Distance measurement in chords and harmony

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..

• How would a C major chord and an A minor chord have a distance of 0 between them? They literally don't share all their notes. Commented Sep 12, 2022 at 14:05
• Sorry that was a blunder of mine, I guess I wanted to refer to scales Cmaj and Amin being the same pitch class set just offset to consider the tonic differently Commented Sep 13, 2022 at 18:39
• What would you do with such a distance metric? For example, with a driving distance metric, taxi drivers try and optimize their routes to consume less time and/or fuel. What is your goal? What are you trying to achieve? Commented Sep 18, 2022 at 0:25
• @piiperiReinstateMonica Good question! I'm actually trying to come up with a framework that would allow people to generate their own "meta-language" (depending on their needs, background, instrument, playing style...) and from that "formalism", they could express all their ideas, It's hard to explain and a bit blurry so far, but think something like mDecks Mapping Tonal Harmony but more open ended, and less biased towards a certain style (i.e jazz) Commented Sep 19, 2022 at 0:40
• To elaborate on your question, this distance is something I'm toying with so far, since I'm trying to create/implement my own meta-language (as a first test) and I wanna "refer to chords in the spectrum of difference/similarity" Commented Sep 19, 2022 at 0:53

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

• This is quite an elegant solution, Yes I'm more interested in pitch classes/notes descriptions (which could be analysed, described through functional or even atonal analysis) This idea is basically setting a fixed orthogonal space, in which dimensions (orthogonality) comes from the fact that we're refering to any harmonic information from the 12dimesional space which is the "chromatic scale", so it becomes our reference point Commented Sep 19, 2022 at 1:01
• The way I was thinking about this so far was flawed in the sense that I was considering any harmonic informaiton through an arbitrary n-dimensional space which makes the whole problem of comparison (difference) have no basis (literally), It gave some results but it didn't feel as solid as a well made mathematical/theoretic model could be Commented Sep 19, 2022 at 1:03

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

• If you've not read it, you might be interested in Emmanuel Amiot's *Music through Fourier Space". Commented Sep 14, 2022 at 15:05
• @Aaron thanks, I didn't read that and looks very interesting to me. I did in the 90's a master thesis on a similar topic using geometric approaches with reverse fourier synthesis to generate sound pattern (not music). Commented Sep 16, 2022 at 8:19
• Also found this, close to the ideas we're approaching from different angles youtube.com/watch?v=UcIxwrZV10A Commented Sep 19, 2022 at 0:56