MDPE.Application:4 - Construct a rhythmic part from a seed, phase and action rule

Use this construction when a performer or arranger needs a playable rhythmic part with specified temporal and ensemble relations, but a promising geometric pattern still leaves its phase, sound and action allocation undecided. It connects generation, arrangement and receiving performance. If an existing part already serves the musical task, use it.

The construction below adapts Toussaint’s The Geometry of Musical Rhythm, second edition (2020): chapter 21, Figure 21.2 for a three-onset/eight-pulse seed; chapter 31’s alternating-hands method; and chapter 32, Figures 32.1–32.2 for pulse-wise alternation and extension. The ensemble, constraints and rehearsal outcomes are constructed examples. The book supplies the mathematical and action constructions, not evidence about these performers or a universal ranking of musical quality.

MDPE.Application:4.1 - Generate the small seed and choose its phase

An ensemble wants three audible attacks in a repeating eight-pulse span. Its pulse 0 is fixed by the accompaniment. Another instrument has a cue at pulse 6, where this new part should remain audibly silent. The percussionist has a low-sounding instrument for the right hand and a contrasting high-sounding instrument for the left. The first musical question is whether a two-cycle timbral exchange works with the accompaniment.

Use x for an audible onset and . for a silent pulse. For this specific seed, begin with three onset columns and five silent pulses. Give each onset one silent pulse, leaving two; append the remaining two to the first two columns. Read each column downwards and concatenate:

initial groups:       [x]   [x]   [x]     remaining: . . . . .
after first sharing:  [x .] [x .] [x .]   remaining: . .
after second sharing: [x . .] [x . .] [x .]
concatenated:         x . . x . . x .
positions:            0 1 2 3 4 5 6 7

The onsets are 0, 3 and 6. Their cyclic gaps are 3, 3 and 2, including the return from 6 to the next cycle’s 0. There are three onsets, eight positions and gaps differing by at most one pulse. These facts establish the selected construction, not its musical value. For a different number of onsets or pulses, use the chapter-21 repeated-distribution algorithm or an already qualified seed; the particular two-sharing calculation above is not a general formula for every pair of counts.

The seed places an attack on the other instrument’s cue at 6. Rotate every onset forward by two positions, wrapping at 8: 0 becomes 2, 3 becomes 5, and 6 becomes 0. The ordered result is 0, 2, 5, with gaps 2, 3, 3. It preserves the cyclic spacing family and attack count, includes the fixed pulse 0 and leaves pulse 6 silent. This rotation changes the relation to the accompaniment even though a comparison of unanchored cyclic shapes treats the two patterns as equivalent. Merely rotating the diagram’s labels would not change the performed phase.

MDPE.Application:4.2 - Turn the onsets into actual hand and sound parts

First alternate hands on audible attacks only. Begin with the right hand and continue alternation across the cycle boundary; do not restart the rule at each bar. Three is odd, so the next cycle starts with the other hand.

Here R and L are audible strokes on the respective instruments. The combined onset pattern repeats after eight pulses, but the hand and timbre assignment repeats after sixteen. In one sixteen-pulse span, right-hand attacks are at 0, 5 and 10, and left-hand attacks at 2, 8 and 13. Those two parts were generated by an action rule; they were not inferred from the onset polygon alone.

The ensemble compares this with playing all three attacks on one sound. The single-sound alternative preserves timing and costs less coordination but removes the proposed timbral exchange. In the constructed rehearsal, the percussionist can perform the alternating-attack version at the agreed 72 pulses per minute with the audible reference retained, and the partner musician reports that pulse 6 remains clear. The ensemble keeps it for the next rehearsal because the two-cycle exchange serves that arrangement. This is a stipulated bounded receiving decision, not a measured long-run reliability or a causal claim that the generation algorithm improved music.

MDPE.Application:4.3 - Change the receiving condition without confusing the two action rules

The accompaniment is revised: the low right-hand sound must now mark pulse 0 of every eight-pulse cycle. The alternating-attack version fails this condition by construction, even when played perfectly. More practice of that same rule cannot fix the alternating first sound.

Compare a second action rule from chapter 32. Alternate the hands on every pulse, including silent motions, and sound only positions 0, 2 and 5. Capitals below mean audible strokes; lowercase letters mean the indicated hand moves without an audible attack, stopping short or using an appropriately silent motion.

pulse:   0 1 2 3 4 5 6 7
motion:  R l R l r L r l
sound:   x . x . . x . .

Eight is even, so this hand allocation repeats at the next cycle. The first two audible attacks are right-hand strokes and the last is left-hand: it is the chapter-32 smooth-toggle seed. The audible spacing and silence at 6 remain; the timbral organization and movement demand change. Instead of three audible strokes with alternation between them, the performer maintains eight alternating pulse motions. That can help one performer and burden another. Silent movement is not a silent instruction to skip the movement.

In the constructed return, the percussionist first sounds an unintended stroke at 4. The ensemble slows the passage, separates audible from silent motions, then returns to the same receiving tempo and accompaniment. Once the required sound pattern and entry are demonstrated under those conditions, it selects this version for that use and retains the two-cycle version for the earlier arrangement. If the performer cannot obtain the required silent motion reliably in the available preparation time, choose a playable allocation or change the instrumentation, entry demand or rehearsal commitment. A correct table supplies no missing capability.

MDPE.Application:4.4 - Extend the phrase while preserving a chosen action relation

Suppose a later arrangement has a twelve-pulse span and still wants a block of right-hand attacks followed by a block of left-hand attacks, with the same three-pulse gaps at the two transitions. Stretching every old interval by 1.5 would move the attacks off the new unit-pulse grid. Adding four rests would keep the old onsets but break the chosen transition spacing. Neither operation supplies the requested relation.

Use chapter 32’s splice instead. Cut the eight-pulse motion after position 3. Copy its four-pulse transition segment at original positions 2–5, then insert that copy between the two halves:

first half       inserted transition    second half
R l R l          R l r L                r L r l

new positions:   0 1 2 3 4 5 6 7 8 9 10 11
new motion:      R l R l R l r L r L r  l

The result has right-hand attacks at 0, 2 and 4 and left-hand attacks at 7 and 9. Its cyclic gaps are 2, 2, 3, 2, 3. Five attacks replace three, the span becomes twelve, and the selected hand blocks and three-pulse transitions remain. Inserting an even number of pulses preserves the parity of the shifted hand motions. This is an explicit preservation claim about this construction, not proof that every stylistic or perceived quality survived.

The new part still needs its relation to the revised accompaniment and the performer’s capability. If the accompaniment actually keeps an eight-pulse landmark, the twelve-pulse construction is not an adequate replacement merely because the arithmetic works. Retain the eight-pulse part or reconsider the joint arrangement. If twelve pulses are intended, try the new part with those receiving landmarks and compare the audible transition, effort and entry rather than declaring a score of five attacks more complex or better in every sense.

MDPE.Application:4.5 - Return the useful part and its limits

The usable result identifies the onset pattern and fixed origin, hand/action rule, instruments and audible differences, receiving accompaniment, demonstrated conditions and selected alternative. Timing geometry, a Method for generating or performing it, understanding the construction and acquiring the performance remain different results. The style and acquisition account explains this distinction when teaching the operation.

MDPE.9 can use the generated alternatives, MDPE.3 their performed combination and MDPE.10 a missing capability. For broader rhythmic comparison, transformation or coordination, use Rhythmics RHY.7, RHY.8 and RHY.11 or the applicable specialist method. Stop at the playable part and bounded receiving result when no wider practice-development decision is waiting.