{
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  "id": "counting-melodies",
  "slug": "counting-melodies",
  "title": "Counting Melodies: Generating Functions, Regular Languages, and the Combinatorics of Pitch, Duration, and Silence",
  "author": "Christopher Yamba",
  "date": "September 2026",
  "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\n$$M_p(q)=\\frac{1-q}{1-(p+2)q+q^2},$$\ngiving the recurrence $a_n=(p+2)a_{n-1}-a_{n-2}$ with $a_0=1$ and $a_1=p+1$. A direct combinatorial 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 subject of Bach's C-minor fugue BWV 847, and outline a Lean 4 formalization. The purpose of the note is both enumerative and conceptual: to show how a musical notation problem turns into a compact piece of combinatorics.\n",
  "version": "v1",
  "status": "author-published-draft",
  "formalization": "incomplete-blueprint",
  "copyright": "© 2026 Christopher Yamba. All rights reserved. Publicly readable; no additional reuse license granted.",
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