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Figure 15: A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and morphetic pitch dimensions.

Figure 15 A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and morphetic pitch dimensions.

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Figure 1: Opening measures from Beethoven’s Quartet, Op.18. No.1 (after Lerdahl and Jackendoff, 1983, p. 51). the visual field into objects, parts of objects, and parts of parts of objects’ (Lerdahl and Jackendoff, 1983, p. 36).
Figure 1: Opening measures from Beethoven’s Quartet, Op.18. No.1 (after Lerdahl and Jackendoff, 1983, p. 51). the visual field into objects, parts of objects, and parts of parts of objects’ (Lerdahl and Jackendoff, 1983, p. 36).
Figure 2: Measures 1-6 of Rachmaninoff’s Prelude in Ci minor, Op.3 No.2.
Figure 2: Measures 1-6 of Rachmaninoff’s Prelude in Ci minor, Op.3 No.2.
In section 6 below, we shall properly formalise the concept of a pattern. However, for now let us say that an object may be called a pattern if and only if it is a set of notes within a piece of music. In general, a repetition involves two patterns. If p; and p2 are the two patterns involved in a repetition then we can fully specify the repetition by specifying p; and a transformation ¢t that transforms p1 into pg. For example, in the case of the repetition indicated in Figure 2, p; would be the pattern consisting of the circled notes and t would be a translation of 7 crotchets along the time axis followed by a tranposition up a minor ninth. In this case, the durations and relative onset times of the notes in p, are preserved in po so this is an example of exact transposed repetition. Figure 3: Measures 1—4 of Barber’s Sonata for Piano, Op.26, showing 5 occurrences of the rising bass pattern Al—A5.
In section 6 below, we shall properly formalise the concept of a pattern. However, for now let us say that an object may be called a pattern if and only if it is a set of notes within a piece of music. In general, a repetition involves two patterns. If p; and p2 are the two patterns involved in a repetition then we can fully specify the repetition by specifying p; and a transformation ¢t that transforms p1 into pg. For example, in the case of the repetition indicated in Figure 2, p; would be the pattern consisting of the circled notes and t would be a translation of 7 crotchets along the time axis followed by a tranposition up a minor ninth. In this case, the durations and relative onset times of the notes in p, are preserved in po so this is an example of exact transposed repetition. Figure 3: Measures 1—4 of Barber’s Sonata for Piano, Op.26, showing 5 occurrences of the rising bass pattern Al—A5.
Figure 4: Measures 14-16 of the Fugue in C major from Book 1 of J.S. Bach’s Das Wohltemperirte Klavier. nearly a quarter of all the notes in the movement. The patterns involved in a perceptually significant repetition may each contain only a few notes or a substantial fraction of all the notes in a piece. If the repeated pattern is just a very small motif, consisting of no more than a few notes, then usually, for such patterns to be perceived as being significant structural units, they have to occur frequently over the course of a piece. Figure 3 shows an example of this occurring in the opening measures of Barber’s Sonata for Piano, Op.26, where a rising left-hand motif in octaves (A1) occurs 5 times within the first 4 bars, both at pitch (Al—A3) and transposed up a perfect fourth (A4 and A5). On the other hand, in a sonata-form movement, it is typical for the whole exposition to be stated twice (as happens, for example, in most of Beethoven’s piano sonatas) and then repeated again in a modified form as the recapitulation. The exposition of a sonata-form movement may contain a significant fraction of all the notes in the movement. In the first movement of Beethoven’s Sonata in F minor, Op.2, No.1, for example, the exposition contains nearly a quarter of all the notes in the movement.
Figure 4: Measures 14-16 of the Fugue in C major from Book 1 of J.S. Bach’s Das Wohltemperirte Klavier. nearly a quarter of all the notes in the movement. The patterns involved in a perceptually significant repetition may each contain only a few notes or a substantial fraction of all the notes in a piece. If the repeated pattern is just a very small motif, consisting of no more than a few notes, then usually, for such patterns to be perceived as being significant structural units, they have to occur frequently over the course of a piece. Figure 3 shows an example of this occurring in the opening measures of Barber’s Sonata for Piano, Op.26, where a rising left-hand motif in octaves (A1) occurs 5 times within the first 4 bars, both at pitch (Al—A3) and transposed up a perfect fourth (A4 and A5). On the other hand, in a sonata-form movement, it is typical for the whole exposition to be stated twice (as happens, for example, in most of Beethoven’s piano sonatas) and then repeated again in a modified form as the recapitulation. The exposition of a sonata-form movement may contain a significant fraction of all the notes in the movement. In the first movement of Beethoven’s Sonata in F minor, Op.2, No.1, for example, the exposition contains nearly a quarter of all the notes in the movement.
Figure 5: Measures 16-19 of the first movement of Mozart’s Symphony in G minor, K.550.
Figure 5: Measures 16-19 of the first movement of Mozart’s Symphony in G minor, K.550.
Figure 7: Measures 1-5 of Contrapunctus VI from Die Kunst der Fuge by J.S. Bach.
Figure 7: Measures 1-5 of Contrapunctus VI from Die Kunst der Fuge by J.S. Bach.
Figure 9: Measures 1—4 of Barber’s Sonata for Piano, Op. 26, showing one reasonably natural way of partitioning the notes into monophonic voices. However, in nineteenth and twentieth century piano music, the voicing is often ambiguous and not explicitly represented in the score. If one wants a ‘neutral’ representation of a score that does not represent just one particular encoder’s interpretation of the score, then one should omit information that is not explicitly represented in the score. A neutral representation of a keyboard work wil therefore often contain no information about voicing. A symbolic representation of what a listener hears when he or she listens to a performance of a work for piano or harpsichord must also not contain any explicit voicing information. Such a representation might be used, for example, as input to a repetition-discovery algorithm forming one part of a computational model of an expert listener. Many music databases contain music in the form of MIDI files but a MIDI representation of a polyphonic keyboard work is often structured so that all the notes are in a single channel. Such a file therefore contains no voicing information. A repetition-discovery algorithm should therefore be capable of finding perceptually significant repeated patterns in a musical representation that does not contain voicing information. Figure 9 shows one reasonably natural way of partitioning the notes in the passage in Figure 3 represent just one particular encoder’s interpretation of the score, then one should omit information
Figure 9: Measures 1—4 of Barber’s Sonata for Piano, Op. 26, showing one reasonably natural way of partitioning the notes into monophonic voices. However, in nineteenth and twentieth century piano music, the voicing is often ambiguous and not explicitly represented in the score. If one wants a ‘neutral’ representation of a score that does not represent just one particular encoder’s interpretation of the score, then one should omit information that is not explicitly represented in the score. A neutral representation of a keyboard work wil therefore often contain no information about voicing. A symbolic representation of what a listener hears when he or she listens to a performance of a work for piano or harpsichord must also not contain any explicit voicing information. Such a representation might be used, for example, as input to a repetition-discovery algorithm forming one part of a computational model of an expert listener. Many music databases contain music in the form of MIDI files but a MIDI representation of a polyphonic keyboard work is often structured so that all the notes are in a single channel. Such a file therefore contains no voicing information. A repetition-discovery algorithm should therefore be capable of finding perceptually significant repeated patterns in a musical representation that does not contain voicing information. Figure 9 shows one reasonably natural way of partitioning the notes in the passage in Figure 3 represent just one particular encoder’s interpretation of the score, then one should omit information
Figure 10: Measures 1-4 of Barber’s Sonata for Piano, Op. 26, showing one plausible way of partitioning the notes into polyphonic voices. sure of discovering all such repetitions, the user would therefore have to use a multiple representatio in which every possible partitioning of the work into polyphonic ‘streams’ were represented. Thi would be completely impractical for a work with many parts such as an orchestral work.
Figure 10: Measures 1-4 of Barber’s Sonata for Piano, Op. 26, showing one plausible way of partitioning the notes into polyphonic voices. sure of discovering all such repetitions, the user would therefore have to use a multiple representatio in which every possible partitioning of the work into polyphonic ‘streams’ were represented. Thi would be completely impractical for a work with many parts such as an orchestral work.
7In SIA and SIATEC there are no such bounds—the patterns discovered may be of any size. Sa ER NN ENTE, STARE RNR mre we | SUA Abe cmns Evenrnemengeeme cee SOK PARE Another approach to discovering approximately repeated patterns in music is presented by Ro und (1999). Like Mongeau and Sankoff (1990), Rolland uses an extended toolbox of editing ope tions. In his method, the length of the patterns involved in the discovered repetitions is limite y setting a minimal and maximal length for them.” Then, all the patterns (factors) falling in tl llowed range are extracted from the musical string. Having created a node for a pattern, it ompared against other patterns (whose length is close enough to that of the considered pattern f their similarity exceeds a given threshold value, an arc between the two nodes corresponding t hese patterns is created and the weight of this arc is set according to their similarity. In this wa he related patterns form stars in a similarity graph. Finally, the patterns are sorted by their prom ence: for every pattern in the similarity graph, the weights of the arcs by which they are connecte o other nodes are accumulated. Then, the patterns are listed in descending order according to tl ccumulations. Rolland claims an overall time complexity of O(n”). However, if the algorithm equired to find patterns of any size this time complexity rises to (at least) O(n*), making it le ficient in the worst case than SIATEC. Also, unlike SIATEC and STA, Rolland’s algorithm is current: mited to the discovery of patterns in monophonic sources.
7In SIA and SIATEC there are no such bounds—the patterns discovered may be of any size. Sa ER NN ENTE, STARE RNR mre we | SUA Abe cmns Evenrnemengeeme cee SOK PARE Another approach to discovering approximately repeated patterns in music is presented by Ro und (1999). Like Mongeau and Sankoff (1990), Rolland uses an extended toolbox of editing ope tions. In his method, the length of the patterns involved in the discovered repetitions is limite y setting a minimal and maximal length for them.” Then, all the patterns (factors) falling in tl llowed range are extracted from the musical string. Having created a node for a pattern, it ompared against other patterns (whose length is close enough to that of the considered pattern f their similarity exceeds a given threshold value, an arc between the two nodes corresponding t hese patterns is created and the weight of this arc is set according to their similarity. In this wa he related patterns form stars in a similarity graph. Finally, the patterns are sorted by their prom ence: for every pattern in the similarity graph, the weights of the arcs by which they are connecte o other nodes are accumulated. Then, the patterns are listed in descending order according to tl ccumulations. Rolland claims an overall time complexity of O(n”). However, if the algorithm equired to find patterns of any size this time complexity rises to (at least) O(n*), making it le ficient in the worst case than SIATEC. Also, unlike SIATEC and STA, Rolland’s algorithm is current: mited to the discovery of patterns in monophonic sources.
Figure 13: 5-dimensional dataset representation of the music in Figure 12.
Figure 13: 5-dimensional dataset representation of the music in Figure 12.
Figure 12: The first two measures of the Prelude in C minor from Book 2 of J.S. Bach’s Das Wohltemperirte Klavier, BWV 871/1.
Figure 12: The first two measures of the Prelude in C minor from Book 2 of J.S. Bach’s Das Wohltemperirte Klavier, BWV 871/1.
Figure 14: A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and chromatic pitch dimensions.
Figure 14: A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and chromatic pitch dimensions.
Figure 16: A simple two-dimensional dataset containing 6 datapoints.
Figure 16: A simple two-dimensional dataset containing 6 datapoints.
Figure 18: A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and chromatic pitch dimensions.
Figure 18: A two-dimensional orthogonal projection of the dataset in Figure 13 onto the plane defined by the onset time and chromatic pitch dimensions.
Figure 20: The first two measures of the Prelude in C minor from Book 2 of J.S. Bach’s Das Wohltemperirte Klavier, BWV 871/1. Figure 19: The set S(D) for the dataset in Figure 16.
Figure 20: The first two measures of the Prelude in C minor from Book 2 of J.S. Bach’s Das Wohltemperirte Klavier, BWV 871/1. Figure 19: The set S(D) for the dataset in Figure 16.
Figure 21: A two-dimensional dataset representation of the music in Figure 20 showing the onset time and morphetic pitch of each note.
Figure 21: A two-dimensional dataset representation of the music in Figure 20 showing the onset time and morphetic pitch of each note.
Table 1: A vector table showing the set V for the dataset shown in Figure 16. We call a table like the one in Table 1 a vector table. Each element in this table is an ordered pair (v,i) where i gives the number of the column in which the element occurs and v is the vector from the datapoint at the head of the column in which the element occurs to the datapoint at the head of the row in which the element occurs. For a k-dimensional dataset of size n, this third step of n(ne =1) the algorithm involves computing vector subtractions. It can be accomplished in a worst-case running time of O(kn?). SIA: Step 2 — Computing the vector table The second step in SIA is to compute the set
Table 1: A vector table showing the set V for the dataset shown in Figure 16. We call a table like the one in Table 1 a vector table. Each element in this table is an ordered pair (v,i) where i gives the number of the column in which the element occurs and v is the vector from the datapoint at the head of the column in which the element occurs to the datapoint at the head of the row in which the element occurs. For a k-dimensional dataset of size n, this third step of n(ne =1) the algorithm involves computing vector subtractions. It can be accomplished in a worst-case running time of O(kn?). SIA: Step 2 — Computing the vector table The second step in SIA is to compute the set
Figure 23: Reading the second column from top to bottom gives V for the dataset shown in Figure 16 The third column gives D[V{i, 2]] for each element V/[i] in the second column. The right-hand side of the third column shows how the non-empty MTPs may be derived directly from V. printed next to it in the third column in Figure 23. For example, V[1] = ((0,1) ,3) in Figure 23, so V [1,2] = 3 and D[V/1, 2]] = (2,1), the third datapoint in the ordered set D for the dataset shown in Figure 16.
Figure 23: Reading the second column from top to bottom gives V for the dataset shown in Figure 16 The third column gives D[V{i, 2]] for each element V/[i] in the second column. The right-hand side of the third column shows how the non-empty MTPs may be derived directly from V. printed next to it in the third column in Figure 23. For example, V[1] = ((0,1) ,3) in Figure 23, so V [1,2] = 3 and D[V/1, 2]] = (2,1), the third datapoint in the ordered set D for the dataset shown in Figure 16.
Table 2: A vector table showing W for the dataset shown in Figure 16.
Table 2: A vector table showing W for the dataset shown in Figure 16.
Figure 26: An algorithm for computing X using V and D. X can be efficiently computed directly from V and D using the algorithm in Figure 26 which exploits the fact that the MTPs in V are already sorted. In Figure 26, the set X is actually represented as an ordered set X. When the algorithm in Figure 26 has terminated, the ordered set X only contains all the elements of X (with no duplicates). In Figure 26, () denotes the empty ordered set and the symbol @ denotes concatenation: if A and B are two ordered sets such that A = (a@1,...@m) and B = (b1,...bn) then
Figure 26: An algorithm for computing X using V and D. X can be efficiently computed directly from V and D using the algorithm in Figure 26 which exploits the fact that the MTPs in V are already sorted. In Figure 26, the set X is actually represented as an ordered set X. When the algorithm in Figure 26 has terminated, the ordered set X only contains all the elements of X (with no duplicates). In Figure 26, () denotes the empty ordered set and the symbol @ denotes concatenation: if A and B are two ordered sets such that A = (a@1,...@m) and B = (b1,...bn) then
Figure 29: An algorithm for printing out J’(D).
Figure 29: An algorithm for printing out J’(D).
Notes:
Notes:
Figure 34: Graphs of running time against input size for SIATEC S is the complete set of datasets used in the experiments, as defined above.
Figure 34: Graphs of running time against input size for SIATEC S is the complete set of datasets used in the experiments, as defined above.
Figure 36: Example of a bounding box in 2 dimensions. We are currently in the early stages of developing algorithms for isolating TECs that contai ‘theme-like” or “motif-like” patterns, since it seems likely that these are the patterns that one wow vish to include in the index of a music database. Most of the perceptually significant patterns show n the musical examples in this paper are theme-like or motif-like (see Figures 1, 3, 5, 6, 11, 12, 1 10 and 35). Each of these patterns can be unambiguously specified by drawing a horizontal rectang round it in the score. In other words, each of these theme-like or motif-like patterns contains 2 he notes that occur within the smallest horizontal rectangle that contains the pattern, whereas yatently non-theme-like pattern like the ones in Figure 2 cannot be specified in this way. In geometr he smallest hypercuboid with edges parallel to the co-ordinate axes that contains a set of points alled the bounding box of that set of points (Cormen et al., 1990, p. 889). In two dimensions, tl pounding box of a set of points S is simply the smallest rectangle with edges parallel to the x-ax nd y-axis that contains S' (see Figure 36). We say that a pattern P in a dataset is bounding-bi
Figure 36: Example of a bounding box in 2 dimensions. We are currently in the early stages of developing algorithms for isolating TECs that contai ‘theme-like” or “motif-like” patterns, since it seems likely that these are the patterns that one wow vish to include in the index of a music database. Most of the perceptually significant patterns show n the musical examples in this paper are theme-like or motif-like (see Figures 1, 3, 5, 6, 11, 12, 1 10 and 35). Each of these patterns can be unambiguously specified by drawing a horizontal rectang round it in the score. In other words, each of these theme-like or motif-like patterns contains 2 he notes that occur within the smallest horizontal rectangle that contains the pattern, whereas yatently non-theme-like pattern like the ones in Figure 2 cannot be specified in this way. In geometr he smallest hypercuboid with edges parallel to the co-ordinate axes that contains a set of points alled the bounding box of that set of points (Cormen et al., 1990, p. 889). In two dimensions, tl pounding box of a set of points S is simply the smallest rectangle with edges parallel to the x-ax nd y-axis that contains S' (see Figure 36). We say that a pattern P in a dataset is bounding-bi
Figure 37: Example of a convex hull in 2 dimensions. polygon P for which each point in Q is either on the boundary of P or in its interior (Cormen et al., 1990, p. 898). More generally, the convex hull of a subset A of a real vector space is the intersection of all convex sets containing A (Borowski and Borwein, 1989, p. 123, s.v. convex hull). Figure 37 shows the convex hull of a set of points in two dimensions. We say that a pattern P in a dataset is conver hull compact if and only if it contains all and only those datapoints in the dataset that occur within the convex hull of P. Cases like pattern A in Figure 18 suggest that it might be interesting to develop an algorithm that searches the MTP TECs generated by SIATEC and selects all and only those TECs that contain convex-hull compact patterns. Alternatively one could isolate those TECs that contain temporally compact patterns (see page 7 above).
Figure 37: Example of a convex hull in 2 dimensions. polygon P for which each point in Q is either on the boundary of P or in its interior (Cormen et al., 1990, p. 898). More generally, the convex hull of a subset A of a real vector space is the intersection of all convex sets containing A (Borowski and Borwein, 1989, p. 123, s.v. convex hull). Figure 37 shows the convex hull of a set of points in two dimensions. We say that a pattern P in a dataset is conver hull compact if and only if it contains all and only those datapoints in the dataset that occur within the convex hull of P. Cases like pattern A in Figure 18 suggest that it might be interesting to develop an algorithm that searches the MTP TECs generated by SIATEC and selects all and only those TECs that contain convex-hull compact patterns. Alternatively one could isolate those TECs that contain temporally compact patterns (see page 7 above).
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