cs.SDMar 27, 2026
SaveAlgo Pärt: An Algorithmic Reconstruction of Arvo Pärt's Summa
Organizations: Institute for Logic, Language and Computation, University of Amsterdam, The Netherlands
Abstract
Arvo Pärt is one of the most popular contemporary composers, known for his highly original tintinnabuli style. Works in this style are typically composed according to precise procedures and have even been described as algorithmic compositions. To understand how algorithmic Pärt's music exactly is, this paper presents an analysis by synthesis: it proposes an algorithm that almost completely reconstructs the score of Summa, his "most strictly constructed and most encrypted work," according to Pärt himself in 1994. The piece is analyzed and then formalized using so-called tintinnabuli processes. An implementation of the resulting algorithm generates a musical score matching Summa in over 93% of the notes. Due to interdependencies between the voices, only half of the mistakes (3.5%) need to be corrected to reproduce the original score faithfully. This study shows that Summa is a largely algorithmic composition and offers new perspectives on the music of Arvo Pärt.
Figures & tables
Figure 1: Excerpt of Für Alina (mm. 2–7). This piano piece was Arvo Pärt’s first work in his tintinnabuli style. The right hand plays a melodic voice ( m -voice) that mostly moves stepwise, and which is freely composed. The left hand has the tintinnabuli voice ( t -voice), that is restricted to notes from the B-minor triad. The relation between the two voices is shown on the right: the t -voice plays the highest triad note below the m -voice, but one octave lower.
Figure 2: Tintinnabuli positions for an A minor triad. Solid notes show the A minor scale as a melodic voice, and open notes show tintinnabuli voices in the five different positions introduced by Hillier (1997) . His terminology is shown above the staff, the numbering used in section 4 below it.
Figure 3: The melodic structure of Fratres is a series of variations of a six-bar theme, of which the first two are shown. The theme leaves and then approaches a pitch center, moving one step further for three consecutive bars. The next three bars repeat the first three but played backward. This results in four types of melodic movement, or modes , that are often used by Pärt to compose m -voices. The first staff also shows the parallel m -voice and the t -voice as small notes.
Figure 4: Four melodic modes commonly used by Arvo Pärt to construct m -voices: two ways to move away from a central pitch, and two to approach it ( Hillier, 1997 ) . All of these are found in Fratres , as shown in figure 3 .
Figure 5: The opening measures of Summa , a setting of the Credo . The piece consists of 16 three-bar sections containing 7, 9 and 7 syllables. The voice distribution is mirrored in every section: sa – satb – tb . The alto and bass have the m -voices, the soprano and tenor the corresponding t -voices. If one skips the small, slurred ‘ornaments’, the m -voices walk up and down an E natural minor scale, while the t -voices are constrained to the E minor triad.
Figure 6: The melodic voices each repeat a 15-note pattern, that walks up and down a scale, throughout the piece. Diatonic pitch is shown vertically and time horizontally, measured in syllables. The notes are shown as dots and ornaments have been omitted. The repetitions of the underlying 15-note pattern are shown in the background. In every section (marked by rehearsal numbers), a voice sings the tonic (E) followed by this pattern, but rotated one step to the left.
Figure 7: Approximate patterns for all voices. These patterns of notes (•) and ornaments (+), when repeated throughout the piece, approximate the melodies of each of the voices. The approximate patterns were constructed manually by comparing all repetitions, so as to make the approximation as good as possible (see Appendix B ). For the t -voices, they however remain approximations: they are better understood as functions of the m -voices (Figure 8 ), not as repetitions of the patterns shown here.
Figure 8: Three tintinnabuli processes for a melody that walks up and down a scale. The constant process (A) remains in the same tintinnabuli position , while the alternating process (B) flips the sign of the starting position every step. The step process (C) only moves to neighbouring triad notes, while keeping a distance of at least triad notes from the melody. This is the process used in Summa .
Figure 9: Evaluation of the algorithmic reconstruction illustrated in (A) by comparing the first bar (bottom staves) with the original (top staves). We encounter four types of reconstruction errors: ornament insertions (red), ornament deletions (blue), duration errors (green), and pitch errors (not shown). In total, we find 86 errors (6.7%) after adjusting ornamented exits (B) . Over half of these are duration errors, resulting from ornament insertions or deletions. And so only 43 ornaments and two pitches need to be corrected (3.5%) to reproduce the original score.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
| Section | Bar | Voices | Syllables | Text | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | sa | 7 | Cre- | do | in | u- | num | de- | um, | ||
| 2 | satb | 9 | Pa- | trem | o- | mni- | po- | ten- | tem, | fa- | cto- | |
| 3 | tb | 7 | rem | coe- | li | et | ter- | rae, | vi- | |||
| 2 | 4 | tb | 7 | si- | bi- | li- | um | o- | mni- | um, | ||
| 5 | satb | 9 | et | in- | vi- | si- | bi- | li- | um, | et | in | |
| 6 | sa | 7 | u- | num | Do- | mi- | num | Je- | sum |
Table 1: Structure of Summa . The piece consists of 16 sections, marked by rehearsal numbers, of each 3 bars. Measures in a section contain 7, 9 and 7 syllables respectively, and use different voices: sa , then satb and finally tb . The next section mirrors this structure. Syllables are distributed across the bars following this scheme, even if this means that a bar line falls in the middle of a word.
Figure 10: Pattern repetitions of the melodic voices. All repetitions of the alto (A) and bass (B) are plotted above one another. We manually identified an approximate pattern of notes and ornaments (shown in the background) that best matches all of the repetitions. In other words, it minimizes the number of deviations. For the t -voices (Figure 11 ), this turned out to be a crucial step in understanding their construction.
Figure 11: Pattern repetitions of the tintinnabuli voices. This is very similar to Figure 10 , but now the patterns are twice as long.
Figure 12: The RepeatPrevious process generates the ornaments for the soprano (A) and tenor (B) by repeating the previous note if this is not equal to the next note. As explained in the main text, this process has several parameters that constrain the range of the ornaments. For the soprano the ornaments have to lie between and , while the next note has to fall below . For the tenor, we use , and .
Figure 13: The TailRotatedPatternProcess process generates the ornaments for the alto (A) and bass (B) parts. The black lines show the respective melodies, and ornaments are indicated by coloured plusses. It essentially repeats a 16-note pattern of ornamentation, but rotates the tail every time to keep the ornamentation in sync with the melody (see main text for details).
Figure 15: Final bars of Summa . The ending of Summa (A) is more freely composed than the rest of the piece and deviates from the patterns observed before. The alto finishes the last repetition of the basic pattern in a four-note melisma, to end on the tonic. Meanwhile, the bass and tenor hold an open fifth on ’Amen’. The reconstruction (B) of course cannot accurately reproduce these measures. We treat the errors as ornament insertions, and count 2 extra insertions for all voices.