Where the sound comes from

I.The anatomy of one note

A vibrating string doesn't produce one frequency; it produces a whole ladder of them — the fundamental and its overtones, the partials. A synthesizer that plays a plain ladder sounds like a doorbell. What makes a piano is a short list of imperfections, and each one is written into this instrument's recipe:

• Where the hammer lands. A grand's hammer meets the string about an eighth of the way along its length, and a string struck at one-eighth cannot vibrate in the modes that have a node there — so roughly every eighth partial is weakened. That missing rung is part of the piano's fingerprint.
• Stiff strings run sharp. An ideal string's partials are exact multiples of the fundamental. Real piano wire is stiff, and stiffness pushes the high partials sharp — the inharmonicity that gives a piano its slightly bell-like top. It grows stronger in the treble, and so does it here.
• Two strings, almost in tune. Most piano notes are sounded by unison strings that are never perfectly identical. The model plays a second string detuned by a few cents; the pair beat gently against each other, which is where the sound's slow shimmer comes from.
• A breath of hammer. The first few hundredths of a second are not string at all — they are felt hitting steel. A short burst of shaped noise opens every note.
• Time moves differently across the keyboard. Treble notes die quickly; bass notes roll on. The decay of every partial is tied to its register, and behind everything sits a small synthesized hall, so the instrument never plays in a vacuum.

II.The pedals, in equations

The sustain pedal does here what it does on a grand: notes released under the pedal keep ringing until it lifts, and the recorder writes it into your MIDI file as real CC64 data. The una corda is modeled by its audible effect rather than its mechanism — there is no action to slide sideways, so instead each note is struck more gently than your dynamics alone would strike it, and its high partials are reined in: the same veiling that softer hammer felt buys on a real instrument.

III.Seven voices, one method

Every voice on the strip is a different recipe over the same synthesis — none of them a recording:

• Piano 1024, the signature voice, and the Concert Grand — two weightings of the struck-string recipe, one tuned for intimacy, one for warmth.
• Bright Grand — the same piano with its upper partials let off the leash, the voice that cuts through a mix.
• Glass E.Piano — the struck-tine sound: sparse partials, long bloom.
• Harpsichord — a pluck instead of a hammer, and true to the instrument, it ignores your dynamics entirely: a harpsichord has no touch velocity, so this one refuses it too.
• Celesta — small, bright, bell-struck; the shimmer decays almost as fast as it arrives.
• Drawbar Organ — the one voice with no decay at all: steady drawbar partials that sustain for exactly as long as you hold the key.

IV.What the equations buy — and what they cost

The buy is speed and weight: sampled pianos are megabytes to gigabytes, and this page opens in an instant because its entire instrument is arithmetic. It also makes the sound infinitely adjustable — the voice has been tuned by ear, one variable at a time, the way you'd voice a real hammer.

The cost is honesty's turn: a real grand is a roomful of coupled chaos — two hundred strings resonating with each other over a wooden soundboard — and no short recipe captures all of it. Which is why the recorder exports your performance as MIDI alongside the WAV: play your take here, then hand the same notes, pedaling and all, to any sampled grand in any DAW. The story of how this instrument was built explains who wrote the first version of these equations — and how long it took.