Since I’ve been deep in this 18edo album project, I thought it’d be a good time to learn more about microtonality and tuning systems in general. So I grabbed Kyle Gann’s The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician off my wishlist and dug in.
I love that title BTW. I certainly can be an impractical musician at times. Not quite as deep into that as some of course, but I do feel that the most efficient path from A to B isn’t always the most creatively inspiring one.
The material in this book more or less divides into two main areas. The first is the quest for a perfect tuning system, which is unfortunately not mathematically possible.
The most consonant sounds — the “sweetest”, purest, least clashing — are, as the Mesoptamians and Greeks proved, the ones with the simplest integer ratios. 2:1 (called the octave for dumb historical reasons), 3:2 (the fifth) and 4:3 (the fourth). It’s downhill from there, but it’s the major and minor thirds that make Western chords what they are.
The problem is, if you insist on using 3:2 as a fifth, the “circle of fifths” used to build a scale is really a spiral. 3/2 to the 12th power is 129.746, which is not a power of 2. Over the centuries, various strategies were invented to cope with this dilemma. Every one of them has its problems, from “you cannot play in that key on this instrument” to “G# is not the same as Ab so I need more than 12 keys for my 12-tone scale” to “absolutely none of these intervals is really in tune.”
Meanwhile in the world of fretted stringed instruments, they never had a choice but to use equal temperament. All of the semitones have the same ratio between each other, though it’s not based on “nice” ratios. Historically when you placed frets, you’d measure 1/18th of the length of the string between the previous fret and the bridge. (This number is more accurate but awkward today.) When you got to the 12 fret you’d be very close to an octave. And coincidentally, the fifth and fourth were pretty close to being right too… it’s just that the thirds and sixths are kind of broken. But we got used to it.
Last night, I was playing pure tones using Bitwig Micro-Pitch to try different intonations and temperaments. This is super easy compared to retuning a piano or building a different organ, of course. In a high enough register it’s shockingly easy to hear these conflicts in tuning that we just take for granted. But then, they can be masked by more complex timbres and arrangements, and the way we compose and play music in the first place.
All of that was not the reason I got into this subject though.
The other aspect of tuning theory is: what do you do if you want to break out of that system and make something that sounds more exotic, eerie, unfamiliar?
There’s a lot of fancy theory here too. Some composers have built scales using other ratio theories. But one of the simpler things is to explore is other equal divisions of the octave (edo) besides 12, or even another basis besides the octave (such as the tritave, the 3:2 ratio). Thus my playing around with 7edo at times, and 18edo in this project. And mostly that started out of curiosity, back when I had the ER-301 Sound Computer in my rack.
In some of those macrotonal scales you just don’t get to play “chords” at all and all of the intervals sound odd. And given the kind of music I make, that’s OK. I don’t really think about Western harmony concepts in my music, though they may happen by accident (I hope that pun doesn’t fall flat…)
In my current project I’m not being super strict about this. I have too many different sources, drones tuned by ear, harmonic series that are inherent a part of the timbre, frequency shifting that throws everything off. But the compositional basis of it is still 18edo.
I’m kind of putting together my thoughts on tuning, scales and quantization options and how to take advantage of them in my hybrid Eurorack/DAW setup. I was planning on writing that up here tonight, but it’s starting to get late and I’m still rearranging the blocks in my head. I’ve tended to do things intuitively without theorizing much, but it’s possible that learning this (and maybe more importantly, finally hearing the differences in different intonations/temperaments) will lead me somewhere. Maybe it’ll just be the occasional dip into a just-intoned triad left bone-dry for the shock factor of zero beating.