Distance measurement in chords and harmony

Question

I've been searching and thinking about this a lot since my mind got opened up to this possibility.

Is there any musical theory "formalism"/framework that deals with measuring the distance between two chords, note clusters or any 2 harmonies.

The premise is that such a notion of distance exists quasi universally in all domains, in different forms, but in harmony I haven't come across anything trying to define what the notion of distance would even mean, except maybe within the circle of fifths (distance between keys) or in modulations. Certain musician do talk about distance, but it's still blurry, and it is always based on some more fundamental assumptions.

I have come across nothing in particular that would describe such a notion at a "low-level", first principles derived, I've been toying with the idea of relative distance based on pitch alone and seeing everything from a chromatic scale POV, let's say we have a CMaj triad, Both Cmaj and Amin would represent "distance = 0", altering any tone within the triad up or down a semitone, for example augmenting the fifth would yield a distance of 1/3 (we altered 1 tone of the 3 possible) but then again how would such computation be if the distance is greater than a semitone? I have some ideas about how I would elaborate on this, but it still feels too arbitrary, and maybe not very informative in the long run concerning what could be done with the results... And how they're interpreted...

I hope anyone can help me with this query since I'm not the biggest theory oriented person, I'm pretty sure someone somewhere has already done some research on a similar topic but I couldn't find anything no matter how much I searched.

Ps: completety forgot about the closest thing (maybe) to what I'm looking for, which is Tonnetz grid, but it doesn't feel like the most encompassing general proposition which I'm lookign for..

Answer 1

This is one of the typical questions that "Music Information Retrieval" deals with, and there are a number of possible approaches. I guess that you are thinking of a symbolic representation of chords, not the acoustics (which would make the chord distance dependent on the temperament). A simple approach is to represent a chord as a 12 dimensional vector and use metrics for vector spaces. See section 1.2 of the following article for a limited overview over some literature:

Maršík, L.: "Using chord distance descriptors to enhance music information retrieval: student research abstract." SAC '17: Proceedings of the Symposium on Applied Computing, pp. 963-964 (2017). (There is a selfarchived version accessible on researchgate)

For measuring the distance between entire sequences of harmonies, see the following OpenAccess article:

de Haas, W.B., Wiering, F. & Veltkamp, R.C.: "A geometrical distance measure for determining the similarity of musical harmony." Int J Multimed Info Retr 2, 189–202 (2013). https://doi.org/10.1007/s13735-013-0036-6

Answer 2

Short answer: no

Long answer: yes, but it gets arbitrary complicated the more you know

Am (a c e) -> C (c e g), one note changed in the set, distance 1, but as a triad movement all notes 'changed', if you play it not as a clustered chord but as an arpeggio. So is it 3 or 1? looking at C -> F -> G: all notes changed between F and G, tension is raised, but the sound is neat.

However, what about a cadence? The transitions sound smooth, so the distances should be all 1, which is obviously not true for IV to V in the Am-C model. Or the progression Am -> G -> F -> E. Sounds smooth to me, even if the last chord is one half step distant. Depending on the style of music you will find even a deeper rabbit hole. Jazz progressions like to throw in altered chords to create chromatic lines with circle of fifth lines combined and in counterpoint theory we get absorbed by a black hole...

Worked out solutions:

If you want to follow something pretty much worked out I'd consider to study the approach of mdecks.com: Mapping Tonal Harmony Pro. IMHO the best practical program to get an idea of distances between chords. Here a screenshot of part of the user interface: https://mdecks.com/mapharmony2022assets/images/mapposter.jpg

Also this author has something worked out, more of a practical approach instead of formalising everything: http://stevenmuschalik.com/MusicTheoryPoster-LowRes.png In this case I like the triad structure tension and release model which can be formalised easier.

Deep dive:

The most extensive research of this thematic (AFAIK) is by Guerino Mazzola, in his book "The Topos of Music" https://link.springer.com/book/10.1007/978-3-0348-8141-8 . It is heavy stuff, very theoretical and extensive but it might be what you are looking for if you want to take a real deep dive worth of several years of study. I for myself have to admit to understand less than 10% of it.

An old book of this researcher is much more approachable but I am not aware of an english translation (Geometrie der Töne). He is also widely disputed, just to get an idea check out this paper: https://dmitri.mycpanel.princeton.edu/files/publications/mazzola.pdf .

HTH