Roll Your Own Theory

Last night I started musing about how one might develop a coherent theory of harmony (that is, scales and chords) for a tuning with 31 equal-tempered steps per octave. This tuning, which we can call 31et or 31edo for short, has some very nice properties, but describing those properties in terms of conventional music theory quickly leads into a morass of confusing terminology.

Our terms for intervals, for example (second, third, fourth, and, for that matter, octave), are firmly rooted in the diatonic tradition — the white keys on the piano, in other words. But 31edo provides some scale modes that are not related to the diatonic modes.

If you hang around online, as I do, with people who do microtonal music, you’ll start hearing a lot of very specialized terminology. Things like “porcupine” and “MOS.” Might other theorists have come up with ideas that I could borrow to describe scales and chords in 31edo? This morning I jaunted over to the xenharmonic wiki to find out.

It soon became apparent that microtonal theory is an intellectual playground, or if you prefer a smoking parlor, in which everybody gets to roll their own. And sometimes the tobacco crumbs leak out the sides. Here, lightly edited, is an attempt to describe MOS, which stands for “moment of symmetry”:

“A Moment of Symmetry is a scale that consists of (1) a generator (of any size, for example a 3/2 or a fifth in 12 equal temperament) which is repeatedly superimposed but reduced within the (2) Interval of Equivalence (of any size, for example most commonly an octave), often called a period, (3) where each scale degree or scale unit will be represented by no more than two sizes and two sizes only (Large = L and small = s).”

The first part of that is clear enough. We’re going to choose an interval of some size and stack it on top of itself, periodically folding it back down so that it stays within the octave (or tritave, or whatever). But what exactly does item (3) in the definition tell us? How can a scale degree be represented by a size? How, indeed, can a single scale degree be represented by two sizes, as that sentence seems to say?

I suspect this writer is trying to tell us that we’re going to end up with two sizes of steps in our scale. If we look at a conventional pentatonic scale on a piano keyboard, for instance, we’ll find that some of the intervals in it are major seconds, and some are minor thirds.

Really, we have two problems here. The first is, there are hundreds of people developing their own concepts and terminology, most of which apply only to their own music. The second is, they don’t always explain their concepts very clearly.

Can all of this intellectual ferment be distilled down into anything useful? I have my doubts.

Tiny Everything

A couple of months ago I learned one or two pieces in Book 4 of Bela Bartok’s Mikrokosmos. And recently one of the people on the Xenharmonic Alliance II group on Facebook posted a link to a really nice piece for solo piano (digital, of course) in 17-note equal temperament.

Inspired by that piece, I figured I’d try my hand at a Mikrokosmos-style piece — short, melodic, and harmonically modern, in 17et. It only took a few hours to whip something up:

I enjoyed the process so much that a few days later I tried again:

In case you’re curious: No, these weren’t played in real time. There’s a lot of hand-editing of note lengths and velocities, quantizing, trying different harmonies by dragging the notes up and down, and manually adjusting the tempo here and there.

If I do more of these, I’m going to have to call the series “Tiny Everything,” so the first piece needs its own name. How about “Antic”? The second piece, with those heavy minor chords in the lower register, seems to be called “Gloom.”

I’m still foodling with the question of how best to notate a 17-note scale on a conventional five-line staff. Or even whether to bother. If you look at the Wikipedia article on 17et, you’ll find Easley Blackwood’s method. Easley obviously put a lot of thought into questions of this sort, but I find it odd that his chromatic scale zigzags. The first three notes, for instance, are C, D-flat, and then C-sharp. Also, with his method the interval of a neutral third (which divides the perfect fifth evenly) is spelled either as an augmented second or as a diminished fourth, never as a third. Read more

Stepping Out (Csound/JI)

Found this little experiment on my Mac tonight. I had forgotten I did it.


It isn’t really a piece of music — just a texture. I suppose it’s post-minimalist, if you like genre terminology: It’s not intended to go anywhere, but it is intended not to be boring or dull while hovering in one spot.

Inspired by hardware step sequencers, which have a limited number of steps but leave open the possibility of modulating the tone while the sequence plays, I thought I’d try to write a simple step sequencer in Csound and give it an interesting amount of variable playback. The pitch sequence table is 16 steps long, but quite often the sequencer resets to the start of the pitch table before all of the steps have been used. The pitches are defined as ratios in just intonation, and after being read they’re multiplied by some factor (usually 1.0, but sometimes 1.5 or 0.75). The table of rhythm values is separate from the table of pitches, and is a different length, so you’ll hear the same rhythm over and over, but in different parts of the pitch sequence. Variable modulation is being applied to the panning (obviously), the portamento rate, and the filter cutoff and resonance. The final few lines of code add a stereo delay.

When Is a Volt Not a Volt?

Regular readers of this space (all five of you) will be aware that I’m fussy about intonation. And yet, I’ve acquired an analog modular synthesizer. Go figure. Analog synthesis is good at many things, but precise intonation is not one of them.

In order to integrate a computer with the modular synth, I’ve acquired Expert Sleepers ES-3 and ES-6 modules. These nifty devices can talk to a computer via ADAT lightpipe, with a very nice PreSonus 1818VSL interface shuttling the 24-bit audio back and forth.

The computer happens to be running Csound. One of my bright ideas is that I’d like to be able to write a fairly complex step sequencer in Csound (not especially difficult to do) and have it play the modular synth.

The oscillators in a modular system of this type are calibrated, in theory, so that when an incoming voltage rises by 1 volt, the oscillator’s frequency rises by one octave. This is called the 1-volt-per-octave standard. My oscillators have 1v/oct inputs.

The output of the ES-3 has a range of +/- 10 volts, and Csound’s audio signals are defined as having a maximum amplitude of +/- 1.000. From this, it’s easy to see Read more

Modular Mojo: More About Pitch

After struggling a bit earlier today (see the previous post) with intonation in my modular synth, I felt I should do a few more tests. There’s some good news to report, and some bad news.

Bad news first: The output of the Toppobrillo Quantimator simply doesn’t match the desired 1v/oct input of my analog oscillators across more than an octave or so. Nor does there appear to be a calibration trimpot on the Quantimator’s circuit board. At least, if there is, I haven’t found a document where it’s mentioned. The quantizer for the Make Noise Rene is pretty much the same. Across three octaves, it just doesn’t produce a reliable 8:1 increase in frequency.

The only oscillator in my system that produces perfect octaves is the Mutable Instruments Braids. This is a digital oscillator, and does its own quantizing of the input CV internally. No surprise that it’s perfect; digital audio is all numbers. The Intellijel Cylonix Shapeshifter, on the other hand, is also a digital oscillator — yet it suffers Read more

Temper, Temper

I’ve composed upwards of a dozen pieces using microtonal equal temperaments — dividing the octave into 17 steps, or 19, or 20, or 31. What I like about these tunings, beyond the fact that you can modulate freely, is that the set of intervals is small enough that you can wrap your brains around it. In a given equal temperament, an interval of, say, seven small-steps always exactly the same sound, no matter what note you start on.

But there’s a price to be paid for this easy-to-grasp uniformity: Even the best, most desirable intervals don’t sound very good.

The two most important intervals are the perfect 5th and the major 3rd. Perhaps this is not surprising; we all know that a major triad sounds stable. The stability arises because the ratios in this chord are found at the very start of the harmonic series. The 5th has a ratio of 3:2 and the major 3rd a ratio of 5:4. At least, in just intonation that’s the case. But it’s not the case in equal temperament.

Our conventional 12-note equal temperament has a very good perfect 5th. It’s within 2 cents (a cent being 1/100 of a semitone) of the ideal 3:2 ratio. The other intervals in 12ET, however, are poor approximations. Our major 3rd is 14 cents sharp when compared to an ideal 5:4 ratio — a difference that is extremely audible, once you know how to listen for it. This out-of-tuneness gives our scale its characteristic edgy sound.

In case you want to whip out your calculator, a 3:2 ratio is about 701.95 cents. A 5:4 ratio is 386 cents and some change.

Several other equal temperaments have better major 3rds, but you have to go clear out Read more


For the past week I’ve been on a Reply All email list in which experimental musicians are being loudly abusive of those who don’t like their music. (I’m one of the latter, which is how I got on the list — the exchange started with a concerted attack on me, but it has now broadened its scope, and I’m mentioned only occasionally.)

I don’t mind the bashing too much, but it does strike me as a curious pastime. Today’s crop included an email (I won’t say from whom) that included this gem: “These hateful assholes should ultimately be ignored – after they’ve had their legs broken….;)”

The question that needs to be asked is not, I think, “Why are these people so angry?” Some people are angry at the world, for one reason or another. Some of the angry people make music. This is not surprising. Their anger does seem to spill over into their music; it quite often sounds angry. That’s okay too.

A more appropriate question might be, “Why do angry musicians insist that their music should be admired?” If you’re going to make angry music, shouldn’t you just sort of take it for granted Read more

In the Lurch

One of the things I like about microtonal tunings is that their exotic flavor sort of invites you to try a few other unusual ideas as well. This piece is in 17-note equal temperament.

Both the melodic mode and the lopsided rhythm were inspired by a piece that was uploaded by one of the other participants in the Xenharmonic Alliance Facebook group. I no longer remember whose piece it was, but credit where due.

Bela Bartok wrote a piece called “Limping Dance,” so I couldn’t use that title. Instead, I think I’ll settle for “Lurching Dance.”

This was done in Cubase, with a variety of software synths — u-he Zebra, Cakewalk Z3ta, Camel Audio Alchemy, Rob Papen Predator and Blue, and KV331 SynthMaster. The drums are from Native Instruments Battery 3.

Jelly Side Down

Toward the end of last year I posted a little essay (“Bent or Broken”) in which I expressed dismay at some of the music being composed, performed, and uploaded by avant-garde microtonalists. I was careful to suggest that my own tastes in music are somewhat conservative. This disclaimer was, of course, an invitation to those who might disagree with me to simply shrug and ignore what I had to say.

Unfortunately, one of the artists whose work I criticized has taken rather extreme exception to what I wrote. I have now removed his name from that post, though I left my unflattering characterization of his recording intact. Nor will I mention his name here. Not content to email me privately, he has now taken the step of sharing his opinions of what I wrote with others in the microtonal community.

This is his right, of course. I feel very bad about upsetting him, and I have apologized for it, but that doesn’t seem to have mollified him. I suggested to him, in an email, that different people have different tastes in music, and that both he and I are entitled to our own disparate tastes. As well as I can figure out, however, he seems to be taking the position that I’m an idiot because my tastes don’t agree with his.

He hasn’t used the word “idiot.” He has, however, referred to me as “lazy” and “an empty shirt,” and to my opinions as “sadistic” and “comical.” In an email that was apparently sent to at least one other person and cc’d to me, he said this: “…until we purge this bullshit out of our Read more

Public Rituals

Here’s a brand new piece of microtonal music — “The Triumphal Procession of Nebuchadnezzar“:

I may make a few dozen more tiny edits — I usually do — but basically it’s ready for public consumption. The tuning is 26-note equal temperament (26EDO for you tuning geeks). I didn’t know much about Nebuchadnezzar when I thought of the title, I just knew I wanted a reference to a Babylonian king. Various things happen during the course of the procession, none of them reassuringly familiar. If you imagine teams of slaves carrying an enormous golden idol past the cheering crowds, you won’t be far wrong.