Tonnetz

Since Euler, and especially since Hugo Riemann, the tonnetz has been thought of as being generated by a combination of major thirds and perfect fifths. Reading an excellent paper by Yale's Richard Cohn, I have suddenly realized how this traditional approach could be generalized using the Matrix/ThumMusic paradigm.

It is much more general to think of the tonnetz as being generated by octaves and (tempered) perfect fifths, just like everything else in the Matrix/ThumMusic paradigm.

Here's a portion of the Matrix's two-dimensional note-space expressed in the ThumMusic System's isomorphic note-layout:



Each note is of the form [alpha, beta] where alpha is the number of octaves (each of width P8) and beta is the number of perfect fifths (each of width P5) which, when added together, give the width of the indicated interval. For example:

• the origin note[0, 0] is zero octaves and zero perfect fifths away from itself, (0 * P8) + (0 * P5);
• note[1, 0] is one octave higher in pitch than the origin, (1 * P8) + (0 * P5);
• note[0, 1] is one P5 higher than in pitch the origin, (0 * P8) + (1 * P5);
• note[-2, 4] is two octaves lower, but four P5's higher, than the origin, (-2 * P8) + (4 * P5).

Assuming that the P8 is 1200 cents wide and the P5 is 700 cents wide, the notes of the note-matrix would have these widths:
Now, let's build a portion of the tonnetz on the note-matrix, following Cohn's paper:


The minor triad Q is surrounded by three major triads P, L, and R.

• P: Parallel;
• L: Leading-Tone Exchange;
• R: Relative.

The above construction of the Q, P, L, & R triads from octaves and tempered perfect fifths is much more general than the traditional construction, because these intervals are the generators of the syntonic temperament, so the tonnetz's properties are invariant across the syntonic tuning continuum, no matter what the specific width of the P5 (within the range 686-720). This continuum includes an infinite number of individual tunings, not just the small number of N-edo tunings (in which N mod 3 = 0) over which Cohn's paper generalizes the tonnetz' traditional construction.

Cohn's paper makes much of the toroidal topology of such equally-tempered tunings (as do many neo-Riemann theoreticians). This emphasis overlooks the syntonic temperament's general topology, which is cylindrical. The tonnetz' octave axis forms a closed loop around the cylinder; its axis of major thirds runs parallel to the cylinder's inifintely-long axis; and its axes of minor thirds and perfect fifths form spirals around the cylinder's inifintely-long axis. Many common chord progressions, such as the IV-vi-ii-V-I, require only the syntonic temperament's cylindrical topology (without which the ii below the vi would differ from the ii above the V by a syntonic comma).

At those points along the tuning continuum that correspond to an equal division of the octave, such as 12-edo, 17-edo, 19-edo, 31-edo, etc., the cylinder snaps into a torus. Each n-edo's toroidal tonnetz has (a) all of the properties of the cylindrical tonnetz, (b) all of the properties shared by all toroidal tonnetzs, and (b) the properties specific to that unique n-edo's tonnetz. These points of equal temperament are like beads on a string -- but what's really interesting is not the beads, but the string.

From Thumtronics' perspective, the potential of the neo-Riemannian PLR operations to provide an invariant basis for music theory across the whole syntonic tuning continuum is very exciting (I think). Or, to express the same thought from the neo-Riemannian perspective, the Matrix/ThumMusic paradigm may give neo-Riemannian theory the opportunity to expand its scope to embrace the entire syntonic tuning continuum, and perhaps also the tuning continua (and tonnetz') of other rank-2 temperaments (e.g., magic, hanson, schismatic, etc.). These other temperaments temper out different commas, so their tonnetz' will be different from the syntonic tonnetz, but the same general principles ought to apply (at some level of abstraction, anyway).

Cohn's paper (like Riemann himself) makes a number of statements regarding the relationship between the tonnetz and "acoustics" that are only true if one assumes that "acoustics" means "the Harmonic Series." Yet the Matrix/ThumMusic paradigm generalizes "acoustics" -- by dynamically aligning a timbre's partials with a tuning's notes, as specified by a temperament's defining intervals -- such that the relationship between the tonnetz and "acoustics" is 1:1. The Matrix/ThumMusic tonnetz is a direct embodiment of generalized musical reality.

I think I'd read something about the PLR approach to chord relationships, chord progressions, and the like before reading Cohn's paper, but it hadn't clicked. Now, it has definately clicked. I suspect that the PLR approach to chord relationships will prove to be a very powerful tool in the Matrix/ThumMusic System.

Cool bananas! :-)

[Update, Thur Sep 25th: A couple of prominent neo-Riemannians have (very) informally agreed (a) that the proposed application of neo-Riemannain theory to the syntonic tuning continuum appears to be both novel and interesting, and (b) that they would read the relevant Matrix/ThumMusic papers and get back to me.]

Matrix

One of the coolest things about working on the Thummer has been using it to discover new things about music. Its isomorphic button-field (keyboard) is like an X-Ray lens that lets people see the deep structure of music.

Bill Sethares and Andy Milne did the heavy mathematical work to prove that what we were seeing was really there. Their proofs can be found here and here (with more papers in the pipeline). These papers, while appropriate for their purpose and venue, are mathematically impenetrable to people with tiny little heads like mine.

Therefore, I've recently posted a draft paper that presents our new musical paradigm, the Matrix, in language that is nearly math-free. You need to know what prime numbers are, and that any natural number can be factored into an unique combination of prime numbers, and that a two-dimensional array of numbers is a matrix (hence the name of the proposed paradigm), but that's about it.

Although I do not claim to be an expert in the history of science, I do know a thing or two about it, and the Matrix model of music theory has all of the hallmarks of a paradigm-shifter. For example, it solves old problems, explains previously-anomalous experimental results, makes predictions that are falsifiable, and has enabled the discovery of new properties (e.g., tuning invariance, which is the basis of Dynamic Tonality).

The Matrix paradigm accomplishes all this as a result of questioning a single key assumption of Western music theory: that musical sounds are those that follow the Harmonic Series. This assumption is embedded so deeply into Western music theory that most musicians and many theorists don't even realize that they are making it. It has been received wisdom since Pythagoras first plucked a string 2,500 years ago.

The Matrix paradigm, in brief, uses a temperament to temper both tuning and timbre in real time. It's the tempering of timbres that's new (building on Bill's previous work). This is, in effect, a generalization of the relationship between Just Intonation and the Harmonic Series that forms the core of Western music theory.

It is hard to imagine a more fundamental alteration of the theoretical basis of music than this. Hence, paradigm shift.