Geometric Representations for Transformed Pattern Matching in Music

David Meredith

cs.SDOct 6, 2026

Geometric Representations for Transformed Pattern Matching in Music

Authors: David Meredith

Organizations: Aalborg University, Aalborg, Denmark

Abstract

We review the notion of representing music using point sets and argue that such representations are better adapted than sequential representations for matching patterns in unvoiced, polyphonic music, such as keyboard music. Musical transformations such as transposition, inversion, diminution, augmentation and retrograde can be modelled by geometric transformations in pitch-time representations that combine translation with scaling parallel to and reflection in the time axis. We identify eight types of geometric pitch-time representation that use chromatic pitch, morphetic pitch, morph or chroma to represent pitch and either onset time or midtime to represent time. We illustrate how these types of representation allow us to characterise different types of musical transformation. For example, by using midtime instead of onset time, we can precisely characterise certain retrograde relationships; and by using morph and chroma pitch representations we can characterise transformations involving octave displacements and duplications. We present the concept of a transformation class and consider the three specific classes, $F_{\mathrm{2STR}}$ , $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$ . We introduce the notion of an inter-pattern transformation graph for a set of patterns, $S$ , and a transformation class, $F$ . Each vertex in such a graph represents a pattern in $S$ and there is an edge in the graph from pattern $P_1$ to pattern $P_2$ if and only if $P_1$ can be mapped onto $P_2$ by a transformation in $F$ . We show, with the aid of such graphs, that the musical relationships between the occurrences of the HAYDN theme in Ravel's Menuet sur le nom d'Haydn can be precisely described in terms of transformations in $F_{\mathrm{2STR}}$ , $F_{\mathrm{2STRMod7}}$ and $F_{\mathrm{2STRMod12}}$ within the pitch-time representations considered.

Figures & tables

Figure 1: Bars 0 to 8 of Maurice Ravel’s Menuet sur le nom d’Haydn , showing one way of partitioning the notes into four “voices”, indicated by the colours of the noteheads.

Figure 2: Bars 33 to 36 of Maurice Ravel’s Menuet sur le nom d’Haydn , showing three examples of a note in one voice being tied to a note in another.

Figure 3: Bars 52 to 54 of Maurice Ravel’s Menuet sur le nom d’Haydn , showing an occurrence of the HAYDN theme (in red) that includes notes that are from more than one notated “voice” and that overlap in time.

Figure 4: Explanatory note on page 2 of the 15 January 1910 issue of the Revue musicale mensuelle de la Société Internationale de Musique , preceding the collection of six pieces constituting the “Hommage à Joseph Haydn”. Translated into English, the note reads as follows: “These pieces have been composed on the following theme … in which the notes B, A, D have been supplied by German musical nomenclature, and the notes D and G have been obtained by applying the sequence of letters in the alphabet to the diatonic series of the musical scale.”

Figure 5: Occurrences of the HAYDN theme in Maurice Ravel’s Menuet sur le nom d’Haydn , identified by Soucy, [10] . Occurrences 1 (bars 0–2), 11 (bars 16–20), 12 (bars 19–21), 14 (bars 25–26), 22 (bars 42–44) and 28 (bars 52–53) were explicitly annotated by Ravel in the original edition of the work, published in the RSIM [ 3 ] .

Panel of Figure 6 — Figure 6: Examples of added-fourth chords (in lilac) and added-ninth chords (in blue) from other works by Maurice Ravel, suggesting that such chords in the Menuet sur le nom d’Haydn may not have been intended to be occurrences of the HAYDN theme.

Panel of Figure 7 — Figure 7: Three further occurrences of the HAYDN theme in the Menuet , that were not identified by either Ravel or Soucy: (a) a partial, inverted occurrence of the HAYDN theme in bars 21–23; (b) an inversion of the HAYDN theme at bars 32–34; and (c) a statement of the HAYDN theme at bars 52–54 that includes notes from different voices, some of which overlap in time.

Figure 8: A chromatic-pitch–onset (or CPO) representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents a distinct untied note or sequence of tied notes in the score. The $x$ -coordinate of a point gives the onset time of the corresponding note in tatums (in this case, quavers) and the $y$ –coordinate gives the chromatic pitch of the note. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 9: A chromatic-pitch–midtime or CPM representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents a distinct untied note or sequence of tied notes in the score. The $x$ -coordinate of a point gives the midtime of the corresponding note in tatums (in this case, semiquavers) and the $y$ –coordinate gives the chromatic pitch of the note. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 10: A morphetic pitch–onset (or MPO) representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents a distinct untied note or sequence of tied notes in the score. The $x$ -coordinate of a point gives the onset of the corresponding note in tatums (in this case, quavers) and the $y$ –coordinate gives the morphetic pitch of the note. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 11: A morphetic-pitch–midtime (or MPM) representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents a distinct untied note or sequence of tied notes in the score. The $x$ -coordinate of a point gives the midtime of the corresponding note in tatums (in this case, semiquavers) and the $y$ –coordinate gives the morphetic pitch of the note. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 12: A chroma–onset or CO representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents one or more tones in the score. The $x$ -coordinate of a point gives the onset time in tatums (in this case, quavers) of the tones represented by the point, and the $y$ –coordinate gives the chroma of the tones represented by the point. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 13: A chroma–midtime or CM representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents one or more tones in the score. The $x$ -coordinate of a point gives the midtime in tatums (in this case, semiquavers) of the tones represented by the point, and the $y$ –coordinate gives the chroma of the tones represented by the point. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 14: A morph–onset or MO representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents one or more tones in the score. The $x$ -coordinate of a point gives the onset time in tatums (in this case, quavers) of the tones represented by the point, and the $y$ –coordinate gives the morph of the tones represented by the point. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

Figure 15: A morph–midtime or MM representation of Ravel’s Menuet sur le nom d’Haydn . Each point represents one or more tones in the score. The $x$ -coordinate of a point gives the midtime in tatums (in this case, semiquavers) of the tones represented by the point, and the $y$ –coordinate gives the morph of the tones represented by the point. The occurrences of the HAYDN theme identified in Figs. 5 and 7 are shown, labelled with the same numbers and letters and coloured using the same colours as in these previous figures. When a point occurs in two patterns, it is drawn twice with different diameters and colours so that its presence in both patterns is visible.

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