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

Abstract

The 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 reflection of the informational society and more specifically – of the logic used in modern programming languages, such as Java, C++, Python, Processing etc. The 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 sufficient to correctly display these algorithms. Thus, a new method of music representation formalization will be presented. This 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. The 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.

Full text

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Lietuvos muzikologija, t. 15, 2014
Introduction
In previous studies of Arvo Pärt’s creative style dierent
approaches conrming the close relation between external
graphical ideas and sound implementation within his com-
positions 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.
e 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. us,
the traditional musicological approaches are not suciently
adequate for Pärt’s compositions analysis.
Instead, the computational musicolog y approaches may
be used. Computational musicology appeared with the
appearance of technology in the 20th century and consists
of search for a computational aspect of music and devel-
opment 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 eve-
rything 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 g enera l form structure, in formation of me-
lodic patterns, in polyphonic relation between voices. us
the aim of the present article ma inly consists of showing these
mathematical reg ularities, especially of the form construction,
Anna SHVETS
Mathematical Bases of the Form Construction
inArvo Pärt’s Music
Arvo Pärto kūrinių struktūros matematinis pagrindimas
Abstract
e 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 reection of the informational society and more specically– of the logic used in modern programming
languages, such as Java, C++, Python, Processing etc. e existing approaches of computational musicology or basically mathematical ap-
proaches partly allow to discover and to represent the algorithms under which the compositions were created. However in particular cases
it is not sucient to correctly display these algorithms. us, a new method of music representation formalization will be presented. is
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. e 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 Ar vo Pärto kūrybos technika paremta tiksliais visų kūrinio parametrų apskaičiavimais. Toks kūryboje naudojamas skai-
čiavimo 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 nepa-
kanka šių algoritmų vien tik parodyti. Straipsnyje pateikiamas naujas muzikos formalizavimo metodas. Naudojamos programavimo kalbų,
kuriomis siekiama logiškai pavaizduoti vykstančius pro cesus, 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.

89
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, allow-
ing 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). e main reason
for the instrumental compositions selection is their freedom
from text and human voices limits, which can add some sort
of restrictions. e selection of the specied compositions
is made under their constructional comparability, essential
for comparative analysis.
Mapping principle
Computational analysis requires the mapping of notes
to numbers. is 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. e 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– va lue of number “2” with minus sign etc.
Fig. 1. Mapping method illustration
Such mapping method was used to Pärt’s composi-
tions in previous works for analysis of Mein Weg structure
(Shvets, 2012), computer modelling of Spiegel in Spiegel
(De Paiva Santana, 2012). e attempts of this method’s
application have been used to the Fratres structure analysis
(Zamornikova, 2011), but inappropriate values assignment
caused the absence of signicant result of the method ap-
plication: all the steps above the “0” note were noted as
number “1” and all the steps below– as number “1” with
“-” sign. us, the ranges aecting the structure congura-
tion were not received.
Form building algorithms
In numeric representation
e construction of the form in Pärt’s compositions is
far from traditional musica l forms organization and relies on
the main melodic voice development. Sometimes the com-
poser 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. e symmetry laws also
play a signicant 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). ese
algorithms are visible on the micro level of each piece con-
struction 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.
e micro level of each piece composition is constructed
from larger and smaller structures, subordinated hierarchi-
cally between them. ese larger and smaller structures are
classied 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 con-
tains 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.
e use of mathematical operations within each struc-
ture has several levels of interactions:
•Stanza-stanza level;
•Phrase-phrase level;
•Element-element level.
e implementation of logic operations will be consid-
ered now in concrete examples. For this implementation
only a few rst stanzas of the main M-voice of each piece
will be shown to set the principle of the form algorithm
formation. e whole ranges in graphical representation
will be shown later.
e interactions on stanza-stanza level are inherent in
the structure of Fratres. In this case the replacement of ele-
ments within the phrases of neighbouring stanzas occurs
(-1 +1 towards +1 -1).
Fratres
1st stanza 2nd stanza
1st p. 0 - 1 + 1 0 0 + 1 - 1 0
2nd p. 0 - 1 - 2 + 2 + 1 0 0 + 2 + 1 - 1 - 2 0
3rd p. 0 - 1 - 2 - 3 + 3 + 2 + 1 0 0 + 3 + 2 + 1 - 1 - 2 - 3 0
e interactions on the phrase-phrase level are inherent
in the Mein Weg and Spiegel im Spiegel structure. ese cases
are rich of mathematical transformations such as:
Vertically mirrored ranges within the phrases (+2+1
towards -2 -1) of the same stanza;

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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).
Mein Weg
init 0 +1 -1 0
1st st. 1st p. +2 +1 0 -2 -1 0
2nd p. +1 +2 +3 0 -1 -2 -3 0
2nd st. 1st p. +4 +3 +2 +1 0 -4 -3 -2 -1 0
2nd p. +1 +2 +3 +4 +5 0 -1 -2 -3 -4 -5 0
Spiegel im Spiegel
1st st. 1st p. -1 0 +1 0
2nd p. -2 -1 0 +2 +1 0
2nd st. 1st p. -1 -2 -3 0 +1 +2 +3 0
2nd p. -4 -3 -2 -1 0 +4 +3 +2 +1 0
e interactions on the element-element level are in-
herent in both parts of Tabula Rasa, in Arbos and Cantus
in Memory of Benjamin Britten. ese 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.
Ludus
1st phrase
1st element 2nd element
1st stanza: 0 +1 0 -1 0
2nd stanza: 0 +1 +2 +1 0 -1 -2 -1 0
3rd stanza: 0 +1 +2 +3 +2 +1 0 -1 -2 -3 -2 -1 0
2nd phrase
1st element 2nd element
1st stanza: 0 -1 0 +1 0
2nd stanza: 0 -1 -2 -1 0 +1 +2 +1 0
3rd stanza: 0 -1 -2 -3 -2 -1 0 +1 +2 +3 +2 +1 0
Silentium
1st phrase 2nd phrase
1st stanza 0 +1 0 -1
2nd stanza 0 +1 +2 +1 0 -1 -2 -1
3rd stanza 0 +1 +2 +3 +2 +1 0 -1 -2 -3 -2 -1
Arbos Cantus
1st stanza 0 -1 0 -1
2nd stanza 0 -1 -2 0 -1 -2
3rd stanza 0 -1 -2 -4 0 -1 -2 -3
4th stanza 0 -1 -2 -4 -3 0 -1 -2 -3 -4
5th stanza 0 -1 -2 -4 -3 -4 0 -1 -2 -3 -4 -5
In graphical representation
All the ranges of numbers presented above can be rep-
resented also with graphics as histograms. e graphical
representation will clear up the similarities of the structure
algorithms between the pieces. With graphical representa-
tion 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
Arbos
It is also obvious that the form structure algorithm of
Mein Weg is a reversed version of Spiegel im Spiegel:
Spiegel im Spiegel

91
Mathematical Bases of the Form Construction in Arvo Pärt’s Music
Mein Weg
e idea of question-answer relations appears in the
Ludus part of Tabula Rasa (element-element level) and
Fratres (phrase-phrase level); Silentium p art of Tabula Rasa
appears as a simplied version of Ludus.
Fratres
Ludus
Silentium
In formulas representation
e application of algebra operations with symmetry
laws allows us to represent the M-voice development as
mathematical formulas or conditions of a computer pro-
gram. e 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 execut-
ing repeated iterations of values. e for loop statement in
programming is usually used when the amount of iterations
is predened before entering the loop:
for ( i = 0 ; i < x; i++ ) {
x = x * (-1)}
is 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.). e “for” operator means
that we want to make the iteration of number i, which ap-
pears in condition of the statement. e assignment of x to x
multiplied by (-1) means that each item of the range that we’ll
receive will be with “-” operator. is co de returns numerical
range of the form development algorithm shown in numeric
representation for Cantus in Memory of Benjamin Britten.
e for loop statements are useful for representation of
the nal numeric values of the range’s development. e
most important x-value within the loop (the nal point of
ranges increase) can be also presented in terms of mathemat-
ics as n-point of range increase.
Earlier the dierences of the form development algo-
rithms on rst few stanzas example were shown. However
the whole form representation requires the nal 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 ecient in the
processing of such amount of information. Only the codes’
representation with the initial and nal set of values could re-
solve the problem of large quantity of information processing
to be used in comparative analysis. e initial set of values for
the main M-voice is especially useful in comparison to dier-
ent M-voices within the same piece and will be shown later.
e graphical visualization of the whole main M-voice
structure allows us to follow farther form a lgorithms’ devel-
opment 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.
e comparison of these graphical representations claries
that similar initial algorithm of development can result in

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Lietuvos muzikologija, t. 15, 2014 Anna SHVETS
Fig. 2. e whole range of the main M-voice from Cantus in Memory of Benjamin Britten
Fig. 3. e whole range of the main M-voice from Arbos
Fig. 4. e whole range of the main M-voice from Spiegel im Spiegel
Fig. 5. e whole range of the main M-voice from Mein Weg
Fig. 6. e whole range of the main M-voice from Fratres
Fig. 7. e whole range of the main M-voice from Ludus
Fig. 8. e whole range of the main M-voice from Silentium

93
Mathematical Bases of the Form Construction in Arvo Pärt’s Music
dierent forms conguration 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 aer reach-
ing the highest n-point of range increase in graphical repre-
sentation. e common feature for all seven main M-voice
forms is aspiration to achieve the n-point of number range,
which also is the nal point in the form development. e
form of the main M-voice from Mein Weg is an exception,
where aer the increase until the highest n-point of ranges,
gradual decrease back to 0 occurs.
e 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. e occurred repetitions
aer 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 rst and
the second stanza (in case of Fratres) x times:
if (n-point = stanza[i])
stanza[] * x
e result of the number of stanzas change is visible on
the example of the comparison of the main M-voice struc-
tures 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.
is parameter is important for comparison of the other
M-voices with the main M-voice within the same piece.
e 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 dierent layers. ese data can be
visualized further.
Types of complementarity
e ways of vertical complementarity in analy zed 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
gures (Spiegel im Spiegel, partly Ludus);
3) Canonic shiing of the M-voices containing the same
rhythmic values (Ludus).
e vertical complementarity is not a single type of
complementarity which can be found in Ar vo Pärt’s music.
e horizontal complementarity also occurs with measure
structures division mentioned above. ere are two cases
of such measure structures in selected compositions, which
dier by its use.
e rst level of use consists of mathematical operations
that could be performed with these structures, but it does
not aect the general structure of M-voices. is level of
measure structures division is used in the Fratres and Ludus
part from Tabula Rasa.
e second level consists of inuencing M-voices devel-
opment because measure structures play the role of frame
slots, which can be lled out only with a special type of
information. Each measure structure has its own set of M-
voices and they are developing independently within separate
measure structures. e 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). e 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”).
e laws of internal and external symmetry are also in-
herent in the measure structures construction. e 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.
e external symmetry consists of the use of meter “6/4” at
the beginning and the end of the piece. e 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
dene its general measure structures construction. Ludus
can be devised into two main parts– before cadenza and
aer. e 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 b efore
culmination has four stanzas (devised into 8 semi-stanzas)
and cadenza-culmination part consists of two internal
1 S ome researches (Zamornikova , 2011) assume the presence of
such level use in Fratres. ey believe that such division occurs
between the S-part (M-voice stanzas) and R-part, consisting of
the two measures lled out with ostinato percussion rhythmic
gure. 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.

94
Lietuvos muzikologija, t. 15, 2014 Anna SHVETS
culminations– rst 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). e
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 rela-
tions. Each semi-stanza contains three measure structures.
uestion-answer relations are inherent in only the third
measure structure (MS3). e measure structures division
is made under meter change: each rst 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
lled with these meters make part of the MS1. e meter of
the second measure structure (MS2) is constantly changing
being aected by the subtraction operator. e third and the
last measure structure (MS3) consist of measures with 4/4
meter. e measure structure principle of division described
is shown in the following table:
Ludus measure structures division
MS1 Comment for
MS1
MS2 Co mm e nt
for MS2
MS3 Comment for MS3
1st stanza 1st s.-stanza 6/4 08/2 G.P. 4/4 (6) +6
2nd s.-stanza 5/4 6/4 17/2 G.P. 1m. unit 4/4 10 +4 +6
2nd stanza 1st s.-stanza 6/4 (3) +3 measures 6/2 G.P. -1m. unit 4/4 12 +2
2nd s.-stanza 5/4 6/4 (4) +3 measures 5/2 G.P. 1m. unit 4/4 16 +4 +6
3rd stanza 1st s.-stanza 6/4 (6) +3 measures 4/2 G.P. 1m. unit 4/4 18 +2
2nd s.-stanza 5/4 6/4 (7) +3 measures 3/2 G.P. 1m. unit 4/4 22 +4 +6
4th stanza 1st s.-stanza 6/4 (9) +3 measures 2/2 G.P. 1m. unit 4/4 24 +2
2nd s.-stanza 5/4 6/4 (10) +3 measures 1/2 G.P. 4/4 30 +6
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 log ic of while loop statement consists of
code execution until the condition is true) with a dierent
initial value for the rst and the second semi-stanzas within
each stanza: for the rst 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 rst semi-stanzas:
x = 0
while ( x <= 9){
x+3}
b) While loop for the second semi-stanzas:
x = 1
while ( x <= 10){
x+3}
e rst while loop will return values 3, 6, 9, which are
the numbers of added measures for the rst semi-stanzas
of the second, third and fourth stanzas consequently. e
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 rst 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-- ) {…}
is code performs gradual decrement (by -1) from
value 8 to 1 and it reects a gradual decrement of quantita-
tive metric units in MS2 in each semi-stanza. us it beg ins
with meter 8/2 in the rst semi-stanza of the rst stanza
and it nishes 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 sym-
metrical laws: the rst 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).
e addition between these rst 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. e same 4 followed by 2 added measures
are repeated yet in the pairs of 4th–5th and 6th–7th semi-
stanzas. e 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 shiing 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 symmetrical ly equal ascending

95
Mathematical Bases of the Form Construction in Arvo Pärt’s Music
order in each semi-stanza. e question-answer relations
between rst and second semi-stanzas occurs on this level
too: the rst semi-stanza contains two phrases with shied
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. e
described regularities can be expressed by means of formula
where the dash signs (“-”) means simultaneity and plus signs
(“+”) complementarity or shiing:
1st phrase 2nd phrase
1st s.-st.: (M-T)*2 + (M-T) | (T-M) + (M-T)*2
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. is time it aects the
quantity of added quarters in bound between phrases. e
bound consist of repeated quarters on the same pitch “A2”
with staccato articulation signs. e quantity of quarters in
the rst semi-stanza equals to 10 and in the second semi-
stanza– to 12, thus the quantity of added quarters between
two semi-stanzas within the same stanza equals to 2. is
regularity will be held for the rest of other pairs of semi-
stanza. e 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 rst stanza was 12, the
quantity of these measures in the rst 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. is regularity can be expressed with nested
for loop, where outer iteration (+6 or i) is executed aer
inner iteration (+2 or j). e 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 aer
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 struc-
ture. e 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
signication: 0.5 = eights, 1 = quarter, 2 = half note, 4 =
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
Cantus in Memory of Benjamin Britten
Arbos
Silentium
Fratres
Spiegel im Spiegel
e described principle of two times rhythmic values
augmentation is used for Mein Weg, Cantus in Memory of
Benjamin Britten and Arbos. e vertical rhythmic con-
struction of Silentium appears as reversed copy of Arbos.
M-voices inherent in Fratres and Spiegel im Spiegel are
guided by equivalenc y principle in their rhythmic cells. e
formula 1*n in Fratres means that the quantity of quarters
varies according to the used meter (7/4, 9/4, 11/4), but
rst half note (2) and last dotted half note (3) remains in
each metric cell, creating the structure of frame with core.
is 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.
e Ludus part is not shown within these representa-
tions because of its horizontal complementarity of the sec-
ond type (which aects the general form structure constr uc-
tion) instead of vertical and will be analyzed in other way.
It is reasonable to analy ze each measure structure separately
by proposing the whole tables of rhythmic development,
because it diers from the rst or second semi-stanza type.

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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
diers a lot– from rhythmic values to rhythmic progres-
sions. Using the same numeric representation of rhythmic
values, let us present the rhythm structure for MS1 with
the following table:
Ludus MS1
1st
stanza
1st s.-st. 6
2nd s.-st. 2 + (3 * 3)
2nd
stanza
1st s.-st. (1.5 * 4) + (3 * 4)
2nd s.-st. 2 + (3 * 3) + (1.5 * 4) + (3 *4)
3rd
stanza
1st s.-st. ((1.5 * 4) + (3 * 4)) * 2
2nd s.-st. 2 + (3 * 3) + ((1.5 * 4) + (3 * 4)) * 2
4th
stanza
1st s.-st. ((1.5*4)+(3*4))*3
2nd s.-st. 2 + (3 * 3) + ((1.5 * 4) + (3 * 4)) * 3
e multiplication sign means repetition of the same
rhythmic values and plus sign– simple neighb ouring of dif-
ferent rhythmic values. If we store the initial rhythmic value
6 (dotted whole note) from the rst 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 simplied version of the same table :
Ludus MS1
1st
stanza
1st semi-stanza x0
2nd semi-stanza y
2nd
stanza
1st semi-stanza x
2nd semi-stanza y + x
3rd
stanza
1st semi-stanza x * 2
2nd semi-stanza y + (x *2)
4th
stanza
1st semi-stanza x * 3
2nd semi-stanza y + (x * 3)
e rhythmic range corresponding to x is present in
every semi-stanza from the second stanza. e range stored
in y plays the role of the beginning of entire rhythmic range
inherent in the second semi-stanza. e formula of rhyth-
mic progression for whole stanza looks as follows:
(x * n) + (y + (x * n))
where n is a number of repetitions from 1 to 3.
e 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).
e MS3 section contains multiple M-voices which
can be divided into two groups regarding used rhythmic
progressions– static and dynamic. e static group contains
rhythmically identical M-voices consisting of quarters (1).
e quantity of M-voices within the static group changes
from three to four upon rst to second semi-stanza’s type
use. e dynamic group contains three M-voices which
rhythmically vary between them and inside themselves.
e rst M-voice generally consists of eighths (0.5) with
the possible insertion of two sixteens on feeble times. e
second M-voice consists of triples (0.3) and the third M-
voice– of sixteens (0.25) only2.
A general table of rhythmic values within the dynamic
group of MS3 from the rst semi-stanza looks as follows:
Ludus MS3 dynamic group
1st M-voice: (0.5 * 2) * 3 + dot
2nd M-voice: (0.3 * 3) * 3 + dot
3rd M-voice: (0.25 * 4) * 3 + dot
e formulas in parenthesis correspond to one quarter
with an appropriate number of repetition of smaller values
in it. e multiplication of values in parenthesis with a
specied number means the repetition within the same
measure where “dot” is included. Each next semi-stanza
adds one more measure to each M-voic e of this g roup before
primarily appeared progression. us, 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 nishing formula ((value*n)*3+dot)
as F, we receive the next formula:
(A * n + F) * n
Considering the dynamic changes in rhythmic progres-
sions 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 aer the regularities are set
already (second stanza):
Ludus
2 e bigger values that can appear at the end of each semi-
stanza for each M-voice are considered as “dots”. us, their
rhythmic values are not taken into consideration.

97
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. e reg ularit y of
two times augmentation/reduction was described already,
but within this piece the combinatorics plays decisive role
in domain of rhythmic development. At rst sight the
combinatorics nishes 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 rst 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 (aer four stanzas), a new c ycle with the same
order of combinations begins:
Arbos rhythmic structure combinatorics
st stanza ba
2nd stanza aa
3rd stanza ab
4th stanza bb
Described rhythmic combinatorics is inherent in all
M-voices of the piece. e values of a and b in the three
M-voices dier from the applied law of vertical two times
augmentation/reduction regularity.
Similar mathematical regulations are used in develop-
ment 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. Aer one measure of primer initialization
of movement, the regularities are set. Each stanza contains
three rhythmic sections. ese rhythmic sections are sepa-
rated by a pause equal to one rhythmic value inherent in
specied 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
1st section 2nd section 3rd section
init. (2 4)*2 2
1st st. 3 (2 4) 1 (4 2)*2 4 1 (4 2) 4
2nd st. 3 (2 4)*2 1 (4 2)*4 4 1 (4 2)*2 4
3rd st. 3 (2 4)*3 1 (4 2)*6 4 1 (4 2)*3 4
Repetition structure in the second and third sections
consists of the frame (1 4) and core (4 2). e number
of repetition of the core in the second section equals to
the formula: x*2, in comparison to the third section. e
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 (aer 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;
Aer 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
Aer reduction of repeated elements, the next formula
appears:
c (b d)*N | a (d b)*E/N d |
where the slash sign “/” means “or aer”. e reversed
succession of elements within the core in the rst section
compared to next two sections to the use of reversion
method for numeric rang e of pitches iterations widely used
in general M-voice construction in Mein Weg.
e 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 be-
long to quarter (1) and three times repeated (plus 8th pause)
or four times repeated eighths belong to a bigger rhythmic
value– half note (2). e 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 repeti-
tions as a and the (2 1)*E/N as b, we receive the following
formula view:
2 a / 1 b 2 / 1 b 2
Clear alternation of bigger value with smaller appears:
2 1 2 1 2
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) corre-
spond to the rst 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

98
Lietuvos muzikologija, t. 15, 2014 Anna SHVETS
split together in the third section. e formalization of the
regularity described looks as follows:
1st section +Eg => +Os
2nd section –Eg =>–Os | +Os => +Og
3rd section –Os =>–Og
Certain combinatory regularities are visible in this for-
malization too, if we consider a few restrictions:
•e numbers of one numeric range movement have the
same sign as the rst number of the range;
•e 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
rst and last numeric range movements. Storing +/-Eg in
A, +/-Os in B and +/-Og in C we receive the following
formalization view:
1st section A B
2nd section A B U B C
3rd section B C
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:
•e same for each layer (Cantus in memory of Benjamin
Britten);
•Diers from layers, but is held within each layer (Mein
Weg );
•Diers from layers and within each layer (Fratres, Arbos,
Silentium, Ludus, Spiegel im Spiegel);
2. Rhythmic correlation:
•Rhythmic values of M-voices are duplicated in cor-
responding 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):
Mein Weg
Cantus in Memory of Benjamin Britten
Arbos
Silentium
Fratres
Spiegel im Spiegel
Ludus
e quantity of T-voices diers from the piece, and usu-
ally equals to one T-voice per one M-voice, but two pieces
make an exception– Arbos with two T-voices per one mid-
dle M-voice and Fratres with one T-voice for two M-voices.
e 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).
e third group of pieces according to the formula
correlation criterion can be ranged regarding the diculty
of the algorithm used. e simplest algorithm is a literal al-
ternation used for both parts of Tabula Rasa. e mirrored

99
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 dicult algorithm is used in the higher T-voice
related with middle M-voice from Arbos. e formula for
T-voice addition depends on the specied 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 fol-
lowed by tintinnabuli-tone g uided 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).
e 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. Etc.
e lowest T-voice e middle T-voice
e formalization of the lowest T-voice addition al-
gorithm 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 -Os => -Og z | +Os => +Og z
2nd phrase -Eg => -Os z | +Eg => +Os z
e 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
aer range from a greater even number with the minus sign
(-Eg) to the smallest odd number with the minus sign (- Os).
e middle T-voice algorithm consists of alternation
and symmetry principles combination: phrases always be-
gin 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) b eginning the
transformation period in each second semi-phrase.
e algorithm for T-voice addition in the highest T-
voice is the simplest compared to two lower T-voices and
consists only of the T-1 formula for T-voice addition. is
T-voice appears as dispersed in accompaniment gure and
can be also considered in the context of horizontal cor-
relations too.
Horizontal correlations
Horizontal correlations of T-voice with M-voice are
more dicult to perceive at rst sight, because they are inte-
grated into the structure which can be treated as melody, but
knowing the range of M-voice, T-voice within its structure
can be found. ere 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 specied
algorithm (preamble to M-voice of violin solos from MS3
of Ludus);
3) T-voice alternation with two dispersed in accompani-
ment M-voices (Spiegel im Spiegel);
4) M-voice germination into T-voice (T-voice of violin
solos from MS3 of Ludus);
e dynamic changes in the third measure structure
from Ludus expressed earlier in rhythmic progressions
(two violin solos) within each stanza nd its application
on the level of T and M-voices horizontal correlations. In
this case the dynamic changes aect 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.
e combinatory eect is augmented when formally
M-voice (alternated M-voice with T-voice) is followed by
separate T-voice. e visual representation of four stanzas
looks as follows:
MS3 from Ludus
In this visualization each column signies a new stanza.
Horizontal space inside the column stands for semi-stanzas
separation. Shades of grey colour express dierent M-voices.
e dierence consists of one of two types of preamble use
before the main range of numbers in M-voice. e rst type
of preamble is with the use of sixte ens and eighths rhythmic
values and the second type is with the use of eighths. e
dierence 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 M-
voice. e formula for T-voice addition undergoes change
of the step from -/+1 to -/+2, which appears in the third
stanza. e relation of the T-voices added to dierent semi-
stanzas within each stanza has a reversed character.

100
Lietuvos muzikologija, t. 15, 2014 Anna SHVETS
e relations of horizontally added T-voices within the
structure of M-voices belonging to dierent semi-stanza
within each stanza also have a reversed character. us, it
will be sucient to understand the log ic of T-voice belong-
ing to the rst semi-stanzas for understanding the logic
of T-voice within the second semi-stanzas. T-voice from
the rst semi-stanza of the rst 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. e
mirroring principle is inherent in the last two stanzas and
resembles the same principle used for vertically added T-
voice to the middle M-voice in Spiegel im Spiegel. If we rep-
resent 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 rst 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 rst formula:
B B BUA AUB
Preambles formation
e 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. at means that
the general formula was created initially and placed at the
very end of the whole MS3 form. en its elements where
gradually reduced to the beginning of the form. e initial
formula can be visually presented as follows:
Full algorithm of preambles from MS3 of Ludus
Two ranges from 7 to 0 reect the structure of two types
of preambles. e range with the plus sign is inherent in the
rst type of preambles and the range with the minus sign–
in the second type. e range inherent in the second type is
guided by simple alternation principle, but the rst typ e 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
the end of stanza. In Fratres the percussion’s “separator” has
an expression of two measures with repeated rhythmic for-
mula. e 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 excep-
tion 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. Aer being repeated 11
times the bell voice stops. en aer 22 measures of pauses
one last sound appears, becoming the end “dot” of the whole
piece. e 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 lled with pauses and 1s to the one sound.
e 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
e 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 math-
ematical analysis (mathematical unity of sets). Such high
level of algorithmization of creative pro cess a llows position-
ing the compositions of Pärt in a context of generative art.
e new methodology of musical processes representa-
tion with the use of computer languages semantics was de-
veloped. e 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 rep-
resentation of reg ularities for the measure grid construction;
nested for loop statement was applied for rhythmic values
addition regularities. e application of new methodol-
ogy allowed us to represent the algorithms inherent in the
development of Pärt’s compositions in an appropriate way,
considering the used creative strategies. e 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.

101
Mathematical Bases of the Form Construction in Arvo Pärt’s Music
Andreatta, A. Méthodes algébriques en musique et musicologie du
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PhD thesis, École des Hautes Études en Sciences Sociales,
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Santrauka
Iki šiol Arvo Pärto kūrybos tyrimuose buvo naudoja-
mi skirtingi metodai, nagrinėjantys glaudų granių 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 kūry binį pro cesą. Pavy zdžiui, Leopoldo Brau-
neisso (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 musicolog y) 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 mu-
zikologijos metodų yra algebrinė muzikos analizės kryptis,
siūlanti matematinius metodus (Andreatta, 2003).
Nors Pär to estetinė pozicija ir teigia, kad tai, ka s gali būti
išreikšta matematiškai, neturi nieko bendra su muzika (Pärt,
1990), savo kūriniuose kompozitorius pasitelkia matema-
tiką, 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 formu-
luotes 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 apribo-
jimų. Pasirinktus kūrinius lengva palyginti jų konstrukcijos /
struktūros aspektu, o tai labai s varbu komparatyvinei analizei.
Panaudojant kompiuterinės kalbos semantiką sukurta
nauja muzikos procesų pavaizdavimo metodolog ija . 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 dėsnin-
gumus. 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 opera-
cijų 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.