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I was reading an article an the author gave the following sequence of chords: C A7 Dm G C

I would like to measure the "distance" if there is such a thing between C and A7. Can this be done?

In other words, I am asking if there are any prominent models that somehow quantify relationships between chords in terms of a metric space (i.e. with a distance function).

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    Closest I know of is Hindemith's Unterweisung im Tonsatz, but be prepared for those distances to be extended to everything in harmony (including chord structure and melodic structure), and also bear in mind that the only person who ever used Hindemith's theories to any great effect was Hindemith himself, which strongly suggests that his theories were inadequate to explain his own music. (Hindemith defines degrees of relatedness, but doesn't formalise it into a distance function.) – user16935 Aug 13 '15 at 1:51
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    It's a good question and just asking it is a sign that you're taking a nice approach to viewing harmony. It's not only the difference between pitch frequencies, but also their interdivisibility (how well pitches divide into one other), which can be viewed as largely a matter of what low whole-number ratios can be made, or nearly made, from them. – Epanoui Aug 13 '15 at 2:31
  • Can you elaborate on the latter half of your remark? I didn't quite follow what you meant by pitches "dividing into"? Do you mean like 440Hz / 2 = 220Hz? – Stan Shunpike Aug 13 '15 at 2:33
  • What exactly do you want to measure by your distance? For example two C maj chords in two registers far apart would be close or not? What about two chords like C 5+ 7 and Db min 6? – dtldarek Aug 13 '15 at 8:02
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The simplest metric, and probably the most frequently used (even if only implicitly), is to count the number of steps between the chords' roots along a one-dimensional line of fifths (or the circle of fifths, if you permit enharmonics and modular arithmetic). I say this is the most frequently used because chord progressions where the root ascends or descends by a fourth or a fifth (which have a distance of one by this metric) are the most frequently used in many styles of Western music, indicating that these chords are "close" in some sense. In your progression, this metric would yield distances of 3, 1, 1, 1. This metric has the property that it plays nicely with tonal music, since the tonic/dominant relation that defines tonality has a distance of one. And even if you start to stray slightly from chords that are in your current key, you will end up visiting "closely-related" keys. Also, because this metric looks only at roots, it doesn't inherently care about chord quality (major vs. minor) or extensions (sevenths, ninths, etc).

Dom has already mentioned a second possible metric: the number of common tones (or more precisely, the number of uncommon tones). The more tones two chords have in common, the "closer" they are considered. This works especially well if you plot chords as shapes in a Tonnetz grid. In this case, all of your triads are seen as triangles. The "closest" chords by this metric are those that share two common tones, which results in graphically "flipping" the triangle along one of its three edges. This would imply, for example, that the C major chord is equally close to C minor, E minor, and A minor (a single flip will always turn a major into a minor, and vice versa). In Neo-Riemannian theory, these types of transformations are even named: Parallel (P), Leading Tone (L), and Relative (R), respectively. There are more complex transformations between chords containing only a single common tone. Ignoring the 7th for simplicity, this metric would give your progression the following distances: 1, 2, 2, 2. This metric is less restricted by tonality, and more focused on voice leading. It can more easily explain the "closeness" of chords like C and A♭ which would traditionally be far-removed. As such, it is more suited to Romantic music, where these traditionally-distant progressions are more common. This metric also inherently accommodates different chord types.

There is an even more complex metric that has been developed by Dmitri Tymoczko in A Geometry of Music, involving n-dimensional orbifolds, but I cannot claim to be very familiar with it. It is well-suited to making you forget about music, and focus on mathematical abstractions.

  • I have A Geometry of Music and I'm pretty sure he discusses the similarity between chords in terms of lattices labeled with combinations of notes similarly to a Tonnetz, but in a 3D space. I'd have to double check though especially since the material in the book is rather hard to digest even with an extensive musical and mathematical background. – Dom Aug 13 '15 at 15:54
  • Wouldn't this be a great application for the tonnetz? – Johannes Dec 25 '15 at 22:18
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This is a big part of voice leading specifically where you look for common tones between chords in the harmony and how you can take advantage of them when transitioning between them.

It's not really a formula as much as it is just assessing how related the two chords are. The basic idea is to just look what notes if any are common and if the notes move by how much.

In your example, C which has the notes C-E-G and A7 which has the notes A-C#-E-G has 2 common tones and is related. If you were voicing these chords 4-part choral style, you may see the two chords voiced like this:

enter image description here

As you can see, 2 of the notes don't move. One note, in this case the tenor, moves chromatically upwards which is very little movement and the the other to move is the bass note by 3rd which is a little more in distance. However you should note that there is typically more movement in the bass line.

Like I said, it's not a formula, but it's a very good way to assess what you are talking about.

  • I had something a little more like this in mind dept-info.labri.fr/~rocher/pdfs/RRHD_icmc10.pdf but I am not sure if this one is any good – Stan Shunpike Aug 13 '15 at 3:39
  • Or rather, I was hoping for something in a language like this, but with your general idea. Your idea about common tones makes sense, but I want a mathematically formalized version – Stan Shunpike Aug 13 '15 at 3:40
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    @StanShunpike Don't get confused between "music" and "mathematics". Both are excellent subjects to study, but it's best to keep a clear head about which one you are studying. Also, keep in mind that most music theories were invented with the benefit of hindsight, by people who didn't create much memorable music themselves. (Of course there are a few exceptions to that generalization) – user19146 Aug 13 '15 at 20:12
  • "I had something a little more like this in mind dept-info.labri.fr/~rocher/pdfs/RRHD_icmc10.pdf but I am not sure if this one is any good." I would say the most significant thing about it musically is this: the amount of comment or evaluation about what anything in the paper sounds like is zero. I don't feel inclined to spend time evaluating it as a piece of mathematics (though I do have a math degree). – user19146 Aug 13 '15 at 20:20
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    @alephzero that last line is one of the biggest misconceptions that keeps going around. Don't confuse studying music theory formally as the only music theory out there. Every composer since the dawn of time writes with certain intentions and some kind of theory in mind even if it is just "I like the way this sounds and I don't like how this sounds". They abide by and follow these ideas and use it to create music and especially in the context of modern music is almost always well within the typical theory studies for a good reason. – Dom Aug 13 '15 at 20:21
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Putting on my mathematician hat, the notion of distance is dependent upon many factors. The real question is what is it that you are looking for when you say distance? Do you mean that you are looking for harmonic similarity? Do you mean some sort of measure of auditory similarity? Do you mean how they relate on the circle of 5ths?

The construct of a Euclidean distance is not really relevant as it is meant to measure a physical distance. Group theoretic arguments are meant to talk about relationships between chordal progressions but are built on a network or lattice structure. When we use lattices we have many different measures of distance each of which has a meaning relevant to the network under examination.

We could also look at the construct of temporal similarity. Here we write each chord in terms of its mathematical form (sine waves, etc) and then look at distances between the signals over time. So, in the end, the use of the term distance needs more definition before we can really answer the question.

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    Would you mind adding a few examples of what you think Stan means? – Jacob Swanson Aug 15 '15 at 5:19
  • It's hard to understand exactly what is meant by distance. That is why I gave a number of constructs. I was hoping the original poser would jump in and add further information. – tmwitten Aug 15 '15 at 19:09
  • I was sorta hoping posters would know what kinds of musical metric space applications exist and could tell me which models were most useful. I deliberately left it open ended – Stan Shunpike Aug 21 '15 at 2:30
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There are other ways to define harmonic "distance". For instance, one could use the 5-limit lattice as a "map" on which the harmonic landscape unfolds by fifths (W-E) and thirds (N-S). Then we could track the chord sequence as it moves, quite literally, on the map.

lattice

The only problem is that of the ii chord, which serves a dual function and actually transports us across the map in an instant (from subdominant "west" to dominant "east"). (See 'Harmonic Experience' by W. A. Mathieu, for example.)

Another, albeit closely related, way to view the distance is to use neo-Riemannian theory and the tonnetz. Then the distance could be tracked in much the same way and perhaps defined as the number of transformations necessary to go from one chord to the next.

tonnetz

In the case of C -> A we would need two transformations: R and P.

0

A few scholarly articles discuss this very question; I'll summarize two here.

In "Square Dances with Cubes" by Richard Cohn, the author discusses what he calls "directed voice-leading sums" (or "DVLS"). In order to find the DVLS between two chords, you simply add together the pitch classes of the first chord (mod 12), the pitch classes of the second chord (also mod 12), and subtract the first sum from the second.

For instance, from C major to G major, we have C E G ({0 4 7}, which adds to 11) moving to G B D ({7 11 2}, which adds to 20, or 8 mod 12). We then subtract 11 from 8, which is -3, or 9 mod 12; the directed voice-leading sum from C major to G major is thus 9. As expected, this is a larger distance than from, say, C major (11) to D major (5), which has a DVLS of 6.

In a similar article by Seth Monahan titled "Voice-Leading Energetics in Wagner's 'Tristan Idiom,'" the author talks about "kinetic displacement metrics," or "KDMs." What's important here (and what is different from Cohn's DVLS) is that Monahan measures direction, as well.

Let's go back to the example where C major moves to G major. In this case, we can understand that there is a common-tone G between the two chords (thus a move of 0 half steps); the other voices move from C to B and from E to D. Since C to B is down one half step (-1) and E to D is down a whole step (-2 half steps), we add these distances together to see that this progression has a KDM of -3. As with Cohn's DVLS, a larger absolute value suggests a greater distance between two chords.

For anyone interested in the cognitive realities of these distances, I suggest "Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations" by Carol Krumhansl.

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