1. Czekanowska, See A., Narrow Melodies in Slavic Countries (Krakow, 1972). The following excerpt from Czekanowska-Kulińska, “Polish Mathematical Methods in Classification of Slavic Folk Song” (Report of the Tenth IMS Congress, 1967-70, p. 440–41) may clarify the basic principles of the applied method. This “study has been based on an analysis of 1250 Slav melodies of the narrow range of which 1107 had been published and 143 are not published yet. …Google Scholar

The following criteria were the basis of the selection:Google Scholar

1. dimension of ambitus (from second to fifth)Google Scholar

2. type of system—modus—genus (diatonic forms exclusively)Google Scholar

3. position of finalis (lower, middle) and the number of subtonic sounds (no more than two)Google Scholar

4. method of treatment of the text (melodies tending to melisma)Google Scholar

5. general character of the metro-rhythmical structure (melodies of free and additional metrics)Google Scholar

6. type of cadence relation (not differentiated, relative to second, third)Google Scholar

7. principles of construction (repetition, alternative, variant, supplementation, answer)Google Scholar

8. kind of texture (monophony, primitive, multipart singing.)Google Scholar

Thus selected material was designated by means of symbols and then arranged using the method of “Wroclaw taxonomy”: individual items were classified into a system and the latter was divided into parts. This made it possible to estimate conventional distances between designated melodies and further, to present it graphically i.e. to draw a dendrite.Google Scholar

Elaboration of such extensive material (1250) made it necessary to introduce a number of conventions and simplifications.Google Scholar

(1) The material has been divided into three separate collections according to the external, non-musical criteria, namely: South-Slav melodies (947), West-Slav melodies (142) and East-Slav melodies (151).Google Scholar

(2) Another convention is used in establishing the extent of differences and similarities between melodies. All kinds of combinations of various sets of features have been listed and marked, with conventional and logically arranged number designation. This ultimately gives the scale of changes.Google Scholar

On this basis, for n item of each collection, the n-1 smallest distances as occurring between the items within the given collection could be defined. After these calculations it was possible to prepare a table illustrating distances for each of the three collections distinguished.Google Scholar

The table of distances was then projected on the Elliot-2 computer in order to obtain sequences of the groups of consecutive ranges. The results supplied by the Elliot computer made it possible to draw a dendrite.Google Scholar

The theoretical explanation of what was done above is as follows: The most perfect arrangement of dendrites is to systematize them according to the smallest distance. The sum of distances of all the sides of a dendrite is called its length. The side of the dendrite is the portion joining its two points.Google Scholar

Particular melodies were considered as points of n-dimensional space with axes respectively named. The interval d(A,B) melodies is defined by the [quoted] figure:Google Scholar

^d(A,B) = (a₁ - b₁)+ (a₂ - b₂) … (a_n - b_n)/n in which a₁, a₂ …, a_n are number values of the features referring to the melody A b₁,b₂ …, b_n are number values referring to the features of the melody B.Google Scholar

The construction of the dendrite depends substantially on the proper definition of the distances between melodies. In this case the distances were defined in such a way that those between similar melodies were small and those between different melodies were large. That this could be done was due to the transformation of quality features into numerical (metric) ones.”Google Scholar

2. See H. Steinhaus, Mathematical Snapshots (New York, 1966). Also: “The starting point for this method is Czekanowski's table of distances of individuals (in this case, individual = a folk melody). Based on this table we arrange the individuals in this way that we join each individual with the nearest one. If this does not bring about the junction of all individuals in one group, we go on joining each group of individuals with the nearest group. Here the distance between two groups of individuals is understood as the distance between the nearest individuals of whom one individual belongs to one of these groups and the second individual to another. The joining of groups is being repeated until all the individuals are arranged in one group. This method of classification here described does not lead on the whole to a linear arrangement of individuals, as is the case in Czekanowski's method, but to a dendritic arrangement that allows ramifications (resembling a genealogical tree). This method is quick, objective and unambiguous. Besides it leads to an arrangement that has the smallest total length.Google Scholar

We divide a set of individuals into k parts by removing from the dendritic arrangement described above the k-1 longest rods. The division into k parts thus obtained has among all possible divisions into k parts the smallest total length of the shortest dendritic arrangements of particular parts.Google Scholar

In forming Czekanowski's table of distances to individuals the distance between two individuals is counted as the sum of absolute differences of features. We may change here the interpretation to treat the features as individuals, the individuals as features, and thus to form a table of distances between features counting the distance between two features as the sum of absolute differences of these features in particular individuals. For such a table the dual problem may be set of classification of features with individuals as their background. The task of choosing representative features on the basis of such a dual table of distances to features seems to be the key to the problem of selection of features. (from Florek, K., Lukaszewicz, J., Perkal, J., Steinhaus, H. and Zubrzycki, S., “On Wroclaw Taxonomy” [Summary], Przeglad Antropologiezny, 17 (1951), p. 21.)Google Scholar

3. See example 1. (Excerpt from a large-scale chart published in A. Czekanowska, Narrow Melodies in Slavic Countries.)Google Scholar

4. See Figure 2; A and B.Google Scholar

5. See Figure 2; A, B, C, D classified as A: D II I 1/2a 1/2; B: D II I 2/4 a 1/2; C: D II I 1/2a 1/3;D: D II I 1/3 a 1/2 (where 3 = adding the elements of contrasts, 4 = different kinds of melisma.)Google Scholar

6. See Florek, K., Lukaszewicz, J., Perkal, J., Steinhaus, H. and Zubrzycki, S., “Sur liason et division des points d'un ensemble fini,” Colloquium Mathematicum, 2/3-4 (1951), p. 282–85.Google Scholar

7. See Figure 1 where the circles described as A and B symbolize the melodies A and B from Example 2 (mentioned above), and the circles described as C and D symbolize the melodies C and D from Figure 2, which actually show the comparative material from other regions.Google Scholar

8. Balkanic group; see Figure 1.Google Scholar

9. See Steinhaus, H., Mathematical Snapshots. Google Scholar