Measuring Melodic Similarity using Transportation Distance

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Measuring Melodic Similarity using Transportation
Distance

• Ref Typke et al. 2003

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Content

• The problem
• Methodology
• Evaluation
• Indexing

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Introduction

• Measuring melodic similarity is the integral part
  of QBH (Query-by-humming) systems

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Introduction

• Represent music as weighted point set in a two
  dimensional space (time-pitch)

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Objective Measure Similarity

• Measuring the (edit) distance between two incipits

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Challenges

• Length of incipits
• Partial matching
• Transposition
• Variation forms

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Earth Movers Distance (EMD)
(Roslin Castle)
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EMD (cont)

• Signature of an incipit a set of points in E2
  where each point has a weight associated with it
• One point per note (rests are implicit)
• Details
• Time coordinate
• Pitch coordinate
• Weight assignment

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Time Coordinate

• Skip leading rests subtract the very first
  notes time coordinate from all time coordinates
• No sense of time signature
• Properly scaled so that distance is not biased

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Pitch Coordinate

• Number-line representation of musical pitch
• MIDI
• Hewletts (1992) Base-40

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Weight

• Increase the importance of it having a
  counterpart of similar weight at the same
  position in the compared melody
• Choices
• note duration (default)
• stress
• Note number making beginning notes more important

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Example
Partially matched
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EMD Revisited
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EMD as Linear Program
Solve by simplex algorithm
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EMD for Melodic Similarity

• Ground distance Euclidean distance
• Scale time coordinate (so that they are
  comparable)
• Transposition transpose one of the melodies so
  that the weighted average pitch is equal (not
  optimum but acceptable)

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Indexing

• Query the database using vantage objects (2002)
• Randomly select k vantage objects from data
  points. Compute the distance to these vantage
  objects to form a vantage space.

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Vleugels and Veltkamp 2002

• was about vector graphic matching

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V-and-V Ideas

• distance function
• measure resemblance between A1 and A2 by
  comparing respective distances from A (vantage
  object)
• Given a query object A?, select all objects
  achieving similar distance as d(A,A?)

Caution of one-way logic
d(A1, A)
A1
A
d(A2, A)
A2
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Ideas

• Problem is relieved by increasing the number of
  vantage objects
• For each object in point set, compute distance to
  vantage objects coordinate in vantage space

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Ideas

• Paper shows
• Object-space neighbor
• Vantage-space neighbor
• and

Proof depends on triangular inequality
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Back to Melodic Similarity

• EMD does not have triangular inequality property
• Proof by construction A, B, AB
• However, seems to work without distortion
• The authors devises another distance metric that
  has TI property