Mathematical Bases of the Form Construction in Arvo Pärt’s Music

Lietuvos muzikologija, t. 15, 2014 Anna SHVETS Mathematical Bases of the Form Construction in Arvo Pärt’s Music Arvo Pärto kūrinių struktūros matematinis pagrindimas Abstract he creative method of the Estonian composer Arvo Pärt is based on a rigorous and strict calculation at all levels of composition. Such a calculative approach of creation is a relection of the informational society and more speciically – of the logic used in modern programming languages, such as Java, C++, Python, Processing etc. he existing approaches of computational musicology or basically mathematical approaches partly allow to discover and to represent the algorithms under which the compositions were created. However in particular cases it is not suicient to correctly display these algorithms. hus, a new method of music representation formalization will be presented. his method is based on the use of statements originating from programming languages to logically represent the occurring processes, including form building, in Pärt’s compositions. he described methodological approaches will be applied to the instrumental compositions such as Cantus in Memory of Benjamin Britten, Arbos, Tabula Rasa, Mein Weg, Fratres and Spiegel im Spiegel. Keywords: Arvo Pärt, computational musicology, algebraic approach, mathematical algorithms of the musical form building, comparative analysis. Anotacija Estų kompozitoriaus Arvo Pärto kūrybos technika paremta tiksliais visų kūrinio parametrų apskaičiavimais. Toks kūryboje naudojamas skaičiavimo metodas yra informacinės visuomenės atspindys, konkrečiau tariant, logikos, naudojamos šiuolaikinėse programavimo kalbose, tokiose kaip „Java“, C++, „Python“, „Processing“ ir pan., atspindys. Dabartiniai skaitmeninės muzikologijos metodai arba daugiausia matematiniai metodai leidžia iš dalies atrasti ir pavaizduoti algoritmus, kuriuos naudojant buvo sukurti muzikos kūriniai. Tačiau tam tikrais atvejais nepakanka šių algoritmų vien tik parodyti. Straipsnyje pateikiamas naujas muzikos formalizavimo metodas. Naudojamos programavimo kalbų, kuriomis siekiama logiškai pavaizduoti vykstančius procesus, tarp jų ir Pärto kompozicijų formos konstravimo procesą, formuluotės. Aprašytas tyrimo būdas taikomas analizuojant instrumentines kompozicijas („Cantus Benjamino Britteno atminimui“, „Arbos“, „Tabula Rasa“, „Mein Weg“, „Fratres“ ir „Spiegel im Spiegel“). Reikšminiai žodžiai: Arvo Pärtas, kompiuterinė muzikos analizė, algebrinis metodas, muzikos formos sudarymo matematiniai algoritmai, komparatyvinė analizė. Introduction In previous studies of Arvo Pärt’s creative style diferent approaches conirming the close relation between external graphical ideas and sound implementation within his compositions were used (Shenton, 2012). Such approaches as style analysis, musical hermeneutics, Schenkerian analysis, set theory, triadic transformation and others (Robinson, 2012) could make sense only partially and do not give the key for the entire understanding of Pärt’s creative process. he idea of the musical archetypes proposed by Brauneiss (Brauneiss, 2012) fails when the same rhythmic structure organization could be found not only in Arbos, for which as the author assumes the structure was designed, but in Mein Weg, Cantus in Memory of Benjamin Britten and in a mirrored version in Silentium from Tabula Rasa. hus, the traditional musicological approaches are not suiciently adequate for Pärt’s compositions analysis. Instead, the computational musicology approaches may be used. Computational musicology appeared with the 88 appearance of technology in the 20th century and consists of search for a computational aspect of music and development of theoretical models of the interactions between the levels of representation, which use this computational aspect (Ahn, 2009). One of the approaches of computational musicology is the algebraic approach to music theory which proposes mathematical methods for music analysis (Andreatta, 2003), which have been used for present analysis. Aim of the article Despite the aesthetic position of Pärt, assuming that everything that could be mathematized has nothing to do with music (Pärt, 1990), within his own works the composer uses mathematics and any sort of calculation at all levels of music composition – in general form structure, in formation of melodic patterns, in polyphonic relation between voices. hus the aim of the present article mainly consists of showing these mathematical regularities, especially of the form construction, Mathematical Bases of the Form Construction in Arvo Pärt’s Music on several examples of Pärt’s music. An additional task lies in the development of a new methodological approach, allowing representation of music processes as logical statements with the use of semantics coming from modern computer languages, such as Java, C++, Python or Processing. For the analysis six instrumental compositions were chosen – Cantus in Memory of Benjamin Britten (1977), Arbos (1977), Tabula Rasa (1977), Fratres (1977), Spiegel im Spiegel (1978) and Mein Weg (1989). he main reason for the instrumental compositions selection is their freedom from text and human voices limits, which can add some sort of restrictions. he selection of the speciied compositions is made under their constructional comparability, essential for comparative analysis. Mapping principle Computational analysis requires the mapping of notes to numbers. his mapping consists of the numerical values assignment to the each step of the scale, also sign “+” or “-” assignment depending on where these steps are situated – above the starting note or below. he note considered as the starting point of voice movement receives the value of number “0”, the note of one step above receives the value of number “1” with sign “+” and the note of one step below – the value of number “1” with sign “-“, similarly the note of two steps above – the value of number “2” with “+” sign and the note of two steps below – value of number “2” with minus sign etc. Fig. 1. Mapping method illustration Such mapping method was used to Pärt’s compositions in previous works for analysis of Mein Weg structure (Shvets, 2012), computer modelling of Spiegel in Spiegel (De Paiva Santana, 2012). he attempts of this method’s application have been used to the Fratres structure analysis (Zamornikova, 2011), but inappropriate values assignment caused the absence of signiicant result of the method application: all the steps above the “0” note were noted as number “1” and all the steps below – as number “1” with “-” sign. hus, the ranges afecting the structure coniguration were not received. Form building algorithms In numeric representation he construction of the form in Pärt’s compositions is far from traditional musical forms organization and relies on the main melodic voice development. Sometimes the composer uses the measure structures grid, guided, in turn, by mathematical logic. Within this grid the composer inserts the main music material (Fratres), which can be alternated with secondary-level music material (Ludus). All these cases will be described in detail. The main Melodic voice’s (M-voice) development consists of linear algebra operations, such as addition, subtraction and multiplication. he symmetry laws also play a signiicant role in the M-voice construction and can be applied in vertical (change of sign for the same range) and horizontal senses (reversed order of numbers). hese algorithms are visible on the micro level of each piece construction and on the macro level between pieces, where the structure of the one piece could appear as a reversed or altered version of another piece. he micro level of each piece composition is constructed from larger and smaller structures, subordinated hierarchically between them. hese larger and smaller structures are classiied as stanzas, phrases and elements. In some cases intermediate structures as semi-stanzas or semi-phrases can appear, but it is not obligatory and depends on each case. Stanza is a completed event within Part’s music. It contains the beginning and the end of the structural numerical range on which the mathematical operations are executed. Stanza can or cannot be divided into smaller structures – the quantity of used numerical operations and transformations decide for it. he use of mathematical operations within each structure has several levels of interactions: • Stanza-stanza level; • Phrase-phrase level; • Element-element level. he implementation of logic operations will be considered now in concrete examples. For this implementation only a few irst stanzas of the main M-voice of each piece will be shown to set the principle of the form algorithm formation. he whole ranges in graphical representation will be shown later. he interactions on stanza-stanza level are inherent in the structure of Fratres. In this case the replacement of elements within the phrases of neighbouring stanzas occurs (-1 +1 towards +1 -1). Fratres 1st stanza st 1 p. 0 - 1 + 1 0 2nd p. 0 - 1 - 2 + 2 + 1 0 3rd p. 0 - 1 - 2 - 3 + 3 + 2 + 1 0 2nd stanza 0+1-10 0+2+1-1-20 0+3+2+1-1-2-30 he interactions on the phrase-phrase level are inherent in the Mein Weg and Spiegel im Spiegel structure. hese cases are rich of mathematical transformations such as: Vertically mirrored ranges within the phrases (+2+1 towards -2 -1) of the same stanza; 89 Lietuvos muzikologija, t. 15, 2014 Anna SHVETS Horizontally mirrored ranges of numbers with the use of linear addition: +1 +2 +3 towards (+4) +3 +2 +1 (or the same range of numbers with “-” sign). init 1st st. 2nd st. 1st st. 2nd st. Mein Weg 0 +1 st 1 p. +2 +1 0 2nd p. +1 +2 +3 0 1st p. +4 +3 +2 +1 0 2nd p. +1 +2 +3 +4 +5 0 -1 0 -2 -1 0 -1 -2 -3 0 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 0 Spiegel im Spiegel 1st p. -1 0 2nd p. -2 -1 0 1st p. -1 -2 -3 0 2nd p. -4 -3 -2 -1 0 +1 0 +2 +1 0 +1 +2 +3 0 +4 +3 +2 +1 0 he interactions on the element-element level are inherent in both parts of Tabula Rasa, in Arbos and Cantus in Memory of Benjamin Britten. hese interactions contain the following mathematical transformations: Mirrored order or numbers within the elements of phrases (+1+2+1) and vertically mirrored phrases (with change of sign); Replacement of the numbers’ order with linear addition of the elements to each new stanza. In graphical representation All the ranges of numbers presented above can be represented also with graphics as histograms. he graphical representation will clear up the similarities of the structure algorithms between the pieces. With graphical representation it becomes obvious that the structure algorithm of Arbos is a slightly altered version of Cantus in Memory of Benjamin Britten: Cantus in Memory of Benjamin Britten Ludus 1 stanza: 2nd stanza: 3rd stanza: st 1st stanza: 2nd stanza: 3rd stanza: 1st phrase 1st element 0 +1 0 +1 +2 +1 0 +1 +2 +3 +2 +1 2nd phrase 1st element 0 -1 0 -1 -2 -1 0 -1 -2 -3 -2 -1 2nd element 0 +1 0 0 +1 +2 +1 0 0 +1 +2 +3 +2 +1 0 1st phrase 0 +1 0 +1 +2 +1 0 +1 +2 +3 +2 +1 2nd phrase 0 -1 0 -1 -2 -1 0 -1 -2 -3 -2 -1 2nd element 0 -1 0 0 -1 -2 -1 0 0 -1 -2 -3 -2 -1 0 Arbos It is also obvious that the form structure algorithm of Mein Weg is a reversed version of Spiegel im Spiegel: Silentium 1 stanza 2nd stanza 3rd stanza st 1st stanza 2nd stanza 3rd stanza 4th stanza 5th stanza 90 Arbos 0 -1 0 -1 -2 0 -1 -2 -4 0 -1 -2 -4 -3 0 -1 -2 -4 -3 -4 Cantus 0 -1 0 -1 -2 0 -1 -2 -3 0 -1 -2 -3 -4 0 -1 -2 -3 -4 -5 Spiegel im Spiegel Mathematical Bases of the Form Construction in Arvo Pärt’s Music Mein Weg he idea of question-answer relations appears in the Ludus part of Tabula Rasa (element-element level) and Fratres (phrase-phrase level); Silentium part of Tabula Rasa appears as a simpliied version of Ludus. Fratres Ludus Silentium In formulas representation he application of algebra operations with symmetry laws allows us to represent the M-voice development as mathematical formulas or conditions of a computer program. he code presentation for the formalization of music representation is a new methodological approach that can be used as alternative to verbal or graphical representations, but in conditions of information society appears as the most compact and appropriate (Lyotard, 1979). It is a logical level of music representation. For example, the form of Cantus in Memory of Benjamin Britten can be represented as the for loop statement, the main logic of which allows executing repeated iterations of values. he for loop statement in programming is usually used when the amount of iterations is predeined before entering the loop: for ( i = 0 ; i < x; i++ ) { x = x * (-1)} his simple code will add negative numbers in ascending order from 0 (i = 0) to x (i < x) until the value of x is reached (i++ means proceeding with a natural range numbers in ascending order – 1,2,3,4,5 etc.). he “for” operator means that we want to make the iteration of number i, which appears in condition of the statement. he assignment of x to x multiplied by (-1) means that each item of the range that we’ll receive will be with “-” operator. his code returns numerical range of the form development algorithm shown in numeric representation for Cantus in Memory of Benjamin Britten. he for loop statements are useful for representation of the inal numeric values of the range’s development. he most important x-value within the loop (the inal point of ranges increase) can be also presented in terms of mathematics as n-point of range increase. Earlier the diferences of the form development algorithms on irst few stanzas example were shown. However the whole form representation requires the inal value to be considered in its entirety. If we consider that we’ve been analyzing only the main M-voice development of one layer and that each piece contains at least more than one layer (but not more than 5 in analyzed works), it becomes obvious that traditional descriptive methods are not eicient in the processing of such amount of information. Only the codes’ representation with the initial and inal set of values could resolve the problem of large quantity of information processing to be used in comparative analysis. he initial set of values for the main M-voice is especially useful in comparison to diferent M-voices within the same piece and will be shown later. he graphical visualization of the whole main M-voice structure allows us to follow farther form algorithms’ development until the formation of the entire structure. Figures 2, 3, 4, 5, 6, 7 and 8 present graphical visualizations of the whole main M-voice from selected Pärt’s compositions. he comparison of these graphical representations clariies that similar initial algorithm of development can result in 91 Lietuvos muzikologija, t. 15, 2014 Anna SHVETS Fig. 2. he whole range of the main M-voice from Cantus in Memory of Benjamin Britten Fig. 3. he whole range of the main M-voice from Arbos Fig. 4. he whole range of the main M-voice from Spiegel im Spiegel Fig. 5. he whole range of the main M-voice from Mein Weg Fig. 6. he whole range of the main M-voice from Fratres Fig. 7. he whole range of the main M-voice from Ludus Fig. 8. he whole range of the main M-voice from Silentium 92 Mathematical Bases of the Form Construction in Arvo Pärt’s Music diferent forms coniguration according to the following parameters: 1) Number of stanzas; 2) Number of repetitions of the same structure. Let us present now the ways of development ater reaching the highest n-point of range increase in graphical representation. he common feature for all seven main M-voice forms is aspiration to achieve the n-point of number range, which also is the inal point in the form development. he form of the main M-voice from Mein Weg is an exception, where ater the increase until the highest n-point of ranges, gradual decrease back to 0 occurs. he exact repetitions of stanzas in case of Fratres or Arbos don’t give such clear visual representation of this main feature, however previous numeric representations have shown the algorithms with the use of linear addition until the n-point of ranges is reached. To formalize the whole structure of these pieces part of a code can be used. he occurred repetitions ater the primer algorithm execution can be presented as if statement. It means that we will follow the primer algorithm until it reaches n-point of range and if it reaches that point, we will repeat the last stanza (in case of Arbos) or the irst and the second stanza (in case of Fratres) x times: if (n-point = stanza[i]) stanza[] * x he result of the number of stanzas change is visible on the example of the comparison of the main M-voice structures from Spiegel im Spiegel and Mein Weg: the n-point of the main M-voice from Spiegel im Spiegel equals to +3/-3 and the n-point from Mein Weg equals to +15/-15. his parameter is important for comparison of the other M-voices with the main M-voice within the same piece. he set of these data allows us to take into consideration the entire form of the piece and to consider the relations between M-voices in diferent layers. hese data can be visualized further. Types of complementarity he ways of vertical complementarity in analyzed pieces is reached by means of three methods: 1) two times rhythmic values augmentation of each next M-voice within the same piece (Cantus in Memory of Benjamin Britten, Arbos, Mein Weg, Silentium); 2) Dispersion of others M-voices in accompaniment igures (Spiegel im Spiegel, partly Ludus); 3) Canonic shiting of the M-voices containing the same rhythmic values (Ludus). he vertical complementarity is not a single type of complementarity which can be found in Arvo Pärt’s music. he horizontal complementarity also occurs with measure structures division mentioned above. here are two cases of such measure structures in selected compositions, which difer by its use. he irst level of use consists of mathematical operations that could be performed with these structures, but it does not afect the general structure of M-voices. his level of measure structures division is used in the Fratres and Ludus part from Tabula Rasa. he second level consists of inluencing M-voices development because measure structures play the role of frame slots, which can be illed out only with a special type of information. Each measure structure has its own set of Mvoices and they are developing independently within separate measure structures. he second level of measure structures division is inherent in Ludus part from Tabula Rasa.1 Mathematical operations performed on measure structures in Fratres consisting of addition of the number “2” in order to receive metric quantity units (“9” as result of 7+2 and “11” a 9+2 additions) were described earlier (Zamornikova, 2011). he number “2” appears also in multiplication operations on micro (repetition of one meter “6/4”) and macro levels (repetition of a number, or a list of meters “7/4 9/4 11/4”). he laws of internal and external symmetry are also inherent in the measure structures construction. he internal symmetry consists of the number “9”, which is situated as a metric quantity unit in the middle of the list of metric units and farther is used as the number of repetitions for stanzas. he external symmetry consists of the use of meter “6/4” at the beginning and the end of the piece. he whole measure structure can be presented as a mathematical formula with described operations: { [ 6/4*2 (7/4 9/4 11/4)*2 ]*9 6/4*2 } Before describing mathematical operations on measure structures used in Ludus part from Tabula Rasa, let us deine its general measure structures construction. Ludus can be devised into two main parts – before cadenza and ater. he part before cadenza in a dramaturgical sense is a gradual increase of tension before cadenza-culmination. Each of these greater parts have internal division: part before culmination has four stanzas (devised into 8 semi-stanzas) and cadenza-culmination part consists of two internal 1 Some researches (Zamornikova, 2011) assume the presence of such level use in Fratres. hey believe that such division occurs between the S-part (M-voice stanzas) and R-part, consisting of the two measures illed out with ostinato percussion rhythmic igure. But as we will see later in comparison with other works by the same composer, the percussion insertion usually plays the role of “points” or “separators” at the end of stanzas and is not treated as independent material, pretending to be divided from the main M-voice. 93 Lietuvos muzikologija, t. 15, 2014 Anna SHVETS culminations – irst with fast decay in a-moll ambience and second with fast dynamic increase in ambience of the diminished seventh chord (VII7 to g-moll, the subdominant function to d-moll, tonality of the next part Silentium). he mathematical operations on measure structures are used in the part before culmination. Mathematical operations in Ludus consist of addition and subtraction. Addition is used for adding measures and subtraction for subtracting meter units. Each stanza consists of two semi-stanzas related by question-answer relations. Each semi-stanza contains three measure structures. Ludus measure structures division MS1 1st stanza 2 stanza nd 3 stanza rd 4th stanza 1st s.-stanza 2nd s.-stanza 1st s.-stanza 2nd s.-stanza 1st s.-stanza 2nd s.-stanza 1st s.-stanza 2nd s.-stanza 6/4 5/4 6/4 6/4 (3) 5/4 6/4 (4) 6/4 (6) 5/4 6/4 (7) 6/4 (9) 5/4 6/4 (10) Comment for MS2 MS1 0 8/2 G.P. 1 7/2 G.P. +3 measures 6/2 G.P. +3 measures 5/2 G.P. +3 measures 4/2 G.P. +3 measures 3/2 G.P. +3 measures 2/2 G.P. +3 measures 1/2 G.P. Comment sections show the regularities, which can be formalized as pieces of codes. Addition of measures in MS1 can be displayed as a while loop statement (the logic of while loop statement consists of code execution until the condition is true) with a diferent initial value for the irst and the second semi-stanzas within each stanza: for the irst semi-stanzas initial value of x before addition is “0” and for the second semi-stanzas the initial value of x equals to “1”: a) While loop for the irst semi-stanzas: x=0 while ( x <= 9){ x+3} b) While loop for the second semi-stanzas: x=1 while ( x <= 10){ x+3} he irst while loop will return values 3, 6, 9, which are the numbers of added measures for the irst semi-stanzas of the second, third and fourth stanzas consequently. he second while loop will return values 4, 7, 10, which are the numbers of added measures for the second semi-stanzas of the same stanzas as in case of the irst semi-stanza’s addition. Subtraction operation used in MS2 section can be expressed with the already described for loop statement: for (i = 8; i >= 1; i-- ) {…} 94 Question-answer relations are inherent in only the third measure structure (MS3). he measure structures division is made under meter change: each irst semi-stanza of each stanza begins with 6/4 meter and each second semi-stanza of each stanza – with 5/4 followed by 6/4 meter. Measures illed with these meters make part of the MS1. he meter of the second measure structure (MS2) is constantly changing being afected by the subtraction operator. he third and the last measure structure (MS3) consist of measures with 4/4 meter. he measure structure principle of division described is shown in the following table: C o m m e n t MS3 for MS2 4/4 (6) -1m. unit 4/4 (10) -1m. unit 4/4 (12) -1m. unit 4/4 (16) -1m. unit 4/4 (18) -1m. unit 4/4 (22) -1m. unit 4/4 (24) 4/4 (30) Comment for MS3 +6 +4 +2 +4 +2 +4 +2 +6 +6 +6 +6 his code performs gradual decrement (by -1) from value 8 to 1 and it relects a gradual decrement of quantitative metric units in MS2 in each semi-stanza. hus it begins with meter 8/2 in the irst semi-stanza of the irst stanza and it inishes with meter 1/8 in the second semi-stanza of the fourth stanza, which is the last semi-stanza of the form structure before cadenza. Addition regularities in the MS3 are guided by symmetrical laws: the irst semi-stanza contains six measures, the last semi-stanza has number six as the quantity of added measures (comparing to the previous seventh semi-stanza). he addition between these irst and eighth semi-stanzas can be expressed with (4+2) * 3, because the quantity of measures added to the second semi-stanza equals to 4 and to the third – to 2. he same 4 followed by 2 added measures are repeated yet in the pairs of 4th–5th and 6th–7th semistanzas. he whole formula for addition regularities in the MS3 has the following expression: {6 [(4+2) * 3] 6} It is easy to perceive that 4+2 also equals to 6, thus the divided internal number “6” is surrounded by external whole number “6”, creating perfect symmetry. Symmetry laws are inherent in the canonic shiting of the M-voices (with aligned T-voices) containing the equal rhythmic values in the same MS3 of Ludus: descending order is followed by mirrored symmetrically equal ascending Mathematical Bases of the Form Construction in Arvo Pärt’s Music order in each semi-stanza. he question-answer relations between irst and second semi-stanzas occurs on this level too: the irst semi-stanza contains two phrases with shited pairs of M and T voices, while the second semi-stanza contains only two pairs of M and T voices, but instead additional M-voice within each of its two phrases. he described regularities can be expressed by means of formula where the dash signs (“-”) means simultaneity and plus signs (“+”) complementarity or shiting: 1st phrase whole note, 8 = double whole note, 16 = quadruple whole note, 32 = octuple whole note; 1.5 = dotted quarter, 3= dotted half note, 6 = dotted whole note. Mein Weg 2nd phrase 1st s.-st.: (M-T)*2 + (M-T) | (T-M) + (M-T)*2 Cantus in Memory of Benjamin Britten 2nd s.-st.: (M-T)*2 + (M+M) | (M+M) + (M-T)*2 Going deeper into phrase-phrase relations within each semi-stanza of MS3 from the Ludus part of Tabula Rasa other addition regularity appears. his time it afects the quantity of added quarters in bound between phrases. he bound consist of repeated quarters on the same pitch “A2” with staccato articulation signs. he quantity of quarters in the irst semi-stanza equals to 10 and in the second semistanza – to 12, thus the quantity of added quarters between two semi-stanzas within the same stanza equals to 2. his regularity will be held for the rest of other pairs of semistanza. he number of added quarters between each of stanzas is equal to 6, thus if the quantity of bound measures in the second semi-stanza within the irst stanza was 12, the quantity of these measures in the irst semi-stanza within the second stanza is 18. It results in 4 times 2 quarters added in pair of semi-stanzas and 3 times 6 quarters added between four stanzas. his regularity can be expressed with nested for loop, where outer iteration (+6 or i) is executed ater inner iteration (+2 or j). he general structure of nested for loop is as follows: for ( i = 0; i < num1; i++) { for ( j = 0; j < num2; j++) {…}} In our case each outer i (+6) addition will be done ater the inner j (+2) addition is executed. Rhythm of M-voices Form organization depends on rhythmic organization in both vertical and horizontal senses. Vertical rhythmic organization mentioned earlier as the first method of vertical complementarity relies on 2 times rhythmic values augmentation of each next M-voice within selected structure. he horizontal organization also uses the principle of 2 times augmented/reduced rhythmic values if they are not homogeneous. For visualization we will apply such values signiication: 0.5 = eights, 1 = quarter, 2 = half note, 4 = Arbos Silentium Fratres Spiegel im Spiegel he described principle of two times rhythmic values augmentation is used for Mein Weg, Cantus in Memory of Benjamin Britten and Arbos. he vertical rhythmic construction of Silentium appears as reversed copy of Arbos. M-voices inherent in Fratres and Spiegel im Spiegel are guided by equivalency principle in their rhythmic cells. he formula 1*n in Fratres means that the quantity of quarters varies according to the used meter (7/4, 9/4, 11/4), but irst half note (2) and last dotted half note (3) remains in each metric cell, creating the structure of frame with core. his representation does not negate the second type of vertical complementarity used in Spiegel im Spiegel, only it shows that two dispersed in accompaniment M-voices have the same harmonic rhythm duration, despite their non-synchronic entry. he Ludus part is not shown within these representations because of its horizontal complementarity of the second type (which afects the general form structure construction) instead of vertical and will be analyzed in other way. It is reasonable to analyze each measure structure separately by proposing the whole tables of rhythmic development, because it difers from the irst or second semi-stanza type. 95 Lietuvos muzikologija, t. 15, 2014 Anna SHVETS First measure structure (MS1) contains only one M-voice which consists of the same numeric ranges that M-voices from MS3. However the rhythmic organization difers a lot – from rhythmic values to rhythmic progressions. Using the same numeric representation of rhythmic values, let us present the rhythm structure for MS1 with the following table: Ludus MS1 1st s.-st. 1st stanza 2nd s.-st. 1st s.-st. 2nd stanza 2nd s.-st. 1st s.-st. 3rd stanza 2nd s.-st. 1st s.-st. 4th stanza 2nd s.-st. 6 2 + (3 * 3) (1.5 * 4) + (3 * 4) 2 + (3 * 3) + (1.5 * 4) + (3 *4) ((1.5 * 4) + (3 * 4)) * 2 2 + (3 * 3) + ((1.5 * 4) + (3 * 4)) * 2 ((1.5*4)+(3*4))*3 2 + (3 * 3) + ((1.5 * 4) + (3 * 4)) * 3 he multiplication sign means repetition of the same rhythmic values and plus sign – simple neighbouring of different rhythmic values. If we store the initial rhythmic value 6 (dotted whole note) from the irst semi-stanza of the 1st stanza in x0 and its following rhythmic series corresponding to the formula (1.5 * 4) + (3 * 4) in x, then if we store the range of rhythmic values from the second semi-stanza of the 1st stanza and corresponding to the formula 2 + (3 * 3) in y, we receive a much more simpliied version of the same table: Ludus MS1 1st semi-stanza 1st stanza 2nd semi-stanza 1st semi-stanza 2nd stanza 2nd semi-stanza 1st semi-stanza 3rd stanza 2nd semi-stanza 1st semi-stanza 4th stanza 2nd semi-stanza x0 y x y+x x*2 y + (x *2) x*3 y + (x * 3) he rhythmic range corresponding to x is present in every semi-stanza from the second stanza. he range stored in y plays the role of the beginning of entire rhythmic range inherent in the second semi-stanza. he formula of rhythmic progression for whole stanza looks as follows: he MS3 section contains multiple M-voices which can be divided into two groups regarding used rhythmic progressions – static and dynamic. he static group contains rhythmically identical M-voices consisting of quarters (1). he quantity of M-voices within the static group changes from three to four upon irst to second semi-stanza’s type use. he dynamic group contains three M-voices which rhythmically vary between them and inside themselves. he irst M-voice generally consists of eighths (0.5) with the possible insertion of two sixteens on feeble times. he second M-voice consists of triples (0.3) and the third Mvoice – of sixteens (0.25) only2. A general table of rhythmic values within the dynamic group of MS3 from the irst semi-stanza looks as follows: Ludus MS3 dynamic group (0.5 * 2) * 3 + dot 1st M-voice: (0.3 * 3) * 3 + dot 2nd M-voice: (0.25 * 4) * 3 + dot 3rd M-voice: he formulas in parenthesis correspond to one quarter with an appropriate number of repetition of smaller values in it. he multiplication of values in parenthesis with a speciied number means the repetition within the same measure where “dot” is included. Each next semi-stanza adds one more measure to each M-voice of this group before primarily appeared progression. hus, the development of all rhythmic progressions for all M-voices of dynamic group can be formalized as follows: ((((value*n)*4)*n) + ((value*n)*3+dot)) *n Or if represent the formula for added measures ((values*n)*4) as A and inishing formula ((value*n)*3+dot) as F, we receive the next formula: (A * n + F) * n Considering the dynamic changes in rhythmic progressions from three analyzed measure structures of Ludus, the visual representation of its horizontal rhythmic structure will be done with use of values in discovered formulas and on example of one semi-stanza ater the regularities are set already (second stanza): (x * n) + (y + (x * n)) where n is a number of repetitions from 1 to 3. he rhythmic organization of MS2 directly depends on metric units subtraction presented earlier and consists only of pauses (Grand Pauses), thus the rhythmic values substruction is exactly the same as in case of metric units (one metric unit equals to one half note). 96 Ludus 2 he bigger values that can appear at the end of each semistanza for each M-voice are considered as “dots”. hus, their rhythmic values are not taken into consideration. Mathematical Bases of the Form Construction in Arvo Pärt’s Music Dynamic changes in rhythmic progressions based on mathematical laws are inherent also in the horizontal rhythmic structure development of Arbos. he regularity of two times augmentation/reduction was described already, but within this piece the combinatorics plays decisive role in domain of rhythmic development. At irst sight the combinatorics inishes on a simple regular replacement of bigger rhythmic value by smaller rhythmic value, creating alternation of iambic and choreal rhythmic structures in each new stanza. However the irst and the last rhythmic values of each stanza create all the possible combinations of two elements – a and b, when the combinatory possibilities are exhausted (ater four stanzas), a new cycle with the same order of combinations begins: Arbos rhythmic structure combinatorics st stanza ba aa 2nd stanza ab 3rd stanza bb 4th stanza Described rhythmic combinatorics is inherent in all M-voices of the piece. he values of a and b in the three M-voices difer from the applied law of vertical two times augmentation/reduction regularity. Similar mathematical regulations are used in development of the rhythmic structure of Mein Weg. In this case we will count the number of repetitions of uniform rhythmic values for each new number from pitch range, present in each M-voice. Ater one measure of primer initialization of movement, the regularities are set. Each stanza contains three rhythmic sections. hese rhythmic sections are separated by a pause equal to one rhythmic value inherent in speciied M-voice (eighth pause for the highest M-voice, quarter pause for the middle M-voice and half note pause for the lowest M-voice): Mein Weg rhythmic structure development 2nd section 3rd section 1st section init. (2 4)*2 2 3 (2 4) 1 (4 2)*2 4 1 (4 2) 4 1st st. 3 (2 4)*2 1 (4 2)*4 4 1 (4 2)*2 4 2nd st. rd 3 (2 4)*3 1 (4 2)*6 4 1 (4 2)*3 4 3 st. Repetition structure in the second and third sections consists of the frame (1 4) and core (4 2). he number of repetition of the core in the second section equals to the formula: x*2, in comparison to the third section. he structure of the frame with varied core is familiar to Fratres where it was inherent in the horizontal proportions of rhythmic values. Let us formalize the numbers received using following assignments: • for rhythmic values repetitions: 1 = a, 2 = b, 3 = c; 4 = d; • for core repetitions (ater multiplication sign): N = natural range of numbers from 1 to 7 and in back order (increment up to the n-point of ranges occurs until 7th stanza with gradual decrement until 14th stanza); E = even range of numbers, which are two times taken a natural range of numbers from 1 to 7; Ater assignment the next formula for homogeneous rhythmic values within each M-voice of Mein Weg appears: c (b d)*N | a (d b)*E d | a (d b)*N d Clear symmetrical succession of core multipliers can be extracted from the previous formula: N–E–N Ater reduction of repeated elements, the next formula appears: c (b d)*N | a (d b)*E/N d | where the slash sign “/” means “or ater”. he reversed succession of elements within the core in the irst section compared to next two sections to the use of reversion method for numeric range of pitches iterations widely used in general M-voice construction in Mein Weg. he number of repetition creates rhythmic groups to which the repeated rhythmic values belong. It means that one time (plus 8th pause) or two times repeated eighth belong to quarter (1) and three times repeated (plus 8th pause) or four times repeated eighths belong to a bigger rhythmic value – half note (2). he formula of rhythmic groups for repeated homogeneous rhythmic values looks as follows: 2 (1 2)*N | 1 (2 1)*E 2 | 1 (2 1)*N 2 Representing the core part (1 2)*N with their repetitions as a and the (2 1)*E/N as b, we receive the following formula view: 2a/1b2/1b2 Clear alternation of bigger value with smaller appears: 21212 Rhythmic sections correspond to the numeric range movements: movement from a greater even number with the plus sign to the smallest odd number (+2 +1) correspond to the irst section, movement from the smallest odd number with the minus sign to a greater odd number (-1 -2 -3) corresponds to the third section, movements from a greater even number with the minus sign to the smallest odd number (-2 -1) and from the smallest odd number with the plus sign to a greater odd number (+1 +2 +3) are 97 Lietuvos muzikologija, t. 15, 2014 Anna SHVETS split together in the third section. he formalization of the regularity described looks as follows: 1st section 2nd section 3rd section +Eg => +Os –Eg => –Os | +Os => +Og –Os => –Og Certain combinatory regularities are visible in this formalization too, if we consider a few restrictions: • he numbers of one numeric range movement have the same sign as the irst number of the range; • he change of sign between numeric range movement are guided by strict alternation; • Even greater number always moves to the smallest odd number and the smallest odd number to a greater odd number. An explanation for two movements split in one (second) rhythmic section is evident with mathematical union – the second section contains the mathematical union of the irst and last numeric range movements. Storing +/-Eg in A, +/-Os in B and +/-Og in C we receive the following formalization view: 1st section 2nd section 3rd section AB ABUBC BC Mein Weg Cantus in Memory of Benjamin Britten Arbos Silentium Fratres Spiegel im Spiegel T-voice Tintinnabuli-voices (T-voices) in the analyzed pieces are dependent on its vertical and horizontal correlations with M-voices. Vertical correlations 1. Formula correlation: • he same for each layer (Cantus in memory of Benjamin Britten); • Difers from layers, but is held within each layer (Mein Weg); • Difers from layers and within each layer (Fratres, Arbos, Silentium, Ludus, Spiegel im Spiegel); 2. Rhythmic correlation: • Rhythmic values of M-voices are duplicated in corresponding T-voices (Cantus in Memory of Benjamin Britten, Arbos, Mein Weg, Fratres, Ludus); • Rhythmic values of M-voices are not duplicated in corresponding T-voices (Silentium, Spiegel im Spiegel); Let us present the graphically described cases combining formula (in white) and rhythmic correlations (in yellow): 98 Ludus he quantity of T-voices difers from the piece, and usually equals to one T-voice per one M-voice, but two pieces make an exception – Arbos with two T-voices per one middle M-voice and Fratres with one T-voice for two M-voices. he pieces regarding formulas of T-voice additions can be summarized to the following cases: Entirely based on the formula T-/+1 for the T-voices addition (Ludus, Cantus in Memory of Benjamin Britten); Based on the formula T-/+1 for the T-voices addition with T+2 incrustations (Mein Weg, Spiegel im Spiegel); Entirely based on the formula T-2 (in relation to the highest M-voice) for the T-voice addition (Fratres); Based on the formula T-/+2 for the T-voices addition with T-/+1 incrustations (Silentium, Arbos). he third group of pieces according to the formula correlation criterion can be ranged regarding the diiculty of the algorithm used. he simplest algorithm is a literal alternation used for both parts of Tabula Rasa. he mirrored Mathematical Bases of the Form Construction in Arvo Pärt’s Music order of T+1/T-1 succession in the question-answer related phrases within semi-stanzas from MS3 of Ludus or the change of step in formula for T-voice addition (T-/+1 / T-/+2) in the highest T-voice of Silentium don’t change considerably the settled alternation principle. A more diicult algorithm is used in the higher T-voice related with middle M-voice from Arbos. he formula for T-voice addition depends on the speciied numerical value from the range of M-voice. Mainly it consists of the T-1 addition formula, but until and including the -5 number from M-voice range, all the odd numbers (-1, -3, -5) are followed by tintinnabuli-tone guided by T-2 addition formula. For the rest of range (until -9) the last addition formula is related with all even numbers (-6, -8*2). he most sophisticated algorithms of T-voice addition are used in Spiegel im Spiegel in two lower T-voices, thus they must be visualized together with M-voice ranges for better understanding: Etc. he lowest T-voice Etc. he middle T-voice he formalization of the lowest T-voice addition algorithm in Spiegel im Spiegel can be made with the use of logical formula for M-voice ranges transformations within one stanza, expressed as follows: 1st phrase 2nd phrase T-voice appears as dispersed in accompaniment igure and can be also considered in the context of horizontal correlations too. Horizontal correlations Horizontal correlations of T-voice with M-voice are more diicult to perceive at irst sight, because they are integrated into the structure which can be treated as melody, but knowing the range of M-voice, T-voice within its structure can be found. here are four cases of such correlations: 1) Simple alternation of M-voice with T-voice (M-voice of violin solos from MS3 of Ludus); 2) Alternation of M-voice with T-voice on speciied algorithm (preamble to M-voice of violin solos from MS3 of Ludus); 3) T-voice alternation with two dispersed in accompaniment M-voices (Spiegel im Spiegel); 4) M-voice germination into T-voice (T-voice of violin solos from MS3 of Ludus); he dynamic changes in the third measure structure from Ludus expressed earlier in rhythmic progressions (two violin solos) within each stanza ind its application on the level of T and M-voices horizontal correlations. In this case the dynamic changes afect the whole MS3 form development (but within the same two violin solos) and are expressed in combinatory order change of T and M-voices alternation. he combinatory efect is augmented when formally M-voice (alternated M-voice with T-voice) is followed by separate T-voice. he visual representation of four stanzas looks as follows: -Os => -Og z | +Os => +Og z -Eg => -Os z | +Eg => +Os z he T-voice with T-1 formula of addition is added to every 0 note (z) within each stanza, except the third 0 note (z) with T+1 formula of T-voice addition, which appears ater range from a greater even number with the minus sign (-Eg) to the smallest odd number with the minus sign (-Os). he middle T-voice algorithm consists of alternation and symmetry principles combination: phrases always begin with T-tone additions according to the formula T+2, which is farther alternated with T-1 formula of T-voice addition. Symmetry with alternation order change occurs on the smallest odd numbers with the plus sign (+Os) and greater even numbers with the plus sign (+Eg) beginning the transformation period in each second semi-phrase. he algorithm for T-voice addition in the highest Tvoice is the simplest compared to two lower T-voices and consists only of the T-1 formula for T-voice addition. his MS3 from Ludus In this visualization each column signiies a new stanza. Horizontal space inside the column stands for semi-stanzas separation. Shades of grey colour express diferent M-voices. he diference consists of one of two types of preamble use before the main range of numbers in M-voice. he irst type of preamble is with the use of sixteens and eighths rhythmic values and the second type is with the use of eighths. he diference between two types of preambles lies in formula construction too, which is described farther in the article. Vertically added T-voice appears only in the second and third stanzas and in the last stanza it is replaced with Mvoice. he formula for T-voice addition undergoes change of the step from -/+1 to -/+2, which appears in the third stanza. he relation of the T-voices added to diferent semistanzas within each stanza has a reversed character. 99 Lietuvos muzikologija, t. 15, 2014 he relations of horizontally added T-voices within the structure of M-voices belonging to diferent semi-stanza within each stanza also have a reversed character. hus, it will be suicient to understand the logic of T-voice belonging to the irst semi-stanzas for understanding the logic of T-voice within the second semi-stanzas. T-voice from the irst semi-stanza of the irst stanza follows the logic of T-1/T+1 succession and it is repeated in the second and beginning of the third stanzas, but in the last fourth stanza the succession that begins the semi-stanza is reversed. he mirroring principle is inherent in the last two stanzas and resembles the same principle used for vertically added Tvoice to the middle M-voice in Spiegel im Spiegel. If we represent the primer succession T-1/T+1 as A and its reversed version as B, we receive the following formalization for the horizontally added T-voice within each irst semi-stanza: A A AUB BUA Naturally the logical formalization for the horizontally added T-voice within each second semi-stanza will be the reversed version of the irst formula: B B BUA AUB Preambles formation he algorithm of two types of preambles formation is more sophisticated than the simple alternation of M and T voices. Also it is developed from the end. hat means that the general formula was created initially and placed at the very end of the whole MS3 form. hen its elements where gradually reduced to the beginning of the form. he initial formula can be visually presented as follows: Full algorithm of preambles from MS3 of Ludus Two ranges from 7 to 0 relect the structure of two types of preambles. he range with the plus sign is inherent in the irst type of preambles and the range with the minus sign – in the second type. he range inherent in the second type is guided by simple alternation principle, but the irst type has two reduced elements (T+1) between the pair of even-odd numbers and one added element (also T+1) between the last pair of odd-even numbers. Percussion Percussion usually plays the role of a separator between stanzas in forms of Pärt’s compositions. It is the case of Arbos and Fratres. In Arbos three bells with a single sound with proper value (rhythmic values 1, 2, 3.5) for each of M-voices (with rhythmic values for each layer 1, 2, 4) mark 100 Anna SHVETS the end of stanza. In Fratres the percussion’s “separator” has an expression of two measures with repeated rhythmic formula. he organ version of Mein Weg has quasi percussion “dots” – eights for the main highest M-voice and quarters for the middle M-voice (Shvets, 2013), separated by pauses of the same rhythmic value from two sides. Cantus in Memory of Benjamin Britten makes an exception to this rule where the bell voice is not related to the M-voice structure and follows its proper mathematical logic: 3 sounded measures (alternated with 2 measures of pauses) and 3 measures of pauses. Ater being repeated 11 times the bell voice stops. hen ater 22 measures of pauses one last sound appears, becoming the end “dot” of the whole piece. he behaviour of the bell voice could be expressed by the following formula: (s_m + p_m) * 3) + 2 p_m) * 11) + 22 p_m) + 1s where s_m corresponds to the measure with sound, p_m to the measure illed with pauses and 1s to the one sound. he relation between number 11 (the times of bell metric structure repetition) and number 22 (number of measures of pauses after those repetitions), is evident, because the number 22 is the number 11 multiplied by 2. Conclusion he presented analysis of the form development in Pärt’s compositions has shown that his music uses a wide range of mathematical operations – from linear algebra (algebraic operations and combinatorics) to the elements of mathematical analysis (mathematical unity of sets). Such high level of algorithmization of creative process allows positioning the compositions of Pärt in a context of generative art. he new methodology of musical processes representation with the use of computer languages semantics was developed. he if statement, for loop and while loop statements were applied to the form structure representation in both senses – as expressions for M-voice development and as representation of regularities for the measure grid construction; nested for loop statement was applied for rhythmic values addition regularities. he application of new methodology allowed us to represent the algorithms inherent in the development of Pärt’s compositions in an appropriate way, considering the used creative strategies. he compactness of representation according to this methodology corresponds to the conditions of information society and allows farther processing of received algorithm in big data context. Literature Ahn, Yun-Kang. L’analyse musicale computationnelle : rapport avec la composition, la segmentation et la représentation à l’aide des graphes, PhD thesis, Université Paris VI – Pierre et Marie Curie, Decembre 2009. Mathematical Bases of the Form Construction in Arvo Pärt’s Music Andreatta, A. Méthodes algébriques en musique et musicologie du XXe siècle: aspects théoriques, analytiques et compositionnels, PhD thesis, École des Hautes Études en Sciences Sociales, Decembre 2003. Brauneiss, L. Musical archetypes: the basic elements of the tintinnabuli style. In: he Cambridge companion to Arvo Pärt, Cambridge University Press, 2012, pp. 49–75. De Paiva Santana, Ch.; Bresson, Jean. Vers la Modélisation des Pensées Musicales: Le cas du “Tintinabuli” d’Arvo Pärt. Conférence sur la Modélisation Mathématique et Informatique des Systèmes Complexes (COMMISCO 2012), Paris, France, 23–25 Octobre 2012, http://repmus.ircam.fr/_media/ depaiva/commisco2012.pdf. Greenbaum, S. Arvo Pärt’s “Te Deum”: A compositional watershed, PhD thesis, he University of Melbourne, June 1999. Lyotard, J.-F. La condition postmoderne: le rapport sur savoir, Editions de Minuit, 1979. Pärt, А. he truth is very simple. In: Sovietskaja muzyka (Russian translation by E. Michaltchenkova, in Russian), 1990, No. 1, pp. 130–132. Robinson, T. Analyzing Pärt. In: he Cambridge companion to Arvo Pärt, Cambridge University Press, 2012, pp. 76–110. Shenton, A. he Cambridge companion to Arvo Pärt, Cambridge University Press, 2012. Shvets, A. “Mein Weg” of Arvo Pärt in the context of postmodernism. In: Scientiic almanac of the Lviv University. Series: Art Studies (in Ukrainian), 2012, Issue 11, pp. 31–44. Shvets, A. Interactive application for visualization of the form of written postmodern music. In: Proceeding of EVA London 2013. British Computer Society (BCS), London, 29th–31st July 2013, p. 302–309. Zamornikova, K.; Katunyan M. Arvo Pärt’s “Fratres”: he Prayer in Music. In: Lietuvos muzikologija, Vol. 12, 2011, pp. 111–136. Zhdanovich, K. “Tabula Rasa” of Arvo Pärt as signiicant degree of creativity and example of tintinnabuli – style. In: Scientiic almanac of the National Musical Academy of P. Tchaikovsky (in Ukrainian), 2008, Vol. 76, pp. 39–48. Santrauka Iki šiol Arvo Pärto kūrybos tyrimuose buvo naudojami skirtingi metodai, nagrinėjantys glaudų grainių idėjų / eskizų ir garso įgyvendinimo muzikos kūriniuose ryšį (Shenton, 2012). Tokie metodai kaip stiliaus analizė, muzikinė hermeneutika, Schenkerio analizė, aibių teorija, triadinė transformacija ir kiti (Robinson, 2012) gali būti prasmingi tik iš dalies – jie nesuteikia galimybės visiškai suprasti Pärto kūrybinį procesą. Pavyzdžiui, Leopoldo Brauneisso (Brauneiss, 2012) pasiūlyta muzikos archetipų idėja nepasiteisina, kai tą pačią ritminę struktūrą galima rasti ne tik kompozicijoje „Arbos“, kuriai, kaip teigia muzikologas, struktūra buvo sukurta, bet ir kituose Pärto kūriniuose – „Mein Weg“, „Cantus Benjamino Britteno atminimui“, ritminės struktūros veidrodinę versiją – „Silentium“ iš „Tabula Rasa“. Taigi tradiciniai muzikos metodai nėra visai tinkami analizuojant Pärto kūrinius. Vietoj jų gali būti naudojami skaitmeninės / kompiuterinės muzikos analizės (angl. computational musicology) metodai. Šis su technikos atsiradimu siejamas analizės būdas atsirado XX a. ir atlieka muzikos skaitmeninio aspekto ir sąveikų tarp pavaizdavimo lygių, kurie naudoja šį skaitmeninį aspektą, teorinių modelių kūrimo paiešką (Ahn, 2009). Vienas iš skaitmeninės muzikologijos metodų yra algebrinė muzikos analizės kryptis, siūlanti matematinius metodus (Andreatta, 2003). Nors Pärto estetinė pozicija ir teigia, kad tai, kas gali būti išreikšta matematiškai, neturi nieko bendra su muzika (Pärt, 1990), savo kūriniuose kompozitorius pasitelkia matematiką, visokeriopą visų muzikinių parametrų skaičiavimą, taikomą bendrai struktūrai, melodijos modelių kūrimui, polifoniniam santykiui tarp balsų. Remiantis keliais Pärto muzikos pavyzdžiais, straipsnyje parodomi matematiniai dėsningumai, ypač susiję su formos konstrukcija. Taip pat šiuo straipsniu keliamas tikslas sukurti naują metodą, leidžiantį pavaizduoti muzikinį procesą kaip logines formuluotes panaudojant tokias modernias kompiuterines kalbas kaip „Java“, C++, „Python“ ar „Processing“. Analizei atrinkti šeši instrumentiniai kūriniai – „Cantus Benjamino Britteno atminimui“ (1977), „Arbos“ (1977), „Tabula Rasa“ (1977), „Fratres“ (1977), „Spiegel im Spiegel“ (1978) ir „Mein Weg“ (1989). Kūrinių pasirinkimą lėmė tai, kad muzikinės jų kalbos nevaržo literatūrinis tekstas ir žmogaus balsas, kurie gali sukelti tam tikrų analizės apribojimų. Pasirinktus kūrinius lengva palyginti jų konstrukcijos / struktūros aspektu, o tai labai svarbu komparatyvinei analizei. Panaudojant kompiuterinės kalbos semantiką sukurta nauja muzikos procesų pavaizdavimo metodologija. Tokios formuluotės kaip if, for loop ir while loop buvo taikomos vaizduojant formos struktūrą dviem prasmėmis – kaip M-balso plėtojimo išraiška ir kaip bendros konstrukcijos tinklelio dėsningumų pavaizdavimas; formuluotė nested for loop panaudota analizuojant adityvinio ritmo verčių dėsningumus. Naujos metodologijos pritaikymas leido pavaizduoti algoritmus, glūdinčius Pärto kūrinių plėtotėje ir susijusius su panaudota komponavimo technika / strategija. Šiam analizės metodui būdingas vaizdavimo glaustumas atitinka informacinės visuomenės sąlygas ir suteikia galimybę gautą algoritmą vėliau apdoroti plačiame duomenų kontekste. Pärto muzikoje konstruojamos formos analizė parodė, kad šio autoriaus kūriniuose panaudotas platus matematinių operacijų spektras nuo linijinės algebros (algebrinių operacijų ir kombinatorikos) iki matematinės analizės elementų (matematinio aibių vieningumo). Tiriant nustatyta, kad Pärto kūrybiniam procesui būdinga algoritmizacija šio kompozitoriaus kūrinius leidžia priskirti generatyvinio meno sričiai. 101