ilyas_a graph theoretic model for music information retrieval

8/22/2019 Ilyas_A Graph Theoretic Model for Music Information Retrieval

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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing

MusicalThemes UsingGraphs

SimilarityFunction

Further Work

A Graph Theoretic Model for Music

Information Retrieval

Amna Ilyas

Bates College

April 6, 2013
http://find/http://goback/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

Presentation Outline

1 Representing Musical Themes Using Graphs

2 Similarity Function

3 Further Work


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model
http://find/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model

Proposed by Alberto Pinto and Goffredo Haus in A NovelXML Music Information Retrieval Method Using GraphInvariants (2007)
http://find/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model

Proposed by Alberto Pinto and Goffredo Haus in A NovelXML Music Information Retrieval Method Using GraphInvariants (2007)Converts a musical theme in a query (audio or scorefragment) into its representative graph.
http://find/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model


Representative graph is compared with the graphs ofmusical themes in a database.
http://find/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model


Representative graph is compared with the graphs ofmusical themes in a database.Comparison is done after filtering and then using asimilarity function.
http://find/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

The Graph Model


Representative graph is compared with the graphs ofmusical themes in a database.Comparison is done after filtering and then using asimilarity function.

The result then is a list of graphs containing the matchingtheme and its variations by the addition of new notes betweenpairs of notes of the query graph.
http://goforward/http://find/http://goback/


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A Graph

TheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

A Musical Theme and its Representative Graph
http://find/


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A Graph

TheoreticModel forMusic

InformationRetrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


A melodic theme is a musical phrase that is comprised of one

complete recognizable melody.


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




Each note becomes a vertex


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




Each note becomes a vertexDirected edges between the vertices denote the progressionof the melody from one note (vertex) to another.
http://find/


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




Each note becomes a vertexDirected edges between the vertices denote the progressionof the melody from one note (vertex) to another.The last note is sent back to the first note using a directed

edge.
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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work

Themes and Graphs
http://find/


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work

Themes and Graphs

These graphs that are built from musical themes are Eulerian.


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work

Themes and Graphs


A trail in a graph is an alternating sequence of vertices anddirected edges, with no repeated edges (vertices may berepeated).


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work

Themes and Graphs


A trail in a graph is an alternating sequence of vertices anddirected edges, with no repeated edges (vertices may berepeated).

A trail with the same starting and ending vertex is called a

circuit.


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work

Trails, Circuits and Eulerian Graphs

For instance, one trail in G, that represents a theme from

Tchaikovskys 6th Symphony, Pathetique is
http://find/


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work



http://find/


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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




de3c#e2de7b
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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




de3c#e2de7b

One circuit in the graph G is
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A Graph



Amna Ilyas

Representing


SimilarityFunction

Further Work




de3c#e2de7b

One circuit in the graph G is

de3c#e2de7be8d
http://find/


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A GraphTheoreticModel for

MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

An Eulerian graph is a graph that contains an Eulerian circuiti.e. a circuit that uses each edge of the graph exactly once.
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


The above graph is Eulerian because it contains the Euleriancircuit


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work



be1c#e2de3c#e4be5ee6de7be8de9c#e10b
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work




All graphs that are constructed from a musical theme areEulerian.

Th h i t ti h
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work

Themes having same representative graphs

It is possible for two different themes to have the same

representative graph.

Th h i t ti h
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


It is possible for two different themes to have the samerepresentative graph. For instance,

This theme has the same representative graph as G.

Th h i t ti h
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


It is possible for two different themes to have the samerepresentative graph. For instance,

This theme has the same representative graph as G.

This is just a different Eulerian circuit of that same graph.

de7be8de3c#e2de9c#e10be1c#e4be5ee6d

The Eulerian circuit corresponding to the Pathetique theme is


Musical Themes and Graphs
http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


However, not every Eulerian circuit of a graph gives a different

musical theme. For instance,

http://find/


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MusicInformation

Retrieval

Amna Ilyas

Representing


SimilarityFunction

Further Work


However, not every Eulerian circuit of a graph gives a different

musical theme. For instance,

These two different Eulerian circuits of this graph correspondto this same melodic theme

de7be8de3c#e2de9c#e10be1c#e4be5ee6d

de

7be

8de

9c#e

2de

3c#e

4be

1c#e

10be

5ee

6d



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A GraphTheoretic

Model forMusic


Amna Ilyas

Representing


SimilarityFunction

Further Work




3 Further Work

Transitive Closure
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A GraphTheoretic

Model forMusic


Amna Ilyas

Representing


SimilarityFunction

Further Work

Transitive Closure

The transitive closure of order i of graph P is the graph Pi

such that V(Pi

) = V(P) and E(Pi

) contains all edges ofPand edges joining any pair of vertices which have a trail of iedges between them.

Transitive Closure
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

Representing


SimilarityFunction

Further Work

Transitive Closure


such that V(Pi

) = V(P) and E(Pi


We will ignore any trails that include loops because loops onlymean that we are staying on the same note.

Transitive Closure
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusicalThemes UsingGraphs

SimilarityFunction

Further Work

Transitive Closure


such that V(Pi

) = V(P) and E(Pi


We will ignore any trails that include loops because loops onlymean that we are staying on the same note.

For example, the graph on the right below represents P3 of thegraph P given on the left

Difference Graph
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

Difference Graph

Difference Graph, H\G is obtained by deleting those edges

from H that are in common with G.

Difference Graph
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

Difference Graph

Difference Graph, H\G is obtained by deleting those edges

from H that are in common with G.

Similarity Function
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

Similarity Function

IfV(G) = V(H) and |H\G| = 0 then it means that G and Hare exactly the same.

Similarity Function
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

S a ty u ct o

IfV(G) = V(H) and |H\G| = 0 then it means that G and Hare exactly the same. So such an H should appear at the top of

our list of graphs in the result.

Similarity Function
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

y



IfV(G) V(H) then the similarity function is

Similarity Function
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

y



IfV(G) V(H) then the similarity function is

(G,H) = |G||G\(H\G)2|

|G| +|G||G\(H\G)3|

|G| +|G||G\(H\G)4|

|G|
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

Consider this theme and its graph A
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

A variation of this theme and its graph B is


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A GraphTheoretic

Model forMusic


Amna Ilyas


SimilarityFunction

Further Work

(B\A) is obtained by deleting those edges from B that are incommon with A
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

(B\A) is obtained by deleting those edges from B that are incommon with A
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

(B\A)2 is the transitive closure of order 2 on the graph (B\A)
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work



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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


A\(B\A)2
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

|A\(B\A)2| = 8
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

Similarly, (B\A)3 is the transitive closure of order 3 on thegraph (B\A)
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

Similarly, (B\A)3 is the transitive closure of order 3 on thegraph (B\A)
http://find/


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A GraphTheoretic

Model forMusic


Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

|A\(B\A)3| = 8
http://find/


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A GraphTheoretic

Model forMusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

|A\(B\A)4| = 9 because there are no trails of 4 edges betweenany two vertices in (B\A)
http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

|A\(B\A)4| = 9 because there are no trails of 4 edges betweenany two vertices in (B\A)Then applying the similarity function on B and A, we get
http://find/


55/72

A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


(A,B) = |A||A\(B\A)2|

|A| +|A||A\(B\A)3|

|A| +|A||A\(B\A)4|

|A|

(A,B) = 989 +98

9 +99

9

(A,B) = 29
http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


(A,B) = |A||A\(B\A)2|

|A| +|A||A\(B\A)3|

|A| +|A||A\(B\A)4|

|A|

(A,B) = 989 +98

9 +99

9

(A,B) = 29

If we get 0 for a B whereV

(A

) V

(B

), we do not includethat B in the list of results.
http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


(A,B) = |A||A\(B\A)2|

|A| +|A||A\(B\A)3|

|A| +|A||A\(B\A)4|

|A|

(A,B) = 989 +98

9 +99

9

(A,B) = 29

If we get 0 for a B whereV

(A

) V

(B

), we do not includethat B in the list of results.Greater this fraction for a graph H, the more different it is fromthe query graph. Hence, the results should be given inincreasing order of this fraction.

http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work



3 Further Work

Short-comings of our approach
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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

It does not consider transformations of the music themesuch as transposition to another key.



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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


It does not involve rhythm.



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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work



It only allows for new notes between pairs of notes in thevariation of the theme.

http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work



It only allows for new notes between pairs of notes in thevariation of the theme.

The list of results is large because of the possibility of onegraph representing more than one theme. The maximum

number of themes represented by a graph is equal to thetotal number of Eulerian circuits in the graph.

Further Work


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

This procedure has been implemented and tested on Maple.

Further Work
http://find/


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A GraphTheoretic


Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work


Further work can be done.

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work



How to allow for variations with notes between pairs of

notes that are already in the original theme?

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work




notes that are already in the original theme?Incorporating isometries into the similarity function.

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work





Understanding the similarity function proposed by Pintoand Haus.

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work






Figuring out a way to represent rhythm in a graph.

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work






Figuring out a way to represent rhythm in a graph.How to make the results more precise?

Further Work
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work






Figuring out a way to represent rhythm in a graph.How to make the results more precise?

How to get from the graphs to the musical themes?
http://find/


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MusicInformation

Retrieval

Amna Ilyas

RepresentingMusical

Themes UsingGraphs

SimilarityFunction

Further Work

Thank you for your attention today.
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ilyas_a graph theoretic model for music information retrieval

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