Dapper '1' and Dreadful '0' (Armstrong, Peter McKenzie)
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Peter McKenzie Armstrong (Rosegarden & Pianoteq)
Edition Ottaviano Petrucci
Peter McKenzie Armstrong (Rosegarden)
|Work Title||Dapper '1' and Dreadful '0', for sequenced MIDI piano|
|Composer||Armstrong, Peter McKenzie|
|Year/Date of Composition||2011|
|Average Duration||78 seconds|
Tinkering one day with the Fibonacci series, I wondered what might happen if, instead of adding always just the most recent two terms to get the one following, I were to add every pair of terms to get for each pair a new term in between:
- starting with 1 2
- one iteration giving 1 3 2
- the next 1 4 3 5 2
- and so on.
I wrote a short generator (J-language script) and ran it to the point (in iteration #9) where it had output all integers between 1 and 88 -- the piano range -- at least once. Except for this sampling's unique final term, all its terms within range had occurred from 2 to 10 times each, with a few dozen others exceeding it. Given these circumstances, I shaped a progression of equal durations, as follows:
- at each instance, a term within key range is allocated to one of 10 unique-volumed tracks, according to the term's occurrence tally at that instance;
- for each term (MIDI key number) so assigned, a rest is placed in corresponding position on all other tracks;
- terms above range are realized as rests on all tracks (i.e., their time component is preserved);
- an additional track, a copy of #1 but with its rest slots filled in by extension of their just-previous notes, integrates and highlights the pattern of first occurrences.
Then, comeuppance! Browsing at the "Online Encyclopedia of Integer Sequences" (OEIS), I encountered for the first time Stern's biatomic array (). When run for two iterations beyond the series I had improvised, this one clearly subsumed the latter's output. The difference, Stern seeds "0 1" replacing my hazarded "1 2", exposes something extraordinary at work: with this "0 1" start, every generator iteration first replicates the just-previous one, before appending then a continuation of its own. OEIS presents several offshoots. My improvised script output the sequence as follows:
- starting with 0 1
- one iteration giving 0 1 1
- the next 0 1 1 2 1
- and so on.
Run this way well unto its 11th iteration (to build up from 0), the procedure now filled the 88-slot range only after 1276 terms, accumulating an occurrence-frequency maximum of 42. Musical realization here, to be conceptually as before, called for selective re-specifying, as follows:
- there are now 42 unique-volume-specific tracks, necessarily at much narrower volume differences;
- tempo is now 4-fold, to put so many more events into a time span compatible with Dapper '1' ;
- a 3-second coda cascades the first occurrences.
Dapper '1' is written in full score, as it has few enough tracks to fit a 11x17-inch page. Dreadful '0' , with too many to fit, is written instead as separate parts. In any case, neither score is intended to facilitate human performance. The music is for auto-sequencer. I did, however, want to give its overall patterning visual realization. Hence this style, with alto clef exclusively (Middle-C in the middle!) to spare the eye an incessant disruption of clef changes. The LilyPond files rework drafts I had initially exported from Rosegarden.
These pieces are named for what strikes me as their "character" -- ultimately their comfy vs jagged patterns of volume distribution.