2013:Discovery of Repeated Themes & Sections Results

Introduction

The task: algorithms take a piece of music as input, and output a list of patterns repeated within that piece. A pattern is defined as a set of ontime-pitch pairs that occurs twice (i.e., is repeated at least once) in a piece of music. The second, third, etc. occurrences of the pattern will likely be shifted in time and/or transposed, relative to the first occurrence. Ideally an algorithm will be able to discover all exact and inexact occurrences of a pattern within a piece, so in evaluating this task we are interested in both:

(1) to what extent an algorithm can discover one occurrence, up to time shift and transposition, and;
(2) to what extent it can find all occurrences.

The metrics establishment recall, establishment precision and establishment F1 address (1), and the metrics occurrence recall, occurrence precision, and occurrence F1 address (2).

Contribution

Existing approaches to music structure analysis in MIR tend to focus on segmentation (e.g., Weiss & Bello, 2010). The contribution of this task is to afford access to the note content itself (please see the example in Fig. 1A), requiring algorithms to do more than label time windows (e.g., the segmentations in Figs. 1B-D). For instance, a discovery algorithm applied to the piece in Fig. 1A should return a pattern corresponding to the note content of $P_{1}$ and $P_{2}$ , as well as a pattern corresponding to the note content of $Q_{1}$ . This is because $Q_{1}$ occurs again independently of the accompaniment in bars 19-22 (not shown here). The ground truth also contains nested patterns, such as $P_{1}$ in Fig. 1A being a subset of the sectional repetition $S_{1}$ , reflecting the often-hierarchical nature of musical repetition. While we recognise the appealing simplicity of linear segmentation, in the Discovery of Repeated Themes & Sections task we are demanding analysis at a greater level of detail, and have built a ground truth that contains overlapping and nested patterns.

Figure 1. Pattern discovery v segmentation. (A) Bars 1-12 of Mozart’s Piano Sonata in E-flat major K282 mvt.2, showing some ground-truth themes and repeated sections; (B-D) Three linear segmentations. Numbers below the staff in Fig. 1A and below the segmentation in Fig. 1D indicate crotchet beats, from zero for bar 1 beat 1.

For a more detailed introduction to the task, please see 2013:Discovery_of_Repeated_Themes_&_Sections.

Ground Truth and Algorithms

The ground truth, called the Johannes Kepler University Patterns Test Database (JKUPTD-Aug2013), is based on motifs and themes in Barlow and Morgenstern (1953), Schoenberg (1967), and Bruhn (1993). Repeated sections are based on those marked by the composer. These annotations are supplemented with some of our own where necessary. A Development Database (JKUPDD-Aug2013) released in March enabled participants to try out their algorithms. For each piece in the Development and Test Databases, symbolic and synthesised audio versions are crossed with monophonic and polyphonic versions, giving four versions of the task in total: symPoly, symMono, audPoly, and audMono. Algorithms submitted to the task are are shown in Table 1.

Sub code	Submission name	Abstract	Contributors
Task Version	symPoly
NF2	motives_poly	PDF	Oriol Nieto, Morwaread Farbood
DM10	SIATECSegment	PDF	David Meredith
DM9	SIATECCompressRaw	PDF	David Meredith
DM8	SIATECCompressBB	PDF	David Meredith
DM7	COSIATECSegment	PDF	David Meredith
DM6	COSIATECRaw	PDF	David Meredith
DM5	COSSIATECBB	PDF	David Meredith
Task Version	symMono
NF1	motives_mono	PDF	Oriol Nieto, Morwaread Farbood
DM10	SIATECSegment	PDF	David Meredith
DM9	SIATECCompressRaw	PDF	David Meredith
DM8	SIATECCompressBB	PDF	David Meredith
DM7	COSIATECSegment	PDF	David Meredith
DM6	COSIATECRaw	PDF	David Meredith
DM5	COSSIATECBB	PDF	David Meredith
Task Version	audPoly
NF4	motives_audio_poly	PDF	Oriol Nieto, Morwaread Farbood
Task Version	audMono
NF3	motives_audio_mono	PDF	Oriol Nieto, Morwaread Farbood

Table 1. Algorithms submitted to DRTS.

Results

For mathematical definitions of the various metrics, please see 2013:Discovery_of_Repeated_Themes_&_Sections#Evaluation_Procedure.

In Brief

To avoid a bias toward the more numerous submissions of Meredith (2013), DM10 was preselected for comparison with Nieto and Farbood's (2013) submissions, based on reported performance for the Development Database. Figure 2 shows establishment recall results on a per-pattern basis for the symbolic-polyphonic version of the task. DM10 outperforms NF2 according to Friedman's test ( $\chi ^{2}(1)=12.7368,\ p<.001$ ), suggesting the former is preferable for discovering at least one occurrence of each ground truth pattern. This result addresses point (1) from the introduction.

Figure 3 shows occurrence recall results on a per-pattern basis for the symbolic-polyphonic version of the task. Again, DM10 outperforms NF2 according to Friedman's test ( $\chi ^{2}(1)=11.2667,\ p<.001$ ), suggesting the former is preferable for retrieving all occurrences of a discovered ground truth pattern. This result addresses point (2) from the introduction.

It should be noted, however, that according to Friedman's test, algorithm NF2 is significantly faster than DM10 ( $\chi ^{2}(1)=4.000,\ p<.05$ ), suggesting the former is preferable for rapid summarisation.

Results were similar but somewhat closer for the symbolic-monophonic version of the task. According to Friedman's test ( $\chi ^{2}(1)=2.9512,\ p=.0858$ , ns), NF1 and DM10 show no significant difference for establishment recall on a per-pattern basis (please see Figure 14). This result relates to point (1) from the introduction. Figure 15 shows occurrence recall results on a per-pattern basis for the symbolic-monophonic version of the task. DM10 outperforms NF1 according to Friedman's test ( $\chi ^{2}(1)=11.6364,\ p<.001$ ), suggesting the former is preferable for retrieving all occurrences of a discovered ground truth pattern. This result relates to point (2) from the introduction.

For the audio versions of the task, algorithms NF3 (audMono) and NF4 (audPoly) were the only submissions. Nieto and Farbood (2013) are credited with being the first to define an algorithm for discovering repeated note content in a synthesised audio file that is time-locked to a symbolic representation. We include some results from these task versions, with future comparisons in mind (Figs. 26-49). Evidently the audio task was more challenging (cf. Figs. 2 and 26).

symPoly

Figure 2. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 3. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 4. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 5. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 6. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 7. Occurrence recall ( $c=.75$ ) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 8. Occurrence precision ( $c=.75$ ) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 9. Occurrence F1 ( $c=.75$ ) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 10. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 11. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 12. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 13. Log runtime of the algorithm for each piece/movement.

symMono

(Poor performance here for algorithm NF1 on piece 2 is likely due to rounding errors in the discovery phase; not anything musically interesting. The task captain tried in vain to identify a workaround.)

Figure 14. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 15. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 16. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 17. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 18. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 19. Occurrence recall ( $c=.75$ ) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 20. Occurrence precision ( $c=.75$ ) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 21. Occurrence F1 ( $c=.75$ ) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 22. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 23. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 24. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 25. Log runtime of the algorithm for each piece/movement.

audPoly

Figure 26. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 27. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 28. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 29. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 30. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 31. Occurrence recall ( $c=.75$ ) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 32. Occurrence precision ( $c=.75$ ) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 33. Occurrence F1 ( $c=.75$ ) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 34. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 35. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 36. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 37. Log runtime of the algorithm for each piece/movement.

audMono

Figure 38. Establishment recall on a per-pattern basis. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 39. Occurrence recall on a per-pattern basis. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 40. Establishment recall averaged over each piece/movement. Establishment recall answers the following question. On average, how similar is the most similar algorithm-output pattern to a ground-truth pattern prototype?

Figure 41. Establishment precision averaged over each piece/movement. Establishment precision answers the following question. On average, how similar is the most similar ground-truth pattern prototype to an algorithm-output pattern?

Figure 42. Establishment F1 averaged over each piece/movement. Establishment F1 is an average of establishment precision and establishment recall.

Figure 43. Occurrence recall ( $c=.75$ ) averaged over each piece/movement. Occurrence recall answers the following question. On average, how similar is the most similar set of algorithm-output pattern occurrences to a discovered ground-truth occurrence set?

Figure 44. Occurrence precision ( $c=.75$ ) averaged over each piece/movement. Occurrence precision answers the following question. On average, how similar is the most similar discovered ground-truth occurrence set to a set of algorithm-output pattern occurrences?

Figure 45. Occurrence F1 ( $c=.75$ ) averaged over each piece/movement. Occurrence F1 is an average of occurrence precision and occurrence recall.

Figure 46. Three-layer recall averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment recall), three-layer recall uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 47. Three-layer precision averaged over each piece/movement. Rather than using $|P\cap Q|/\max\{|P|,|Q|\}$ as a similarity measure (which is the default for establishment precision), three-layer precision uses $2|P\cap Q|/(|P|+|Q|)$ , which is a kind of F1 measure.

Figure 48. Three-layer F1 (TLF) averaged over each piece/movement. TLF is an average of three-layer precision and three-layer recall.

Figure 49. Log runtime of the algorithm for each piece/movement.

Discussion

If an occurrence of a ground-truth pattern contains forty or more notes then, according to Fig. 2, it is likely that SIATECSegment (DM10, Meredith, 2013) and motives_poly (NF2, Nieto & Farbood, 2013) will return a pattern rated as at least 75% similar. When we restrict attention to these successful discoveries and ask to what extent can the algorithms retrieve all exact and inexact occurrences, we find that SIATECSegment performs relatively well (see nonzero entries for the black line in Fig. 3), with the exception of the first patterns in pieces 1 and 2. We conclude, therefore, that the discovery of repeated sections has been addressed well by the current submissions, but that the discovery of themes and motifs requires more attention in future iterations of this task.

When assembling the ground truth, it was remarkable that most often a motif occurs as a subset of a theme or repeated section, which is not surprising given Drabkin’s (2001) definition of a motif as ‘the shortest subdivision of a theme or phrase that still maintains its identity as an idea’. One suggestion for future work is to apply a discovery algorithm to find repeated sections, and then apply the algorithm again but to the output sections only, in order to retrieve these nested and important musical motifs.

(More detailed, per-piece feedback for each submission to follow.)