Play two notes together. Some pairs sit still and some fight, and anyone can hear which is which. The question here is whether you can also see it. The sound is drawn as it plays, and the difference between a pleasant pair and a harsh one turns out to be a difference in shape.
cooking · work in progressA low tone and a second one some distance above it. The distance is measured in cents, hundredths of a semitone, and the named buttons jump to the intervals a piano offers. Press play, then drag. Headphones or a real speaker help; turn the volume down a notch first.
The thin lines are the low and high tones on their own. The thick line is what reaches your ear: their sum. If the sum repeats inside the window, the bracket marks where.
The same sound from far enough away to see it swell and fade. That swelling is beating, and beating too fast to count is what roughness is made of.
The low tone moves the pen left and right, the high tone up and down. A simple ratio draws a closed figure; an awkward one draws a tangle.
A pure tone is a single sine wave. Every real instrument carries harmonics, fainter tones at two, three, four times the note, and their mix is what makes a flute sound unlike a trumpet. Each slider is one harmonic. Both tones share the voice. Watch what it does to the pictures, and to the tritone in particular.
Pitch runs left to right, an octave per equal step. A red arch joins two harmonics close enough to beat against each other; the darker the arch, the rougher the pair.
Instead of one interval, all of them. The curve is the roughness of this voice against itself at every distance from unison to the octave, added up over every pair of harmonics. The dips are where the harmonics line up. Click anywhere on it to jump the dial there.
Computed, not asserted: the curve is the sum of the beating between every harmonic of one tone and every harmonic of the other, using the fit Sethares made to Plomp and Levelt's listening experiments. Set the voice to Pure and most of the hills go away.
A tone is a wave, and the picture of a wave is pressure over time. Two tones played together simply add. When the second is a simple multiple of the first, three to two for a fifth, the sum lines up again after a couple of cycles and the picture holds still. When the ratio is awkward, the sum takes many cycles to come round, and the eye reads it as a scribble. That is the first thing to see: pleasant pairs repeat quickly, harsh pairs do not.
The second is beating. Two tones a little apart drift in and out of step, so the sound swells and fades at the rate of the difference: 220 and 226 hertz beat six times a second. A few beats a second is a wobble. Twenty or thirty a second is too fast to count, and the ear hears it as roughness, which is most of what harsh means. Move the tones further apart than about a whole tone at these pitches and the ear separates them again; the roughness fades and you hear two notes.
The third is the voice. A pure sine is only rough against a tone very close to it, so with the voice set to Pure most intervals are fine, the tritone included. Real instruments carry harmonics, and every harmonic of one tone can beat with every harmonic of the other. With the Bright voice, the tritone's third harmonic lands about forty hertz from the other tone's second, and that clash is the red arch above the spectrum. Consonance is not a property of the interval alone. It is a property of the interval and the voice together.
The last is tuning. A piano divides the octave into twelve equal steps so it can play in every key, and the price is that nothing but the octave is an exact simple ratio. Its fifth is two cents flat of three to two, which beats slowly, less than once a second. Its major third is fourteen cents sharp of five to four, and at this pitch the fifth harmonic of one note beats against the fourth of the other nine times a second, fast enough to hear. Switch between equal and just on a third and the zoomed-out picture shows the difference.
The curve in step four is a model of a lab result, not a measurement of you. Plomp and Levelt asked listeners in 1965 to rate pairs of pure tones and found that roughness peaks at about a quarter of a critical band and fades by the band's width. Sethares fitted a formula to that curve in 1993 and summed it over the harmonics of real voices, which is what is drawn. It reproduces the ranking musicians already had, octave then fifth then fourth, and that agreement is its evidence, not a proof.
It also says nothing about why a diminished chord in a film score feels like a threat. That part is learnt, and the curve does not know it. The word harsh does two jobs, and only the physical one is drawn here.
Two tones, one voice for both, no room, no loudness. Real chords have three or more notes, the notes usually have different voices, and how loud a sound is changes what the ear tolerates. Laptop speakers lose the low tone and much of the beating; the pictures are the same on any machine, the sound is not.
Everything is computed in this tab from the same recipe the speakers are given: the drawing does not listen to the audio, it is the audio, worked out ahead of time. No recording, no upload, no model.
Counted, not judged. Nothing here says an interval is wrong. The tritone was banned by some and beloved by others, and the picture is the same for both.