The Scale of /u/?/u/ is the IPA symbol for the vowel sound in "boot" or "full". The slashes mean "this is a vowel sound, not a letter".

A vowel is not a single tone.

When you sing the vowel ooh, your vocal folds produce a rich, buzzing tone — dozens of frequencies stacked on top of each other. The vowel is not in those frequencies. It's in what your mouth does to them.

The buzz from the larynx is approximately a sawtooth wave: a fundamental pitch plus integer multiples — the harmonic series. Every harmonic is present at the source. What changes between ooh and ee and ah is not which harmonics exist, but which ones are amplified by the resonant cavities of your throat, mouth, and lips.

Those resonance peaks have a name: formants. Three of them — F₁, F₂, and F₃ — are enough to distinguish almost every vowel a human voice can make. They are not frequencies the voice generates. They are windows of amplification, sculpted by the shape of the tract above the vocal folds.

The same buzz, the same fundamental — only the windows have moved. Move F₁ up and the vowel opens. Spread F₁ and F₂ apart and it brightens. You are not changing pitch. You are changing what gets through.

Vowels cluster at corners.

If you plot every vowel a language uses on a grid of F₁ against F₂, a pattern appears. Vowels don't scatter evenly. They cluster, and most cluster at the corners.

This isn't accidental. A language has to be heard over rain, over crying children, over the next room. The vowels that survive are the ones that contrast most. The corners of the F₁/F₂ plane — /i/, /a/, /u/ — are the three points most acoustically distant from each other, and almost every language on earth uses them.

The vowel chart is, in this sense, a map of perceptual distance. It tells you which sounds the ear can keep apart. It does not, by itself, suggest any musical structure. But it might.

What if we listened to the filter?

So far we've used formants the way they're meant to be used — as filters. They shape the buzz of the larynx into recognisable speech.

Now set that aside, and consider only where the formants are. Each vowel has three salient resonance peaks: F₁, F₂, F₃. Three frequencies, three positions in pitch space. What happens if we treat those positions not as filter parameters, but as pitches in their own right — played as tones, simultaneously?

This is a category move, and it should be named as such. A resonance peak doesn't "produce a tone" in any physical sense: it amplifies whatever harmonic falls near it. There is no oscillator at 300 Hz when you say ooh. But nothing stops us from playing a soft tone at the resonance frequency, knowingly, as a compositional act. The premise is that the geometry of the filter is musically interesting — that the where of resonance is as worth listening to as the what of the source.

The result is stretched, slightly strange, and not quite tonal. It is not the sound of someone saying ooh — that requires a buzz and a filter, not three independent tones — but it has a kind of acoustic family resemblance to the vowel that produced it. Three points in pitch space. The seed of a scale.

Not every scale comes from math.

Western music has settled on twelve-tone equal temperament — the octave divided into twelve mathematically equal steps. It's one system among many. Gamelan ensembles in Java and Bali tune their bronze instruments to scales — pelog and slendro — that do not divide the octave equally, and in some cases do not honour the octave as a perfect ratio.

These tunings are not summoned by the metal. They are inherited, regionally variable, and shaped by centuries of cultural convention; the physics of bronze sets a range of possibilities, not a fixed result. What's worth noting is that beating between paired instruments — the shimmering ombak — is intentional and structural, not a defect. A scale whose degrees beat against each other when sounded together is a feature of the tradition, not a tuning error. I'll come back to that later: the formant-derived scales here have their own version of it.

From three points to a scale.

Three pitches are not yet a scale. To turn them into one, we need a rule that takes us beyond them.

Here is the rule we'll use. Place F₁, F₂, F₃ on a logarithmic frequency axis. Three points on a curve. The simplest curve through them is an exponential — and we'll fit one, anchored at F₁ and F₃, and use it to generate as many further degrees as we want. Its growth ratio has a closed form:

For /u/: r = √(2500 / 300) = √8.33 ≈ 2.887.

That single number is the fingerprint of the vowel. It captures how widely the formants spread on a logarithmic scale. /u/ clusters them low and close (moderate r). /i/ pushes F₂ far up (largest r). /a/ spaces them more evenly and at smaller spread (smallest r). Different vowels, different r, genuinely different scales.

Once we have r, the scale is generated by stepping along the curve. We need somewhere to start; A440 is the convention Western tuning already settled on, so we'll use it.

n = 1 is the root (A4). n = 2 is the root times r. n = 3 is root × r². Negative n goes the other way: n = 0 is root / r; n = −1 is root / r²; and so on, downward into the audible range below the root.

The line above is the generator. F₁, F₂, and F₃ (the solid dots) are the data points it was fitted to. The open dots above and below are extrapolations — implied by the geometry, not literally present in the vowel. The model treats F₂ as roughly the geometric mean of F₁ and F₃; for /u/ and /e/ the actual F₂ lies close to that prediction, for /i/ it doesn't. We've absorbed that mismatch by anchoring on F₁ and F₃ only.

Now playable.

The raw generator climbs and descends exponentially — each step covers most of an octave or more. To turn that into a familiar 8-note instrument we fold every degree back into a single octave above the root, by halving it until it lands between 440 and 880 Hz. This is the same trick Western tuning uses to wrap any scale into a single playable octave.

Here's the math, degree by degree:

The right column is the playable scale. Try it. The keys are labelled by their degree number, not by Western note names — partly because the pitches don't sit on equal-tempered notes, and partly because mapping a non-Western scale onto Western letters smuggles in assumptions that don't belong here.

This is the scale of /u/ by this particular method. Different formant values would give slightly different scales; different generator rules (regression instead of anchored fit; bandwidths instead of points) would give different ones again. The choice we've made is one defensible compositional move, not the only one.

r is the fingerprint.

If /u/ gives one scale, every vowel gives one. Switch r and the topology of the scale changes — sometimes drastically.

/a/ has the smallest r — its formants are bunched closer in log-frequency space — so its eight degrees spread more evenly across the octave; the scale sounds the most "scale-like" of the five. /i/ has the largest r; its degrees, after octave reduction, fold back onto each other and produce tight beating clusters around a few centres — that's the ombak from Stage 4 showing up here. /u/ sits between, denser at the lower end of the octave. The contour of each scale follows the geometry of the vowel that produced it.

The same shape, different scales.

The clearest way to hear what these scales actually do to music is to take a melody that engages with multiple scale degrees and play it through each scale, unchanged in shape. Ode to Joy — Beethoven, public domain, recognisable everywhere — uses five scale degrees in a clear, simple phrase. Below is the melody on a piano roll: time on the horizontal, pitch on the vertical. Switch scales on the left and watch the notes glide to their new pitches.

In Major the melody sounds the way you remember it. In Minor, three changed degrees turn it from triumphant to wistful. In the vowel scales — particularly /u/ and /i/, where the degrees pack tightly — the same shape compresses; the melody loses its sweep. /a/, with its more even degree spacing, holds the melodic contour but recolours the intervals.

Where the shimmer lives.

Melodies are about intervals in sequence. Chords are about intervals in parallel — multiple pitches sustained together, beating against each other. That's where the ombak from Stage 4 actually emerges in these scales, because beating only happens when notes overlap in time.

Here's the classic four-chord pop progression — I, V, vi, IV — built as triads on the scale degrees 1, 5, 6, and 4. In diatonic scales (Major, Minor) these resolve into familiar shapes. In the vowel scales, the same triad pattern produces stacks that beat against themselves; the irregular tuning makes the sustain shimmer.

Each chord is strummed — the three notes enter in quick succession rather than struck together — and panned across the stereo field: bottom note to the left, middle in the centre, top note to the right. The voice is a soft pad with a medium attack and a long release, so each chord swells in and bleeds into the next. Watch the envelope as a chord sustains: when the simultaneously sounding notes are close enough to beat against each other (especially in the vowel scales), it visibly oscillates. That oscillation is the shimmer.

What's left.

This isn't a finished musical system. The generator is one rule among many; the formant values are population averages, not the vowel of any particular speaker; the scale has been forced into octave equivalence for familiarity rather than necessity. Most of the interesting questions are still open.

1. What if the scale didn't octave-repeat? The raw exponential generator produces a non-octave system whose steps each cover several octaves. That's strange but coherent — and closer to the spirit of pelog and slendro, which don't honour the octave either.
2. Formants are not points; they are bands. F₁ has a bandwidth of perhaps fifty hertz around its centre. What if a scale degree weren't a pitch but a zone — a range you bend within? That's closer to how the voice actually works.
3. What would harmony look like in this system? Two simultaneous vowels would mean two simultaneous scales — modulation as articulation, chord changes as words.
4. And: is there a scale of a sentence? Of a whole conversation? The formants drift constantly when you speak. A melody is hiding in there.

That's the article. The scale exists by choice, not by necessity. What you do with it is open.