Part of The Beauty of Physics, Mathematics, and Music (and Maybe Cats).

A tuning table has two numbers where an instrument has a sound. Write down 200 Hz and 300 Hz and the ratio is a perfect fifth, 3:2. The table does not say whether those notes come from sine-wave oscillators, bowed strings or metal bars. It does not specify what else vibrates above either frequency.

That missing information matters. A note can contain many frequency components, called partials. When two notes sound together, the ear receives the combined collection. Changing the partials can change the interactions without changing the two frequencies listed in the tuning table.

The tempting slogan is that consonance lives in timbre. That goes too far. Consonance concerns the relationship between sounds, and the listener who hears them. Timbre changes that relationship; it does not remove the interval or the listener from it.

Two Notes Contain More Than Two Frequencies

For an ideal harmonic tone with fundamental frequency $f_0$, the partials occur at

$$ f_n=nf_0,\qquad n=1,2,3,\ldots $$

The positive integer $n$ labels a partial. Real instruments also have attacks, decays, noise and other departures from this idealisation. Even within the harmonic model, amplitudes matter: a barely audible tenth partial contributes differently from a strong second one. Spectrum is therefore one important part of timbre, rather than its complete definition.

Take a deliberately simple pair of harmonic sounds, with fundamentals of 200 Hz and 250 Hz. Their ratio is 5:4, a just major third. The fifth partial of the lower tone and fourth partial of the upper tone coincide:

$$ 5(200\ \mathrm{Hz})=4(250\ \mathrm{Hz})=1000\ \mathrm{Hz}. $$

Retune the upper fundamental to 251 Hz, keeping its spectrum harmonic. That particular pair becomes 1000 Hz and 1004 Hz. Their difference is 4 Hz.

For two equal-amplitude sinusoids, the identity

$$ \cos(2\pi f_a t)+\cos(2\pi f_b t) =2\cos\!\bigl(\pi(f_a-f_b)t\bigr) \cos\!\bigl(\pi(f_a+f_b)t\bigr) $$

shows how a rapid oscillation acquires a changing envelope. Here $f_a$ and $f_b$ are the two partial frequencies and $t$ is time. The magnitude of the envelope repeats at $|f_a-f_b|$, giving four beats per second in this example.

This calculation establishes an interaction between one pair of components. It does not calculate the pleasantness of the whole chord. There are other partial pairs, their amplitudes need specifying, and slow beating is not automatically unpleasant.

Roughness Has a Frequency Scale

Reinier Plomp and Willem Levelt’s 1965 experiments asked listeners to judge pairs of pure tones. The result was not a row of special dips at every familiar musical ratio. Near unison, increasing the separation first increased dissonance; larger separations then reduced it. The troublesome separation depended on frequency and was related to the auditory system’s critical bandwidth [[1]].

The underlying distinction is between components that interact strongly within the ear’s frequency-selective processing and components sufficiently separated to be better resolved. Critical bandwidth provides a scale for that distinction, rather than a sharp universal border between beautiful and ugly sounds.

For complex tones, a musically large interval between the fundamentals can still leave nearby upper partials. Our 200 Hz and 251 Hz notes are separated by 51 Hz, yet the selected upper components differ by only 4 Hz. The interval name alone conceals that pair.

Nor is every fluctuation in amplitude heard as roughness. Slow beats can sound like a fluctuation in loudness. Faster interactions can produce a rough sensation. The physical superposition and its perceptual description are related, but they are not the same measurement.

A Spectrum Can Suggest a Tuning

William Sethares made the timbre–tuning relationship computational in 1993. His local-consonance method assigns a dissonance contribution to pairs of partials and sums those contributions for a specified spectrum. Varying the frequency ratio between two notes produces a curve. Its local minima identify candidate intervals with less modelled dissonance than their neighbours [[2]].

The relationship can also be run backwards: specify desired intervals, then search for spectra whose minima fall near them. Sethares gave examples for ten-tone equal temperament as well as stretched and compressed spectra. These are predictions under a psychoacoustic model. The paper explicitly called for controlled listening tests; its informal tests with friends and colleagues did not provide that level of evidence [[2]].

A simple calculation explains the stretching idea without requiring the full optimisation. Replace harmonic partial frequencies with

$$ f_n=f_0 n^p,\qquad p=\log_2(2.1)\approx1.0704. $$

Here $p$ is a dimensionless stretching exponent chosen for illustration. With $p=1$, this would be the harmonic series. With the chosen value, the second partial is $2.1f_0$, rather than $2f_0$.

For a lower fundamental of 200 Hz, that second partial lies at 420 Hz. Put a second, identically shaped spectrum at a fundamental of 420 Hz and its first partial coincides with the lower note’s second partial. An ordinary 400 Hz octave instead leaves a 20 Hz separation for that pair.

The matching interval has moved because the components moved. Whether the entire resulting sound is pleasant still requires the rest of the spectrum and a listening experiment. A coincidence is a mechanism to investigate, not a beauty certificate.

Listening Tests Make the Claim More Precise

Raja Marjieh and colleagues put the idea to a large experimental test in 2024: five studies containing 23 experiments and 235,440 human judgements. They sampled continuous ranges of intervals, rather than restricting the choices to the piano’s keys, and manipulated partial frequencies, amplitudes and presence [[3]].

For stretched spectra, the preferred octave-like interval shifted upwards. The reported peak was 12.78 semitones, compared with 12.04 for the harmonic baseline. The stretching manipulation also shifted preferences in three-note chords and replicated in a South Korean participant group. These results support a causal effect of spectral changes on the measured preferences [[3]].

They also exposed the limits of a roughness-only account. Reducing upper partial amplitudes did not flatten interval preferences as the tested interference model predicted. Removing upper partials still left preferences near the fifth and octave. A composite account, combining harmonicity with interference and a contribution from pleasurable slow beating, explained more of the findings [[3]].

Harmonicity concerns how well a sound’s components fit an integer-multiple pattern. It differs from counting close neighbours in a spectrum: a sound can fit that pattern despite being widely spaced, or contain little roughness without fitting it. Distinct mechanisms can therefore make similar predictions for ordinary harmonic instruments and disagree for carefully constructed alternatives.

That is the purpose of the unusual sounds. They separate explanations that familiar instruments allow to travel together.

Pleasantness Is a Judgement With a Listener Attached

The 2024 experiments asked about pleasantness. This is an accessible way to study consonance, but it does not directly measure a composer’s judgement of harmonic function or the emotional force of a passage. The tones were mostly synthetically constructed, the chords were isolated, and online testing limited control over listening equipment [[3]].

A dissonant chord can be exactly the sound a piece needs. Making every moment individually smooth would not establish that the sequence works as music. Context, expectation and learned style enter at that level.

Culture also matters before a sequence is involved. Josh McDermott and colleagues found that Tsimane’ participants with little exposure to Western music did not share the strong consonance preference of their US comparison groups, despite responding to acoustic roughness and discriminating relevant sounds [[4]]. Detecting a distinction does not require liking one side.

The South Korean replication in the 2024 study therefore broadens the evidence without establishing a universal ranking. Its authors explicitly acknowledge exposure to Western music and the need for wider cultural comparisons [[3]].

There is a productive middle ground here. Acoustics constrains the signal; experiments establish what particular listeners do with it. Neither task becomes unnecessary because the other exists.

Thirteen Steps Can Span a Tritave

The Bohlen–Pierce system gives the timbre–scale relationship a particularly unfamiliar setting. Its equal-tempered form divides a 3:1 frequency ratio, called a tritave, into thirteen equal logarithmic steps. Mathews, Pierce, Reeves and Roberts investigated its harmonic properties in 1988 [[5]].

The step ratio and size are

$$ q=3^{1/13}\approx1.0882, \qquad 1200\log_2 q\approx146.30\ \mathrm{cents}. $$

The factor 1200 converts an octave to cents. Thirteen steps multiply the frequency by three, not two. Calculating $1200/13$ would therefore describe a different scale.

Odd-partial spectra make the choice intelligible. Sets beginning with $f,3f,5f,7f$ can share components when shifted by a factor of three; a factor of two instead puts those odd components at even multiples of $f$. This is an elementary spectral observation, not proof that everyone should prefer tritaves or stop hearing octave relationships.

My earlier discussion of twelve-tone tuning starts from the usefulness of certain ratios and approximations. This adds the missing question: useful for which sounds?

The instrument belongs in the tuning table.

Sources and illustrative calculations checked on 2026-10-12.

References

[1] Plomp, R., & Levelt, W. J. M. (1965). Tonal consonance and critical bandwidth. Journal of the Acoustical Society of America, 38(4), 548–560. DOI: 10.1121/1.1909741.

[2] Sethares, W. A. (1993). Local consonance and the relationship between timbre and scale. Journal of the Acoustical Society of America, 94(3), 1218–1228. DOI: 10.1121/1.408175. Full text.

[3] Marjieh, R., Harrison, P. M. C., Lee, H., Deligiannaki, F., & Jacoby, N. (2024). Timbral effects on consonance disentangle psychoacoustic mechanisms and suggest perceptual origins for musical scales. Nature Communications, 15, 1482. DOI: 10.1038/s41467-024-45812-z. Full text.

[4] McDermott, J. H., Schultz, A. F., Undurraga, E. A., & Godoy, R. A. (2016). Indifference to dissonance in native Amazonians reveals cultural variation in music perception. Nature, 535, 547–550. DOI: 10.1038/nature18635. Author-hosted full text.

[5] Mathews, M. V., Pierce, J. R., Reeves, A., & Roberts, L. A. (1988). Theoretical and experimental explorations of the Bohlen–Pierce scale. Journal of the Acoustical Society of America, 84(4), 1214–1222. DOI: 10.1121/1.396622.

Changelog

  • 2026-10-12: Research and calculation review. Explained partial interactions with worked examples, distinguished model predictions from listening evidence, and bounded the claims about consonance, culture and Bohlen–Pierce tuning.