COUNTING MELODIES V2 · INTERACTIVE COMPANION

Every rhythm has a shape.

Cut time into events. Turn a rhythm into a ribbon, a Young diagram, and a melody. Discover what each representation remembers.

Read the revised article ↗Open the counting lab ↗

ONE RHYTHM, FOUR VIEWS

2 + 1 + 3 = 6

6 CELLS · 3 EVENTS

Ribbon

Event order retained

Young

λ = (3, 2, 1)

Ferrers

λ = (3, 2, 1)

Conjugate

λ′ = (3, 2, 1)

The k-th part of the conjugate counts the events lasting at least k units. A partition represents an inventory of durations, not a unique melody.

Build the rhythm, one gap at a time

Click a gap: 0 continues an event →; 1 begins the next event ↓. All parts stay positive.

123456

01100

4 / 6

Rhythm in this inventory

SEE IT · CHANGE IT · HEAR IT

The musical representation

Original 3/8 rhythm notation for 2 + 1 + 3; gap word 01100 — Rhythm notation from the supplied v2 catalogue. Pitches and rests are edited below; the engraving shows the duration skeleton.

Event 1 · 2 unitsEvent 2 · 1 unitsEvent 3 · 3 units

Each rectangle is an event; width is duration and height is pitch. Rests sit below the pitch grid. Adjacent rests are disabled to preserve the manuscript’s canonical model.

TWO GENERATING-FUNCTION VIEWS

What does q count?

Duration · the manuscript’s melody series

M₅(q) = (1 − q) / (1 − 7q + q²)

1 + 6q + 41q^2 + 281q^3 + 1926q^4 + 13201q^5 + 90481q^6 + …

[q⁶] M₅(q) = 90,481

This coefficient counts every legal melody of duration 6. Your chosen rhythm alone admits 205 pitch/rest assignments; its partition admits 1,230.

Inversions · an exploratory q-analogue

F_λ(q) = Σ_{rhythms r with inventory λ} q^inv(r)

1 + 2q + 2q^2 + q^3

Here q tracks inversions of the duration sequence, not elapsed time. An inversion is a pair i < j with dᵢ > dⱼ. This companion experiment does not add a claim to the manuscript. F_λ(1) = 6 = ℓ! / ∏ m_k!.

PUZZLE MODE · BUILD BY LISTENING AND LOOKING

Find the hidden rhythm.

Reconstruct a duration sequence from its Young diagram and inversion count. The representations above update together as you try each cut.