Nick Vuci's Fundamentals of Xen: Difference between revisions

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: [[harmonic_series_2-8 1.png]] 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.

[[File:Subharmonic series 7-1.png|thumb|Subharmonic Series segment 7-1]]

[[File:Wilsonic lattice.mp4|thumb|Wilsonic lattice]]

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. [[harmonic_series_2-8 1.png]] 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).

 

 

Then extrapolate it infinitely. [[Pasted image 20260223170725.png]] 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. [[Pasted image 20260223170748.png]]

 

 

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. [https://generalizedtonnetz.nickvuci.com/ 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.[[Pasted image 20260223171154.png]]