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 INTERVALMETRIC 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 intervalmetric characteristics (IScharacteristics) as a primary description. General properties of IScharacteristics are studied, including the alphabet of IScodes, combinations of elements with each other. As a secondary description, repeated chains of IScodes 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 IScharacteristics Invariant to the
National Belonging
5. Nationally specific Features of Melodies
6. Nationally specific Intonation Fragments
Conclusion
Notes
References
Acknowledgment
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 intervalmetric 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 intervalmetric characteristics. The text consisting of notes is replaced by the sequence of IScodes. The IScode in the place characterizes the transition from the to the tone of melody and is represented by a triplet: is the absolute value of the interval (the number of degrees between the and tones in the melody); the sign of ("+" corresponds to an ascending motion of melody pitch line and "–" to descending one; if =0, then the plus sign is used); 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 be the ISrepresentation of the melody where in the position is the triplet . The llong fragment of text will be called lgram. In a text of length N – 1 there are exactly (N – l) lgrams separated by a sliding frame of width l. The number of different lgrams will be denoted by (it is obvious that ). Let us call the set the frequency characteristics of the order of the text T, where is a pair < lgram (2) and F (frequency of its occurrence in the text) > . The elements are conveniently organised in descending order of F. The full frequency spectrum of the text T is defined as a set of frequency characteristics where is the length of maximum repetition in the text T.
Thus, the frequency characteristic of the order is just a set of all possible repetitions of the length l in the text, which is added to by uniquely occurring lgrams. The full frequency spectrum contains information on all the repetitions of the length 1, 2, …, in the text.
The full frequency spectrum can be calculated both for a single text and for a set of texts , where m is the number of texts. When we deal with a set of texts, the concatenation is formed as , where * is the separation sign between different texts and is calculated. All lgrams that have the separator are eliminated from . 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 multiclass recognition, each class is represented by its individual learning set of texts. For simplicity, the number of classes is taken to be 2, and are the learning samples for classes 1 and 2, respectively, and and are the frequency characteristics of the order for each class. Let a set of l–grams common for and be denoted as . 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 settheoretic complements and to the intersection of two sets: and . This is schematically shown in figure I.

Figure I
Formally, and .
If the number of classes k > 2, is defined as a totality of l–grams common to at least a pair of samples from . Complements are denoted, respectively, , .
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 can be used for classification. The "contrast" property assumes that an lgram 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 (, 219 melodies), French (, 338 melodies), and American ( , 140 melodies) folk songs of different genres were analyzed. The total length of melodies in IS–representation for the first sample is =9197, for the second sample, =18641, and for the third, =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 , their intersection , and complements . 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 shows the frequency of occurrence in sample of the BIS–codes with , which corresponds to the repetition of a same pitch (or ). Formally, for the sample with the total number of IS–codes , , where is the frequency of occurrence of the code in.
(2) The Mean (for all melodies from) length of the maximum (for each melody) recitative chain indicates the clusterability level of recitative elements inside the melody . Formally, , where is the length of maximum series of IS–codes with =0 in the melody and m is the number of melodies in .
(3). The coefficient of asymmetry of the pitch line 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, , where is the mean value of the interval for ascending motion and is the mean value of the interval for the descending motion. They are calculated as follows:
,
where is the number of IS–codes of the sample with values and , respectively, and is the number of IS–codes with values and .
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.