Peter McKenzie Armstrong
                                Pre-Compositional Algorithms

Given a pitch-class series, IntLens generates an exhaustive catalogue of registrations via selective octave offsets per interval class.  Its Version 2 was included with Krikos (this as a program name, see below) among curricular materials at National Chiao University, Taiwan ('89);

Version 1, 1986 (Pascal) IntLens octave-complements &/or compounds the intervals of an input pitch-class series in all combinations of selection by interval class.  The selection cycle iterates for each of six ranges of expansion, defined as <starting level for all ICs> to <target level for selected ICs>.  These are: class to complement, class to compound, class to both; complement to compound, complement to both; and compound to both. Within each range and combination, series registrations are calculated separately per mirror form (P,I,R,RI) and coordinate order (X/Y or Y/X).  The registrations total (48 times (2 to the power of half the octave modulus)). Each is output as a pitch series array of signed integers beginning with 0.

Version 2, 1988 (Forth) Generation is expanded to specify for each Version 1 array all multiples within a range of nine octaves.  The registrations are now examined as simultaneities, i.e., as chromatic saturation chords.  Output for each is a half-page chart with graph and associated statistics, as for example:



A separate SORT available also outside the main loop details relationships among these otherwise arbitrarily sequenced registrations. Up to sixteen criteria keys are applied, independently ascending or descending and in user-specified priority. Output is a vector-graph cascade with a sidebar of key values. The criteria are:

    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

This version was optimized for prime-number octave dimensions through routines incorporated from SetTrans_P. AllDeg, one of IntLens's secondary functions, was fully developed later and separately as Krikos, score generator for several pieces of that name. Its algorithm is fully described with them under Compositions. IntLens 2 had a separate student manual.

Version 3, 1989 (APL) Initially a development environment port, producing the first dependable output graphs from function SORT, this version expanded IntLens's filing of intermediate data for subsequent retrieval by independent compositional software.

Version 4, 2001 (J) A final port, more fully absorbing its precursors and in a contemporary APL incarnation, this version anticipates its use as an external function with Csound-score building and related facilities incorporated from Krikos.



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As a wished contribution to the Darmstadt conference then already decades past, I once drafted an article called "Belated Serial Goodies.  It focused on the opening thrust of postwar atonality--a reenvisioning of the serial idea towards extension of its domain to all compositional parameters.  The article first reviewed schemes for subsuming serialist tenet and processes under some higher, usually geometric, abstraction.  While noting gains in this respect, I suggested that such empowered functionality, though innate in the serial notion per se, had been sidelined by preconditions in its early specification. Two such were identified and corresponding revisions offered.  One is described here.

Suppose that the basic serial forms (inversion &/or retrogression) were derived, not from the concept of axial "mirror" reflection, but from that of rotation in the plane.  Tracked as a turning graph of Cartesian point coordinates, these forms are seen collectively to represent only ONE orientation on the compass--180 degrees, cardinally &/or ordinally expressed.  Pitch/time identity swaps (coordinate exchanges) at 90 or 270 degrees, together with the origin itself, account for the remaining cardinal points.  All other orientations the mirrors ignore.  Considered instead, they are seen to evolve a serial functionality which implicity(!) embraces simultaneity, permutation, omission and recurrence.

Algorithm. "KRIKOS.WS" is an APL workspace whose core embodies this process. Given (any) initial point coordinates, it generates all possible alignments, converts their specifications to musical parameter arrays (as I wish), and formats scores for Csound. Its intermediate data is independently stored and may be so exploited.



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An integer summation program modelled on the Fibonacci/Lucas series.  Given any Markov-Order number of seeds (maximally 6), the program proceeds to generate each next number by summing as many just- previous terms.  Each generation is constrained by an input modulus, so ends when the modular cycle is complete.  Two moduli are used: initially 12, yielding values for interpretation as pitch classes; then 6 against the result, giving values for a parallel cycle of octave registrations.

For each pitch-class/octave-registration pair, a tally is kept.  Each tally total, as a reciprocal, then becomes the constant duration value for its corresponding pitch.  Consequently within any cycle, the collective duration for all instances of any given pitch is exactly 1.

The program, which generated the data for my Cycles for Synclavier, is quoted with sample output in Winsor, P., The Computer Composer's Toolbox, Windcrest '90.



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Otherwise a standard PIRRI-matrices generator, this program assumed any octave modulus (from the dimension of its input array) and issued separate outputs per coordinate order or interval-multiplicative factor, to explore and document the following perspective:


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.



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An old utility written in eight screens of Forth.  Many fewer screens (or an APL one-liner) could deliver all that the subtitle promises.  But its logic emphasizes the application of further constraints.  Quoting the documentation:


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.
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.

Three arrays are output per example found: the pitch-class series, its intervals as signed classes, and the position separations per interval-class pair.



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Note: My coding languages have been, chronologically: Forth, Pascal, C, APL, J.
Most programs have been data generators for immediate composition projects.