For background to this post, check out this thread:
I do like a good diagram! The Tonnetz fits that description, but just how good is it? That’s what I wanted to work out, and that’s why my posts on that thread were preoccupied with 13th chords. I was trying to find the limits of the Tonnetz. I think the 13th is the limit (the 15th degree is simply a return to the tonic), so any system that works for chords up to the 13th is good enough for me.
Why I’m fine with limits is that there cannot be a perfect system for all types of music. I think I recall that Allan Holdsworth took the approach that if we accept that the octave is a fundamental non-negotiable element of music, the next step is to decide how to deal with intermediate notes. He figured that the 12 semitones was a good place to start for a guitarist, without necessarily being the only game in town. He then considered the many ways that intermediate notes could be selected, and took a binary approach: there are 12 semitones, each of which can be in one of two states (included or excluded from a scale), therefore there are 2 to the power of 12 unique scales. That’s 4,096 scales, ranging from all notes off (a bit of John Cage there, if you know what I mean) to all notes on (Schoenberg perhaps?). Holdsworth added a rule that there could be no continuous sequences of five excluded notes, or something like that. This would eliminate a fair few of the 4,096 options. He apparently tried all the ones that fit his system. This has turned into a major digression, all I really wanted to illustrate is that in music you can do anything, but some things are more useful/common/fundamental/simple than others, and the value of the Tonnetz is primarily related to how strongly it supports these U/C/F/S things. It cannot hope to do everything.
Ah, that’s the other thing I was trying to get at. Even for the 13th there are variations, some more common than others, and there seems to be some difference of opinion in terms of which is the most basic 13th. To start with, I thought that chords based on C, for example, would follow this rule:
Chord Notes Thirds
C C-E-G major, minor
C7 C-E-G-B major, minor, major
C9 C-E-G-B-D major, minor, major, minor
C11 C-E-G-B-D-F major, minor, major, minor, minor
C13 C-E-G-B-D-F-A major, minor, major, minor, minor, major
That is, all added notes would remain in the key of C (all naturals). But when I looked at the Tonnetz, zigzagging down and up while moving to the right from C, the sequence is:
Tonnetz C-E-G-B-D-F#-A major, minor, major, minor, major, minor
Almost, but not exactly the same. Why an accidental on the F? Is it just because the zigzagging thirds sequence (major-minor) is a given? So I started checking around. That’s when things got shaky. It turns out that what appears in the Tonnetz is actually a Cmaj13. A C13 is like this, with a flattened 7th:
C13 C-E-G-Bb-D-F#-A major, minor, minor, major, major, minor
But other sources (example Chord Calculator) seem to keep the F natural, so I’m at an impasse. I don’t know enough to decide, and I am not sure it’s that important.
So that’s where I decided to stop the analysis and start drawing pictures! Here’s one I cooked up, with the explanation below.
I realised that the pattern of the Tonnetz could extend infinitely in all directions, repeating along all three of its axes, and that it could be rendered as an unbound surface of a torus. I wanted to 3D print one, or at least get someone to make one for me, and began making design decisions, one of which was to keep the scale uniform, so that the lines connecting notes were actually proportional to the number of semitones they represented. That’s why we have 3-4-5 triangles and the ghost of Pythagoras instead of the perhaps prettier isometric layout and the ghost of Buckminster Fuller. Getting the circle of fifths (the horizontal lines in the Tonnetz) on the main (toroidal) axis of the torus will involve a bit of skew because that is made up of the hypotenuse of the triangles, which I have on the diagonal of my design. I resolved the question of whether to show sharps or flats by inventing yet another form of notation. So AB, for example, is intended to stand for A# or Bb, that is, something between A and B. I removed all the extra bits of information like triad names and added instead a line for the chromatic scale which is also present in the Tonnetz, amazingly. This is shown in dotted pink.
This leads to a problem. If I want a single chromatic line to wind around the torus continuously, like a helix, I need to skew the proportions a bit more, and the thing that has to give is the right-angled 3-4-5 triangle. It doesn’t even help to go back to the equilateral triangles, because that would result in a set of many parallel chromatic rings around the torus, not a single winding line. That isn’t wrong, it’s just different to what I had had in mind. So I’ve reached a pausing point, unsure which direction to go in or even whether it is worth continuing at all.
Meanwhile, here is a near-minimal pocket version of the information in the Tonnetz. It could be minimised still further by removing the repetition of notes around the edges, but I didn’t do that, because I wanted to make it obvious that if you want to go further right from a note on the right edge, for example, you just pick up its counterpart on the left edge and go right from there, or vice versa. Same for top and bottom.
So that’s as far as I got with this. I’ll leave it to marinate for a while.
EDIT: Nothing new under the sun, as they say. I should have image-googled ‘tonnetz torus’ before writing all this. Plenty of people have already been down the same road. Actually, I am really glad about this.