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

A Neurological basis for tonal space

I recently thought of a rough hypothesis for the neurological basis of tonal music: the application of place cells and grid cells to the navigation of tonal space. A lack of relevant hits on the hypothesis' key phrases using Google suggests that it has not previously been proposed.

To simplify egregiously, place cells, found in the hippocampus, remember a "place" in the spatial environment, while grid cells, found in the entorhinal cortex, form a hexagonal grid, and remember the relationships among objects in the spatial environment.

From an evolutionary perspective, having good spatial memory confers a considerable survival advantage on an individual, so it's no surprise that humans have neurological hardware that is optimized for this purpose. Such optimized hardware is often borrowed for related task. I hypothesize that the mind borrows this spatial-memory hardware to process musical information, thereby enabling music to be processed as "movement through tonal space."

Tonal space can also be represented as hexagonal grid, for example as a isomorphic keyboard, of which a subset of notes form a hexagonal tonnetz. If such a tonnetz is tied to specific pitches, then it maps to a pitch space.

However, only one fixed pitch is necessary to map a interval-based tonal space to a pitch space. Consider, for example, the hexagonal isomorphic note-layouts (keyboards) described in the spreadsheet JIMS_Note.xls. These all describe intervals, not pitches. To describe pitches, the origin note [0, 0] needs to be associated with a specific reference frequency (e.g., 440Hz).

With a grid cell describing an isomorphic interval-layout, any given interval, sequence of intervals (melody), or stack of intervals (chord) could be described/recognized by a specific pattern of points on that grid. If any one such point were associated with a specific frequency via a place cell, then the same interval-pattern could be described/recognized by the same grid cells, despite its transposition by octave, key, or tuning.

I have no evidence whatsoever for this hypothesis, nor any counter-evidence. I only thought of it earlier this week.

Many studies have shown that musical training stimulates the development of those parts of the brain dedicated to spatial-temporal processing. The grid & place cell system may be the mechanism by which this stimulation is effected.

But, what do I know? ;-)

Notational Load

I define notational load as the cognitive load imposed on a learner by the notation in which a concept is expressed, rather than by the concept itself.

As examples of notational load, consider:

• Roman numerals vs. Arabic numerals: Arithmetic operations such as long division are much easier to teach, learn, and apply using Arabic numerals than Roman numerals. Indeed, the Romans’ numeric notation is often given as the main reason why the Romans contributed almost nothing to mathematics per se, whereas the Arabs made great advances in mathematics.
• Chinese writing vs. Korean writing: The use of Chinese ideograms for Korean writing restricted literacy to Korean elites until the invention of the Korean-specific Hangul writing system, which made it possible for “a bright Korean-speaking student to become literate in one day, and a slow student in ten.” Smilarly, the Cherokee-specific writing system produced a similar jump in literacy rates within the Cherokee-speaking population of the early 1800's.
• The musical staff vs. neumes: Guido d’Arrezo’s invention of sight-singing, including the musical staff and solmization, is credited with reducing the training time of Church singers from ten years to “one, or at most two” – thus reducing the cost of music education by between 80% and 90% without sacrificing quality.

The development of other notational systems, such as calculus, the Periodic Table, and Feynman diagrams, have similarly contributed to significant increases in the efficiency of education.

Perhaps of even greater importance, the development of new notations has often led to new discoveries that were literally “inconceivable” using the previous notations, because notations invariably constrain, in addition to reflecting, patterns of thought. Examples include the Arabs’ use of their numerals in developing algebra and algorithms (the names of which reflect their Arabic roots), Mendeleev’s prediction of new elements based on “holes” in his Periodic Table, and Feynman’s use of his own diagrams in making major contributions to quantum mechanics.

Cognitive load theory recognizes three kinds of cognitive loads (from an educational perspective): intrinsic load (inherent to the subject being studied), extraneous load (irrelevant to the subject being studied), and germane load (which is extraneous to the current lesson in isolation, but which reduces of the overall load of the lesson-sequence as a whole).

Notational load seems to me to be an entirely extraneous load. In opposition to this position, one could argue that the mastery of a given domain’s traditional notation is required for communication with other professionals within that domain, and that this “communication conformity requirement” makes mastery of a domain’s traditional notation intrinsic. For example, without the ability to read traditional music notation, musicians cannot read the works of other composers.

Or…can they? Using modern music notation software programs, musicians can convert any given piece of written music to alternative, non-traditional notations such as guitar tab. Many of these programs support a “plug-in” architecture that enables the developers of alternative notations to retroactively upgrade the software to support new notations The potential availability of such notation-translation software, in any given domain, and the ease of distributing it over the Internet, significantly reduces the communication conformity requirement, supporting the claim that notational load is extraneous.

Changes in notation are not easy to effect in any domain, especially among tightly inter-connected professionals. However, a dramatic increase in learning efficiency may make it possible for non-professionals to rapidly gain knowledge previously restricted to a domain’s professionals. For example, more people in the USA now read Guitar Hero’s scrolling tablature than read traditional music notation, and are, as a result, learning more about music than they otherwise would.

Musical Intuition

We denote this primary wisdom as Intuition, whilst all later teachings are tuitions.
(Ralph Waldo Emerson)

Intuition is a funny thing. Mostly, intuition means that “something new corresponds with your expectations,” which means that it corresponds with your experience.

But what if your experience was misleading?

Consider, for example, a caveman observing the Sun. It is "intuitively obvious" to this caveman that the Sun is moving around a fixed Earth, because that’s what he experiences every day. Or consider the incidence of infectious diseases. In a unsanitary city of foul water, tainted food, and ubiquitous disease-vectors like mosquitoes, fleas, lice, and cockroaches – that is, in almost any city in the world, until very recently – it would have been “intuitively obvious” that illness, health, death, and survival were all essentially random, or in the hands of the Gods. The underlying patterns were hidden by the experience of randomness.

So it is with music. Most people’s experience with music-making misleads them into thinking that music is about pitch, because pitches are what’s notated, pitches are what are controlled by traditional instruments’ interfaces, and pitches are what musicians talk about among themselves. It seems intuitively obvious that music is about pitch.

However, this experience is misleading. Music is not about pitch. It’s about intervals – i.e., the gaps between pitches. At this level of abstraction, any given musical structure – an interval, a melody, a chord, a chord progression, or even an entire musical piece – is the same in any octave or key. This is not at all obvious to most instrumental musicians, for whom the ability to transpose “on sight” is rare and awe-inspiring. Those who’ve learned music by singing using tonic solfa are more likely to recognize this higher-level abstraction, because their key-independent experience prepares their intuition to recognize the “invariance” of musical structures across keys and octaves.

Likewise, the experience of most musicians is misled by their implicit assumption that musical timbres are, and must be, harmonic – i.e., follow the spectral pattern defined by the Harmonic Series. This assumption is so deeply ingrained in Western music theory – dating at least from Pythagoras, 2,500 years ago – that most music theorists assume it without even recognizing that an assumption has been made. When the music of some indigenous cultures – in Indonesia, Thailand, and Mandinka Africa – was discovered to be inharmonic, this physical basis for music theory was challenged. Many people just threw up their hands and said that musical structure had to be “just cultural; just experience” – i.e., intuition.

However, if you abstract music to the next higher level – i.e., to patterns of relationships among intervals, as defined by a comma sequence – then it becomes clear that the music of the above-listed cultures and that of the West all share the same deep structure, and that the sonic spectra (timbres) of the instruments used by each culture bears an invariant relationship to its characteristic tuning within that deep structure. Yet this “tuning invariance” – first described just last year (2007) – is so non-intuitive that it had been overlooked by generations of music theorists, arguably because their experience was so firmly grounded in the Harmonic Series that their intuition misled them.

It is remarkable that so many of the world’s musical cultures use combinations of tuning & timbre that share the same deep, invariant structure. Why this one structure, and not others?

It is entirely possible (but entirely speculative at present) that the human brain contains a hard-wired isomorphic note-layout which reflects this deep structure. Such a note-layout presents any given musical interval, chord, chord progression, etc., with invariant geometry in all tunings of such a deep structure. The findings of many recent studies in music cognition can be interpreted as supporting this hypothesis. Like everything else in Western music theory, those studies have tended to be pitch-based, and to assume the use of 12-tone “equal temperament” tuning, but Occam’s Razor suggests that this one entity – a hard-wired isomorphic note-layout of interval-detecting brain cells – can explain their findings very simply. No studies have yet been performed to determine whether such a hard-wired note-layout exists, in part because the discovery of tuning invariance is so recent, and was made by relative outsiders to the music cognition community (as is so often the case).

Which brings me back to the tonic of this piece: musical intuition. The only possible source of “intuition” that’s deeper than personal experience is the hard-wired physical reality of the human brain. If the brain did indeed contain a hard-wired isomorphic note-layout, then that note-layout would be the ultimate source of musical intuition – invariant across octaves, keys, tunings, and cultures.

For the experience of music-making to be deeply and truly intuitive, the tools of music-making – music notation, music control interfaces, music synthesizers, etc. – would need to reflect this hard-wired geometry of music. This hasn’t been technically feasible until recently, nor commercially feasible until even more recently, but it is entirely feasible today.

If music-making were to be made truly and deeply intuitive, in a culturally-invariant way, then the percentage of the world’s population that could afford to successfully gain a self-sustaining level of musical competence could increase dramatically. Furthermore, it would elevate music to being truly a single universal language, with lots of interesting regional dialects.

I submit that this would be a maifestly Good Thing.