From local kernels to global form: modeling the emergence of musical content

Francesco Vitucci; Michele Lorusso; Francesco Scagliola

cs.CLAug 25, 2026

From local kernels to global form: modeling the emergence of musical content

Authors: Francesco Vitucci, Michele Lorusso, Francesco Scagliola

Organizations: Conservatorio di Musica “N. Piccinni” di Bari

Abstract

Markov models are established tools for symbolic music, including non-homogeneous formulations. The narrower contribution examined here is an observation-driven estimation mechanism: overlapping sliding windows derive a trajectory of local transition kernels from one symbolic sequence rather than from an exogenous formal partition. We test this mechanism on 273 logical note events from Debussy's Syrinx (1913), using the often-proposed A-B-A' reading as a reference rather than ground truth. We apply the same validation to absolute-pitch and notated-duration kernels. At $L=6$ , both reference boundaries attain the Jensen--Shannon maximum in both dimensions; the duration plateau is substantially narrower (64 of 267 comparisons) than the pitch plateau (210 of 267). Because the theoretical maximum for consecutive sliding-window comparisons is set by window geometry and equals $1/\sqrt{L-1}$ for maximal turnover of the entering/leaving transition, the pitch value at $L=6$ and its broad plateau are not, by themselves, strong evidence. Their cross-dimensional alignment is consistent with boundary sensitivity, while the broad plateaus preclude treating either curve alone as a unique automatic segmenter. Five-hundred-draw re-synthesis experiments quantify departure from the source in both dimensions and expose an exact-copy degeneracy at $L=2$ .

Figures & tables

Panel of Figure 1 — Figure 1: Sliding window ( $L=4$ , step $=1$ ). Each row is one window position; from every $w_{m}$ a local transition matrix $P^{(m)}$ is estimated. Successive windows overlap by $L{-}1$ symbols, so the estimated kernel changes smoothly as the window advances.

Panel of Figure 2 — Figure 2: Trajectory of transition matrices. Colour-coding by formal region shows how the estimated kernel evolves as the sequence moves from one section to another.

Figure 3: Eight successive matrices from the trajectory ( $L=6$ , $t=25,\ldots,32$ ). Darker cells indicate higher transition probability. With only five transitions per window, the support is sparse and changes whenever one transition enters or leaves.

Panel of Figure 4 — Figure 4: Working A–B–A’ reference for Syrinx . The two sectional coordinates are comparison targets, not ground-truth labels.

Figure 5: Successive transition-distribution JS distance between absolute-pitch and notated-duration transition distributions. Dashed lines mark the measure-9 and measure-26 references. A point on a high plateau is not an isolated boundary peak.

stream	$L$	mismatch	global JS	sectional JS
P	2	0 [0, 0]	0 [0, 0]	0 [0, 0]
P	6	.678 [.462, .912]	.167 [.116, .213]	.196 [.128, .255]
P	20	.839 [.711, .945]	.303 [.251, .358]	.366 [.299, .431]
D	2	0 [0, 0]	0 [0, 0]	0 [0, 0]
D	6	.344 [.293, .392]	.129 [.091, .172]	.177 [.126, .231]
D	20	.588 [.509, .663]	.222 [.175, .276]	.320 [.249, .397]

Table 1: Re-synthesis departure: mean [2.5%, 97.5%], 500 draws; P=absolute pitch, D=notated duration.

Figure 6: One illustrative draw for each

L

. Quantitative results are reported in Table 1 ; visual similarity is not used to select a window.

Markov Models

Markov models describe systems that evolve probabilistically, with future behavior determined by the current state and transition rules.

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Abstract

Markov Models

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