Chord Shape

Frequencies

Every key is a note in octave 4: C4 is MIDI note 60 and B4 is 71. In equal temperament a note with MIDI number m has frequency f = 440 × 2^((m − 69) / 12) Hz, so A4 (m = 69) is exactly 440 Hz and D4 is about 293.66 Hz.

Axes

The selected notes are sorted by frequency. The lowest drives x, the next y and the highest z, whatever order you click them in. Adding a note below the others moves every existing note along one axis.

The trace

Each axis follows coord_i(u) = sin(2π · r_i · R · u + phase_i), where r_i = f_i / f_lowest, R is the number of cycles of the lowest note on show, and u runs from 0 to 1 across that window. Every axis has the same amplitude. The phases 0, 0.6 and 0.2 radians for x, y and z are artistic starting choices that open the curve up; they are not a property of the chord. One note gives a line traveled back and forth, two give a flat (planar) trace, and three give a curve in space that may cross or retrace itself.

Why equal temperament stays open

Equal-tempered ratios between different notes in one octave are irrational, so the three motions never line up again exactly. The page shows 4 cycles of the lowest note and leaves the two ends of the trace open. They are not joined, and the trace is not a closed loop.

Just ratios

The page starts in just ratios, but they are one chosen palette, not the unique or historical tuning of every chord; equal temperament is always one click away. The lowest selected note keeps its equal-tempered frequency, and each higher note is tuned by how many semitones it sits above that note:

• 0 1/1
• 1 16/15
• 2 9/8
• 3 6/5
• 4 5/4
• 5 4/3
• 6 45/32
• 7 3/2
• 8 8/5
• 9 5/3
• 10 9/5
• 11 15/8

So D–F♯–A becomes 4 : 5 : 6, about 293.66, 367.08 and 440.50 Hz. Its A sits slightly above 440 Hz because D is the anchor, which is one way this differs from equal temperament. Other chords get other ratios: C–E–G♯ is 1 : 5/4 : 8/5, not 4 : 5 : 6.

Complete and partial repeats

With whole-number ratios the curve does repeat. The repeat takes P cycles of the lowest note, where P is the least common multiple of the ratios’ reduced denominators. When P is 24 or less the whole repeat is drawn and the trace closes on itself. Longer repeats are cut at 24 cycles and labelled partial: C–C♯–F♯ would need 480 cycles.

The swings

The swinging dots under the trace are the x, y and z coordinates themselves: each dot sits exactly where its note puts the point along that axis. The ticks count the swings each note completes in one pass. In just ratios the counts are whole numbers, such as 4, 5 and 6 for D major; in equal temperament they are not (D major gives about 4, 5.04 and 5.99), which is why that trace never closes.

Moving point and sound

The moving point is slow motion: it crosses the drawn window in 12 seconds and then starts over, again and again, while the real window lasts only milliseconds. When the trace closes, the end of one pass meets the start of the next. When it does not (equal temperament, or a partial just repeat), the point jumps back to the start at the end of each pass; no line is drawn to join the two ends. The point moves on its own from the start unless your device asks for reduced motion, and Pause holds it still. Sound is separate. It plays plain sine tones at exactly the frequencies in the Selected table, at equal levels.

What this is not

These are coordinate motions computed from frequencies, not the physical shape of sound and not a microphone recording. Because every ratio is measured from the lowest note, playing the same intervals in another key gives the same shape, so a shape does not identify a key or a composition. Nothing here claims to reconstruct any composer’s voicing, tuning or timbre.