UFPr Arts Department
Electronic Musicological Review
Vol. 6 / March 2001

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A QUANTITATIVE ANALYSIS OF NATIONALLY SPECIFIC FEATURES OF SONG MELODIES WITH THE USE OF INTERVAL-METRIC CHARACTERISTICS

 

Irina V. Bakhmutova, Vladimir D. Gusev, Tatiana N. Titkova

 

This paper addresses the problem of differentiating song melodies by their "nationality" with the use of interval-metric characteristics (IS-characteristics) as a primary description. General properties of IS-characteristics are studied, including the alphabet of IS-codes, combinations of elements with each other. As a secondary description, repeated chains of IS-codes with lengths of 1, 2 or more symbols are considered. An easily interpreted system of integral quantitative factors is suggested that includes the factor of recitativity, the coefficient of asymmetry of the pitch line. This framework enables the formulation of nationally specific features of Russian, French, and American songs in terms of repetitions. Fragments of melodies that are most informative for the classification are thus found. A method of search for the most and least typical representatives of each class of samples is also proposed. The latter serves as basis for outlining boundaries between classes, which are displayed in characteristics of corresponding melodies as "the most French among Russian melodies" or "the most American among French melodies", for example. The most typical representatives of each class can be considered as centers of clusterization of melodies. The presence of such a clusterization in the form of an unconscious adaptation had been explored in a previous work.

Introduction
1. The System of Melody Representation
2. Representation of Texts in Terms of Repetitions
3. Description of the Experiment
4. Properties of IS-characteristics Invariant to the National Belonging
5. Nationally specific Features of Melodies
6. Nationally specific Intonation Fragments
Conclusion
Notes
References
Acknowledgment

 

Introduction

The essential structural components of any kind of text are repetitions. The role of repetitions (melodic, rhythmic, metric) in musical composition is of special significance. The repetition of separate fragments (intonations) in a melody facilitates its learning whilst the melody is enriched by variation. Repetitions are very important for the classification of melodies by genre, style, composer, etc. The representation of musical texts in terms of repetitions proposed by Bakhmutova, Gusev & Titkova (1990) seems to be very convenient for the purpose.

The aim of this paper is to illustrate the classificatory capability of this method of representation with examples that reveal the characteristics which allow the differentiation of melodies by their "national" belonging. This problem has already caught the attention of musicologists and has been studied at the level of rhythmical repetitions (Boroda, 1990). The basis of our work are the interval-metric characteristics of melodies.

The authors’ interest was drawn to this topic through previous work that suggested adaptations (probably unconscious) in sets of Russian and French folk songs. Bakhmutova, Gusev & Titkova (1997) found that, in sets of practically the same number of Russian and French folk songs, the number of "close" (similar to ear) melodies in French songs is much larger than that in Russian songs. This peculiarity suggested the need to find out the specific features of French melodies which distinguish them from Russian and explain such phenomenon. For the present study we extended the material to include also a set of American folk songs. This enabled a more precise differentiation of the common and specific regularities in the sets of melodies.

 

1. The System of Melody Representation

Musical texts are multidimensional by nature, as every sound can be characterized by its pitch, duration, and metric accent. It is very difficult to analyze variations for all three dimensions simultaneously. Following Zaripov (1983) we represent musical texts in the form of interval-metric characteristics. The text consisting of notes is replaced by the sequence of IS-codes. The IS-code in the {short description of image} place characterizes the transition from the to the tone of melody and is represented by a triplet: {short description of image} is the absolute value of the interval (the number of degrees between the {short description of image} and {short description of image} tones in the melody); the sign of {short description of image}("+" corresponds to an ascending motion of melody pitch line and "–" to descending one; if {short description of image}=0, then the plus sign is used); {short description of image} is a metric accent of sound ("+" corresponds to transitions from a metrically stronger to a metrically weaker tone; "–" corresponds to transitions in the opposite direction). For example, the code 3 + – is interpreted as a jump of a fourth upwards with simultaneous increase in the metric accent.

The description suggested represents a desirable compromise between two contradictory requirements. On the one hand, it is comprehensive enough in that the melody does not lose its individuality. On the other hand, it is not excessively detailed; in particular, it does not take into account either the duration of sounds or the qualitative (tonal) characteristic of the interval.

 

2. Representation of Texts in Terms of Repetitions (1)

Let {short description of image} be the IS-representation of the melody where {short description of image} in the {short description of image} position is the triplet {short description of image}. The l-long fragment of text will be called l-gram. In a text of length N – 1 there are exactly (N – l) l-grams separated by a sliding frame of width l. The number of different l-grams will be denoted by {short description of image} (it is obvious that {short description of image}). Let us call the set {short description of image}the frequency characteristics of the {short description of image} order of the text T, where {short description of image} is a pair <{short description of image} l-gram (2) and F (frequency of its occurrence in the text) >{short description of image} . The elements {short description of image} are conveniently organised in descending order of F. The full frequency spectrum of the text T is defined as a set of frequency characteristics {short description of image} where {short description of image} is the length of maximum repetition in the text T.

Thus, the frequency characteristic of the {short description of image} order is just a set of all possible repetitions of the length l in the text, which is added to by uniquely occurring l-grams. The full frequency spectrum contains information on all the repetitions of the length 1, 2, …,{short description of image} in the text.

The full frequency spectrum can be calculated both for a single text and for a set of texts {short description of image}, where m is the number of texts. When we deal with a set of texts, the concatenation is formed as {short description of image}, where * is the separation sign between different texts and {short description of image} is calculated. All l-grams that have the separator are eliminated from {short description of image}. l–grams can be ordered by the number of texts in which these l–grams are presented. This is essential for the choice of the most informative features that characterize the given class of objects (texts).

In a problem of multi-class recognition, each class is represented by its individual learning set of texts. For simplicity, the number of classes is taken to be 2, {short description of image} and {short description of image} are the learning samples for classes 1 and 2, respectively, and{short description of image} and {short description of image} are the frequency characteristics of the {short description of image} order for each class. Let a set of l–grams common for {short description of image} and {short description of image} be denoted as {short description of image}. The l–grams presented exclusively in one sample are the most interesting for the purpose of classification. These l–grams can be interpreted as the set-theoretic complements {short description of image} and {short description of image} to the intersection {short description of image} of two sets: {short description of image} and {short description of image}. This is schematically shown in figure I.

 

 

Figure I

Formally, {short description of image} and {short description of image}.

If the number of classes k > 2, {short description of image} is defined as a totality of l–grams common to at least a pair of samples from {short description of image}. Complements are denoted, respectively, {short description of image}, {short description of image}.

If the classes of texts under consideration are close enough, the complements may be weak (with a small number of l–grams). In this case, some l–grams with a "contrast" property taken from the intersection {short description of image}can be used for classification. The "contrast" property assumes that an l-gram should have its maximum representation in one of samples (in different melodies) and its minimum representation in all the remaining samples. Algorithms for calculations of the frequency spectrum and intersections of two and more spectra and their complements are based on hashing procedures (Gusev, Kosarev & Titkova, 1975; Gusev & Titkova, 1994).

 

3. Description of the Experiment

Samples of Russian ({short description of image}, 219 melodies), French ({short description of image}, 338 melodies), and American ({short description of image} , 140 melodies) folk songs of different genres were analyzed. The total length of melodies in IS–representation for the first sample is {short description of image}=9197, for the second sample, {short description of image}=18641, and for the third, {short description of image}=7779 symbols (each symbol is a triplet corresponding to the IS–code).

In the course of the experiment full frequency spectra were obtained for the samples {short description of image}, their intersection {short description of image}, and complements {short description of image}. Based on the frequency spectra some integral numerical values characterizing each of three samples on the whole were obtained. Some examples are given below.

(1) The Recitativity factor {short description of image} shows the frequency of occurrence in sample {short description of image} of the B-IS–codes with {short description of image}, which corresponds to the repetition of a same pitch ({short description of image}or {short description of image}). Formally, for the sample {short description of image} with the total number of IS–codes {short description of image}, {short description of image}, where {short description of image} is the frequency of occurrence of the code {short description of image} in{short description of image}.

(2) The Mean (for all melodies from{short description of image}) length of the maximum (for each melody) recitative chain {short description of image} indicates the clusterability level of recitative elements inside the melody . Formally, {short description of image}, where {short description of image} is the length of maximum series of IS–codes with {short description of image}=0 in the melody {short description of image} and m is the number of melodies in {short description of image}.

(3). The coefficient of asymmetry of the pitch line {short description of image} indicates an averaged difference in steepness of the growing and drop of individual peaks forming the melodic contour. Averaging is done for all the peaks inside the melody and for all the melodies of the sample. Formally, {short description of image}, where {short description of image} is the mean value of the interval for ascending motion and {short description of image} is the mean value of the interval for the descending motion. They are calculated as follows:

{short description of image} {short description of image},

where {short description of image} is the number of IS–codes of the sample {short description of image} with values {short description of image} and {short description of image}, respectively, and {short description of image}is the number of IS–codes with values {short description of image} and {short description of image}.

These and a number of other integral characteristics convey important information on the general differences amongst the samples. Local characteristics that enable judgements on the "national" belonging of certain melodies will be considered in Section 6.

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