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.]

Tuning Invariance and the Brain

I had a great "first contact" meeting with Bob Duke yesterday. He’s the Director of UT/Austin's Center for Music Learning, and Google suggests that he's very well regarded by the music education world, with an international profile.

We're meeting again next week.

Bob wanted more information on two points I raised in my presentation, so I sent him links to two papers: the first describing Bill Sethares' work on the relationship between tuning and timbre, and the second (Burgoyne, 2005) showing the brain's perception of tonal pitch-space. This posting is an extended answer to the issues Bob raised.

Tonal Pitch Space & the ThumMusic Note-Layout
Figure 3d in Burgoyne's paper is the result of using Maximum Variance Unfolding (MVU) instead of Multi-Dimensional Scaling (MDS) to measure & display the relationships in Weber, Krumhansl, Kessle, & Lerdahl's tonal pitch space.

Why use MVS? To quote Burgoyne:

Like MDS, this algorithm produces an embedding from a matrix of pair-wise distances, but while maximizing the variance of the output embedding, it seeks to preserve only the distances between nearest neighbors. This subset of distances is locked, and a nonlinear optimization technique is used to expand the data as much as possible given these locks, analogous to stretching a ball-and-stick model in which the balls correspond to harmonies and the sticks correspond to the locked distances.

What Figure 3d shows, then, is one slice through the relationships among nearest neighbors in tonal pitch space – and along that slice, the relationships match those of the ThumMusic note-layout.

Relationship of Tuning & Timbre
The Indonesian gamelan, Thai renat, and Mandinka balafon are all traditionally tuned in an inharmonic manner. Bill’s research shows that the tuning of these instruments is closely "related" (his term) to the timbres produced by those instruments. Clearly, then, the human ear/brain/mind can accept a wide range of tunings as being "musical," as long as those tunings are "related" to the timbres in which they are played (or vice versa – same thing). The X_System's use of X_Spectra is based on this insight.

Bill's work supports the argument that the ear/brain/mind's hardware and software can process, as tonal music, a wider set of tuning relationships than has been investigated by Krumhansl, Lerdahl, etc. as above, so long as the tuning and timbre are "related."

Tuning and the Brain
Importantly, the geometry of the ThumMusic note-layout is tuning invariant – i.e., the pattern of notes is the same no matter what the tuning (with some caveats). Since the perception map shown in Burgoyne’s Figure 3d is identical to the tuning invariant ThumMusic note-layout, then it seems likely that the brain's perception of tonal relationships ought to be tuning-invariant (with related timbres), too.

I hadn't made this connection before.

Cool!