Musical metrics and manifolds

1 General musically-motivated metric spaces

Chords etc.

Consider grouping chords played on a piano with some known (idealised) 16-harmonic spectrum. How can we understand the harmonic relations of chords played upon this keyboard?

We know a-priori that this projection can be naturally embedded in a Euclidean space with something less than $16\times 12=192$ dimensions, since there are at most that many spectral bands. In fact, since there are only 12 notes, we can get this down to 12 dimensions. The question is, can we get to an even smaller number of dimensions? How small? By “small” here, I mean, can I group chords on a lower number of dimensions than this in some musically interesting way? For bonus points, can I group chords into several different such maps and switch between them?

Without explanation, here is one similarity-distance embedding of all the chords using an ad hoc metric based on thinking about this question. (Brightness: The more notes in the chord, the darker. Hue: I forget.)

Question: can we use word-bag models for note adjacency representation? Since I asked that question I discovered the Chord2vec model of Walder’s lab, whose answer is “yes”. See Madjiheurem, Qu, and Walder (2016).

2 To understand

• Basic group theory: What are the symmetries of the group of all pitch class co-occurrences in the 12-tet? The most common pitch class co-occurrence is a 12-vector over the Boolean algebra where a “1” in the first entry means “the note C was played”, in pos 2, “C-sharp was played” etc. Symmetries we care about are, for example, that (under equal temperament) chord harmonic relationships should be invariant under transposition — i.e. rotation of the entries of the vector. (this last is not so in the case of Sethares/Terhardt style dissonance theory, which considers only unwrapped harmonics.)

• Dmitri Tymoczko’s Geometrical Methods in Recent Music Theory

• Henjan Honing’s Musical Cognition: any good?

3 Dissonance

a.k.a. “Spectral roughness”. A particular empirically-motivated metric, with good predictive performance despite its simplicity, and wilful lack of concern for the actual mechanisms of the ear and the brain, or modern nuances such as masking effects and the influence of duration on sound perception etc. Invented by Plomp and Levelt (Plomp and Levelt 1965), and developed by variously, Sethares, Terhardt and Parncutt and others.

Some sources seem to distinguish roughness in the sense of Sethares from the Plomp and Levelt sense, although they use qualitatively equations. I suspect therefore that the distinction is philosophical, or possibly pointed failure to cite one another because someone said something rude at the after-conference drinkies.

An overview by Vassilakis might help, or the app based on his work by Dr Kelly Fitz.

Plomp and Levelt’s dissonance curves ((Plomp and Levelt 1965; Sethares 1997)).

Juan Sebastian Lach Lau produced some actual open-source software (DissonanceLib) that attempts to action this stuff in a musical setting. There is a slightly different version below attributed to Parncutt.

A convenient summary of both is in the DissonanceLib code.

It’s most useful for things where you are given the harmonics a priori; I’m not especially convinced about the tenability of directly inferring this metric from an audio signal (“how dissonant is this signal?”). We should be cautious about the identifiability of this statistic from signals nonparametrically e.g. windowed DTFT power-spectrogram peaks, just because beat frequency stuff is complicated and runs into the uncertainty principle. (Terhardt, Stoll, and Seewann 1982; Sethares 1998b) give it a go, though. Inferring dissonance between two signals known to be not dissonant might work though, or perhaps one might need parametric approaches, as in linear system identification

Dissonance is an interesting measure, despite these problems, though because it is very much like a Mercer kernel, in that it constructs a distance defined on an (explicit) high-dimensional space; Also, the “nearly circular” geometry it induces is interesting; For harmonic spectra, you recover the equal-tempered 12-tone scale and the 2:1 octave by minimising dissonance between twelve notes with harmonic spectra (i.e. plucked string spectra), which is suggestive that it might do other useful things.

Also, it’s almost-everywhere differentiable with respect to your signal parameters, which makes fitting it or optimising its value easy.

David E Verotta’s thoughtfully illustrated an essay on this.

Anyway, details.

4 Induced topologies

🏗 Nestke (2004) and Mazzola (2012). Tymozcko.

5 References