Arvo Pärt’s Second Symphony and XJI-Invariant Twelve-Tone Rows

Mart Humal

The originality of Arvo Pärt’s music is manifest not only in his works written in the tintinnabuli-style but also in his early twelve-tone compositions, original both in their expression and techniques. His Second Symphony (1966) is based on the row, which, following Milton Babbitt (1960), can be represented by the ordered number couple succession: (0, 0) (1, 3) (2, 1) (3, 2) (4, 4) (5, 7) (6, 5) (7, 6) (8, 8) (9, 11) (10, 9) (11, 10). Its structure is somewhat similar to that of Webern’s Op. 30. The latter has invariance under operation X – “exchange operation”, according to Michael Stanfield (1984) – consisting of the exchange of the order and pitch class (pc) numbers in each couple. However, unlike Webern’s row, this row results, under operation X, in its order number and pc inversion JI (operation J being Rot1R): (0, 0) (1, 2) (2, 3) (3, 1) (4, 4) (5, 6) (6, 7) (7, 5) (8, 8) (9, 10) (10, 11) (11, 9). Therefore, Pärt’s row represents an X/JI-invariant twelve-tone row.

Similarly to the Webern row, such a row can be generated by a certain manipulation on the chromatic scale. But whereas in Webern, this manipulation consists of the exchange of two pairs of pitch classes, in Pärt it consists of the formation of three rotational groups (RG) containing three pitch classes each. Having a normal ascending order (Rot0) in a chromatic scale, these pitch classes are transformed under Rot1 in the prime form of an X/JI-invariant row, and under Rot2 – in its X/JI form.

X/JI-invariant rows are a special row type which probably has never been discussed in the theoretical literature. Moreover, when Pärt wrote this symphony, he was not aware of the special properties of its row and did not use them, since these aspects of the twelve-tone theory have not been discussed before 1970s. On the other hand, at the end of the symphony there is a very prominent quotation from Tchaikovsky’s piece “Sweet Day-Dream”. The programmatic meaning of this quotation is quite obvious and has been repeatedly discussed (e.g. in Klotyn’ 1969 and Aranovsky 1979). However, it can be shown that probably the row of the symphony is derived from this melody. In the paper, the possible derivation of this row (as well as the structure of X/JI-invariant rows in general), and its use in each of the symphony’s three movements will be discussed.

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