Mapping Tintinnabuli Transformations

My project, Tintinnabuli Mathematica, involves the process of trying to understand Arvo Pärt’s tintinnabuli method of composition in order to use this knowledge for my own music-making. This post summarizes what I’ve learned so far about the compositional rules for transforming a sequence of notes in a melody (the M-voice) into tintinnabuli harmonies (T-voices).

The tintinnabuli method transforms the notes of the M-voice into notes in the scale’s triad, either above or below the original pitch. For example, the first T-voice above the M-voice (T+1) takes the first note in the triad above the M-voice pitch. The T+2 voice takes the second note above the M-voice, and so on. In Pärt’s compositions, T-voices may use these transformation  rules consistently or may alternate between them. My aim in this project is to code this generative process into a program to create musical sequences, and to use the code to explore and extend the process. For this purpose, I drew up a couple of charts that map the transformation rules. These simple charts allowed for an analysis of the process, and revealed that there are six different forms of T-voice. The following charts use scientific pitch notation, and are based on the A natural minor scale (or A Aeolian mode), which is often used by Pärt and is easy to show because it doesn’t contain sharps or flats. Other scales or modes produce quite different sounds, and the amount of consonance/dissonance varies not only with the chosen scale but also as the T-voices are closer or further away from the M-voice.

The first chart is a concise representation of the tintinnabuli rules. The colours represent pitches: the scale forms a spectrum of colours in which the triad (A-C-E) is red-yellow-blue. Each row represents a voice, with the M-voice in the middle. The chart shows which notes each of the T-voices take for each of the M-voice notes, by looking at the columns. For example, if the M-voice is sounding the note D5, the corresponding T+1 pitch will be the first note in the triad above it, which is E5. The pattern repeats in both directions, such that T+4 is equivalent to T+1, etc. This chart also demonstrates that there are six unique T-voices, i.e. that pairs of the upper (+) and lower (-) T-voices are dissimilar. For example, T+3 is identical to T-1 except when the M-voice is sounding one of the notes in the triad. The same goes for the pairs T+2 & T-2 and T+1 & T-3.


The second chart looks at the same transformation rules in another way. This time, the rows represent notes in the scale rather than the voices. A range of 3 octaves is shown here, with higher pitches towards to top. The M-voice is shown in the middle again, but in this arrangement it forms a diagonal line. The six T-voices are coloured differently. because of this, you can see how the pattern of voices repeats, such that T+1 comprises the same pitch classes as T+4. Like the previous chart, it also shows that the six voices are unique because they don’t align with each other.


These charts have been useful for the purpose of my musical projects, but since they reflect my personal understanding, they may be an inaccurate representation of Pärt’s approach. My understanding is largely informed by Paul Hillier’s biography of Arvo Pärt and by the analyses in the Cambridge Companion to Arvo Pärt edited by Andrew Shenton. Those studies suggest that Pärt uses only four of the six T-voices, to avoid octave transformations and unisons. This occurs in the T+3 and T-3 voices only when the M-voices sound one of the triad notes.

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One Response to Mapping Tintinnabuli Transformations

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