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!