The (Isomorphic) Cortical Topography of Tonal Structures

At the request of Daniel Levitin, I added this post to his Facebook page's discussion board, which I will also paste below.

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Gentlepersons,

Most work in music cognition assumes 12-tone equal temperament, which is a perfectly reasonable starting point. However, I suspect that the findings thereof can be easily generalized to alternative tunings, using some recent discoveries.

The first discovery is the two-dimensional syntonic temperament. Its tuning continuum includes nearly all of the tunings ever used by humankind in the real world, from the from the 7-tet ("7-tone equal temprament," hence "7-tet") tunings related to the timbres of the Thai ranat and African balafon to the 5-tet tunings related to the timbres of the Indonesian gamelan, with 17-tet, Pythagorean, 12-tet, the meantones, and an infinite number of other tunings in between.

The second discovery is the relationship between two-dimensional temperaments, such as the syntonic temperament, and two-dimensional isomorphic keyboards.

If the pattern of notes on a two-dimensional keyboard is generated by the same two intervals that generate a two-dimensional temperament (such as the syntonic temperament), then the keyboard will be "isomorphic" with that temperament. What this means is that any given interval in that temperament will have the "same shape" in every tuning of that temperament. Therefore, any given combination or sequence of intervals also has the "same shape" everywhere on an isomorphic keyboard, in every tuning of that temperament. This is tuning invariance.

For a demonstration of tuning invariance on an isomorphic keyboard (with embarrassingly-over-the-top commentary), please see this video.

This tuning invariance applies to all syntonic tunings, including tunings that are equal and non-equal, regular and irregular (such as “well-temperaments”), and also "rank-2, 5-limit Just Intonation" tunings (see proofs here or here).

Alternatively put, syntonic tunings include Western (Pythagorean, 12-tet, 1/4-comma meantone, 31-tet, “circulating”) and non-Western (Indonesian, Thai, Mandinka African) tunings, and the JI tunings used both in the West and in non-Western cultures (which rarely exceed 5-limit; the blues is, arguably, 7-limit, but that case is also well-handled by an isomorphic note-layout).

The one non-syntonic temperament which I can find to have been used by humankind in the real world is the (Turkish) schismatic temperament. Because its generators are the same as those of the syntonic temperament, it is compatible with the syntonic temperament's isomorphic keyboards, and hence with the conclusions of this posting—but it is a special case, beyond the scope of this posting, so I won't mention it again.

The third discovery—at least, we haven't been able to locate any prior art yet—is that such isomorphic keyboards include within their pattern of notes a tonnetz, as described by Euler/Oettingen/Riemann etc. (see Figures 7 and 8 in this paper.) An important point is that such a tonnetz is tempered; that is, it is not based on "ratios of small whole numbers" (i.e., Just Intonation), but rather on a mapping from these "just" intervals to intervals of the syntonic temperament.

Such a "tempered" tonnetz has the same tuning invariance as the isomorphic keyboard from which it is drawn. Hence, the relationships among the notes on such a tonnetz are tuning invariant, too.

The tonnetz is (I believe) well-known to be the dual graph of the "chart of the regions" described by Schoenberg and others (see this book, p. 105). Hence, any such tempered "chart of the regions" is likewise tuning invariant.

The map of perceptual tonal space described by Krumhansl, Janata, and other cognitive psychologists, is precisely such a tempered "chart of the regions." Hence, this map *ought* to be tuning invariant, too.

The hard-wiring of a tuning invariant map of perceptual tonal space could help explain both

1. The diversity of real-world tunings, in that an infinity of syntonic tunings are compatible with such a perceptual space, and
2. The limitations on that diversity, in that
• only the tunings of the syntonic (and perhaps schismatic) temperament fit this perceptual space, and
• a culture’s dominant instruments must produce a timbre that is closely "related" to such a tuning (wherein "related" has the meaning described here).

The latter point must not be overlooked in any related experiments. For example, using harmonic timbres for all tunings will produce invalid results.

If perceptual tonal space were indeed found to be tuning invariant, then this could would be an important scientific step towards a truly universal theory of music.

Neither I nor my collaborators have the skills or knowledge of musical cognition sufficient to execute the kinds of experiments needed to explore this issue further. We would be delighted to help, though. Ping me at jim@iGetItMusic.com.

Thanks!

Jim Plamondon
Unaffiliated Musical Heretic