Classifier of leveling of sound intensity level between violin strings

Classifier of leveling of sound intensity level between violin strings Piotr Wrzeciono Warsaw University of Life Sciences - SGGW; Nowoursynowska 159; 02-776 Warszawa;Poland Summary The sound intensity leveling between violin strings is one of the most important property of this instrument. This property determines a musical utility of violin. The leveling is also evaluated by jurors during violin making competitions, but those mentioned evaluations are subjective. The pa per presents a new classifier of leveling of sound intensity level between violin strings. The estim ated evaluation of this violin’s property is based on objective parameters. The AMATI multimedia database, which contains recordings and evaluations of violins from 10 th International Henryk Wieniawski Violin Making Competition, was used in this research. The classifier was made using a genetic algorithm. Each individual represents the different classifier. A genotype of the individual consists of four chromosomes. Two chromosomes encode the basic mathematical functions. The other chromosomes encode the factors of those functions. The number of individuals of population was constant. A tournament selection was chosen as the selection method. A set of basic functions was made, where each function had its own binary code. This set contains logarithm function, power function, the polynomials of various orders and other functions. A fitness function was the arithmetic mean of differences between the calculated evaluations of leveling and the evaluations made by jurors. The values of fitness function were calculated for each individual separately. The best of obtained classifiers was consistant with the evaluation of leveling made by jurors for 79% instruments from the AMATI multimedia database. PACS no. 43.75.+a

1. Introduction

1

One of the most important property of the violin is the sound intensity leveling between strings. This parameter describes differences of sound intensity level during changing the string by violinist. It means, that the leveling is good, if the sound intensity of violin is independent from the string. This idea is mentioned by musicians and violin makers [1][2], however the measurement of leveling between strings is a difficult problem to resolve. The subjective evaluation of this property is usually made by the musicians or violin makers, e.g. during different violin making competitions [3]. This kind of evaluation based on the personal experience of jurors, which include a playing technique and sound timbre preference. The objective measurement of this analyzed property is much more complicated, because it is not possible to measure leveling between string 1

(c) European Acoustics Association

using the direct methods. However, the objective parameters which describe the sound intensity leveling between string can be calculated by an analysis of violin sound.

2. Objective measurement of string leveling 2.1. The measurement method The measurement’s method of leveling between string bases on the method of searching violin modes in the sound [3][4][5]. The following parameters of leveling are calculated: BDG, BAG, BEG, BAD, BED and BAE [6][7]. The “B” letter was used as a symbol of leveling between the strings. The subscript describes the pair of the violin strings: D-G, A-G, E-G, A-D, E-D and A-E. The unit of measure of string’s leveling is the decibel.

2.2. The parameters of leveling To calculate BDG, BAG, BEG, BAD, BED and BAE [6][7], the four recordings of chromatic scale played on violin were used. Each recording has to be made

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Wrzeciono, Classifier of leveling of sound intensity level between violin strings

7–12 September, Krakow

in the near field and contains one chromatic scale per string. There are chromatic scales played on the G string (G), the D string (D), the A string (A) and the E string (E). In the next step, the energy of whole recording is calculated: the EG for the G string, the ED for the D string, the EA for the A string and the EE for the E string. After the calculating of the energy of chromatic scale, the searching for violin modes from 101 Hz to 198 Hz is made. In the next step, the sum of the found modes’ energy is calculated, for each string separately. The sums of modes’ energy are marked as E0G, E0D, E0A, E0E. The next parameters to calculated are the relations between the energies of whole chromatic scales and the energies of modes. Those relations are marked as ΘG, ΘD, ΘA and ΘE, which are calculated by formula (1).

( ( ( (

ΘG=10log 10 ΘD =10log10 ΘA =10log 10 ΘE =10log 10

EG E 0G ED E 0D EA E 0A

) ) ) )

EE E0 E

[dB ] for the Gstring [dB] for the Dstring

(1) [dB ] for the A string [dB] for the E string

The values of parameters BDG, BAG, BEG, BAD, BED, BAE are obtained by formula (2). B DG =∣ΘD −ΘG∣ B AG=∣Θ A−ΘG∣ B EG=∣ΘE −ΘG∣ B AD=∣ΘA−Θ D∣ B ED =∣Θ E −Θ D∣ B AE =∣Θ A−Θ E∣

land), every four years. During this event, only the new violins are evaluated in many categories. On the second stage of this competition, the leveling between violin’s string is evaluated, as one of properties of the instrument. In this research, the AMATI multimedia database was used as a source of data [8]. This database contains recordings of violins and their evaluations from 10 th edition of International Henryk Wieniawski Violin Making Competition (2001).

3.2. Evaluations made by jurors During the second stage of the 10 th International Henryk Wieniawski Violin Making Competition, the four jurors evaluated instruments in following categories [5][7][8]: loudness of sound (per string – from 4 to 20 points), timbre of violin sound (per string – from 4 to 20 points), ease of sound generation (from 1 to 20 points), leveling between strings (from 1 to 15 points), correctness of the instrument’s assembly (from 1 to 10 points) and individual properties of the violin sound (from 1 to 15 points). The violins were marked by numbers [5][7][8]. The identifiers of instruments were different on each stage of competition. Each juror worked independent. The final evaluation were the arithmetic mean of evaluations made by jurors.

4. Creating of classifier 4.1. Values of objective parameters

[dB] [dB ] [dB ] [dB ] [dB ] [dB ]

(2)

The parameters from formula (2) are the objective description of the leveling of sound intensity level between violin strings [6][7]. The values of those parameters range is from 0.04 dB to 14.28 dB [6].

3. The subjective evaluation 3.1. Violin Making Competition The evaluation of leveling between string is often made on several violin making competition. One of the world most famous competition is the International Henryk Wieniawski Violin Making Competition, which takes place in Poznań (Po-

The values of objective parameters of leveling between string of instruments from AMATI multimedia database are published in paper [6]. The unit of those values is decibel. The values of leveling evaluation are points. It means, that they are dimensionless. From this reason, to calculate the evaluation by classifier, it is necessary to convert values of BDG, BAG, BEG, BAD, BED and BAE. The dimensionless parameters was calculated by formula (3). The new parameters are following: B̂ DG - dimensionless value of BDG, B̂ AG ̂ EG - dimensionless dimensionless value of BAG, B value of BEG, ̂B AD - dimensionless value of BAD, B̂ ED ̂ AE dimensionless value of BAD and B dimensionless value of BAE. The number of instruments in AMATI multimedia database is 54, but 53 violins took part in the competition. The one extra instrument is property of the violinist, who played during recording

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Wrzeciono, Classifier of leveling of sound intensity level between violin strings

7–12 September, Krakow

session. This instrument was build in 19 th century. Because it was an old violin, this instrument was not used during described research.

Table I. The set of functions for genetic algorithm

B DG

̂ DG = 10 10 B B̂ AG = 10 ̂ EG = 10 B B̂ AD = 10

B AG 10 B EG 10 B AD 10

(3)

BED

̂ ED = 10 10 B ̂ B AE = 10

BAE 10

4.2. The classifier model The classifier was designed as an general equation (4). In this equation, f10, f20, f30, f40, f50, f60, f11, f21, f31, f41, f52 and f61 are the general function from the Table I. The symbols o1, o2, o3, o4, o5, o6, o7, o8, o9, o10 and o1 represent the operators :addition (+), subtraction(-), multiplication(*) and division(/). The operators and functions are chosen by genetic algorithm. E lbs = ̂ ̂ EG) o3 f 10 ( B DG ) o1 f 20( B̂ AG) o2 f 30 ( B ̂ AD) o4 f 50 ( B ̂ ED) o5 f 60 ( B̂ AE) o6 f 40 ( B ̂ ̂ AG ) o8 f 31 ( B ̂ EG ) o9 f 11 ( B DG ) o7 f 21 ( B ̂ ̂ ̂ ) f 41( B AD) o10 f 51 ( B ED) o11 f 61 ( B AE

base for the other more complicated function [9] [10]. According to these rules, the four basic functions was used (Table I).

(4)

The very important part of formula (4) is the order of arithmetic operation. Because in this formula the operators are present only in general, the special rule of order was created. First of all, the result of operator o 1 is remembered. This remembered result is the first argument for the next operator o2. The next argument for operator o 2 is the value of f30 function. The result of operator o 2 is remembered as the first argument for the o 3 operator. This rule is the same for the all operators. It means, that the result of former operator is always the first argument for current operator.

4.3. The set of mathematical functions On the beginning, the function of classifier is unknown. It is possible to write a general formula (4), but in this way it can be used any mathematical function. Because of that, the set of basic functions has to be limited [9][10]. One of the methods of creating of this type of classifier is the choosing of the functions, which are the

Function (binary code)

Type

00

ax + b

01

axb

10

a logbx

11

a ebx

The first column “Function (binary code)” is the binary code of the kind of function. This code is using in genetic algorithm. The second column “Type” presents the type of function. In general, the parameters are real. Only third function (10) has the limits for the b parameter. In this case, b has to be greater than zero. In the program, if the 10 function was chosen, the module of b parameter is used as the base of logarithm.

5. Genetic algorithm 5.1. Multi-chromosomal algorithm The genetic algorithms base on the genetic theory of evolution [9][10]. From this reason, there are a lot of common parts with the genetic science in general. One of them is the division for the organisms which have only one chromosome and the organisms with more then one. In the most applications of genetics algorithm, the individuals has only one chromosome [9][10][11], but if the properties has very different meaning, e.g. numerical parameters and symbolic functions, the multi-chromosomal individuals are used [9][10] [11]. In presented algorithm of searching for the classifier of leveling between string, the three different kinds of properties have to be coded: functions, numerical parameters for those functions and operators. Because of that, the multi-chromosomal algorithm was used to find the best classifier of leveling between string. In genetic algorithm, each individual represent one solution of problem [9][10]. In this research, each individual represents one classifier. Each individual has its own evaluation, which is calculated by fitness function [9][10]. The set of individuals is a population. In each generation, the most important operations are: the crossover of individuals and mutations. In this research the tournament method [9][10] was used to choose the individuals for crossover. The crossover was

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Wrzeciono, Classifier of leveling of sound intensity level between violin strings

7–12 September, Krakow

made in a random point of chromosome. The mutation is a random change in the chromosome. This is necessary for avoiding the local minima. After those mentioned operations, the evaluations of individuals are calculated. In this research in each generation, the same number of new individuals were created. For this reason, the number of individuals was constant.

5.2. Properties coding The properties of classifier were coded on the four chromosomes (Table II). Table II. The genotype of an individual C.0

f10

f20

f30

f40

C.1 a b o1 a b o2 a b o3 a b C.2

f11

f21

f31

f50 o4 a b

f41

f51

f60 o5 a b o6

leveling with the evaluations given by the jurors of the 10th International Henryk Wieniawski Violin Making Competition. The fitness function is given by the formula (6). N

QI =

∣E lbs ( k) − E J ( k)∣ N k=1

∑

(6)

In formula (6) the QI is the value of the fitness function. N - is the number of analyzed violins. In this research, the N value was 53 (according to the AMATI multimedia database). The k represents kth instrument. The Elbs(k) is the calculated evaluation of leveling for k-th instrument. The EJ(k) is the evaluation of leveling, which was made by juror for the k-th instrument. The best value of QI is zero.

6. Analysis of results

f61

C.3 a b o7 a b o8 a b o9 a b o10 a b o11 a b

6.1. Input data First chromosome (C.0) coded the kind of functions from f10 to f60. Each function was represented by two bits, according to the Table I. The parameters a and b of those functions were coded in the second chromosome (C.1). The number of bits of a and b parameters were coded on M bits. The number of bits of parameters were one of the initial values for program. The U2 coding was used to coding the value of parameters a and b. The values of parameter were calculated by formula (5). value =

Maxvalue 2

M −1

binaryvalue

(5)

The “value” is the value of a or b parameter. The “Maxvalue” is the maximal value of parameters (a and b). The “binaryvalue” is the U2 coded number. The operators were also coded in the C.1 chromosome (Table II). Each operator was coded on two bits. The coding was following: 00 is the adding, 01 is the subtraction, 10 is the multiplication and 11 represents the division. The C.2 chromosome contained the gens of the functions from f11 to f61. The rules of coding were the same as in the C.0 chromosome. The genes from C.3 chromosome coded the operators and parameters of the functions from the C.2 chromosome.

5.3. Fitness function The fitness function calculates the evaluation of the quality of the individuals. In this research, the value of the fitness function represents the similarity of the calculated evaluations of

The classifier was calculated for the each juror separately, for the median (the median was calculated for each instrument individually) and for the arithmetic mean of the juror’s evaluation. The initial number of individuals was 300. The number of the new individuals was 150. The probability of the one value in the binary gen was 0.5. The probability of mutation was 0.02. The Maxvalue of parameters was 18. The M number (see the chapter number 5.2) was 35 bits. The values of objective parameters of leveling came from the paper [6]. The evaluations made by jurors came from the AMATI multimedia database.

6.2. Convergence of algorithm The minimum value of QI was 1.1575602193. It was obtained for the median of juror’s evaluations. The minimal values of QI for the other subjective evaluations (juror 1, juror 2, juror 3, juror 4 and the arithmetic mean) were about 1.6. This result was obtained after 37 successful corrections of population. The number of generations in the successful iterations were following: 6, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 3, 1, 2, 6, 1, 2, 1, 6, 16, 5, 1, 26, 1, 41, 124, 66, 882, 482, 1277, 466. The program worked about three hours to obtain this result. The configuration of computer was: Intel Pentium B950 2.1 GHz with 6 GB RAM. The implementation of the described genetic algorithm was written in Java.

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Wrzeciono, Classifier of leveling of sound intensity level between violin strings

7–12 September, Krakow

6.3. Obtained functions

(8)

E lbs = 13.1−0.0871⋅B̂ AE

For the minimal value of QI the following functions were obtained as components of classifier (Table III). The “Op.” means operator.

Only for a few instruments, the x1 is greater than zero. For those cases, the correction calculated from the formula (7) was added.

Table III. The function found by program

7.2. Formula of the best classifier

Function

Type

a

b

Op.

f10 f20

a logbx

3.4851974992

6.2479867449

-

a logbx

-13.1087149977

2.9192793213

-

f30

ax + b

-0.6181720414

5.9893674413

+

f40

ax + b

-9.7172829093

11.2526321083

-

f50

ax + b

-14.6396264776

0.0881265557

*

f60

a logbx

-6.6320589351

8.1169634017

-

1.013982319

16.759947121

*

b

f11

ax

f21

a ebx

9.2869665491

-3.9270484624

+

f31

bx

-15.1777604716

17.857060393

*

f41

ax

b

-5.9649469602

4.1272668442

*

f51

axb

-4.1823956219

-17.810969989

-

f61

ax + b

0.0871283098

-13.1183135917

ae

6.4. General form of classifier The obtained formula of classifier (7) without the analysis of the values of functions from f10 to f61 is quite complicated.

E lbs

̂ )⋅f ( B ̂ ) x1 = f 41 ( B AD 51 ED ̂ ̂ ̂ AG ) x 2 = f 10 ( B DG )⋅ f 60 ( B AE)⋅ f 21 ( B ̂ AG )⋅ f 60 ( B ̂ AE )⋅f 21( B̂ AG) x3 = f 20 ( B ̂ ̂ AE )⋅f 21( B̂ AG) x4 = f 30 ( B EG )⋅ f 60 ( B ̂ AD)⋅ f 60 ( B ̂ AE )⋅f 21 ( B̂ AG) x5 = f 40 ( B ̂ ED)⋅ f 60 ( B ̂ AE )⋅f 21 ( B̂ AG ) x6 = f 50 ( B ̂ ̂ AG ) x 7 = f 11 ( B DG )⋅f 21 ( B x8 = f 31( B̂ EG) ̂ AE ) x 9 = f 61 ( B = (( x 1)⋅( x 2− x 3− x 4+ x 5− x6 − x7 +x 8))− x 9

(7)

7. The formula of classifier 7.1. Formula of simplified classifier The very important part of the analysis of results obtained by using a genetic algorithm is finding the most important components of the solution [9][10]. In the formula (7) the most important part is x1 and x9. The values of the x1 parameter are in the most cases only a little greater than zero. The typical value of x1 is about 10-14 and the smallest value if this parameter is 10 -127. It means, that only the last part of solution is important. As a result of this analysis, the obtained simplified classifier of leveling between string is a simple linear function (8).

In the formula (7) the parameter x8 is almost zero. The maximum value of this parameter was 10 -23. After reducing the formula (7), the formula of the best classifier was calculated (9). E lbs = 4.127 ̂ −17.811 ̂ ̂ ED +0.088)⋅ −24.946 B AD ⋅B ED (−14.64 B −3.927 B̂ (6.632log 8.117( B̂ AE )(9.287e )) + −(0.0871⋅B̂ AE +13.1) AG

(9)

8. Final results table The final results of searching for classifier are showed in Table IV. The title’s columns are following: “No” is the number, which the instrument was marked; “Jur. 1”, “Jur. 2”, “Jur. 3”, “Jur. 4” are the evaluations of leveling made by jurors (first, second, third and fourth); “Med.” is the median value from the jurors’ notes; “Calcul.” is the calculated evaluation by classifier (7); “Round” is the rounded value of calculated evaluation; “Correct” is the result of the comparison the calculated evaluation with the evaluations made by jurors. The evaluations made by jurors, which are written in bold style, are congruent with the calculated evaluation. The rules of correctness of calculated evaluation was following: if the module of the difference between the calculated evaluation and median was less the best QI value, the calculated evaluation was correct, and if the rounded calculated evaluation was the same as any evaluation made by the juror (for each violin separately), the calculated evaluation was also correct. Table IV. Final results table No

Jur. 1 Jur. 2 Jur. 3 Jur. 4 Med. Calcul. Round Correct

10

12

16

12

12

12

12.89

13

YES

11

11

11

10

11

11

12.59

13

NO

15

10

7

8

8

8

13.01

13

NO

17

11

13

11

11

11

13.01

13

YES

18

11

12

10

11

11

13.03

13

NO

20

13

14

13

13

13

12.97

13

YES

21

12

12

10

12

12

12.87

13

YES

23

10

10

9

10

10

12.95

13

NO

24

10

10

10

10

10

12.59

13

NO

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Wrzeciono, Classifier of leveling of sound intensity level between violin strings

7–12 September, Krakow

No

Jur. 1 Jur. 2 Jur. 3 Jur. 4 Med. Calcul. Round Correct

30

13

10

14

13

13

12.92

13

YES

31

9

9

8

9

9

9.52

10

YES

32

14

14

14

14

14

12.99

13

YES

33

14

14

13

14

14

12.94

13

YES

35

14

13

12

14

13.5

13.02

13

YES

36

12

12

12

12

12

13.02

13

YES

37

8

11

9

8

8.5

13.02

13

NO

39

11

12

10

12

11.5

12.96

13

NO

40

10

10

11

10

10

12.78

13

NO

41

13

14

13

13

13

13.02

13

YES

43

8

8

7

8

8

12.23

12

NO

46

13

10

15

13

13

12.71

13

YES

49

13

13

13

13

13

12.36

12

YES

56

10

12

9

10

10

12.39

12

YES

58

13

12

13

13

13

12.74

13

YES

60

13

12

12

13

12.5

12.85

13

YES

65

14

13

13

14

13.5

12.72

13

YES

66

13

15

13

13

13

13.01

13

YES

72

14

15

13

14

14

12.82

13

YES

74

13

11

12

13

12.5

12.97

13

YES

76

13

12

10

13

12.5

12.47

12

YES

77

14

15

13

14

14

12.97

13

YES

78

14

15

14

14

14

12.91

13

YES

79

15

16

15

15

15

12.94

13

NO

80

12

17

12

12

12

12.83

13

YES

84

13

14

14

13

13.5

12.87

13

YES

85

14

13

13

14

13.5

13.03

13

YES

88

13

14

12

13

13

12.97

13

YES

89

12

19

14

12

13

12.82

13

YES

91

13

13

12

13

13

12.89

13

YES

92

14

13

14

14

14

12.90

13

YES

93

13

14

14

13

13.5

12.99

13

YES

100

14

10

10

14

12

12.99

13

YES

104

10

10

9

10

10

12.93

13

NO

105

11

13

11

13

12

13.03

13

YES

108

11

11

12

11

11

10.58

11

YES

109

13

13

11

13

13

12.94

13

YES

111

8

13

12

8

10

12.98

13

YES

112

10

15

15

10

12.5

12.51

13

YES

113

13

8

9

13

11

13.01

13

YES

115

12

10

10

12

11

11.75

12

YES

116

13

13

12

13

13

12.95

13

YES

117

14

11

12

14

13

13.03

13

YES

118

15

12

12

15

13.5

12.99

13

YES

Conclusion The obtained classifier has a very high percent of correctness. The calculated evaluation of leveling of sound intensity level between violin strings is correct for the 42 (79%) instruments from the AMATI multimedia database. The all calculated evaluation fulfill the competition regulations, because all those values are greater than 5 and smaller than 15. The obtained classifier corresponds very well to the opinions of the musicians and violin makers [1][2]. References

[1] N. Harnoncourt: Baroque Music Today: Music As Speech: Ways to a New Understanding of Music, Amadeus Press, Cambridge 1995. [2] W. Kolneder: The Amadeus Book of the Violin: Construction, History, and Music, Amadeus Press, Cambridge 2003. [3] P. Wrzeciono: A New Method of Searching for Violin Modes, The IEEE Region 8 Eurocon 2007 Conference, Warszawa, 189-193. [4] P. Wrzeciono, A method of detecting the C4 violin mode in the energy spectra of chromatic scales, Archives of Acoustics (2007), vol 32, nr 4, 197-201 [5] P. Wrzeciono, K. Marasek, Violin Sound Quality: Expert Judgements and Objective Measurements, Advances in Music Information Retrieval (Studies in Computational Intelligence), Springer, New York, 237260, 2010. [6] P. Wrzeciono: Metoda pomiaru wyrównania natężenia dźwięku pomiędzy strunami skrzypiec, Proc. (2011) 58th Open Seminar on Acoustics, Gdańsk – Jurata (Poland), Vol. II, 397-408. [7] P. Wrzeciono: The relationship between the subjective and objective measurement of sound intensity leveling between the violin strings, Proc. (2012) 59th Open Seminar on Acoustics, Poznań-Boszkowo, 281-284. [8] E. Łukasik: AMATI: multimedia database of musical sounds. Proc. 2003 Stockholm Music Acoustics Conference, KTH [Kungliga Tekniska Hogskolan], 79-82. [9] Z. Michalewicz: Genetic Algorithms + Data Structures = Evolution Programs, Springer, New York 2011 [10] D.E. Goldberg: Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Professional, 1989 [11] J. A. Walker, K. Völk, S.L. Smith, J. F. Miller: Parallel evolution using multi-chromosome cartesian genetic programming, Genetic Programming and Evolvable Machines, December 2009, Volume 10, Issue 4, 417445.