Using Transportation Distances for Measuring Melodic Similarity Pichaya Tappayuthpijarn Qiang Wang

Outline Introduction Melodic Similarity Measuring The Earth Mover’s Distance (EMD) The Proportional Transportation Distance (PTD) Results & Conclusion

Introduction Weighted notes are well suited to represent music. Measuring melodic similarity by comparing weighted notes. Using EMD or PTD to measure the similarity.

Melodic Similarity Measuring Weighted Notes represented by (Time, Pitch, Weight). The weights only reflect the note durations.

Properties of a Metric **For pseudo-metric, only properties i, iii and iv holds.

The Earth Mover’s Distance (EMD) measures the minimum amount of work needed to transform one into the other by moving weight.

Minimize the overall cost with constraints

W, U are the total weights of A, B respectively,

Example : a weight flow with EMD

The Proportional Transportation Distance (PTD) EMD v.s. PTD PTD is a modified version of EMD. In the case of unequal total weight, triangle inequality does not hold for the EMD. By PTD, triangle inequality holds, making database search more efficient by using indices.

The Proportional Transportation Distance for both point set, divide every point’s weight by its point set’s total weight then calculate the EMD for the resulting point sets.

Properties of PTD A pseudo-metric. Triangle inequality holds true a+b>=c

Gives clear distinction of groups. The fact that the EMD allows partial matching, while the PTD matches all notes, leads to a clear distinction of these groups by the PTD, but not the EMD.

Can recognize augmented versions of the same melody as similar.

Result & Conclusion PTD leads to the identification of about 3.9% of 100 randomly chosen search ( for r = 1). For r = 5, we need less than 1000 calculations instead of 476,000. PTD Indexing reduces the query running time.

Thank you!