Equal Divisions of the Blog

This morning, as I typed a very long reply with several paragraphs to someone on Reddit (a pastime I indulge in fairly often), I had an epiphany. Not a world-changing epiphany, mind you, but the sort of highly insignificant epiphany you get when you realize that today is actually Wednesday and not Tuesday like you initially thought. (Unfortunately I didn’t get the pleasure of that specific epiphany today, as I knew it was Wednesday from the start.)

My epiphany was that when I am on Reddit, which most of the time involves me refreshing the synthesizers subreddit every five seconds or browsing the latest prequel memes, I frequently type very long comments and comment replies at a very rapid pace as I did this morning. I then realized that this feverish pace could possibly be a perfect candidate for bottling up, driving to a top-secret laboratory, and carefully releasing in a treated room onto a test subject called “posting on my blog more than twice a year”.

There is a catch though: the things I type about very quickly are usually obscure things I’m interested in, usually related to music or instruments. But, based on the readership of my biannual life updates, I don’t think that’s such a stretch for this blog and I figure if at least one person gets something out of a post then it’s done it’s job. Also, it’s my blog so why wouldn’t I write about what I want to write about? It’s almost as if that’s why this blog exists in the first place!

So, allow me to explore the latest object of my fascination. It’s something I’ve been interested in for a while, but I don’t think I’ve ever discussed it with anyone but a select few who I thought would enjoy the Illuminati-level of crazy this particular subject emits. It’s a subject that speaks to my obsession with new sounds and textures.

I’ll begin by asking a question: what is something that 99.9% of music has in common?

There are many answers to that question (it uses notes and rhythms!), but the one I’m interested in is the fact that 99.9% of all music uses the same twelve notes. This is treated as a given; it is a common trope among musicians that “there are only twelve notes”, and what follows is either a) “it should be easy to find one of them that sounds good”, or b) a debate on whether or not truly original music is still possible, because how many different ways can you arrange twelve notes? (That’s not a rhetorical question, by the way – more on that later.)

Allow me to ask a follow-up question: why does 99.9% of all music use the same twelve notes?

This question, too, has many answers – answers that have been documented, theorized, and pontificated about. People have graduate degrees in answering that question and there are countless academic publications written about it. I liken it to a question of similar gravity to the business of music as the Ultimate Question (as posed in the Hitchhiker’s Guide to the Galaxy) is to the business of existence. (And the answer to this question is a bit more interesting – but less funny – than “42”.)

The basic answer as far as I can work out is that these twelve notes in equal temperament provide a decent approximation of the purest harmonic intervals with the least amount of effort, in a way that works in any key center. Certainly the aggregate sonorities of C major and F# major are much more preferable in equal temperament than they are in C major just intonation (even though the justly-tuned C major chord is far superior to the equal temperament one).

By now you’re probably thinking that there’s inevitably a “however” coming, and you’d be right.

(Okay, I’ll say it: however.)

The problem is that twelve equal divisions of the octave (shorthand 12EDO, also written 12TET or 12ET for “12-tone equal temperament”) is not a perfect system. Far from it, in fact. Its most practical use is on fixed-pitch keyboard instruments that need to be able to play in any key center equally well. Or, put another way, that need to be able to play in any center equally unwell. But for instruments with the freedom to individually adjust individual pitches, “equal temperament” doesn’t really exist. For example, it is standard practice for an orchestral brass section to tune every chord to be harmonically pure as it would be in just intonation in that key. If they didn’t, those powerful brass chords everyone loves (if you don’t, you’re wrong) would sound terrible.

Yet keyboard instruments for the most part soldier on with 12EDO, and nobody listens  to a Beethoven piano sonata and complains about its harmonic impurity. This is why 12EDO works and is the standard, but it also only works because our ears are used to it. If you play a C major chord in just intonation and then play a C major chord in equal temperament, you suddenly hear just how out of tune the equal temperament chord really is. (As an aside, there is actually a keyboard instrument that can play any chord in just intonation, called the Tonal Plexus (with up to 1688 keys!).)

Before I go further, I’d like to point out that many people have written about and made videos about this subject much better and in much more detail than I can. If you’re interested in learning more from people who really know what they’re talking about, just Google or YouTube some of this stuff and you’ll find dozens of articles and videos about everything I’m talking about. Maybe, just maybe, you’ll become a little bit obsessed like I have. To get you started, here’s a video about 12EDO from an excellent YouTube channel, “12tone”. It discusses some things I’m about to, so if you don’t want spoilers you can save this rabbit hole for later. Secondly, here’s a simple video audibly demonstrating the difference between pure intervals and their impure equal temperament equivalents.

We haven’t even reached the actual subject of this post yet, so let’s get closer with another question.

Can there be more than twelve notes?

For many musicians, this question also has an easy answer. Of course you can have more than twelve notes, you can use quarter tones (using the notes equally spaced in between each chromatic half-step). The problem is that quarter tones have a pretty bad reputation, mostly because the vast majority of compositions using them are unlistenable esoteric exercises in sounding excessively unpleasant. For those with a little more taste, they do have musical use. Jacob Collier, a paragon of modern harmony and musicianship, uses them very effectively as passing tones or even to modulate everything up by a quarter tone to give a different mood such as in this example. For Jacob Collier there are infinite notes as he uses different tunings (A=432 instead of the standard 440 for example, which is an entirely different subject (if you would like to know my personal opinion on the matter, A=442 for life)) and even slowly glissandos between them (including in the linked example). In short: he’s a beast.

Fangirling aside, even quarter tone microtonality is not really what I’m getting at here. After all, the 24EDO quarter tone scale is still based on 12EDO, just with another note in each of the spaces. It has the same tuning issues as 12EDO with the helpful (?) addition of some alien-sounding notes.

By now we’re over 1200 words into this post and you probably really want me to get on with it. Fortunately, we’ve covered everything we need to get to the main subject of my fascination and explain it well. To do this, I’ll start by asking one more question:

What if you divide the octave by something other than twelve?

That is what this blog post is really about. While the harmonic series and harmonically pure intervals and ratios are fixed as they are based on mathematics and the physics of sound, temperaments and systems of tuning are artificial creations that can be anything you want. While we’ve explained the good reasons for 12-tone equal temperament being the standard, it seems a lot more arbitrary when you look at it as just a number with EDO after it. So think about it: what if instead of 12 tones per octave, you used 13? Think about the implications of that. Suddenly everything you know gets thrown out the window. The distance between adjacent notes is completely different, the scales and modes you can use are completely different, and most importantly: the sound is completely different. Playing anything using 13EDO creates harmonies not possible with 12EDO, and vice versa. 13EDO is an entirely new world of music waiting to be written.

And the best part? That’s just the tip of the iceberg.

There are hundreds of xenharmonic tunings, and everything about them challenges the standards of music. Everything from 5EDO to 313EDO and beyond, each with their own sound. You don’t even have to be restricted to using the octave (ratio 2:1, or the second partial in the harmonic series) as the dividing interval. You could instead use the tritave (ratio 3:1, or an octave and a fifth and the third partial in the harmonic series) as used in the Bohlen-Pierce scale (13 divisions of the tritave, or 13ED3), the perfect fifth (ratio 3:2) as used in the Carlos Alpha scale (9 divisions of the perfect fifth), or even the pentave (ratio 5:1, the fifth partial in the harmonic series, which is so large that there are fewer than five pentaves within the range of human hearing) as used by Stockhausen (25ED5). Furthermore, there are xenharmonic scales which are not equal divisions, scales and modes created within each temperament, and anything else you can think of. There’s an entire Xenharmonic Wiki which you can use (like I have) to really dive into the black hole of information. It has articles on every tuning and which ones have good pure intervals to use.

So what’s the point, apart from “oh that’s pretty cool”?

There is a YouTuber called Sevish who puts out electronic music that’s entirely xenharmonic, and it sounds great. Not only does it sound great, but in the midst of the unfamiliar sonorities it’s still pleasing to listen to like any good 12EDO composition would be, rather than the dissonant noise most microtonal music (and, it must be said, a lot of contemporary 12EDO music) tends to be. Microtonality (and as an aside, these scales are properly grouped together as xenharmonic rather than microtonal because some scales, like 7EDO for example, are macrotonal, as they have more space between the notes rather than less) is for the most part not very well accepted for this (valid) reason, but Sevish proves in my mind that xenharmony is a valid musical avenue to pursue. And that means something very profound.

Remember when I said at the beginning of this post that “how many different ways can you arrange twelve notes?” is not a rhetorical question? Well, the exact number for arranging twelve notes (using each only once) is 479,001,600. Add repeated notes and note lengths and harmonies and different instruments and the amount of possibilities for music using 12EDO is practically infinite. Millions of pieces have been written throughout human history using just 12EDO, but 12EDO is only one star in an entire galaxy of temperaments. In a nutshell, we haven’t even scratched the surface of what is musically possible.

There are, of course, some problems. Not only is 12EDO the established standard, not only is nearly all existing music written and performed in 12EDO, and not only are peoples’ ears used to 12EDO, but nearly every instrument that exists is built with 12EDO in mind. Instruments that can glissando like fretless string instruments, the trombone, the theremin, and the human voice can of course play/sing in any temperament possible, but the hurdle there is doing it in tune with ears unfamiliar with every note in the scale. So what we need are relatively fixed-pitch instruments that can play in xenharmonic temperaments. There are a few that do exist; people have made Bohlen-Pierce (13ED3) guitars, the woodwind maker Stephen Fox sells three sizes of Bohlen-Pierce clarinets, and with extended techniques even some traditional instruments can be played to a much higher degree of accuracy (such as the saxophone, which Phillip Gerschlauer can play with 128 notes per octave). But essentially if you want to play in a non-standard temperament you need a custom-built instrument.

Another problem is notation. All standard Western notation is specifically for 12EDO (or sometimes 24EDO) music. For every xenharmonic temperament a new notation system would need to be used, but thankfully many of them could be based on the standard system with only a few changes.

Interestingly, keyboard instruments could be the champion of xenharmonic tunings. While the least flexible regarding traditional just intonation, they are already the most flexible with xenharmonic tunings. More and more software synthesizers are being designed with xenharmonic support in mind by including alternate tunings and/or having the ability to import scale files from a software called Scala, which is a program specifically designed to create xenharmonic scales to use with MIDI instruments. This is how Sevish is able to create his xenharmonic tracks for YouTube. Using these softsynths as live (rather than sequenced) performance instruments is more difficult, because you need some way to play them and using a typical 12EDO keyboard is not very intuitive.

Then there’s the simple problem that very few people know about xenharmonic tunings at all and much fewer actually use them. For such a vast undiscovered world of sound, the amount of people exploring it is infinitesimally small. Perhaps this post can get a few more people to join the expedition, but I have solutions for the other two problems as well. Here is a page out of my sketchbook that has designs for keyboards in other tunings, with the typical 12 note per octave keyboard at the top left for reference. And here’s one with ideas for notation systems for certain xenharmonic scales. This is obviously far from complete, and many tunings will require more complicated notations than that (such as only certain note letters having accidentals), but it’s a proof of concept demonstrating that notation for tunings outside of 12EDO doesn’t have to be completely foreign.

Although it is much more difficult to attempt to compose notated music in xenharmonic tunings than it is to load up a software synthesizer that uses Scala files to play in different tunings, it is one of my goals to persevere and write a collection of concert etudes for a xenharmonic keyboard in order to create accessible xenharmonic classical repertoire. But as I am me, I have many other composition goals besides that, so we will see. In the mean time, I challenge any and all musicians to learn about and experiment with the world of xenharmonic temperaments. I’m sure there’s lots of wonderful xenharmonic music waiting to be created.