This is the final part of a series introducing xenharmonic music theory. In the first part, I talked about musical perception, especially the perception of microtones. In the second part, I explained roughness theory, which is an empirical theory of dissonance independent of musical tradition. The first two parts overlap with conventional music theory, but in this third part, I finally reach the music theory that is more particular to the xenharmonic tradition.

I’m just going to scratch the surface here, with an eye towards how you would actually use it in practice, if you were a composer. Most readers, I imagine, are not composers. It’s okay if it’s just a hypothetical for you, as long as you learn something.

**Tuning systems**

A tuning system is a set of pitches to be included in your music. You could, of course, just say that any frequency goes–and that’s what we call free tuning. But such unconstrained freedom can be difficult for composers and performers alike. Also many instruments can’t play just any pitch, you have to pick a tuning system with a finite set of frequencies, and design your instrument around that. These frequencies aren’t strictly adhered to–some small variation in frequency is used for color, texture, and expressiveness–but the tuning system defines the ideal.

**Octave equivalence**

Most tuning systems have what’s called octave equivalence. Each note is considered equivalent to the notes an octave below or an octave above. Tuning systems with octave equivalence are periodic with a period of 1200 cents. For example, if your tuning system includes 300 cents, then it would also include 1500 cents, 2700 cents, 3900 cents, and all these intervals would be treated as equivalent.

In part 1, I gave the cent values for the first few harmonics: 1200, 1902, 2400, 2786, 3102, 3369, and 3600 cents. Using octave equivalence, we can translate all of these harmonics into the 0-1200 cents range. The results are: 0, 702, 0, 386, 702, 969, and 0 cents.

I won’t say much about tuning systems without octave equivalence, but they definitely exist. The best-known tuning system without octave equivalence is the Bohlen-Pierce scale. Instead of being periodic in 1200 cents, it’s periodic in 1902 cents, the 3:1 ratio. So, theoretically, the interval at 300 cents would be equivalent to the interval at 300+1902=2202 cents–but I suspect the typical listener will not hear it that way just because the tuning system declares it so.

**JIs and edos**

One common type of tuning system is just intonation, or JI. Just intonation contains exact integer ratios. Most just intonation systems will have 0, 386, and 702 cents exactly, along with many other integer ratios. Different JI tunings make different choices about which integer ratios to include.

Another common type of tuning system is an edo, which stands for equal divisions of the octave. Importantly, this means equal divisions of the octave on a *logarithmic* scale. For example, the 10edo tuning system includes 0, 120, 240, 360, 480, 600, 720, 840, 960, 1080, and 1200 cents. The standard western tuning system is 12edo, so it includes 0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, and 1200 cents

There are, of course, other tuning systems, but I’ll spend most of my time talking about edos. If you want to learn about other tuning systems, sorry my intro is too basic for you.

**The most obvious effect of tuning**

Even if we just stick to edos, that introduces a whole lot of tuning systems. Can you tell the difference between 10edo and 19edo? If you’re a composer, how do you decide which one to use?

Here’s a very simple answer to get you started. Look at how well the tuning system approximates the 3/2 ratio. 3/2 is 702 cents, and the nearest interval in 10edo is 720 cents. That’s an 18 cent difference, which will have a huge effect on how the tuning system sounds. Poor approximations will sound off-kilter, and good approximations will sound more consonant. We’re relatively sensitive to the tuning of the 3/2 ratio, because it’s such a musically important ratio, and because we’re used to 12edo, whose 3/2 approximation is only 2 cents off.

Beyond that, you can also look at how well the tuning systems approximate 5/4. And then 7/4, and 11/8. But we’re used to a tuning system that already has poor approximations for these ratios, so in my opinion the impact isn’t as large.

**Temperaments**

In the context of xenharmonic music theory, a temperament is a mapping from (a subset of) the rational numbers onto musical intervals. The purpose of a temperament is to determine how to approximate integer ratios within your chosen tuning system.

To understand how this works, let’s walk through an example. We want to map 3/2 into the 10edo tuning system. 3/2 is 702 cents, which is pretty close to 720, which is 6 steps in 10edo. So our temperament maps 3/2 -> 6 steps. Next we want to map 5/4. That’s 386 cents, which is close to 360 cents or 3 steps. So we have 5/4 -> 3 steps.

Next, let’s map 25/16. 25/16 is the square of 5/4. And the rule is that if you take the square of a number, the temperament maps it to an interval which is twice as large as the first one. So we have 25/16 -> 6 steps.

Did you notice that 3/2 and 25/16 both map to 6 steps? This implies that if we look at the ratio 25/24, the temperament will map it to an interval of zero steps.

**Tempered commas**

What I just showed in the above example is a temperament that “tempers out” the 25/24 comma. A **comma** is a small musical interval. We say that the temperament tempers out the 25/24 comma, because it treats the comma as zero steps, which is to say it approximates 25/24 ~ 1.

The reason we talk about temperaments, is that a lot of music is based on stacking approximate integer ratios. In western music theory, we have the major chord, which is an approximation of the 1:5/4:3/2 frequency ratio. The most basic chord progression is I-V, two major chords in succession, with the second chord having 3/2 times the frequency. So the I-V chord progression moves from 1:5/4:3/2 to 3/2:15/8:9/4. So we end up stacking these frequency ratios, getting strange ratios like 3/2*5/4=15/8. And that’s just the most basic chord progression! If you stack enough of these ratios, you end up with commas. And then the question is, does your temperament temper out the comma, or does it not?

**Comma pumps**

Commas suggest a particular idea that you can insert into your composition. You can write a chord progression that deliberately stacks ratios until they generate a comma. This is called a comma pump.

If you’re using a temperament that does *not* temper out the comma, then the comma pump will cause an upward or downward shift. If you repeat the same comma pump over and over, the music will slowly drift upwards or downwards, like a spiral. On the other hand, if your temperament tempers out the comma, that might produce a sort of tension release, as notes unexpectedly align. Or maybe not? It will sound like what it sounds like, so I suppose you’d have to test it out.

A very famous example of a comma pump is the jazz song “Giant Steps”. This song has a chord progression from 0 cents to 800 cents to 400 cents, back to 0. 400 cents is an approximation of 5/4, so it’s a bit like we’re stacking 5/4 three times, pumping the 125/128 comma. The 12edo tuning system tempers out this comma! So Giant Steps makes use of a chord progression that actually isn’t possible in every tuning system. (Although, you could just let the song spiral out, that might be cool too.) Likewise, other tuning systems contain chord progressions that aren’t possible in our tuning system.

**Temperament theory**

Temperaments don’t map every rational number to a frequency. Usually, a temperament will only map rational numbers consisting of a certain set of prime factors. For example, the usual temperament for 12 edo only uses the prime factors 2, 3, and 5, because 12edo has decent approximations for those prime factors, but has an awful approximation for the 7:1 ratio. Once we decide which prime factors are included in a temperament, and how the temperament maps those prime factors, that’s sufficient to define the temperament.

There’s even some temperament notation based on how the temperament maps prime factors… although the notation is very mathy. Basically you list out the number of steps corresponding to each prime factor. So the 12edo standard temperament is notated <12 19 28]. You can probably do things with this; I don’t know what you can do, I’m just tickled by the fact they’ve borrowed bra ket notation from quantum mechanics.

And by the way, you have multiple temperament options for a single tuning system. Usually there’s a natural choice of temperament, just based on the nearest approximation of each prime factor. But you can arbitrarily include or exclude different prime factors. And you could assign a prime factor to the second best approximation instead of the best, if it pleases you.

**Compositional advice**

So let’s return to my advice about understanding a tuning system. My first suggestion was to look at how well the tuning system approximates the 3/2 ratio, and then maybe the 5/4, 7/4, and 11/8 ratios. My second suggestion is to look at possible temperaments, and understand which commas get tempered out. Those temperaments suggest musical possibilities, such as comma pumps.

Or you could not do any of that. You can just find a microtonal keyboard and try things out. A lot of composers just compose something that sounds good to their ear, and then it can be quite difficult to analyze the music after the fact. Temperament theory is just a direction to explore. Personally, I’ve written music in 18edo, which has two very bad approximations of the 3/2 ratio at 667 and 733 cents. The neat thing you can do with 18edo is to alternate between the two bad approximations. But that doesn’t really fit into temperament theory, and it doesn’t have to.

There are other directions to explore too! For instance, scales. In the western musical system, even though there are 12 notes, often people only use 7 at a time. So if you’re using 19edo, you really don’t need to use all 19 notes just because they’re there. You could use a subset, a scale. There’s some scale theory out there, which suggests which notes to include. I’m not going to go into it here.

I’m reaching the limit of my music theory knowledge here, and that’s okay. Past some point, hardly anyone knows what they’re talking about, and they’re just doing whatever they think works. Xenharmonic music is a lot of legitimately new territory–sometimes you need to draw your own map, or do without one.

consciousness razor says

It’s funny, I think about all this differently … none of the business with rational numbers like 3/2 or 5/4, commas, etc.. It’s not like I don’t know about it, but I guess it’s just not an issue for me.

One way to put it is that, in order for that augmented triad to be in a system using n-EDO, it must be that n = 3k for some natural number k, because that type of chord/set uniquely divides the octave into 3 equal parts. So in 12-EDO, you can just think of it as 3*4 = 12, and k is the four-semitone interval, the (equal temperament) major third. And of course that interval itself doesn’t change, no matter which n you might choose. It just may or may not be among the intervals that are “available” or “allowed” with some n. Or you may think of it as being some other multiple of whatever your smallest interval is, rather than four semitones. But that’s just about calling it different names (or not using it).

Twelve is kind of a nice choice if you care about these things, because the only numbers less than it which are relatively prime are 1, 5, 7, and 11. (Note those are the generators, since the up/down chromatic scale and the circle of fifths/fourths will give the entire collection of 12 notes.)

You can also think about it in terms of rotational symmetry. If you’re visualizing the set of pitch-classes as a circle (or, say, only 12 points equally spaced around it), then the shape of the augmented triad is basically an equilateral triangle inscribed on it. (We really just care about the vertices of the triangle, where it intersects the circle – there isn’t actually anything else to talk about inside/outside the circle.) So, you can rotate that shape by 1/3 of a turn, and you will have the same points again. If you’re not “allowed” to do that rotation because you don’t have that interval, then don’t.

That’s also why there are only four distinct transpositions of the chord in 12-EDO, not the usual twelve. So, you can have one with (for example) C, D-flat, D, or E-flat, but when you move up to E, it is the same set again (C, E, G-sharp/A-flat, in whatever order) which you got by only rotating/transposing the thing and not some other way.

In the case of four-element sets (tetrachords, if you like), things get a little more interesting. There is the one you might think of immediately: the fully diminished seventh (e.g., 0369 in 12-EDO), which divides the octave into four equal intervals. It’s shaped like a square (or the points are 1/4 of a turn around the circle), much like the equilateral triangle before. This is also a nice feature of the octatonic scale (e.g., 0134679T), because the diminished seventh is the complement of it (i.e., 258E is equivalent to 0369 — just adding/subtracting 2222), meaning it’s the same thing when looking at the non-notes or the “gaps” between the notes.

But there are others among the tetrachords: the sets 0167 and 0268 only have six transpositions each because of their symmetry. If we’re thinking about multisets too, there is also 0066. They are all of the form (0, x, 6, x+6), where x can be 0, 1, 2, or 3. The basic reason is that the “6” tritone interval is dividing the circle exactly in half, and 4 is an even number. So we can construct such sets with both halves containing the same subset (such as 00 and 66, 01 and 67, 02 and 68, or 03 and 69), and they will line up with each other when properly rotated (in this case, by 6).

In 12-EDO, there aren’t any like that with five or seven notes, because those are not factors of twelve and don’t have any factors in common with it. But with sets containing 6, 8, 9, and 10 notes, there are once again other nice shapes like that. (The 6-note hexachords are especially interesting.)

After that, my general attitude is that those are fairly large and unwieldy sets, which we should probably consider breaking down into smaller chunks whenever we come across them, if we’re trying to analyze what’s going on in a piece of music. And at twelve notes or more, working with 12-EDO as I’ve been doing, you’re really just talking about multisets anyway (and a huge number of them, with only the one “regular” set that is the chromatic scale).

Siggy says

@CR #1,

In 12edo, the more common comma pump isn’t 125/128, but rather 81/80. The 81/80 comma pump is basically going up 4 perfect fifths, and down one major third. A perfect fifth is 7 steps, and a major third is 4 steps. So that’s 7 + 7 + 7 + 7 – 4 = 24 = 0 (mod 12).

But this property of 12edo really doesn’t have anything to do with the prime factorization of 12, nor the rotational symmetry of chords. For example, the property is shared by 19edo. A perfect fifth is 11 steps, and a major third is 6 steps. So we have 11 + 11 + 11 + 11 – 6 = 38 = 0 (mod 19). But this property is not shared by 15edo, which has a perfect fifth of 9 steps and major third of 5 steps. 9 + 9 + 9 + 9 – 5 = 31 = 1 (mod 15).

Another example, let’s look at 72edo. If we’re using the nearest approximation of 5/4, that’s 23 steps. So if we map the 125/128 ratio onto 72edo, that’s 23 + 23 + 23 = 69. So even though 72 is divisible by 3, 72edo’s standard temperament doesn’t temper out the 125/128 comma. Of course, you don’t have to use the standard temperament, you could use a “warted” temperament that uses the 2nd best approximation for the 5/4 ratio, which is 24 steps. This temperament tempers out the 125/128 comma because 24 + 24 + 24 = 72 = 0 (mod 72).

This all goes to show why, in xenharmonic temperament theory, the prime factorization of the edo is not particularly important.

consciousness razor says

I mean, yes, you can just

stipulatethat these things are “perfect fifths” or a “major thirds” if you want to do that, then carry on from there. But I think that in an n-EDO system per se, that’s not very well motivated and not playing by that system’s own rules. We don’t really have to import those kinds of extraneous concepts into it, but what we do need to look at are the types of shapes/structures you can make (exactly, not approximately) out of your basic unit interval that divides the octave into however many equal parts. That imposes some very definite mathematical constraints, based onexactlywhat’s going on whenever you have this or that collection of pitch-classes. There’s not really a notion of trying to create some extra sort of wiggle room in order say that you got something else out of it too.This is an example of what I’m talking about above, in case it isn’t clear. There’s no real reason in the first place (when you only tell me “72edo”) to go hunting for approximations of a 5/4 ratio. You just have a circular space divided into 72 pieces, and the basic question is about which structures you can make with that.

It’s obviously not impossible to do this sort of thing, but the impulse to try to come up with an approximation like that is coming from other considerations elsewhere, which aren’t really a part of 72-EDO itself. Do we need this? What’s it really giving us that we wanted? Is it making things harder or more complicated than they need to be? Lots of questions like that to worry about, you know?

I guess if you just regard this as a different type of theory with a different set of goals and assumptions, then you’re free to say this or that is unimportant, according to it. I’m honestly not sure what to make of that, but it’s something.

Siggy says

@CR #3,

You seem really skeptical of the goals and assumptions of xen temperament theory, but I’m not really trying to persuade you to adopt this as a framework. I’m just trying to explain what the framework even is. The main thing I will say in favor of the framework, is that people who actually compose music in edo tuning systems seem to find it helpful at least some of the time.

I’ve never heard anyone in xen music talk about musical set theory. It simply isn’t an influential framework in xen circles, so what can I say?

consciousness razor says

To keep things relatively simple, let’s look at 24-EDO (made with quarter tones, instead of semitones). I mentioned before that with the four-element set-classes in 12-EDO, there are four set-classes that have a smaller number of transpositions (and related properties) due to their symmetry. Those were 0066, 0167, 0268, and 0369. The first three have six transpositions, and the last has only 3, since it’s even more symmetric in some sense.

If we translate them into 24-EDO, it’s easy to see that we can just double all of those numbers and use mod 24 arithmetic instead of mod 12. (For clarity, I’ll separate items with commas this time):

(0, 0,12, 12)

(0, 2, 12, 14)

(0, 4, 12, 16)

(0, 6, 12, 18)

But in 24-EDO, there are more symmetric options, put together in the same basic way as the others, because there aren’t just the even numbers:

(0, 1, 12, 13)

(0, 3, 12, 15)

(0, 5, 12, 17)

Those still have the same degree of symmetry as the first three I mentioned above, while the (0369) set or its 24-EDO version is still the most symmetric you can get with four-element sets, no matter what. Because in all cases it’s shaped like a square. And I think it’s very nice that we can understand all this via simple stuff like geometry and such — it just sort effortlessly falls out of the math immediately, without any extra work. At any rate, if we translated those back into 12-EDO, they have some half-integers:

(0, 0.5, 6, 6.5)

(0, 1.5, 6, 7.5)

(0, 2.5, 6, 8.5)

And if you think about putting them into a set-class/configuration space, then of course they’re still in the same space as before, but they would occupy the midpoints between some of the 12-EDO set-classes that we started with. And then, besides the extra sets themselves, there are a lot more transitions from one set to adjacent sets in a space like that, since you can also move from a midpoint to another nearby midpoint (not just along one of the lines that already existed in 12-EDO).

Anyway, I guess the point is that that’s more or less how it happens that you end up with additional possibilities when dealing with microtonal systems like this (or sometimes a new and different set of possibilities). And I feel like it’s not clear anymore what we’re doing, whenever we introduce approximations and the like into the picture.

consciousness razor says

I think it might be based on a belief that you can’t use it for anything but 12-EDO, so microtonal/xenharmonic people would not think it’s of interest to them.

Siggy says

@CR, #6

I think it’s presumptuous to say that they have any beliefs about musical set theory at all. Most people don’t use xen temperament theory either, but it’s not because they have any particular beliefs about it, it’s because nobody is obligated to make use of or even be aware of any particular musical theory framework.

You seem really attached to musical set theory, to the extent that you are trying to explain it in the comments of a series about an unrelated musical framework. I can’t tell what you’re trying to accomplish.