Wednesday, June 20, 2012

Why does a well-tuned modern piano not sound out-of tune?

Karlheinz Stockhausen is listening.
"Neue Musik ist anstrengend", wrote Die Zeit some time ago: "Der seit Pythagoras’ Zeiten unternommene Versuch, angenehme musikalische Klänge auf ganzzahlige Frequenzverhältnisse der Töne zurückzuführen, ist schon mathematisch zum Scheitern verurteilt. Außereuropäische Kulturen beweisen schließlich, dass unsere westliche Tonskala genauso wenig naturgegeben ist wie eine auf Dur und Moll beruhende Harmonik: Die indonesische Gamelan-Musik und Indiens Raga-Skalen klingen für europäische Ohren schräg."

The definition of music as “sound” wrongly suggests that music, like all natural phenomena, adheres to the laws of nature. In this case, the laws would be the acoustical patterns of sound such as the (harmonic) relationships in the structure of the dominant tones, which determine the timbre. This is an idea that has preoccupied primarily the mathematically oriented music scientists, from Pythagoras to Hermann von Helmholtz.

The first, and oldest, of these scientists, Pythagoras, observed, for example, that “beautiful” consonant intervals consist of simple frequency relationships (such as 2:3 or 3:4). Several centuries later, Galileo Galilei wrote that complex frequency relationships only “tormented” the eardrum.

But, for all their wisdom, Pythagoras, Galilei, and like-minded thinkers got it wrong. In music, the “beautiful,” so-called “whole-number” frequency relationships rarely occur—in fact, only when a composer dictates them. The composer often even has to have special instruments built to achieve them, as American composer Harry Partch did in the twentieth century.

Contemporary pianos are tuned in such a way that the sounds produced only approximate all those beautiful “natural” relationships. The tones of the instrument do not have simple whole number ratios, as in 2:3 or 3:4. Instead, they are tuned so that every octave is divided into twelve equal parts (a compromise to facilitate changes of key). The tones exist, therefore, not as whole number ratios of each other, but as multiples of 12√2 (1:1.05946).

According to Galilei, each and every one of these frequency relationships are “a torment” to the ear. But modern listeners experience them very differently. They don’t particularly care how an instrument is tuned, otherwise many a concertgoer would walk out of a piano recital because the piano sounded out of tune. It seems that our ears adapt quickly to “dissonant” frequencies. One might even conclude that whether a piano is “in tune” or “out of tune” is entirely irrelevant to our appreciation of music. [fragment from Honing, 2011.]

ResearchBlogging.orgJulia Kursell (2011). Kräftespiel. Zur Dissymmetrie von Schall und Wahrnehmung. Zeitschrift für Medienwissenschaft, 2 (1), 24-40 DOI: 10.4472_zfmw.2010.0003

ResearchBlogging.orgHoning, H. (2012). Een vertelling. In S. van der Maas, C. Hulshof, & P. Oldenhave (Eds.), Liber Plurum Vocum voor Rokus de Groot (pp. 150-154). Amsterdam: Universiteit van Amsterdam (ISBN 978-90-818488-0-0).

ResearchBlogging.orgWhalley, Ian. (2006). William A. Sethares: Tuning, Timbre, Spectrum, Scale (Second Edition). Computer Music Journal, 30 (2) DOI: 10.1162/comj.2006.30.2.92


  1. It's an important point, but I wonder if they factor in the extent to which the tones of the dodecaphonic scale actually deviate from what we'd expect in a Pythagorean tuning. For example, a perfect fifth above A4 'ought' to sound at 660Hz, but today we tune it at 659.26Hz. That's a difference, but it is far, far too small for our ears to notice a difference. If I remember rightly, it's only on the seventh degree of a major scale that there is a perceptible difference in expected tones, and those do not have the same important structural function as intervals like fifths and fourths.

    Although music systems around the world demonstrate that ours is arbitrary to an extent, it's important to notice that the human auditory system allows for a vast number of possible scales, yet all the scales of Indonesian, Indian, modal, and pentatonic music have great similarities in their structures and consonance compared to all the ones that we could make up. Of course, there are no rules; no one can dictate what a scale ought to be, and no scale is intrinsically better than another, but this universal tendency towards certain kinds of scales does indicate an innate preference.

  2. I find it surprising how quickly many theorists dismiss the idea of ratios just because of observed deviance. Isn't this just a matter of tolerance? As W. Sethares showed in Tuning Timbre Spectrum Scale, there are complex interactions between timbre and tuning. And there are JND issues. And categorical perception. We surely can dismiss any prescriptive or overly-rigid conception of tuning, but that doesn't mean it is totally relative. Tuning is NOT a matter of complete relativism.

    Can't we accept that the perceptual facts that convinced Pythagorus, Helmholt, Partch et al have an actual musically-relevant basis? There is some degree of adapting to any arbitrary tuning, but there are universal factors in sensory dissonance (i.e. roughness) that shouldn't be ignored.

  3. There are several misunderstandings here.

    First, Pythagoras was not interested in perception (but in numbers). Aristoxenus was. He wrote ‘‘The nature of melody is best discovered by the perception of sense” (cf. Hawkins, 1868).

    Second, low integer ratios (such as those liked by Pythagoras) were shown not to be sufficient to describe perceived consonance. Reinier Plomp and Pim Levelt convincingly showed this in a perceptual experiment (Plomp & Levelt, 1965). Next to the ratio between tones, absolute frequency and timbre play a role (As was suggested earlier by Helmholtz). Setharis incorporated those findings in is nineties model (Setharis, 1993).

    Finally, all these psychophysical models leave out the cognitive component, such as the role that, e.g., learning and mere exposure plays in the perception of consonance (Honing, 2011).

  4. Thanks for the clarification. We are and were in agreement. I have no sympathy for Pythagorus' dogmatic numerology.

    All I wanted to emphasize was just what you now added, "the ratio between tones… play[s] a role" — not a defining role, but not a role which can be completely erased by the other factors. I fully understand and agree about absolute frequency, timbre, and cognition all having substantial impact on the holistic result.

    Incidentally, Sethares 2005 is a much more thorough discussion of this vs his earlier work.