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, 2021; Published here earlier in 2012]

Honing, 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).Honing, H. (2021). Music Cognition: The Basics. Routledge. doi 10.4324/9781003158301Kursell, Julia (2011). Kräftespiel. Zur Dissymmetrie von Schall und Wahrnehmung. Zeitschrift für Medienwissenschaft, 2 (1), 24-40 DOI: 10.4472_zfmw.2010.0003Whalley, Ian. (2006). William A. Sethares: Tuning, Timbre, Spectrum, Scale (Second Edition). Computer Music Journal, 30 (2) DOI: 10.1162/comj.2006.30.2.92

Another one bites the dust?

A Tsimane' man plays the flute (from: McDermott et al., 2016).

The music theory literature has been suggesting it for a long time: the idea that simultaneously sounding tones with frequency relationships that are low integer multiples, like 1:2 (octave) or 3:2 (a perfect fifth), are determinant of how listeners perceive consonance. It is an idea that is often related to the overtone structure of natural sounds (such as the voice or string instruments) suggesting that musical harmony is reflective or even a result of the acoustic structure that is found in natural, harmonic sounds that are surrounding us (see earlier entries).

However, a study that was published in Nature today, makes both ideas quite unlikely (McDermott et al., 2016). The authors conclude that "consonance preferences are unlikely to be innate, and that they are not driven by exposure to harmonic natural sounds such as vocalizations." Instead, consonance preferences seem to depend on exposure to particular types of music, presumably those that feature consonant harmony. In an elegantly controlled study McDermott and colleagues compared the perception of musical, speech and natural sounds in North American listeners (both musicians and non-musicians) and compared them to two groups of Bolivian listeners, of which one group rarely is in contact with Western culture, a tribe named Tsimane' (Chimane).

All participants rated the pleasantness of sounds. Despite exhibiting Western-like discrimination abilities and Western-like aesthetic responses to familiar sounds and acoustic roughness, the Tsimane’ rated consonant and dissonant chords and vocal harmonies as equally pleasant. By contrast, Bolivian city- and town-dwellers exhibited significant preferences for consonance, albeit to a lesser degree than North American listeners. The results indicate that consonance preferences can be absent in cultures sufficiently isolated from Western music, and are thus unlikely to reflect innate biases or exposure to harmonic natural sounds. It seems we can remove 'consonance perception' from our list of candidate constituent elements that might underlie the human predisposition for music, i.e. musicality (see Honing et al., 2015).

UPDATE: Related news article in Dutch.

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, 525, 7611. DOI: 10.1038/nature18635.

Honing, H., ten Cate, C., Peretz, I., & Trehub, S. (2015). Without it no music: cognition, biology and evolution of musicality Philosophical Transactions of the Royal Society B: Biological Sciences, 370 (1664), 20140088-20140088 DOI: 10.1098/rstb.2014.0088

De juiste toon: had Pythagoras gelijk? [Dutch]

In muziek zijn vele wiskundige en natuurkundige wetten te vinden. Liggen die patronen aan de basis van wat we mooi vinden? Hebben onze hersenen een voorkeur voor bepaalde patronen? En hoe zit het met andere culturen, die weer andere patronen waarderen?

Over al deze zaken werd er stevig gediscussieerd tijdens de BètaBreak van 18 november j.l. met Michiel Schuijer (Muziektheoreticus, lector aan Conservatorium van Amsterdam), Henkjan Honing (hoogleraar Muziekcognitie aan de UvA) en Jan van de Craats (hoogleraar Wiskunde aan de UvA). Enkele van de referneties die genoemd worden staan hieronder (Plomp & Levelt, 1965; Savage et al., 2015).



De benadering van muziek als een natuurkundig of wiskundig verschijnsel heeft als mogelijke valkuil om naast geluidsleer een soort getallenleer te worden. Alsof harmonische, mooie of ‘juiste’ muziek door de natuur bepaald of zelfs afgedwongen wordt. Er klinkt iets in terug van het, in steeds wisselende gedaantes terugkerende Oudgriekse idee van een ‘harmonie der sferen’, het idee dat de wiskundige structuur van muziek iets zou kunnen onthullen over de natuur zelf. Of omgekeerd: dat een elegante formule die de code van de muziek van vermaarde componisten (denk aan Bach) weet te kraken en de onderliggende getallenstructuur ervan blootlegt, ons kan laten zien hoe mooi, hoe ‘natuurlijk’ die muziek is. Maar al Pythagoras’ ideeën over consonantie in termen van heeltallige ratio’s ten spijt: een hedendaagse, zorgvuldig maar allesbehalve heeltallig gestemde piano wordt door opvallend weinig mensen als ‘vals’ ervaren. Het is de eeuwenoude tegenstelling tussen muziek opgevat als getal en muziek als empirisch feit (cf. Pythagoras versus Aristoxenus). Muziek huist niet zozeer in het geluid of in het getal, maar eerder in het hoofd van de luisteraar (Honing, 2012).

Honing, 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)
Plomp R, & Levelt WJ (1965). Tonal consonance and critical bandwidth. The Journal of the Acoustical Society of America, 38 (4), 548-60 PMID: 5831012
Savage, P., Brown, S., Sakai, E., & Currie, T. (2015). Statistical universals reveal the structures and functions of human music Proceedings of the National Academy of Sciences, 112 (29), 8987-8992 DOI: 10.1073/pnas.1414495112