Computing Geometric Measures of Melodic Similarity

I4B 15 洪士強

Introduction

Most recognizable element in Western music is melody. how to computing the measures of melodic similarity

Ex : Similarity of two real numbers can computed as their difference, or the square of their difference. Convert into geometric model, use Computational Geometry to arrive a quantifiable measure of their similarity or difference.

Meaningful Measures / Quantifiable Measures One is statistical data from audience who played melodies and asked to rate their relative degrees of similarity.Give quantifiable results so pairs of melodies can be said to be “more similar” or “less similar” than others. Choosing geometric model of melody and to operate on it will give such quantifiable results.

The Orthogonal Chain as a Model for Melody By Ó Maidin in 1998, the “area between orthogonal chains” method presents geometric model of melodies and a difference measure between the two. Melodies converted into alternating chains of horizontal and vertical line segments that describe melody as pitch vs. time function. 

The Orthogonal Chain as a Model for Melody

The Orthogonal Chain as a Model for Melody Difference between two melodies can described as the magnitude of the unsigned area between two orthogonal chains. 

The Orthogonal Chain as a Model for Melody Find the mininum area is diffcult beaucse the melodies can be shifted in the x (time) and y (pitch) directions and still be considered to have the same perceptual similarity.

Computing Difference Between two Repeating Melodies

Melodies are often cyclic and have no defined beginning or end. Better model for these melodies is not a planar 2-dimensional surface but a cylindrical wrapping of such a surface.

Computing Difference Between two Repeating Melodies Pitch variable associated with the z-direction which passes through the cylinder's central axis and the time variable is replaced by the angle measure Θ.

Computing Difference Between two Repeating Melodies Finding a vertical shift Δz and an angular shift ΔΘ that minimize the area between two melodies Ma and Mb.

2003, Aloupis, Fevens, Langerman, Matsui, Mesa, Nuñez, Rappaport, and Toussaint propose 2 algos. : 1.melody shifted only in z direction.2.melody shifted in both z and Θ directions.

Algorithm : Area Minimization along zMa fixed , Mb swept vertically. Z : Difference between first horizontal segments of the melodies. Ci : lines partitions the area between the melodies into quadrangles.

Z-event : The position two horizontal segments overlap , Ai(zi)=0W : the width of the intersecting parts. zbi : the vertical coordinate of Mb in Ci.zai : the vertical coordinate of Ma in Ci.αi : the vertical offset of each horizontal segment in Mb from zb1.

Algorithm : Area Minimization along zA(z) = wi | zbi - zai |zbi = zb1 + αi , A(z) = Σ wi | zb1 - ( zai - αi ) |zai – αi= zi , Ai(zi) = 0A(z) = Σ wi | zb1 - zi |A(z) = Σ wi | zi – zb1 |

Algorithm : Area Minimization along zA(z) = wi | zi – zb1 | :This is a weighted sum of distances from zb1 to all the z-events. Minimum is the weighted univariate median of all zi and can be found in O(n) time

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Algorithm : Area Minimization along zZ-events occur at

z = {-7, -6, -5, -3, -1, 0}Weighted according to the total width of the incident horizontal segments, the following is a minimal distribution list:Zweighted = {-7, -7, -7, -7, -7, -7, -7, -7, -6, -6, -5, -5, -5, -5, -5, -3, -3, -3, -3, -3, -1, -1, -1, -1, 0, 0,

0, 0, 0, 0, 0, 0}

Algorithm : Area Minimization along zMedian of this list is the weighted univariate median of all the z's with weighting done by the respective w's, is

median ( zweighted ) = -3

In accordance with the proof , it is at this z-value that we observe the minimum sum of the areas:

min ( A(z) ) = 20

Useful Industry Applications

Perhaps the most exciting possible use of these measures is Query-by-Humming, or QBH