On Performance Similarity and Structure Segmentation

Introduction

Our objective is to find a way to quantify similarity of performances for which we have a matched score. But first, we have to answer the following questions :

What is performance similarity
How to measure siilarity in match scores?
Which similarity metric to use ?
Why measure performance similarity?
What elements affect our perception of performance similarity?

Self-Similarity Matrices

The concept of self-similarity matrices is fundamental for capturing structural properties of music recordings. Generally, one starts with a feature space L containing the elements of the feature sequence under consideration as well as with a similarity measure $f : \mathcal{L} \times \mathcal{L} \to \mathbb{R}$.

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Interval Vectors

Definition: An interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. There is an equivalence between complementary intervals within the octave for example $2^{nd}$ minor and $7^{nth}$ major intervals, etc.

Examples :

Any Major chord : $ [0, 0, 1, 1, 1, 0 ] $
Any Minor chord : $ [0, 0, 1, 1, 1, 0 ] $

Interval Vectors and Cadences

Typical Cadences
$\textrm{V}$ & - & $\textrm{I}$	authentic
$\textrm{I}$ & - & $\textrm{V}$	half
$\textrm{IV}$ & - & $\textrm{I}$	plagal
$\textrm{V}$ & - & $\textrm{VI}$	deceptive

Interval Vectors are commutative, i.e. $\textrm{V} \to \textrm{I} = \textrm{I} \to \textrm{V} $:

$ (\textrm{V/I})_{maj} = [1, 2, 2, 2, 3, 0] $
$ (\textrm{V/I})_{min} = [2, 1, 2, 3, 2, 0] $

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porridge = "blueberry"
if porridge == "blueberry":
    print("Eating...")

For every window $w_i^j$ we compute the interval vector of all notes in thye window/ Therefore, $\textrm{Int_Vec}(w_i^j)\in \mathbb{N}^6$. $$ \mathcal(W) = \sum_{i=1}^N\sum_{j=1}^\nu \textrm{Int_Vec}(w_i^j) $$ Then, let $\mathcal{X}$ be some matrix decomposition of $\mathcal{W}$. The SSM $\mathcal{S}$ of $\mathcal{X}$ is: $$ \mathcal{S} = \frac{\mathcal{X} \cdot \mathcal{X}}{| \mathcal{X} |^2} $$

Test

\begin{tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] { F & B \ & A \}; \path[-stealth] (m-1-1) edge node [above] {$\beta$} (m-1-2) (m-1-2) edge node [right] {$\rho$} (m-2-2) (m-1-1) edge node [left] {$\alpha$} (m-2-2); \end{tikzpicture}

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One Two Three

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