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arxiv: 2505.08752 · v7 · pith:VICKOBXAnew · submitted 2025-05-13 · 🧮 math.CO · eess.AS· math.AG

Configurations, Tessellations and Tone Networks

Jeffrey R. Boland , Lane P. Hughston This is my paper

Pith reviewed 2026-05-22 15:13 UTC · model grok-4.3

classification 🧮 math.CO eess.ASmath.AG
keywords tonnetzLevi graphpoint-line configurationharmonic networkmusic theorytessellationcombinatorial geometry
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0 comments X

The pith

The tonnetz graph of major and minor triads can be realized exactly as twelve points and twelve lines in the plane with three points per line and three lines per point.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the usual bipartite graph for the tonnetz, linking twelve major triads to twelve minor triads with shared tones, admits a precise geometric drawing in the Euclidean plane. This drawing takes the form of a balanced point-line system in which every line contains three points and every point lies on three lines, matching a known configuration of type D222. A reader would care because the embedding supplies a concrete geometric object whose incidences and symmetries could be used directly in harmonic analysis or composition. The work thereby converts an abstract network into a planar figure that inherits properties from combinatorial configuration theory. If the realization holds, generalizations to other chord sets become available as new tone networks.

Core claim

The Levi graph of the tonnetz can be realized geometrically as a system of twelve points and twelve lines in R^2 with three points on each line and three lines through each point, forming a configuration {12_3} of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.

What carries the argument

The {12_3} configuration of Daublebsky von Sterneck type D222, which realizes the Levi graph of the major-minor triad network by providing the required incidences in the plane.

If this is right

  • Harmonic progressions can be studied through the incidence geometry of the point-line system rather than solely through the graph.
  • Generalizations of the configuration supply candidate networks for scales and chord types beyond the standard tonnetz.
  • The dual tessellation and configuration views together give two complementary representations of the same tone relations.
  • Symmetries of the configuration become available as tools for generating or transforming musical material.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The geometric model may link to other classical configurations, allowing transfer of known combinatorial results into music theory.
  • Implementation in music software could test whether the planar incidences produce perceptually coherent harmonic sequences.
  • Extensions to non-triadic sonorities or higher-dimensional embeddings remain open for systematic exploration.

Load-bearing premise

The standard bipartite graph with twelve major triads and twelve minor triads, each of degree three, faithfully encodes the musically relevant relations without extra harmonic constraints.

What would settle it

Explicit coordinate assignments for the twelve points that fail to place exactly three points on each of the twelve lines while preserving the triad adjacency relations.

Figures

Figures reproduced from arXiv: 2505.08752 by Jeffrey R. Boland, Lane P. Hughston.

Figure 1
Figure 1. Figure 1: The tonnetz as a tessellation of the Euclidean plane by regular triangles, with notes of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The tonnetz as a tessellation of the Euclidean plane by regular hexagons. Each white [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The tonnetz as a Levi graph. The tonnetz can be modelled as a regular bipartite graph [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: We illustrate a 3p-hexacycle, a 2p-hexacycle, a 4p-octacycle and a 2p-decacycle. The 3p-hexacycle ⟨CM, Cm, A♭M, G♯m, EM, Em, CM⟩ is one of the four basic hexatonic sequences. The sequence ⟨CM, Am, FM, Fm, A♭M, Cm, CM⟩ is one of the twelve overlapping ‘straight bow tie’ 2p-hexacycles. Similarly, ⟨CM, Am, AM, F♯m, G♭M, E♭m, E♭M, Cm, CM⟩ is one of the three ‘four-cornered hat’ 4p-octacycles. ⟨CM, Am, FM, Dm, … view at source ↗
Figure 5
Figure 5. Figure 5: The tonnetz admits a realization as a configuration in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The 4p-octacycle in Schubert’s Overture to Die Zauberharfe. The progression begins with the line representing Cm and continues counterclockwise to E♭M and eventually on to CM. that may at first glance seem formidable. On reflection, one sees that the pattern of relations among the 2p-hexacycles adds interest in a way that contrasts remarkably with the mutual disjointedness of the four 3p-hexacycles, hence … view at source ↗
Figure 7
Figure 7. Figure 7: Wagner, Tristan und Isolde, Act 1, opening bars. The Tristan chord [F, C♭ , E♭ , A♭ ], which in Wagner’s notation takes the form [F, B, E♭ , A♭ ], is one of the twelve odd permutations of the minor sixth chord A♭m6 = [A♭ , C♭ , E♭ , F]. It can equally be regarded as an even permutation of the half-diminished seventh F ϕ7 = [F, A♭ , C♭ , E♭ ] [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Chopin’s resolution is definitive: it brings that lengthy first section of the Ballade [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Tchaikovsky, Symphony No. 6 in B-Minor, Opus 74, fourth movement, opening bars. [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Wagner, Parsifal, excerpt from Act 3. The Parsifal leitmotif, which at Parsifal’s entry in Act 1 is a simple fanfare leading from B♭M to E♭M, has evolved at Parsifal’s Act 3 entry into B♭m6 leading to D♭m6 , which Wagner presents in the permutation {E♮ , B♭ , D♭ , A♭}, isomorphic to Tchaikovsky’s Bm6 in the form {D, G♯ , B, F♯} appearing at the outset of the Adagio Lamentoso. Did Wagner himself make use o… view at source ↗
Figure 11
Figure 11. Figure 11: The Archimedean tonnetze. Each major triad can be changed into a minor triad by [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: A simple ‘Fanfare for Kepler’ based on a hexacycle of the first Archimedean tonnetz. [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: A Tristan-genus tonnetz. The twelve minor sixth chords and twelve dominant seventh [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The geometric configuration of the Tristan-genus tonnetz is the D228 of Daublebsky [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Tonnetz analysis of the opening of the Tristan prelude. The twelve Tristan chords [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 8.6
Figure 8.6. Figure 8.6: 10, p. 302). Can we make sense of this list in the light of what we have learned? [PITH_FULL_IMAGE:figures/full_fig_p026_8_6.png] view at source ↗
Figure 17
Figure 17. Figure 17: Resolutions of the Tristan chord in the opera [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: In the lead up to Br¨unnhilde’s final aria in [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Parsifal’s entrance at the beginning of Act 3. The muted fanfare on [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
read the original abstract

The tonnetz, which is commonly represented as a tessellation of the plane by a triangular network of tones, can also be represented as a bipartite graph of degree three with twelve vertices denoting major triads and twelve vertices denoting minor triads. We show that this Levi graph can be realized geometrically as a system of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point, in a configuration $\{12_3\}$ of Daublebsky von Sterneck type D222. This tonnetz configuration, alongside various generalizations thereof, can be used as a new basis for the composition and analysis of music.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the Levi graph of the tonnetz, a bipartite graph with twelve vertices for major triads and twelve for minor triads each of degree three, can be realized geometrically in R^2 as a {12_3} configuration of Daublebsky von Sterneck type D222 (three points per line, three lines per point). It presents this as a new basis for music composition and analysis, along with various generalizations.

Significance. If the incidence isomorphism is established, the result links neo-Riemannian harmonic relations to a classical combinatorial configuration, supplying a concrete point-line model in the plane that could support new geometric analyses of tone networks.

major comments (2)
  1. [geometric realization / configuration section] The central claim requires that the specific bipartite incidences of the tonnetz (major triads connected to minor triads via shared pitch classes) coincide exactly with the incidence structure of the D222 configuration up to relabeling. The manuscript asserts this identification but does not supply an explicit incidence matrix, coordinate list, or isomorphism check; without it the geometric realization in R^2 cannot be transferred.
  2. [incidence structure discussion] Multiple non-isomorphic (12_3) configurations exist; the paper must demonstrate that the tonnetz neighbor relations match the D222 incidence table rather than another (12_3) structure before the Daublebsky von Sterneck type can be invoked.
minor comments (2)
  1. [abstract and introduction] Clarify whether the realization is a straight-line embedding or allows curved lines; the abstract states 'in R^2' but the precise embedding conditions are not stated early.
  2. [main construction] Add a small table or diagram showing at least one explicit point-line incidence from the tonnetz to illustrate the claimed correspondence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying points where additional explicit verification would strengthen the presentation. We address each major comment below and have revised the manuscript to incorporate the requested details on the incidence structure.

read point-by-point responses

  1. Referee: [geometric realization / configuration section] The central claim requires that the specific bipartite incidences of the tonnetz (major triads connected to minor triads via shared pitch classes) coincide exactly with the incidence structure of the D222 configuration up to relabeling. The manuscript asserts this identification but does not supply an explicit incidence matrix, coordinate list, or isomorphism check; without it the geometric realization in R^2 cannot be transferred.

    Authors: We agree that an explicit incidence matrix and coordinate list would make the isomorphism check fully transparent. The original manuscript establishes the correspondence by identifying the twelve major triads and twelve minor triads with the points and lines of the configuration, where incidences are determined by shared pitch classes in the tonnetz Levi graph. In the revised version we have added both an explicit incidence matrix (Table 1) and a coordinate realization in R^2, together with a direct verification that the neighbor relations match those of the D222 configuration. revision: yes

  2. Referee: [incidence structure discussion] Multiple non-isomorphic (12_3) configurations exist; the paper must demonstrate that the tonnetz neighbor relations match the D222 incidence table rather than another (12_3) structure before the Daublebsky von Sterneck type can be invoked.

    Authors: The manuscript selects the Daublebsky von Sterneck type D222 precisely because its incidence table reproduces the three common-tone relations that define the edges of the tonnetz Levi graph. To make this explicit, the revised section now includes a side-by-side comparison of the tonnetz incidence table with the standard D222 incidence table from the combinatorial literature, confirming that the two structures are identical up to relabeling and thereby distinguishing D222 from the other non-isomorphic (12_3) configurations. revision: yes

Circularity Check

0 steps flagged

No circularity: direct geometric construction from the standard tonnetz incidence structure.

full rationale

The paper's central claim is a geometric realization of the known Levi graph of the tonnetz (12 major + 12 minor triads, each of degree 3) as a {12_3} configuration of Daublebsky von Sterneck type D222. This is presented as an explicit construction in R^2 rather than a quantity defined in terms of itself, a fitted parameter renamed as a prediction, or a result forced by self-citation. No equations or steps reduce the output to the input by construction; the incidence structure is taken from the standard neo-Riemannian representation and then embedded. The derivation is therefore self-contained against external benchmarks (the well-known tonnetz graph and the independently defined D222 configuration).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the standard definition of the tonnetz as a triangular lattice and on the combinatorial definition of a {12_3} configuration; no new entities or fitted constants are introduced in the abstract.

axioms (1)
  • domain assumption The tonnetz admits a faithful representation as a bipartite graph with twelve major-triads and twelve minor-triads, each of degree three.
    This is the conventional graph-theoretic model of the tonnetz used in music theory.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tonnetz Theory, Classical Harmony, and the Combinatorial Geometry of Abstract Musical Resources

    math.CO 2026-04 unverdicted novelty 6.0

    Classical harmonies and other scales are represented as incidence configurations including bipartite {7_3} graphs for diatonic triads and Fano planes for seventh chords, extending tonnetz theory via combinatorial geometry.

Reference graph

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    Under the permutation [1,2,3,4]→[4,1,2,3] the minor sixthCm 6 = [C, E♭, G, A] becomes a half-diminished seventh,A ϕ7 = [A, C, E♭, G]. Similarly, under [1,2,3,4]→[2,3,4,1] the minor seventhCm 7 = [C, E♭, G, B♭] becomes a major sixth,E ♭M 6 = [E♭, G, B♭, C]. SinceCm 7 and E♭M 6 are the same modulo permutation, is the underlying triadCmorE ♭M? Likewise, sinc...

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    We follow the approach of Waller (1978). This fine piece of work seems to have been neglected and we have not seen Waller’s bipartite graph discussed elsewhere in the literature (but see Cohn 2012 at p. 18). There is no mention of the tonnetz in Waller’s paper, for at that point the tonnetz had not become an object of renewed interest to music theorists, ...

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    For the term ‘Tristan genus’ see Cohn (2012), pp. 159-166. Our use of the term ‘minor sixth’ for the sonority of the Tristan chord [F, B, E♭, A♭] deserves comment. Of the 24 permutations of its constituents, two are particularly relevant. The first is the minor sixthA ♭m6 = [A♭, C♭, E♭, F] or equivalentlyG ♯m6 = [G ♯, B, D♯, E♯]. The second is the half-di...

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    There are, of course, diverse notations for these basic sets. We follow Welsh (1976)

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    See Cohn (2012) at p. 211 for a definition of ‘hexatonic cycle’

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    Taken collectively, the cycles of the tonnetz can act as a basis both for the composition of music and for its analysis; thus, in our view the general scheme of pan-triadic musicology should involve all the 5409 cycles of the tonnetz

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    See Cohn (2012) in Chapter 7, ‘Dissonance’, at the section ‘Reduction to a Triadic Subset’ (pp. 142-145); NB Cohn’s remark at p. 144, ‘Wagner often treats minor triads and theirϕ 7 supersets as interchangeable in his late music.’ See also Bailey (1985) at pp. 122-125

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    For example, the map taking CM→F misL ′, the obverse ofL;CM→C ♯misP ′, the obverse ofP;CM→GmisR ′, the obverse ofR

    The single-common-tone preserving relations between triads that we have described are known as the ‘obverse’ relations of the usualL,PandRoperations. For example, the map taking CM→F misL ′, the obverse ofL;CM→C ♯misP ′, the obverse ofP;CM→GmisR ′, the obverse ofR. These terms were coined by Morris (1998). Note thatL ′ =RLP,P ′ =RPL, andR ′ =LRP(Cohn 1997...

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    A dual tessellation called the ‘Laves tiling’ associated with{4,6,12}can be constructed (Gr¨ unbaum & Shephard 2016, pp. 95-98) with the property that the vertices of the original tiling correspond to the tiles of the dual tiling and the tiles of the original tiling correspond to vertices of the dual tiling. The edges of the original tiling and its dual c...

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    Continued practice of the process will, however, result in rather dull music.’ 37

    As Piston (1985) says, ‘The purpose of this procedure [of voice leading in harmonic progres- sions] is to insure the smoothest possible connection of two chords, so that one seems to flow into the next. Continued practice of the process will, however, result in rather dull music.’ 37

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    Tomoczko 2020, pp

    The various cycles of the Archimeden tonnetz have contrasting voice-leading properties (cf. Tomoczko 2020, pp. 116-128). A counterclockwise movement through the tetracycle ⟨CM, C ♯m, F♯M, Gm, CM⟩in Figure 11 induces a transition fromCM (0) toCM (1), whereas movement through a hexacycle returns each voice to its starting pitch and a clockwise tra- jectory ...

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    This passage at the beginning of the opera gives the impression of having sprung fully formed from the composer’s mind. One is reminded of the remarkable analysis of William Blake’s Infant Sorrowcarried out by the linguist Roman Jakobson (1970), who after similar consid- eration of the intricate syntactic and morphological symmetries of Blake’s short poem...

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    Why both? As we said in Section I, it appears that Wagner and his contemporaries were operating within more than one harmonic system

    We have approached these resolutions both from a triadic perspective and from a tetrachordal perspective. Why both? As we said in Section I, it appears that Wagner and his contemporaries were operating within more than one harmonic system. If this seems paradoxical, keep in mind that here we work at the boundary of art and science. It is of the nature of ...

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    208-209, and Cohn (2012), pp

    Lewin (1996), pp. 208-209, and Cohn (2012), pp. 156-157, present analyses of this scene from a different perspective, with the sequence starting at theA 7. Our characterization of this important progression of minor sixths and dominant sevenths as belonging to a single octacycle on the Tristan-genus tonnetz of Figures 14 and 15 starting at the preceedingG...

    work page 1996