How Du You Compose Yourself?

TILING SIX-PART DOUBLE CANONS ON TRICHORDS LUIGI VERDI* TO JOHN RAHN very trichord can be superimposed on some of its transpositions and inversions without doubling. In this way, they form various hexachords according to the transposition level of the embedding. Each trichord can also be superimposed on a different trichord, giving rise to mixture. This essay focuses on tiling six-part double canons on trichords, intended as pairs of three-note series that combine by simple, inverse, and retrograde motions without doubling. These canons are created by setting the various trichords at transposition levels determined by the module, a generative numerical sequence that defines the horizontal motion, in relationship with another numerical sequence, the inference, which defines the vertical quantities of a trichord, through the common tones among all its transpositions and inversions. The inference generates the common-tone vector, a sort of “genetic code” that highlights the good inversion entries, placed without common tones on the levels of transposition ; the module generates the base module, showing the levels upon which trichords are transposed, and the differential vector, showing the * Translated from Italian by Deborah Burton E 500 Perspectives of New Music relationship between the base module and all its transpositions, inversions , and rotations, making possible states of “chemusical” aggregation , in a sort of synthesis between chemistry and music, or Chemusic. 1. WELL-TEMPERED TRICHORDS: TWELVE THREE-PART CANONS ON TWELVE TRICHORDS My first objective is to construct twelve canons for three parts, each of which is based on various vertical forms of a single trichord. To this end, I have established a series of intervals by which the trichords are transposed , so as to create canons derived by a periodic repetition of transposition . I chose an all-interval series, 1-2-3-4-5-6-7-8-9-10-11 (N.B.: this is a listing of intervals rather than pitch-class numbers and thus is not a twelve-tone series). This listing is called module.1 (Example 1) The total of numerical elements of the module defines the meter. In this case, the module 1-2-3-4-5-6-7-8-9-10-11 consists of eleven intervals, so it will be realized musically in an eleven-count meter such as 11/8. The sum of the module elements is defined as the base. In this case, 1+2+3+4+5+6+7+8+9+10+11 = 66 ≡ 6 (mod 12), with base class (0,6) of period 2. One will note that the second part of the module is simultaneously a retrograde inversion and transposition at the tritone of the first part, giving rise to a base 6 and a period 2. This creates certain implications from a musical point of view; that is, superimposing the retrograde on the original version of the whole structure will again result in transposition at the tritone. The base module is the succession of all the transpositions up to the return to the original state; in this case it is constituted of two identical parts transposed at the tritone. (0, 1, 3, 6, 10, 3, 9, 4, 0, 9, 7) (6, 7, 9, 0, 4, 9, 3, 10, 6, 3, 1) as in Example 2. If rhythmicized, the period will end after two measures of 11/8, when the structure returns to the original state. Using the base module switches the module from an intervallic representation to a listing of transpositions: it could be summarized in a shortened series, given by all the active transpositions: (0, 1, 3, 4, 6, 7, 9, 10), corresponding in this case to the octatonic collection 01. The base module automatically gives rise to an inverse base module (0, 11, 9, 6, 2, 9, 3, 8, 0, 3, 5) (6, 5, 3, 0, 8, 3, 9, 2, 6, 9, 11) (Table 1) and the consequent active inverse transpositions: (0, 2, 3, 5, 6, 8, 9, 11). Treating the trichords as simultaneities and transposing them via the intervals of the module, it is possible to create twelve canons for three parts, in which every trichord moves toward transpositions of itself according to precise contrapuntal rules, as explained in the following. Tiling Six-Part Double...