Counting Melodies
Generating Functions, Regular Languages, and the Combinatorics of Pitch, Duration, and Silence
The revised manuscript, with engraved scores, ribbon and Young diagrams, the complete catalogue of 32 rhythms, and an interactive companion.
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Counting Melodies: Generating Functions, Regular Languages, and the Combinatorics of Pitch, Duration, and Silence Christopher Yamba September 2026 Revision 2 — 2026-09-20 Abstract We study a deliberately small but musically natural enumeration problem: how many monophonic melodies of fixed total duration can be written when notes carry pitch, rests carry no pitch, durations are integer multiples of a smallest temporal unit, and adjacent rests are identified with the single rest obtained by merging them? The model leads naturally to integer compositions, regular languages, and ordinary generating functions. For a pitch alphabet of size p and unrestricted positive integral event durations, the generating function reduces to a rational function 𝑀𝑝(𝑞) = 1 − 𝑞 1 − (𝑝 + 2)𝑞 + 𝑞2 , giving the recurrence 𝑎𝑛 = (𝑝 + 2)𝑎𝑛−1 − 𝑎𝑛−2 with 𝑎0 = 1 and 𝑎1 = 𝑝 + 1. A direct combi- natorial formula is also obtained. For a 4/4 bar quantized into sixteen sixteenth-note units and two chromatic octaves ( 𝑝 = 24 ), the model gives 41, 006, 928, 746, 285, 492, 812, 225 admissible melodies. We then lift the commutative counting series to a noncommutative formal series that retains individual melodic identity, illustrate the encoding on the sub- ject of Bach’s C-minor fugue BWV 847 with engraved scores, ribbon, Young and Ferrers diagrams, and outline a Lean 4 formalization. The purpose of the note is both enumera- tive and conceptual: to show how a musical notation problem turns into a compact piece of combinatorics. 1. The question A bar of music looks finite. The staff has a fixed width; a time signature fixes a duration; a chosen smallest note value fixes a temporal grid. Yet even in a monophonic line, the number of possible bars becomes enormous as soon as rhythm, silence, and pitch are allowed to interact. The point of this note is not to claim that a melody is exhausted by pitch and duration. It plainly is not. Articulation, dynamics, phrasing, harmony, timbre, voice leading, meter, notation, and musical expectation all matter. The point is narrower and, mathematically, cleaner: Given a fixed amount of time, how many legal monophonic event se- quences are possible under a specified set of notation rules? The problem began from a very concrete representation. Take a 4/4 bar, measure time in sixteenths, and write the bar initially as sixteen atomic temporal units. Grouping consecutive units creates event durations. If order matters, these groupings are compositions rather than partitions. Pitch then colors note events, while rests behave differently: two adjacent rests are not musically distinct from their merged rest in the duration model used here. 1