classification of turkish makam music: a topological approach · 2019-09-06 · classi cation of...

Classification of Turkish Makam Music: A TopologicalApproach

Mehmet Aktas*

University of Central Oklahoma

*Joint work with Esra Akbas, Jason Papyik and Yunus Kovankaya

1 / 33 Mehmet Aktas* Classification of Turkish Makam Music: A Topological Approach

Outline

1 Introduction

2 Network representation of music

3 Diffusion Frechet FunctionDiffusion Frechet Functions in Rd

Diffusion Frechet Function in Networks

4 Classification

5 Experiment

Introduction

Outline

1 Introduction

4 Classification

5 Experiment

Introduction

Makam Music

Makam is a key concept in music of a large geographical region(including north African, middle eastern and Asian countries).

As it is defined in [1], makam is “the feature that is created by therelation of pitches of a scale or melody and the tonic and/ordominant”.

A Turkish music expert may distinguish two songs in two differentmakams by listening, however it is a difficult task to categorize thesesongs computationally.

This research project aims to develop a new method for makamclassification using network representation of music and networktopology.

Introduction

Makam Music

Introduction

Makam Music

Introduction

Makam Music

Introduction

Methodology

Network representation of music

Outline

1 Introduction

4 Classification

5 Experiment

Network representation

Music can be defined as the non-trivial connections and interactionsof sounds. To study this complex structure in music, music pieces canbe represented as networks.

We take n-tuples of consecutive notes as vertices.

For example, if we have a sequence of notes of a song asp1,p2,p3, ...,pk for k ∈ Z+ and use 2-tuples, the vertices arev1 = (p1,p2), v2 = (p2,p3), ..., vk−1 = (pk−1,pk).

We add edges between consecutive n-tuples.

We assign the number of the occurrences of the correspondingn-tuples in the song as the weights to vertices.

We assign the number of co-occurrence of the tuples as the weightsto the corresponding edges.

Example

Figure: The weighted network representation of the song Ber kusayi from theAcemasiran makam.

Diffusion Frechet Function

Outline

1 Introduction

4 Classification

5 Experiment

Main reference

Main goal in data analysis is to obtain informative summaries of data.

Traditional descriptors such as the mean and the variance-covariancegive useful global summaries of data. But not always!

Traditional descriptors such as the mean and the variance-covariancegive useful global summaries of data.

But not always!

Descriptors that can give information on local structures may be moreinformative than global ones for complex data sets.

Frechet Function

Definition

Let α be a probability density function in Rd . The Frechet functionF ∶ Rd → R+ is given by

Fα(x) ∶= ∫Rd

∣∣x − y ∣∣2α(y)dy

Definition

If we have n samples {y1,⋯, yn} in Rd drawn from α, the empiricalestimator of the Frechet function is defined as

Fαn(x) =1

∑i=1

∣∣x − yi ∣∣2

The minimizer of the Frechet function is equal to mean.

Diffusion Frechet Function Diffusion Frechet Functions in Rd

Outline

1 Introduction

4 Classification

5 Experiment

Motivation

Back to the motivating example, the Euclidean distance based FrechetFunction in Rd is not able to detect local behavior.

This behavior can only be detected by analyzing data at different spatialscales. What to do?

Solution: Define a different distance in the domainthat enhances local behavior!

Motivation

Back to the motivating example, the Euclidean distance based FrechetFunction in Rd is not able to detect local behavior.

This behavior can only be detected by analyzing data at different spatialscales. What to do? Solution: Define a different distance in the domainthat enhances local behavior!

Embedding of Rd in L2(Rd)To define this distance, the properties of the Heat (Diffusion) equation willbe used. Let kt ∶ Rd ×Rd → R be the fundamental solution of the Heatequation.

kt(x , y) can be interpreted as the temperature at time t at point ywhen the heat source was at point x at time 0.

If kt(x , y) is large, it means that heat diffuses fast from x to y .

Associate each point x in Rd with the function kt(x , ⋅) = kt,x in thespace of square-integrable functions L2(Rd)If heat diffuses in a similar way from points x , y ∈ Rd to any otherpoint z ∈ Rd , the functions kt,x and kt,y will be close in L2(Rd).

dt(x , y) ∶= ∣∣kt,x − kt,y ∣∣2

Associate each point x in Rd with the function kt(x , ⋅) = kt,x in thespace of square-integrable functions L2(Rd)

If heat diffuses in a similar way from points x , y ∈ Rd to any otherpoint z ∈ Rd , the functions kt,x and kt,y will be close in L2(Rd).

dt(x , y) ∶= ∣∣kt,x − kt,y ∣∣2

Diffusion Distance

kt will define an embedding from Rd to L2(Rd). Let x , y ∈ Rd . TheDiffusion Distance dt ∶ Rd ×Rd → [0,∞), for t > 0, is defined as:

dt(x , y) ∶= ∣∣kt,x − kt,y ∣∣2.

For the heat kernel we have a closed form of the diffusion distance,

d2t (x , y) =

8πtd/2[1 − exp(− ∣∣x − y ∣∣2

8t)] .

Diffusion Distance

kt will define an embedding from Rd to L2(Rd). Let x , y ∈ Rd . TheDiffusion Distance dt ∶ Rd ×Rd → [0,∞), for t > 0, is defined as:

dt(x , y) ∶= ∣∣kt,x − kt,y ∣∣2.

For the heat kernel we have a closed form of the diffusion distance,

d2t (x , y) =

8πtd/2[1 − exp(− ∣∣x − y ∣∣2

8t)] .

Let α be a probability measure in Rd . The Diffusion Frechet function isdefined as

Vα(x , t) ∶= ∫Rd

d2t (x , y)α(dy).

If {y1, ..., yn} are i.i.d. samples, its empirical estimator is defined as

Vαn(x , t) ∶=1

∑i=1

d2t (x , yi).

For small values of t, the function will highlight local information. Varyingt offers a multiscale analysis of data.

Vα(x , t) ∶= ∫Rd

d2t (x , y)α(dy).

Vαn(x , t) ∶=1

∑i=1

d2t (x , yi).

Vα(x , t) ∶= ∫Rd

d2t (x , y)α(dy).

Vαn(x , t) ∶=1

∑i=1

d2t (x , yi).

Example

Multiscale analysis of data composed of a ball and a circle:

For scales between t = 0.5 and t = 3.125, the minima have information onthe topology and geometry of the data (2 clusters, one ball, one circle).

Diffusion Frechet Function Diffusion Frechet Function in Networks

Outline

1 Introduction

4 Classification

5 Experiment

Motivation

Diffusion distances can also be constructed for probability measuresdefined on the nodes of networks, hence, we can define diffusion Frechetfunctions on these domains.

We can use these methods to obtain information on local behavior ofnetwork data!

Diffusion Distance in Networks

To define a heat diffusion process on networks, we must define an analogof the Laplacian.

Let v1, ..., vn be the nodes of a weighted network K and W be then × n weighted adjacency matrix

The graph Laplacian is the matrix ∆ defined by ∆ = D −W where Dis the diagonal matrix with diagonal entries dii = ∑n

k=1 wik .

The heat kernel can be expressed as kt(i , j) = ∑nk=1 e

−λk tφ(i)φ(j)where 0 ≤ λ1 ≤ ... ≤ λn are the eigenvalues of ∆ with orthonormaleigenvectors φ1, ..., φn.

The Diffusion Distance between vi and vj is defined as

d2t (i , j) =

∑k=1

e−2λk t(φk(i) − φk(j))2

k=1 wik .

d2t (i , j) =

∑k=1

k=1 wik .

d2t (i , j) =

∑k=1

k=1 wik .

d2t (i , j) =

∑k=1

k=1 wik .

d2t (i , j) =

∑k=1

Diffusion Frechet Functions in Networks

If ξ = [ξ1, ξ2, ..., ξn] is a probability distribution on the nodes, then wedefine the Diffusion Frechet Function for networks as

Fξ,t(i) =n

∑j=1

d2t (i , j)ξj

The Diffusion Frechet Functions are stable up to p-Wasserstein distance.

Diffusion Frechet Functions in Networks

If ξ = [ξ1, ξ2, ..., ξn] is a probability distribution on the nodes, then wedefine the Diffusion Frechet Function for networks as

Fξ,t(i) =n

∑j=1

d2t (i , j)ξj

The Diffusion Frechet Functions are stable up to p-Wasserstein distance.

Example

Classification

Outline

1 Introduction

4 Classification

5 Experiment

Classification

Algorithm

First, we construct the feature list from all n-tuples seen in the songs.

We assign the reciprocal of the n-tuple’s vertex’ DFF value as thecorresponding feature for the song.

After obtaining the feature vectors of songs, we train a classifier usingthese obtained features and their class values with a machine learningalgorithm.

Classification

Algorithm

Classification

Algorithm

Experiment

Outline

1 Introduction

4 Classification

5 Experiment

Experiment

Dataset

The data set we use is the latest version of the SymbTr, the TurkishMakam Music Symbolic Data Collection, that consists of 2200 piecesfrom 155 makams.

The notes in the dataset are represented by the Turkish Makam scale,which has nine microtones between each whole tone.

The column we use in the score representation is “NoteAE” thatincludes the pitches specified as in the Arel theory.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

NoteAE A4 F5 E5 F5 E5 F5 E5 F5 E5 F5 G5 A5 G5 F5 E5 D5

Experiment

Dataset

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Experiment

Dataset

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Experiment

In our experiments, we use 1035 songs from the 15 most commonmakams with 469797 notes in total.

We first create the weighted network representation of each songusing different number of tuples, i.e. n ∈ {1,2,3,4}.

For each tuple value, we also use different scale parameter t in thediffusion Frechet function, i.e.t ∈ {10−1,10−2,10−3,10−4,10−5,10−6,10−7}.

We apply the Random Forest classification algorithm to build ourprediction model.

We use the 10-fold cross validation process to evaluate our method.

Experiment

Classification Results

n = 1 n = 2 n = 3 n = 4

t = 10−1 84.06 88.60 87.44 86.47

t = 10−2 85.70 88.02 88.02 85.60

t = 10−3 87.63 88.79 88.89 85.51

t = 10−4 88.99 89.66 88.21 85.70

t = 10−5 89.76 88.79 87.63 85.70

t = 10−6 90.14 89.57 88.89 86.09

t = 10−7 89.18 89.18 88.89 85.31

Table: Classification results (accuracy) for different tuple and t values. The cellsfor the best result in each column is colored gray.

Experiment

Confusion Matrix

Acemasiran 49 0 0 0 0 0 0 0 0 0 0 1 0 0 0 98.0

Beyati 0 32 1 0 0 3 0 0 0 0 0 2 0 0 11 65.3

Buselik 0 0 43 0 0 0 0 0 0 0 0 0 0 0 0 100

Hicaz 0 0 0 118 0 0 0 0 0 0 0 0 0 0 0 100

Hicazkar 0 0 0 0 65 0 0 0 0 0 0 0 0 0 0 100

Huseyni 0 1 0 0 0 41 0 0 0 3 0 3 0 0 15 65.1

Huzzam 0 0 0 0 0 0 75 0 0 0 0 0 0 4 0 94.9

K.hicazkar 0 0 0 0 0 0 0 50 0 0 1 0 0 0 0 98.0

Mahur 0 0 0 0 0 1 0 0 75 0 0 0 0 0 0 98.7

Muhayyer 0 1 1 0 0 5 1 0 0 33 0 2 0 0 1 75.0

Nihavent 0 0 0 0 0 0 0 0 0 0 103 1 0 0 0 99.0

Rast 0 2 0 0 0 2 1 0 0 0 0 73 0 0 7 85.9

Saba 0 0 0 0 0 0 0 0 0 0 0 0 46 0 1 97.9

Segah 0 0 0 0 0 0 2 0 0 0 0 0 0 69 1 95.8

Ussak 0 7 0 0 0 5 0 0 0 0 0 16 0 0 61 68.5

Precision 100 74.4 95.6 100 100 71.9 94.9 100 100 91.7 99.0 74.5 100 94.5 62.9

Table: The confusion matrix for n = 1 and t = 10−6

Experiment

Comparison

We compare our algorithm with the n-gram technique, which is themost frequently used makam classification method.

The n-gram technique is basically used to create feature for each songby computing the frequency of n-grams.

Experiment

Comparison

We compare our algorithm with the n-gram technique, which is themost frequently used makam classification method.The n-gram technique is basically used to create feature for each songby computing the frequency of n-grams.

Experiment

Comparison

We compare our algorithm with the n-gram technique, which is themost frequently used makam classification method.The n-gram technique is basically used to create feature for each songby computing the frequency of n-grams.

Experiment

References

[1] Arel, H Sadettin. Turk Musikisi Nazariyat (The Theory of Turkish Music), ITMKDyaynlar, (1968).

[2] Martinez, Diego H. Diaz. Multiscale Summaries of Probability Measure withApplications to Plant and Microbiome Data, Dissertation, The Florida State University,(2016).

[3] Martinez, Diego H. Diaz. Probing the Geometry of Data with Diffusion FrechetFunctions, Applied and Computational Harmonic Analysis (2018)

Experiment

Thank you!

classification of turkish makam music: a topological approach · 2019-09-06 · classi cation of...

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