Nick Vuci's Fundamentals of Xen
The four pillars of Xen theory are:
- Just Intonation (JI)
- Equal Divisions of the Octave (EDO)
- Moments of Symmetry (MOS)
- Temperaments
With EDOs, MOS, and temperaments, it’s important to know that although these concepts are presented within a unified context, they exist as modular components.
Just Intonation (JI)
If the pitch or frequency of a note is the number of times per second it causes the air to vibrate, and an interval is when two notes play at the same time, then Just Intonation (JI) is where the frequencies of the notes form whole-number ratios with each other.
Harmonic Series
A fundamental aspect of JI is the harmonic series. The harmonic series is the series that happens when you keep adding the first frequency over and over, ad infinitum. Which is 1:2:3:4:5:6:7… If we take a segment of it, we get different intervals and chords. For example, 4:5:6 is the major triad. Try it out in Scale Workshop by creating a new scale from the ‘harmonic series segment’!
Clarifaction: something that confused me when I was learning about Just Intonation is the conflation I made between JI intervals and divisions of string vibrations. That’s because it’s two ways of looking at the same thing. See the physics forumla wavelength is proportionate to the speed of sound (or light) over frequency: lambda equals v over f.
Primes
Sometimes intervals can be reduced, for example 6/3 is the same as 2/1, but sometimes they cannot if they include primes. Primes are important because they introduce new flavours of intervals. When we have collections of JI intervals, because they can theoretically go on forever, we use conceptual limits to make things finite and usable.
Prime-Limit
The most common limit is the prime-limit. If we take a scale that has intervals which use prime 7 but no higher primes, we say it is a 7-prime-limit scale, or 7-limit. As example, 3/2 is a 3-prime-limit interval, 5/4 is a 5-prime-limit interval, 7/4 is a 7-prime-limit interval, 11/8 is an 11-prime-limit interval, etc…
Prime Factorization
So far we’ve covered two types of notation, but there’s a third one that’s important called “prime factorization.” Sometimes this is called a “monzo,” but we’ll stick to calling it prime factorization for now. Prime-factorization is when you break an interval down by the primes. This is imperative.
Odd-Limit
But we have another limit, which is the odd-limit. The odd-limit is basically the smallest odd number an interval be broken down into. When the prime-limit and the odd-limit coincide, we call that the n-limit.
Subharmonic Series
Let’s take a step back. You know about the harmonic series, sometimes called the overtone series, since harmony results from the alignment of partials. Now we have to introduce the second-to-last fundamental idea of JI, and that is the subharmonic (or undertone) series. if we go to scale workshop and choose 1 as the lowest number and 8 as the highest, we get this scale: 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1. Note the scale shape:
As we go up the scale, intervals get smaller and smaller. The subharmonic series is the Mathematical opposite of this, and the same 1/8 range gives us this undertone series segment: 8/7, 8/6, 8/5, 8/4, 8/3, 8/2, 8/1. It’s important to know that although we’re calling this the undertone series, the undertone series does not have a physical acoustic basis.
Now remember how I said 4:5:6 is the major triad? Well, the subharmonic version is the minor chord: 8/6, 8/5, 8/4 or 1/(6:5:4). This means 4:5:6 and 1/(6:5:4) are Mathematical inversions for each other, or the otonal and the utonal versions (for overtone and undertone).
Tonality Diamond
Time for the most important concept of JI. When we take an odd-limit, there’s something very interesting you can do. You can arrange it in a way that shows how the intervals form interwoven otonal and utonal chords of that limit. This is called the tonality diamond.
Try it out with Nick Vuci’s Tonality Diamond Explorer!
Lattices
Lattices are another way to visualize JI:
Conclusion
In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.
Some extra things on Just Intonation is that there are other approaches to Just-Intonation besides the basic tonality diamonds. You have primodality that exploits prime modes of the harmonic series and incorporates commas, which are conventionally seen as a musical inconvenience / blight, and then you have Hexany by Erv Wilson which is a way to make JI without a tonal-center using Combination Product Sets. The Wilsonic software makes all these things accessible. What's important to note about high-complexity JI music psycho-acoustically is that extremely high-limit JI isn't going to be perceived as high limit JI, rather it is that super complex JI intervals end up being reinterpreted as approximations of simpler JI intervals. Point of diminishing returns.
Equal Divisions of the Octave (EDOs)
Last time we covered JI and mentioned the harmonic series. 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1 is the segment we looked at last time.
See how the steps get smaller, but in a regular way. Technically speaking this is a type of equal division, the Arithmitic Equal Division, where we instead multiply the base frequency by a set amount. That amount is (2/1)^(1/x), where x is our number of equal divisions. Henceforth, we will refer to logarithmic equally divided octaves as EDOs. Unlike JI intervals that can be described as x/y ratios, the intervals of EDOs will always be irrational numbers when you try to express them as ratios. Instead we decide to use a unit called the cent, which divides the octave into 1200 logarithmically equal parts. That way we can say that the cent values and it’s easier to describe them. With EDOs there are two main categories, prime and non-primes. Non-prime EDOs are composite (by definition), which means you can think of them as being made up of smaller EDOs interwoven within each other. If an EDO is some multiple of a prime EDO, it can be called a superset of that prime EDO (e.g. 22edo is a superset of 11edo, and 11edo is a subset of 22edo).
Subgroups
Extending on the concept of prime-limit, let’s cover subgroups. We could write 7-prime-limit or we could write 2.3.5.7, it’s two ways of saying the same thing. A subgroup is where we remove some numbers from our limit. Say we use intervals with only 2.3.7 and no 5. We call htat a no-5 subgroup. For example, if we remove the 5, that means we have to remove the intervals 5/4 and 6/5 from our possible intervals. We can use these to focus in on specific structures of some limit, or treat them as musical structures unto themselves.
Temperaments
Temperaments in Brief
Temperament means you have notes that work as approximations of just intervals that are discolored or tempered so that one interval can represent multiple different JI notes, which tempers a comma. However, in Xenharmonic theory, a temperament goes further as a reference to regular temperaments (RTT), which are those temperaments that specifically temper out certain commas and also generated by some set of intervals. RTT was quite a feat in math. This is irrelavant to musicians, but to Mathematicians, I recommend checking out D&D’s guide where they worked hard to compile all the resources from online mailing lists on Xenharmonics in order to define the technicalities of RTT. They made it accessible at the level of linear algebra.
Temperaments Explained
Temperaments are based on approximating some JI thing, which can be a subgroup. Standard meantone is a 5-limit temperament, but orgone is a 2.7.11 temperament. A way you can think of it is that meantone uses 4:5:6 and 1/(6:5:4) based harmony and orgone uses 8:11:14 and 1/(14:11:8) based harmony. The limit or the subgroup it’s referring to is the JI structures and harmony the temperament is trying to represent. Most temperaments refer to the JI structure you are approximating and bending in an equal division.
Take the triadic diamond:
Then extrapolate it infinitely.
The further you go, the further you get from 1/1, and no matter how far you go, you’ll never get back to 1/1. But sometimes you’ll get an interval that’s very close to 1/1 in sound, but not in number. Take the meantone comma, 81/80, and draw a line through 81/80 and 1/1 infinitely. Now imagine we take 81/80 and we make it equal to 1/1 and we spread the 21 cents out throughout the whole lattice. The point is that every interval on the line, between 1/1 and hte small interval 81/80, becomes equal to 1/1 when 81/80 does. Now the lattice loops instead of going on forever. Now when you use major and minor chords to get to the spot where 81/80 would be, you instead land on 1/1, and all of the major and minor chords are slightly out of tune. Whenever an EDO approximates some JI intetrvals, you can make the same triangular grid with the EDO intervals instead of the JI ones and it’ll show you what commas it tempers out visually. Now what does all this mean? Let’s take the 7-limit diamond.
We can remove the 6 (i.e. the 3) and we get this:
And if we input A = 1, B = 1, C = 7, you get these intervals.
Same shape, but different harmonic structure: 2.5.7 subgroup.
Now let’s go back to the infinite JI lattice and look at the small interval 50/49. I said any interval that goes thru that line before, but that was just to evoke mental imagery. Really, it’d go through every other interval on the line the same distance apart, e.g.:
That’s essentially a subgroup temperament. Not sure the name, but the names aren’t important.
I took the 7-limit diamond and removed a prime to get the 2.5.7 subgroup, then generated a lattice using the intervals those primes generate and found a small interval I wanted to squash out. That interval is called a comma. Squashing it out is called “tempering it out,” and what results is the temperament.
Now let’s look at 27edo. Let’s take the 27edo tempered representation of the 2.5.7 subgroup intervals. Hmm, if we superimpose the JI lattice together, we see the spot where 50/49 would be 1\27 not 0\27. 27edo does not temper out 50/49, but what does it temper? Whatever JI intetrval the red dots of the tempered lattice overlay are related to the commas that 27edo tempers out.
Here we see that 27 tempers out 128/125 and you can make music in that temperament in 27edo by using 4:6:7 and 1/(7:6:4) chords.
Temperaments Ext.
A temperament is an abstraction, but what is it abstracting? The astraction is that if you take a lattice of symbolic JI intetrvals, you can temper out certain ones, which sort of bends the JI lattice and makes this constant line along where the comma is tempered. However, this is still all symbolic; the temperament isn’t actually sound, it’s a map of potential sound. It’s a map of symbolic internal equivalence, which says that some interval or intervals are now the same, whereas before they weren’t. But beyond that, it doesn’t say anything else. So in Blackwood, 256/243 becomes the same as 1/1. That’s what Blackwood is at its core, that equivalence. But then the question remains: well, what are the actual cent values of the rest of the now altered intervals? You have this infinite line of equivalence, but you don’t have anything else. That’s where the tuning comes in. So then now we can tune the now tempered intervals in a certain way to get some result. But you can’t just tune one in isolation. Everytime you change the cent value of one interval the rest change (except for the unison and the tempered out intervals, they stay constant). So then you have certain tuning targets. For example, we have the unchanged interval (eigenmonzo). Let’s say we have a symbolic Balckwood alttice and we decide we’ll orient the tuning of that lattice so that 5/4 is pure. 5/4 is the unchanged interval of the temperament. It is the targeted interval that is unchanged from pure. However, the thing about temperaments is that error is spread out. And if we have one pure interval interval, that means others are inherently less pure. So sometimes instead we use larger tuning targets, such as tonality diamonds, i.e. odd-limits. That’s what it means when you see things like “9-odd-limit monotone” or whatever that range of the tuning is optimized to target the 9-odd-limit diamond as a whole better than other ranges. So the error is minimized with that target in mind. So in that case you have less error across that tuning target but no pure intervals. People explored all of this decades ago in the ’90s and concluded that there are only a few good temperaments and a few good EDOs for them. As in EDOs that aren’t giant, but also have tolerable mistuning of intervals. With Blackwood [10], it’s a 10 note scale, and it can exist in many tunings. But the best overall tuning for it is generally considered to be 15edo, because it’s not too many notes and does an ok job at what Blackwood musically suggests.
Generators
You can generate any temperament by stacking an interval over and over. Whatever that interval is, that’s the generator of the temperament. A generator is a universal concept in Xenharmonics. The JI lattice the infinite 5-limit diamond, is made using 3/2 and 5/4 as generators. 12edo? You can make using 100 cents as a generator. Generators are intervals you stack on themselves to make some structure.
Basically think of it like this Imagine JI 5-limit So what do you have? You have 2.3.5 JI Which is rank-3
fm’latghor — 2:45 AM 3 generators on the tonality diamond
Nick Vuci — 2:46 AM Like you need 3 generators: 2.3.5 And you get 4:5:6 triads Now why happens with meantone? You temper out a comma and collapse a rank. In this case you temper so that a single generator gives you an approximation of both 3 and 5 So meantone is a 2.3.5 (5-limit) temperament but it’s only rank-2 since you can make it by using an octave and a meantone generator (where a tempered 3 also gives you a 5) Then you can reduce any temperament to an equal tuning, where again you reduce rank so that a single generator gives you the 2, the 3, and the 5 So all equal tunings are rank-1 Lots of this stuff is kinda abstract
MOS
Moments of Symmetry (MOS) are special scales based on a simple principle. Take some generator and reduce in some period (usually the octave). Stack the generator over and over. When you get some collection of notes by doing this that have only two step sizes, you have a moment of symmetry. Another way to think of it is to consider you have some number of large (L) and small (s) steps, say 5L and 2s. Make it so that the 2s are as evenly distributed amongst the 5L as possible. This looks like LLsLLLs or sLLsLLL. This is called maximal evenness.
Step-Size Ratio
The largness or smallness of the steps is called the step size ratio. The quality of the step ratio, whether it’s hard or soft or equalized is categorized according to the Temperament Agonstic Mos NAMing System (TAMNAMS). The ways to modify mosses are documented on the page “Operations on MOSes”. Things like chromatic alterations, subdividing scales (muddling), or neutralizing the scale are all documented. There’s a lot you can do with MOS. Things like splitting the scale into a hexachord or a tetrachord allows you to transpose the same melodic segment. You can morph between scales, say between hard diatonic and soft diatonic. It’s worth looking at the MOS family tree as well for reference, illustrating MOS’s recursive structure. The Math behind MOS is actually quite simple, but that’s not what we’re hear to cover.
Tonnetz
A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. Play with Vuci’s Generalized Tonnetz here. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.