Stretch the octave
Rate two-note chords built from three kinds of tone and find out how the sound of an instrument changes which intervals feel pleasant.
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Why do some pairs of notes sound sweet and others sour? The textbook answer goes back to Pythagoras: simple frequency ratios. An octave is 2:1, a fifth is 3:2, and those intervals should sound good whatever instrument plays them.
In 23 online experiments with 4,272 participants, the lab tested that idea by changing the sound itself: ordinary tones, tones with their overtones pulled slightly apart, and a synthetic version of the Javanese bonang. If the ratios were all that mattered, the same intervals would sound best every time.
Pick a sound, rate 20 chords, then lay your map over the study’s. What the study found is below the experiment, so you can try it first.
You can also press the keys 1 to 7.
What the study found
Harmonic tones. With ordinary tones, the map has eight clear peaks, on average about 0.05 semitones from a note of the Western chromatic scale. The heights follow the old split into consonant and dissonant intervals, and agree with laboratory studies going back to the nineteenth century.
Stretched tones. Stretching every overtone moved the peaks with it. The preferred octave rose from 12.04 to 12.78 semitones, and the other peaks shifted the same way. Squeezing the overtones together had the opposite effect. A replication with listeners in South Korea showed the same pattern.
The bonang. Most of the familiar peaks vanished. The octave stayed (11.98 semitones), and new peaks appeared at 2.6 and 4.8 semitones. All three lie close to degrees of the slendro scale (2.4, 4.8 and 12). The listeners were American and had probably heard little gamelan music. That supports an idea from the engineer and music theorist William Sethares: the slendro scale grew partly out of the sound of the gamelan’s own instruments.
Why. No single classic theory explains all of this. The best model combines three tendencies: we like sounds that resemble a single harmonic tone, dislike fast beating between nearby overtones (roughness), and mildly enjoy slow beats.
How the sounds are made
Your browser builds every chord in the moment, following the paper’s recipe. Each chord holds two tones for 1.3 seconds, with the lower note somewhere between G3 and F4. The harmonic tone has ten harmonics that fall off by 3 dB per octave. In the stretched tone the nth partial (the first is the fundamental) sits at 2.1log₂ n times the fundamental instead of n times. The bonang’s upper tone has partials at 1, 1.52, 3.46 and 3.92 times its fundamental, after Sethares’s field measurements, over a plain four-harmonic lower tone. Each of the 20 chords comes from a different 0.75-semitone slice of the 0–15 semitone range, so your map covers the whole range.
Where this comes from
Related work
- Local consonance and the relationship between timbre and scale Proposed that a tone's spectrum sets its consonant intervals; the source of the bonang tone.
- Tonal consonance and critical bandwidth Dissonance as roughness between partials that fall close together.
- The acoustic component of Western consonance The roughness model behind the paper's interference account.
- Simultaneous consonance in music perception and composition The harmonicity model, and a comparison of 21 consonance models.
- Perceptual fusion of musical notes by native Amazonians suggests universal representations of musical intervals Listeners with little exposure to Western music still hear simple intervals as fused.
The study curves use the public ratings, which cover 165 of the 198 listeners in the harmonic study, 157 of 194 in the stretched one and 152 of 170 for the bonang. As in the paper, each listener’s ratings are z-scored, then smoothed (Gaussian kernel, 0.2 semitones), with 95% bootstrap bands. Your curve uses a wider kernel (0.6 semitones) because it rests on 20 ratings rather than 7,500. To place you among the study’s listeners, each of them is scored the same way on 20 of their ratings, spread across the range like yours.