A Quantitative Study of Local Variation Regularities
on the Base of Russian, French, and American Folk
Songs.
Irina V. Bakhmutova , Vladimir D. Gusev , Lubov A. Nemytikova,
Tatiana N. Titkova.
Institute of Mathematics, Siberian Branch of Russian Academy of
Sciences, Russia, Novosibirsk
Abstract
The general approach to obtaining the quantitative characteristics of
the local variation in music texts is described. The approach was
tested on a set of the Russian, French, and American folk songs. The
general and national specific features of the local variation are
revealed. The results obtained can be used to estimate weights of edit
operations in the musically oriented measures of closeness, for the
classification (by the style, genre, composer etc.) as well as for the
computer composition of melodic variations.
INTRODUCTION
In general the variation seems to be quite an important component of creative
work. It is especially important for musical compositions since, usually, they are
produced by the “varied repetition” principle. The varied (or imperfect) repetition
is considered to mean the pair of fragments from musical text closed in a given
metric. According to quantitative estimates we obtained (Bakhmutova., Gusev,
Titkova 1987), no less than half of all repetitions in musical texts are realized in a
varied form.
The ways of variations of the individual fragments and the melody as a whole are
extremely different. Let the first case be characterized as a local variation and the
second as a global variation. The local variation is realized by using the operations
over single symbols (substitutions, insertions, permutations) and does not lead to
the essential change in length of the repeated fragment. The global variation is a
variation of a higher hierarchy. It concerns to the groups (or blocks) of symbols
(block substitutions, insertions, duplications etc.). In the general case, the global
variation cannot be reduced to a set of local variations. The change in the number
repetitions of any intonation , which often occurs in dancing melodies, can be a
typical example. Another example is the insertion of melodic fragment, which is
sang with one syllable.
The purpose of this work is a study of the quantitative and qualitative
characteristics of the local variation on the base of the Russian, French, and
American folk songs. The problem of global variation is beyond the scope of the
paper since the formation stage of sufficiently representative sample of pairs (or
groups) from closed melodies has to be done prior to their study. To the closed
melodies one can assign the variants of the same songs fixed in musical literature
and also quite numerous cases of “unconscious adoption” of melodies as a whole
detected by computer (Zaripov 1983 and Bakhmutova, Gusev, Titkova 1997)
This work is an outgrowth of the studies described in (Bakhmutova., Gusev,
Titkova 1987). The obtaining of the quantitative characteristics of the local
variation is quite laborious problem and it requires the use of computer. This led to
the introduction of some (possibly bounding) formalisms and the development of
effective algorithms of calculations. Just the quantitative approach distinguishes
this work from the traditional art studies. The approaches similar to that we used
for musical text are unknown to us, if any. Note that the conceptually close
approaches were used in analysis of variation of other language systems, in
particular, sequences of aminoacids. The construction of the closeness matrix for
elements of aminoacid alphabet is a brilliant example.
1 THE BASIS OF PROJECT. FORMALIZATION OF USED
CONCEPTS.
1. 1 The definition of imperfect repetition
The imperfect repetitions detected in song melodies provide the basis for a
quantitative study of the local variation characteristics. These repetitions could be
the arbitrary fragments from musical text close in sense of edit distance and
satisfying some natural limitations. The concept of edit distance (Wagner and
Fisher 1974) seems to be quite an appropriate measure of closeness.
Let U and V be the arbitrary strings of symbols (probably of different length)
composed from the elements of alphabet  ,   empty symbol (    ). The
strings U and V can be transferred into each other by the way of successive
application of admissible (or edit) operations. Usually, a set of three operations is
considered: insertion, substitution and deletion of a symbol. They can have
different weights . Let us denote:  (ai   )  the deletion weight of symbol
ai ;
 (  b j )  the
insertion weight of symbol b j ;
substitution weight of symbol
a i by b j .
 (ai  b j ) 
the
If
S  ( S1 , S 2 ,, S k ) is a sequence of
 (S )    (S i ) . The
i
edit operations, its weight is
edit distance between U and V is the weight of optimal
transfer of one string into another i.e. the magnitude d (U , V )  min
S :u  v
 ( S ) . In
the simplest case (with single weight), d (U , V ) is equal to the smallest number of
operations like “insertion”, “substitution” or ”deletion” of the symbol transferring
one string into another. Below we shall use just this simplest definition since the
weights of edit operations a priori are not known.
The imperfect repetition is called to be a pair of strings U and V such that:
(1) 0  d (U , V )  k ,
(2)
k  integer number;
k  min( U , V ) , where |X| is the length of string X. The limitation (2)
denotes that the distance between U and V is not large thereby enabling one to call
the pair (U,V) the imperfect repetition. Since this condition is not satisfied at small
lengths of U and V, it is reasonable to introduce the limitation from below to the
length of repetition;
(3)
min( U , V )  L , where L is the threshold cutting the strings whose
closeness is of a random character.
1. 2 The definition of variation characteristics
Let
T  T1 , T2 ,, Tm  be the sample of song texts where M is the number of
melodies. We shall distinguish the imperfect repetitions of the first kind which are
formed by strings U and V such that U ,V   Ti (1  i  M ) and the
repetitions of the second kind: U  Ti , V  T j (i  j ) . According to this
division, let the whole set of imperfect repetitions revealed in T be divided into
two groups:
1l ,k (T ) and  l2,k (T ) . The repetitions from 1l ,k (T ) present the
“conscious variation”, the repetitions from
 l2,k (T ) - the “unconscious adoption
accompanied by variation”.
Each repetition from
1l ,k (T ) or  l2,k (T ) is characterized by a certain type of
substitutions (“what” and by “what” is replaced), insertions (what element is
added) and deletions (what is removed). The frequencies of occurrence for
different types of substitutions, insertions and (in the general case) the other edit
operations will be called the quantitative characteristics of variation.
1. 3 The system of melody representation
The musical texts by their nature are multidimensional as every sound is
characterized by pitch, duration, and metric accent. It is very difficult to analyze
the variation for all three dimensions simultaneously. Following, R.Kh.Zaripov
(Zaripov 1983), let us to represent the musical texts in the form of interval-metric
characteristics. The text Ti consisting of N i notes is replaced by the sequence of
( N i -1) IS-codes. The IS-code in k-th place
(1  k  N i  1) characterizes the
transition from the k-th tone to the (k+1) tone of melody and is represented by the
triplet:
I k  the absolute value of interval (the number of degrees between k-th
and (k+1)-th tone in melody); the sign of
I k (“+” corresponds to ascending motion
of melody pitch line, “” to descending one, if
put);
I k  0 , then “+” sign is always
S k is a metric accent of sound (“+” corresponds to the transition from the
metrically stronger tone to the metrically weaker one and “-” on the contrary). For
example, the code 4   , is interpreted as a jump of four degrees down with
simultaneous increase in the metrical accent.
As is seen, the suggested description provides the desirable compromise between
two contradictory requirements. On the one hand, it is full enough so 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 the duration of sounds, ignores
the qualitative (tonal) characteristic of the interval and it is invariant with respect
the sequential transfers. The disregard of these factors is referred to the limitation
of the method.
2 THE DESCRIPTION OF EXPERIMENT
The samples of the Russian -
TR (219 melodies), TF - French (338 melodies), and
American T A (140 melodies) folk songs of different genres were analyzed
separately. The total length of melodies in IS-representation for the first sample,
N R =9197, for the second sample, N F =18641 and for the third, N A =7779.
To search for (L,k) – repetitions instead of the universal but laborious scheme of
dynamic programming the facilitated procedure was used which was based on the
assumption about the smallness of parameter “k” in comparison with L (Gusev
and Nemytikova 1996). If k  L , then at any order of “k” substitutions (or
insertionsdeletions) in the length of fragments forming the (L,k) – repetition there
is rather large nondisturbed string z which is common for both these fragments (for
example, if L=9, k=2, the zlength is equal to 3). This string occurs in the text in
the form of exact repetitions. They are easily distinguished and can be reference
repetitions for finding (L,k) – repetitions. One more simplification was in the
restriction of edit operation set to its minimum in each experiment: first, the only
substitutions were admissible, then – only the insertions and deletions and after
that – the block substitutions of one element by the string of some (usually not
more than 4) elements.
The following combinations of parameters L and k were used in the experiments:
(6,1), (9,2), (12,3), (16,4). If any certain substitution or insertion was occurred in
melody several times it was taken into account only once. The substitutions and
insertions on the boundaries of compared fragments were not taken into account.
For the insertions and deletions the generalized statistics was calculated i.e. there
were made no differences between the insertion and deletion. Along with the
calculation of the number and variety of insertions and deletions, the places of their
disposition inside the compared fragments were analyzed and was estimated the
clustering degree of “distortions”.
3 EXPERIMENTAL RESULTS
The total number of substitutions, insertions and block substitutions fixed in each
experiment is quite great (especially for the repetitions of second kind). The most
widespread varieties of single substitutions and insertions are found several ten
times. Since the sizes of samples TR , TF and T A are different, it is more
convenient to operate not by the absolute frequencies (F) of certain edit operations
but their ranks (r) i.e. their places in frequency ordering (by decrease).
3. 1 The statistics of single substitutions
On the basis of the analysis of the repetitions of first kind we discovered 88
varieties of substitutions in Russian sample, in American sample – 99 (though its
size is less), and in French one (the largest in size) – 154. About a half of all
varieties of substitutions in each sample were met only once. The substitution
(1  )  (1  ) is the most frequent for all samples. If, for example, its
frequency in sample T is equal to F that means that there are F melodies each
containing the (L,k)- repetition with mentioned pair of codes in one of those
positions where the non-coincidence was fixed.
The analysis of other single substitutions used very often shows they are connected
with:
a) the substitution of discending  ascending motion of pitch line by the
recitative one (1  )  (0  ); (2  )  (0  ) ;


b) The change in direction of pitch line motion on retention of the absolute value
and metric characteristic of the interval
(1  )  (1  );
(2  )  (2  );
c) the negligible change of interval value on retention of the pitch line motion and
metric relations (2  )  (1  ); (2  )  (1  ) .


The substitutions like (0  )  (0  ), (1  )  (1  ) , connected with
the change of metric relations do not appear among frequent ones but make a
noticeable contribution into the increase of alphabet of substitutions. Their part in
alphabet amounts 30% for T A , 27% for TF and 16% for TR i.e. the Russian
melodies are differed essentially by this index from the French and American ones.
The samples T A , TF and TR are substantially correlated with types of used
substitutions though the nationally specific features also manifest themselves. So,
the substitution (0  )  (0  ) is more characteristic to the American songs.
T A i.e. its rank rA  2
since rF  20 , rR  27 . The substitution (2  )  (1  ) is typical for the
French songs: rF  11 whereas rA  35 , rR  60 . The substitution
(3  )  (1  ) occurs more often in Russian melodies: rR  4 whereas
rA  15 , rF  87.
It takes the second place by its frequency in
The statistics of substitutions by the repetitions of second kind is strongly
connected with the same statistics by repetitions of the first kind but there are some
exceptions. Their analysis allows to estimate the significance of one another
substitution. In particular, to the most significant substitutions may be referred
those that essentially increased their rank (i.e. dropped low) in the statistics of the
second kind compared to the statistics of the first kind (0  )  (0  ) for

T A , (2  )  (0  ) for TR , (3  )  (1  ) for TF
others .
and some
3. 2 Tandem substitutions
With the non-coincidences k  2 the substitutions in imperfect repetitions show
the tendency to clustering. This tendency has a non-random character and appears
in tandems from 2,3,k non-coincidences following in succession. The degree of
substitution clustering in Russian melodies is essentially lower than that in
American and French ones.
The tandem substitutions show the specific interconnections between the elements
of tandem. For simplicity, we have restricted ourselves to the case of double
tandems presented as ( I1 S1 )( I 2 S 2 )  ( I1 ' S1 ' )( I 2 ' S 2 ' ) . Formally, such a
tandem may be considered as the concatenation of single substitutions
( I1 S1 )  ( I1 ' S1 ' ) and ( I 2 S 2 )  ( I 2 ' S 2 ' ) connected by specific relations.
Let us show the main types of interconnections between the tandem elements.
1) I1  I 2  I1 ' I 2 ' ; S1  S1 ' , S 2  S 2 '. This interconnection can be
characterized as the conservation of the interval balance. Its varieties are:
1a) “filling in” a large interval  the change of big jump by the sequence of
smaller ones retaining the total balance: (5  )(0  )  (3  )( 2  ) . In
the general case, we are dealing with different filling in the same interval:
(3  )( 4  )  (1  )(6  ) ;
1b) sharpening or smoothing a symmetric peak in the melodic line amplifying or
(3  )(3  )  (1  )(1  ) ,
decreasing the
melody dynamism
(2  )( 2  )  (0  )(0  ) and etc.;
1c) movement to the same sound in melody by passing it from above or from
below:
(4  )(3  )  (2  )(3  ) , (4  )( 2  )  (3  )(5  ) ;
2)
as
I1 S1  I 2 ' S 2 ' and I 2 S 2  I1 ' S1 ' . This interconnection can be characterized
the
permutation
of
neighbouring
elements:
(3  )(0  )  (0  )(3  ) ,
(2  )( 2  )  (2  )( 2  ) ,
(0  )(0  )  (0  )(0  ) is characteristic for American melodies and
etc.;
3) sign I1  sign I 2 , sign I1 '  sign I 2 ' , but sign I1  sign I1 '. This
interconnection may be characterized as the change in motion direction of melodic
line: (1  )(1  )  (1  )(1  ) , (1  )(1  )  (1  )(1  ) . The
substitutions of discending or ascending motion in melody by the recitative one can
also be referred to these cases: (3  )(1  )  (0  )(0  ) ,
(2  )( 2  )  (0  )(0  ) ;
4) I1  I1 ' , I 2  I 2 ' , I1  I 2  I1 ' I 2 ' , where sign “” denotes the
closeness of compared interval values. This interconnection generalizes the
mechanism of “sharpening” or “smoothing” (see 1b) when the balance of intervals
is not preserved:
(2  )(5  )  (1  )(7  ) , (2  )( 2  )  (1  )(1  ) etc.
3. 3 The statistics of single insertions  deletions
The possible variety of insertions is potentially defined by the size of alphabet of
IS-codes
   38 . The actual alphabet amounts
2 3 of the potential one for
all samples. In view of the differences in lengths of samples the American one is a
leader in relative number of insertions, the Russian one is the least saturated with
them.
The ranking of insertions by frequency in each ordering reflects the frequency
structure of IS-codes in samples (on the average the frequency is reduced with an
increase in
I ). The “recitative” insertions (0  ), (0  ) are mostly
T A and TF ), then the discending gamma-like elements
(typical for TR ) follow, after that the ascending ones
widespread (especially in
(1  ), (1  )
(1  ), (1  ) and etc. The dominance of recitative insertions is explained by
their relative “ neutrality ”. Usually, they are caused by the difference in lengths of
series from recitative elements in compared fragments (the case “a”) and also they
very often precede or terminate the very significant by value overfalls of pitches of
melodic line (the case “b”). The both cases take place in the alignment of two
fragments that compose the ( L, k ) -repetition suggested below (to the insertions in
one line correspond the gaps in another, the coincidences are designated by “
”),
this sign is absent under substitutions):
0

0 5  2  0  0 1  1  1 1  1 










4  0  1 



0 0  0 5  2  0  0 1  1  1  1  1  0  4  0  1 
case “a”
case “b”
A number of insertions has the nationally specific character. So, the insertion
(3  )  a jump three degrees down with simultaneous increase in sound  is
characteristic to the Russian sample (its rank rR
have correspondingly:
rA  16, rF  19.
 6 ) since to the T A and TF we
3. 4 The local  global schemes of variation
As in the case of substitutions, the insertions have a tendency to clustering forming
the tandems. If the length of tandem is 3 and more, then the interpretation of block
insertions of such length is beyond the packed only local variation. The same
conclusion is valid in respect to the block substitutions when one string of symbols
is replaced by another one not necessarily of the same length. The variation
schemes of such kind will be called the local  global variation. Not having a
chance of detailed consideration of them we indicate only some mechanisms of
their origins:
(1) in melodies of quick dances with a lot of short repetitions the block insertion
can occur because of the different number of short repetitions in two similar
melodic lines;
(2) the music phrase, where the melody reaches its culmination, can be repeated in
a “weaker” form from which just the most spectacular melodic fragments are
removed;
(3) many block substitutions result from the singing of any syllable at repeating the
same verse line. In this case, the block substitution has the form of “filling in” an
interval with saving the balance between the left and right parts (for example,
(4  )  (1  1  1  1  ) ).
CONCLUSION
The general enough approach and effective algorithms for obtaining quantitative
characteristics were developed and tested on quite representative samples of the
Russian, French, and American folk songs. The common and nationally specific
features of local variation were obtained.
The results can be used for the identification of closed melodies, the composing of
melodic variations by computer, the correct estimation of weights in musically
oriented measures of closeness and also for the classification (by style, composer
etc.).
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