Peter McKenzie Armstrong
Pre-Compositional Algorithms
A Quasi-Prismatic Pitch-Class Series Registrator
(1986..2001)
1) ExpLevel 5) Frame 9) MinInt 13) MinFreq 2) RowNum 6) Gaps>OD 10) MaxInt 14) MaxFreq 3) Factor 7) NumClust 11) IntSpread 15) FrqSpread 4) Range 8) MaxClust 12) IntsUsed 16) FrqsUsed
Permutation of a Metered Pitch Series via Incremental X/Y Dot-graph Rotation
(1991)
Generation of Pitch/Duration Cycles via Modular Additive Sequences
(1986)
Generalized Multiplicative Transform via Prime-Only Octave Dimensions
(1985)
If the multiplicative transform is applied to a series whose octave dimen- sion is evenly divisible, a full cycle of permutations preserving the total chromatic cannot be had. It is for this reason alone that our 12-tone system yields not five such transformations (via factors 6,5,4,3,2) but just one (via factor 5). A 6-tone system yields none, but not because of its smallness. The mere 7-tone yields already two (via 3,2) and the 11-tone, four (5,4,3,2). And these occur simply because 7 and 11 are not divisible, but are prime. If octave dimension is prime and the multiplicative transform is applied via all factors (cardinally &/or ordinally, leading to I, R or RI), all interme- diate permutations preserve the full chromatic. Further, each end series is seen integrally with them in a continuous generation. Freed from its special-case constraint, the transform now proves a generative principle. This context integrates simple augmentation as well. Before reduction in- modulo, any series multiple 'refracts' the series as-it-were in expansions ranging another octave &/or durations-cycle for each factor increment.This program's essential functions were later incorporated into IntLens 2, described above.
All-Interval Row Generator
(198?)
Generate a permutation of the integers modulo-n (pitch classes), such that all possible unique successive differences are represented in absolute value (i.e., as interval classes) twice each. If the modulus is even, then signs modi- fying the instances of a given interval class must be opposite; if it is odd, they must be the same, with a final value redundancy.Three arrays are output per example found: the pitch-class series, its intervals as signed classes, and the position separations per interval-class pair.
The number of array positions separating the instance pair of each interval class must be unique for odd moduli and nearly so (one exception) for even.