Read more of our Music Special Issue:
Are animals naturally musical?
The illusion of music – don’t believe everything you hear
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Flexible scales and immutable octaves
Singing in the brain – music can change the way you think
Web exclusive – hear five great auditory illusions
ONE of the stars of last year’s meeting of the Mathematical Association of America was Robert Schneider, lead guitarist of the indie band The Apples in Stereo. Delegates listened enthralled as he played a track from the group’s latest album, New Magnetic Wonder. From the very first notes the music sounds strange and almost eerie, a bit like a record played backwards.
Its ethereal quality stems partly from Schneider’s use of tone generators in place of conventional musical instruments. There is also a deeper reason why the music sounds strange: it uses an unusual musical scale in which the intervals between the notes are based on logarithms. Nevertheless, the more you listen to it, the more pleasing it becomes. “When we experience mathematical functions with our ears, we call it sound,” says Schneider. “When the math is particularly elegant and well ordered, we call it music.”
Schneider is not the first musician to invent a new musical scale, and some long-standing musical cultures, such the gamelan orchestras of Bali and Java, use scales very different from the standard western one. Even the scale used by medieval European musicians was different from the one familiar to modern ears. Which raises the issue of whether our preference for a particular musical scale arises because it’s what we are used to or because of something innate. Arguments have been raging for decades over the extent to which music is nature or nurture, but now some firm evidence has started to emerge.
Both melodies and the musical scales on which they are based are all about differences in frequency – what musicians call the “intervals” – between the notes. Listen to pairs of randomly chosen frequencies played simultaneously and you’ll find that some combinations sound pleasing while others can be unpleasant. What matters as far as our perception is concerned is the ratio of the frequencies: the ones we prefer tend to be simple ratios such as 2:1, 3:2 and so on.
When one note is double the frequency of another – an interval of an octave – the two together do not just sound good, they sound almost the same. Not only do we hear the same note for middle C (usually 261.6 hertz) and the note an octave above (523.2 hertz), we also call them both C. Giving the same name to notes an octave apart is not just a European convention. “It happens in all cultures that have names for notes,” says Martin Braun at the Neuroscience of Music research organisation near Karlstad, Sweden.
Playing with scales
This, however, is where the common ground ends. Most western music is based on a scale in which the octave is divided into 12 notes called semitones, each differing in frequency from the last by the 12th root of 2, or 5.95 per cent. Some cultures use a different system. In Java and Bali, for instance, there are two different scales, one dividing the octave into five more or less equally spaced notes, and the other into seven not so equally spaced notes. Gamelan orchestras play in both scales and sound decidedly strange to the uninitiated.
In Schneider’s log scale, successive notes are closer and closer together and the number of individual tones per octave increases almost exponentially in successive octaves. As a result, it is best suited to computer-based music. To play it on a piano would require an instrument with an infinitely long keyboard, not to mention a pianist with infinitely long arms. Despite its apparently exotic nature, it does sound melodic, even if the intervals between individual notes sound foreign.
So it seems you can mess about with scales, and that the way listeners perceive the resulting music depends at least in part on what they are used to. But when it comes to octaves, things are less flexible. There is a physical reason for this. A string tuned to middle C, say, doesn’t just vibrate at 261.6 hertz (the fundamental), but also at multiples of the fundamental, known as harmonics (see Diagram). In fact, all instruments produce a spectrum of sounds like this.
Immutable octaves
This could explain why rhesus monkeys, who are decidedly unmusical, can nevertheless recognise simple tunes they have learned when these are moved up or down an octave or two, but fail to do so when the tunes are transposed by half an octave (see “Are animals naturally musical?”). Braun says this suggests our brains are hard-wired to recognise octaves, and his own research reveals that the human brain detects octaves even when pitch perception is disrupted.
For his study, Braun enlisted the help of a concert pianist with absolute pitch – meaning she can identify the pitch of any note played in isolation – and a mood-stabilising drug called carbamazepine, with the curious side effect of downshifting her perception of pitch by about a semitone and distorting her perception of musical intervals. Even with the drug, she had no trouble recognising octave intervals. Braun says this shows that two notes separated by one or more octaves share the same neurological pathway, activating a characteristic set of neurons within part of the brain called the auditory thalamus (Hearing Research, vol 210, p 85). So while the ability to recognise octaves appears to be innate, we learn other musical intervals in much the same way as children learn languages.
There’s something liberating about knowing that our appreciation of music is a mixture of both nature and nurture. It means that over time we can come to enjoy sounds that we initially found offensive. It also explains the long list of musicians who were once considered avant garde but are now seen as mainstream.