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).

Existing approaches to music structure analysis in MIR tend to focus on segmentation. 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.

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 paterns; (B-D) Three linear segmentations.

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.

Code	Researcher(s)	Algorithm
Task Version: symPoly
NF2	Nieto and Farbood (2013)	motives_poly
DM10	Meredith (2013)	SIATECSegment
DM9	Meredith (2013)	SIATECCompressRaw
DM8	Meredith (2013)	SIATECCompressBB
DM7	Meredith (2013)	COSIATECSegment
DM6	Meredith (2013)	COSIATECraw
DM5	Meredith (2013)	COSIATECBB
Task Version: symMono
NF1	Nieto and Farbood (2013)	motives_mono
DM10	Meredith (2013)	SIATECSegment
DM9	Meredith (2013)	SIATECCompressRaw
DM8	Meredith (2013)	SIATECCompressBB
DM7	Meredith (2013)	COSIATECSegment
DM6	Meredith (2013)	COSIATECraw
DM5	Meredith (2013)	COSIATECBB
Task Version: audPoly
NF4	Nieto and Farbood (2013)	motives_audio_poly
Task Version: audMono
NF3	Nieto and Farbood (2013)	motives_audio_mono

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.

The results are 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 the results for these versions of the task, with a view to comparison in future years.

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

(To be written.)