Themes
Small whole numbers
Consonance is frequency ratios of small integers, and it has been known since Pythagoras. Almost everything interesting follows from those integers refusing to line up.
How many of the twelve
A second family of small integers, and they are not ratios. Seven notes chosen from twelve, three of those seven stacked, six semitones that split the octave evenly, three beats laid against two. These arguments never mention a frequency: the object is a set of positions on a circle, and the answers are counted rather than measured.
The comma that will not close
Twelve fifths do not equal seven octaves. Every tuning system in history is a different decision about where to hide the difference.
The ear is not a microphone
Loudness is not amplitude, pitch is not frequency, and a note whose fundamental is missing still has one. Perception is a model, not a measurement.
What the listener supplies
A downbeat, a key, a category, a preference — none of them is in the sound, and every one of them can be measured. These essays are about the part of music that is contributed by the person listening, which is also the part that varies between listeners and between traditions.
The limits of the listener
The quietest audible tone, the smallest audible difference, the fastest series that can still be a beat, the delay at which one sound becomes two. Each is a measured boundary with a published source and a spread between listeners, and each one bounds a claim made somewhere else on this site.
The same shape, moved
A chord, a scale and a rhythm are all patterns that keep their identity when shifted. What survives the shift is what the theory is about.
What the room does
No sound reaches an ear unmodified. Reflection, absorption and resonance are part of the instrument, and always have been.
Notation hides things
Staff notation is a five-hundred-year-old compression. What it records beautifully and what it cannot record at all are both worth knowing.
The instrument is not the model
Every argument here starts from an idealised object: a perfectly flexible string, a spectrum of exact harmonics, an ear with unlimited resolution. Real strings are stiff, real spectra are not harmonic, and the ear has a bandwidth. The differences are measurable and they matter.
Somebody measured it
A surprising amount of this subject rests on a published measurement — of four drummers, of seventy-six speakers, of a room full of tuned pianos. Which somebody, on what material, and how far it generalises is part of the claim.
Time is a circle
Rhythm drawn as a line looks arbitrary. Drawn as a cycle, the patterns that recur across continents turn out to be the same few shapes.
Where the ends are
A tube stopped at one end has only odd partials, and it would have them if it were made of glass. A membrane's modes are Bessel zeros and a string's are whole numbers, and neither fact is about the material. The boundary conditions decide the sound; everything else is detail.
The second time is different
Almost everything a listener hears has been heard before, a few seconds earlier, in the same piece. Nothing about the sound changes on a repeat; everything about its status does. These essays are about repetition as the material of form — what a distance matrix can find in it, what it costs in bits, and why a literal repeat is the one boundary a change detector cannot see.
The design was forced
A piano's hammer strikes at a seventh of the string, a bass string is wound, a guitar's saddle is slanted and a clarinet has one register hole doing a dozen jobs. None of those is a tradition. Each is the answer to an arithmetic that admitted no better one, and the leftover error is computable.