Geometry, music and the brain

9 minute read

The lead report in Science this week was this paper by Dmitri Tymoczko, titled "The geometry of musical chords":

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses (Tymoczko 2006:72).

Like a lot of things mathematical, the mathematical description of this is fairly distant from everyday experience. Cosmic Log provides a pretty good summary of the mathematical connections. This is pithy:

For years, string theorists have used music as a metaphor for fundamental particles, and now Tymoczko is using the mathematics of string theory to understand the fundamentals of music.

The next couple of paragraphs capture the essence of the work:

The math makes it easier to understand objectively what great musicians and composers do in their head. "When you sit down to interact with a piano, you're actually interacting with a non-Euclidean space, because there are many different ways you can play a C-major chord on a piano," Tymoczko said.
He said orbifolds capture the multidimensionality of music: how harmony interacts with counterpoint, how chords are connected with each other, even how notes are arranged "to minimize the amount of effort that musicians have to make when moving from chord to chord."

I think it helps to read a few different descriptions, and so I'm also linking the perspective in Science by Julian Hook, which includes some history, showing why Tymoczko's paper is part of a long tradition of mathematical application to music:

Mathematical music theory, although terra incognita to practicing musicians and even to many professional music theorists, has in recent years blossomed into a sizable and multifaceted industry. Pitch-class set theory (3), the study of a discrete 12-note quotient space, was developed as a means of confronting the analytical challenges posed by "post-tonal" music of the 20th century, whose harmonic materials are more varied and complex than those in most earlier music. Diatonic set theory (4, 5) investigates the subtle and beautiful relationship between the 12-note chromatic scale and diatonic scales such as the C major scale, with seven unequally spaced notes per octave (a scale type of great importance in many styles of music). Scale theory (6, 7) studies structural properties of scales and their subscales more broadly, allowing variation in both chromatic and diatonic cardinalities and occasionally engaging considerations of tuning and acoustics.
A particularly active area is neo-Riemannian theory, which synthesizes modern group-theoretic techniques with inspiration drawn from the work of the prolific German musicologist Hugo Riemann (1849-1919) and his contemporaries. In its basic form (9, 10), neo-Riemannian theory investigates certain transformational relationships among the 12 major and 12 minor triads in ways that are algebraically elegant, musically suggestive, and readily visualized in various forms of a graph known as a Tonnetz (tone network), in which the harmonic path traced by a musical composition may be plotted (Hook 2006:49-50).

In other words, musical progressions form paths or shapes in multidimensional spaces. Music that is part of the classical Western tradition actually falls within a fairly restricted set of possible paths; other musical traditions also form paths that to a greater or lesser extent overlap (although the dimensionality of the spaces may be different for different systems).

This seems very interesting from the perspective of how musical abilities emerged in the brain, and what relationship music may have to other mental functions or abilities. Whether by training or innate preference, human minds perceive certain classes of mathematical relationships among chords and note sequences as "special". We may describe this "specialness" in many different terms: "harmonic", "melodic", "musical", "cool", etc. Some of these paths have emotional resonance. Some of them have become loaded with cultural significance. Some of them communicate, either directly (through encoding) or indirectly (through redundantly providing additional context or mnemonics for texts of various kinds, including lyrics).

In all cases, the carriers for these cultural, emotional, or communicative aspects of music are the sequences of sounds themselves, which our minds apparently distinguish in accordance with their mathematical properties.

Science has visited this issue before, with a 2002 paper by Petr Janata and colleagues that attempted to localize the geometry of musical tonal structures in the prefrontal cortex:

Western tonal music relies on a formal geometric structure that determines distance relationships within a harmonic or tonal space. In functional magnetic resonance imaging experiments, we identified an area in the rostromedial prefrontal cortex that tracks activation in tonal space. Different voxels in this area exhibited selectivity for different keys. Within the same set of consistently activated voxels, the topography of tonality selectivity rearranged itself across scanning sessions. The tonality structure was thus maintained as a dynamic topography in cortical areas known to be at a nexus of cognitive, affective, and mnemonic processing.

Reading over that paper, they didn't really have anything concrete about how these brain areas may have functioned to facilitate this ultimately geometric processing, but they apply an explicitly geometric model (this one less string-theory-related!).

There has been quite a bit of work in the last few years attempting to relate these musical abilities (mostly without considering their geometric nature) to the processing of other functions -- particularly language and mathematics. This paper from last year by Koelsch and colleagues relates this point to how syntactic sequences in both music and language are processed in the brain:

Results demonstrate that processing of musical syntax (as reflected in the ERAN) interacts with the processing of linguistic syntax (as reflected in the LAN), and that this interaction is not due to a general effect of deviance-related negativities that precede an LAN. Findings thus indicate a strong overlap of neural resources involved in the processing of syntax in language and music.

Another study by Koelsch et al. last year examined which areas of the brain are associated with music processing in both adults and children, musically trained vs. not musically trained:

Subjects made judgements on sequences that ended on chords that were music-syntactically either regular or irregular. In adults, irregular chords activated the inferior frontal gyrus, orbital frontolateral cortex, the anterior insula, ventrolateral premotor cortex, anterior and posterior areas of the superior temporal gyrus, the superior temporal sulcus, and the supramarginal gyrus. These structures presumably form different networks mediating cognitive aspects of music processing (such as processing of musical syntax and musical meaning, as well as auditory working memory), and possibly emotional aspects of music processing. In the right hemisphere, the activation pattern of children was similar to that of adults. In the left hemisphere, adults showed larger activations than children in prefrontal areas, in the supramarginal gyrus, and in temporal areas. In both adults and children, musical training was correlated with stronger activations in the frontal operculum and the anterior portion of the superior temporal gyrus.

These studies both make use of the regular structure of music to elicit reactions in subjects to expected vs. nonexpected transitions. This is directly using those multidimensional geometric relationships, and probing which parts of the brain are sensitive to them, in a sense.

Music is not an isolated function, in exhibiting temporal structure. For example, there is this paper by Levitin and Menon:

The neuroanatomical correlates of musical structure were investigated using functional magnetic neuroimaging (fMRI) and a unique stimulus manipulation involving scrambled music. The experiment compared brain responses while participants listened to classical music and scrambled versions of that same music. Specifically, the scrambled versions disrupted musical structure while holding low-level musical attributes constant, including the psychoacoustic features of the music such as pitch, loudness, and timbre. Comparing music to its scrambled counterpart, we found focal activation in the pars orbitalis region (Brodmann Area 47) of the left inferior frontal cortex, a region that has been previously closely associated with the processing of linguistic structure in spoken and signed language, and its right hemisphere homologue. We speculate that this particular region of inferior frontal cortex may be more generally responsible for processing fine-structured stimuli that evolve over time, not merely those that are linguistic.

And as long as I am abstract-quoting, there is this paper investigating how musically-trained people may differ from non-musically-trained people in math processing:

The neural correlates of the previously hypothesized link between formal musical training and mathematics performance are investigated using functional magnetic resonance imaging (fMRI). FMRI was performed on fifteen normal adults, seven with musical training since early childhood, and eight without, while they mentally added and subtracted fractions. Musical training was associated with increased activation in the left fusiform gyrus and prefrontal cortex, and decreased activation in visual association areas and the left inferior parietal lobule during the mathematical task. We hypothesize that the correlation between musical training and math proficiency may be associated with improved working memory performance and an increased abstract representation of numerical quantities.

It's not clear to me from this reading whether there is a good case for music being secondary to other related functions like language -- although the commonality between the interpretation of syntactic structures of language and music suggests that one may have followed the other. That raises the interesting possibility that there may be an underlying geometric arrangement to phonetic information. The brain recognizes phonemes by contrast just as these musical sequences are defined by contrasts with each other, so it is plausible that there is a fundamental multidimensional model that includes both. Finding such commonalities would certainly clarify how the brain must handle such information, and would thereby provide much evidence about linguistic and musical (and possibly mathematical) evolution.

I am really less bullish on the possibility that our ostensive mathematical abilities may be related to language or music. There appears to be little in the way of mathematics that is basic to human cognition (such as might be shared cross-culturally, for instance), and much of what is there falls sort of generally into geometry, short counting sequences, and pattern-matching of various kinds. Music might certainly facilitate the learning of patterns -- and I could conceive this may underlie the so-called "Mozart effect" of greater math skills following musical exposure (although the effect itself may be a myth, see Bangerter and Heath 2004; Talero-Gutierrez et al. 2004). But I don't imagine that analytic geometry and music make more than incidental use of the same brain functions.

With music and language, on the other hand, I expect we'll hear a lot more about the connection between them.


Bangerter A, Heath C. 2004. The Mozart effect: tracking the evolution of a social legend. Br J Soc Psych 43:605-623. DOI link

Hook J. 2006. Exploring musical space. Science 313:49-50. DOI link

Janata P, Birk JL, Van Horn JD, Leman M, Tillmann B, Bharucha JJ. 2002. The cortical topography of tonal structures underlying Western music. Science 298:2167-2170. DOI link

Koelsch S, Fritz T, Schulze K, Alsop D, Schlaug G. 2005. Adults and children processing music: an fMRI study. Neuroimage 25:1068-1076. PubMed

Koelsch S, Gunter TC, Wittfoth M, Sammler D. 2005. Interaction between syntax processing in language and in music: an ERP study. J Cogn Neurosci 17:1565-1577. PubMed

Schmithorst VJ, Holland SK. 2004. The effect of musical training on the neural correlates of math processing: a functional magnetic resonance imaging study in humans. Neurosci Lett 16:193-196. PubMed

Talero-Gutierrez C, Zarruk-Serrano JG, Espinosa-Bode A. 2004. Musical perception and cognitive functions: is there such a thing as the Mozart effect? Rev Neurol 39:1167-1173. PubMed

Tymoczko D. 2006. The geometry of musical chords. Science 313:72-74. DOI link