5. On Microtonal Music

So you’re heard about this “microtonal” thing, and you’ve become curious about it. Maybe you even listened to some music, and it sounded weird and kind of… out of key but maybe also simultaneously “locked-in” in the way that a cappella choirs and barbershop quartets are. Or maybe you’re wondering why we would ever want more than 12 notes in an octave at all, or you’re familiar with musical traditions - Arabic, Anatolian, Indian classical, klezmer, and gamelan among many others - that make use of scales and harmonies beyond or totally separate from the modern Western canon. Even the sweetness of old church organs - you know, the ones that sometimes have weird split black keys? - are an example of microtonal music! But why would we ever want to make use of such dissonant intervals as we find crunched beneath the semitone distance of a minor second?


Before we can talk about that, we need to think a little more closely about the nature of musical pitches. While pure sine wave tones are comprised of a single frequency called the fundamental, nearly every other musical implement - from guitars to trumpets to the human voice - is built up out of that fundamental frequency and multiples of that frequency. Why should that be? Because that frequency - like the familiar A440 - is literally some vibration in some medium fixed at some nodes, and that vibration in turn stretches and compresses air, ultimately transmitting that vibration to your ears. But the integer multiples of that frequency - called partials - can often appear as higher-energy excited modes of the same vibrating medium; for complicated physics reasons, wind instruments only ever have the even or else the odd partials contribute to the final sound, while in others, like stringed instruments and voice, all of the partials - both odd and even - appear. Of course, digitally produced audio can have any mix of partials you like.


And it’s not just a mathematical abstraction or physical curiosity! The 2nd partial corresponds to the pitch an octave above the fundamental, a ratio of 2/1; the 3rd partial is a perfect fifth above the second partial, a ratio of 3/2. We go on up the chain of the so-named harmonic series, noting that to add together intervals, we multiply the ratio of pitches, so that the 4th partial is two octaves above the fundamental (4/1), which is also a perfect fourth above the 3rd partial (4/3). The 5th partial is a major third (5/4) above that, and the 6th partial, a minor third (6/5) above that. While we should note that the 8th partial is three full octaves above the fundamental, the 9th partial a whole tone or major second above that, and the 15th partial a semitone below the 16th partial - itself four full octaves above the fundamental frequency - the 7th partial has no good equivalent in the Western 12-tone system. If our fundamental is a C, then the closest we can get to the 7th partial - which famously shows up in barbershop quartets as the “harmonic seventh” - is a B-flat nearly a third of a semitone too flat! The story is even worse for the next prime harmonic, the 11th, a G-flat almost precisely half a semitone too flat as well. (The 10th, of course, can be straightforwardly constructed as an octave above the 5th.)


The problems with the nice neat conception of (e.g.) piano keys as lining up neatly with partials doesn’t end there, because I’ve lied to you a little bit more: the common equal-tempered tuning most common in the modern day doesn’t even give you the kind of tidy small-integer ratios - just-intonation systems, as they’re called - that the above simplified story would suggest form the basis for all music theory. And how could it? If you know some music theory, you know that the circle of fifths would suggest that we should be able to stack 12 perfect fifths to get some number of octaves - 7 of them, in fact, since each perfect fifth is 7 semitones large. But because stacking intervals multiplies the ratios, this would have to mean that (3/2)^12 = 2^7, that is, 3^12 = 2^19! This is clearly impossible - no nontrivial power of two is also a power of three. This is the reason for the “wolf fifth” of older tuning systems; the modern equal-tempered system instead simply starts with the octave defined to be 2/1 and constructs the semitone directly as its twelfth root, which is generally close enough to trick our ears - though we lose some of the beat-free locked-in sound that the human voice can still easily and ringingly achieve.


This is also the reason we might want to pick a number other than 12 for our equal division of the octave: Anatolian music favors an effective 53, which allows for almost flawless perfect fifth, while Arabic music and klezmer prefers quarter-tones (24 notes), which notably still allow for harmonic sevenths and elevenths; in both, the neutral third - halfway between major and minor - plays an important role as well. (You can check out some King Gizzard if modern rock is more your speed.) Indian classical shruti sometimes represent a 22-note uneven division of the octave, while gamelan music goes the other way entirely, splitting the octave into just 5 or 7 notes - though Indonesian tunings vary more widely than in the Western canon. Finally, while baroque-era organs’ “quarter-comma meantone” is outside the scope of this post, one of my favorite tunings these days is its modernized version, using 31 notes and sacrificing a little of the perfection of a fifth to gain sweeter thirds alongside access to harmonic sevenths; numerous artists have both composed new music in 31-note scales, as well as directly porting over old favorites from classical and jazz music. (I like the adaptation of Stevie Wonder’s “I can see clearly now”.)

And it doesn’t end there! Modern artists like Sevish and Brendan Byrnes have used 22-note systems and hand-picked just-intonation scales as well to great effect in electronic music and guitar-centered pop rock respectively; Sevish is all over the place but has also used the alien Bohlen-Pierce scale, which shuns even harmonics and treats the 3rd partial as just as important as classical music theory treats the octave, while Byrnes has employed 27-note scales to great effect, using the sharper fifths to achieve a hyper-bright effect. For my part, I’ve adapted some music: an alarm clock tune and the Price is Right theme into 22-tone, as well as Grandma Kim’s favorite hymn (I’m Pressing On the Upward Way) into church-organ-mimicking 31-tone. In a later post, I’ll go into more depth on recommendations on where to start with microtonal music that sounds relatively pleasant to the lay ear, as well as sharing a theory of how microtonal counterpoint might work.

 

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