 300\(^{\circ }\), 60\(^{\circ }\), 120\(^{\circ }\), 60\(^{\circ }\), 120\(^{\circ }\), 240\(^{\circ }\), 300\(^{\circ }\), 60\(^{\circ }\), and 120\(^{\circ }\) respectively. The sum of all the interior angles is 1800\(^{\circ }\); the same sum as the interior angles of a regular dodecagon.

3. 3.

   When the interior angles are partitioned further based on their rotational symmetry (Item ), they belong to one of eight congruence classes (mod 12). The SUM classes of the consonant triads also belong to the same congruence classes (mod 12).

4. 4.

   These triangles are offset by translation from the equilateral triangles that form chords. For comparison, see pp. 28 and 93 in Hook [14] for standard examples of triangular tessellation in the \(\textit{Tonnetz}\) that represent chords. The geometry of this first-level partitioning will be used to discuss: 2/3-8/9 axis of symmetry, functional music-theoretic boundaries, and shadings within tonal regions.

5. 5.

   In the example, two angles of the G-node calisson from Fig. 2 form interior angles of the I-shaped tile given in Fig. 1: \(\measuredangle {XYZ}=60^{\circ }\) (a defining angle of the calisson) and \(\measuredangle {YZW}=120^{\circ }\).

6. 6.

   Mathematicians describe a polygonal “ear” as a triple of successive vertices \(x_{i-1}\), \(x_{i}\), \(x_{i+1}\) of a simple polygon P such that \([x_{i-1},x_{i+1}]\) is a diagonal that lies in the interior of the polygon. If the diagonal \([x_{i-1},x_{i+1}]\) is in the exterior, then it is called the polygonal “mouth.” Successive ears form tabs; and successive mouths form blanks.

7. 7.

   With this 2/3-8/9 axis of symmetry, the interior angles can be partitioned further into two equivalence classes. By convention, interior angles of simple polygons are measured clockwise, so they are always positive with the arc pointing clockwise as shown in Fig. 2. There are music-theoretic reasons based on the I-shaped tile’s rotational symmetry, to measure the interior angles in reference to the horizontal rays uniquely associated to each interior angle. Such measurements result in some negative-valued interior angles but this paper follows the standard convention.

8. 8.

   Equal tilings are defined by Grünbaum and Shephard [12] in Sect. 1.1 in which there exists a similarity transformation of the plane that maps one tiling onto the other.

9. 9.

   For further discussion, see Sect. 1.3 and Fig. 1.3.3 in Hook [14]. This space is spelling consistent, so generic intervals such as a fifth reflect ordering of letters by a fifth. B moves to F\(\sharp \), not G\(\flat \).

10. 10.

    These twelve representatives are repeated through each tile to produce a conformed spelled pitch-class space which also corresponds to a conformed chromatic space.

11. 11.

    A tiling in which there is a marking or motif on each tile. See p. 28 of Grünbaum and Shephard [12].

12. 12.

    Monomorphic tilings are characterized by tile shapes, not tile labelings.

13. 13.

    But not necessarily from a music-theoretic viewpoint. This paper argues that there is another choice of polygon: one that is concave.

14. 14.

    A result first proved by Fedorov in 1885. See Marjorie Sennechal and R. V. Galiulin [15] for a thorough analysis and summary in French and English of Federov’s 1885 work “Nachala Ucheniya o Figurakh” (An Introduction to the Theory of Figures).

15. 15.

    Recall that the diatonic collection can be partitioned into twelve equivalence classes with respect to key signatures: \(\textit{DIA} /\sim _{k}\) with \(k\in \mathbb {Z}\) representing the number of accidentals (mod 12). See the Table in Example 5–14 on p. 246 in Straus [16]. This paper modifies Straus’ index notation in order to be more mathematical with negative indices (\(k<0\)) indicating the number of flats in the key signature and positive indices (\(k>0\)) indicating the number of sharps. Straus uses the symbols \(\flat \) and \(\sharp \) in his indexing along with positive integers. See Hook [14] for a different mathematical notation underpinned by signature transformations.

16. 16.

    In the general case, this lower part will be designated \(\textit{DIA}_{k}\) with \(k\in {\mathbb {Z}}\), possibly (mod 12), indicating the number of accidentals in the key signature.

17. 17.

    In the musical example (Sect. 7), this upper/lower exchange represents the world of pure imagination (D\(\flat \) major) in contrast to an unspecified scale from \(\textit{DIA}_{0}\) that contains of the Fmaj\(^{9}\) chord, a scale of unadorned reality with no accidentals.

18. 18.

    In the general case, this upper part will be designated \(\textit{DIA}_{k-5}\) with \(k\in {\mathbb {Z}}\), possibly (mod 12), as before.

19. 19.

    These tile regions represent two collections that are maximally even, a property first described by Clough and Douthett [3]. The scales in these collections have Myhill’s property and are said to be a well-formed scale as defined by Carey and Clampitt [2].

20. 20.

    Recall that Myhill’s property describes collections in which every non-zero generic interval has two specific intervals. The property was first described by Clough and Myerson [4] and named after mathematician John Myhill.

21. 21.

    The work of Marek Žabka [18] first explored well-formedness in two dimensions in the \(\textit{Tonnetz}\). The reader should note how parallelogram-shaped tiles are used by Žabka as a simplifying structure in his theory’s use of just intonation with the theorist selecting “representatives in accordance with the tiling implied by the commas.“ (p. 2).

22. 22.

    This is an infinite subset of all \(\textit{Tonnetz}\) nodes with spelled pitch-class D.

23. 23.

    The center in Fig. 1 is a special case of this balanced scale representation discussed by Hook [14] on p. 20; where he identifies the spelled pitch-class D as 0 because D is “centrally located among the seven unsigned notes (white keys) on the line of fifths, so taking this note as the origin for spc [spelled pitch-class] numbers results in a system that exactly balances sharps and flats.” In the general case of the I-shaped tiles, accidentals are not balanced by this center, only the number of fifth-generated pitch classes are balanced.

24. 24.

    As Cohn acknowledges, this viewpoint represents a competing strain of thought, not his own. This viewpoint emerged from eighteenth-century thoroughbass treatises via Gottfried Weber (1821–1824) and Francois-Joseph Fétis (1844) where “tonal regions are primarily represented by scales rather than chords.” Cohn [6] uses this contrasting description on p. 329 to motivate his own theoretic work and discussion of Lerdahl’s tonal pitch space.

25. 25.

    A \(\textit{Tonnetz}\) of regions can be reconciled with Cohn’s \(\textit{Tonnetz}\) of triads but requires a different set of tile shapes, specifically serrated alley tiles in one of three orientations based on homotopic equivalence classes from Tymoczko [17] discussed in Coury-Hall [10, 11].

26. 26.

    These more advanced mathematical ideas are explored by this author (Coury-Hall [10, 11]) in terms of another I-shaped tiling structure in the \(\textit{Tonnetz}\) whose lower structure is the result of a transformational groupoid embedded in the schritt-wechsel group, called the Riemannian groupoid, acting on the consonant triads.

27. 27.

    Any part of the diatonic collection can be organized into either form through tile transposition/translation, or similarly through transposing/translating the twelve consecutive positions in the line-of-fifths of Fig. 4. The matching diatonic counterpart of the collection, however, will be different for each organization. For example, if \(\textit{DIA}_{0}\) is in the lower region, then \(\textit{DIA}_{-5}\) is in the upper region. If \(\textit{DIA}_{0}\) is in the upper region, then \(\textit{DIA}_{5}\) is in the lower region.

28. 28.

    See p. 196 of Carey and Clampitt [2] for the formal definition of well-formed scales in a context separate from tiling.

29. 29.

    Six calissons border this axis (2/3: D and E\(\flat \); 8/9: A\(\flat \) and A) and poles (F and C). The axis-bordering edges of the calissons form a dotted line connecting the boundary of the convex-to-convex fittings (tab-to-tab) of the tiles immediately above and below and can be extended indefinitely.

30. 30.

    The upper extensions of world of “pure imagination" have been reduced at this point from an earlier A\(\flat ^{13}\) to form a less ethereal A\(\flat ^{9}\) in order to match the Fmaj\(^{9}\) that follows. This reduction in the upper extensions places the harmony at the level of his guests’ (lesser) imaginations, referenced as “your imagination" in the song, which is an octave lower than the setting for Wonka’s own “pure imagination.".

31. 31.

    Recall that Willy Wonka is testing the character of the young golden ticket winners unbeknownst to them (or the audience) until the end of the story, so a deceptive cadence is certainly appropriate in the text-setting. The choice of Fmaj\(^{9}\), however, is more subtle and engaging in its deception.

This author wishes to thank the anonymous reviewers for many insightful comments.

Authors and Affiliations

1. New York City, NY, USA

   M. A. Coury-Hall

Corresponding author

Correspondence to M. A. Coury-Hall .

Editors and Affiliations

1. Escola Superior de Música de Catalunya, Barcelona, Spain

   Thomas Noll

2. Georgia State University, Atlanta, Georgia

   Mariana Montiel

3. Universidad Politécnica de Madrid, Madrid, Spain

   Francisco Gómez

4. University of Coimbra, Coimbra, Portugal

   Omar Costa Hamido

5. Complutense University of Madrid, Madrid, Spain

   José Luis Besada

6. University of Coimbra, Coimbra, Portugal

   José Oliveira Martins

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