How to measure the tension or resolution of a chord?

Question

Let me begin..

A chord is fundamentally built off the selection of multiple intervals. Those intervals hold a position in a 12TET interval ranking from most consonant to most dissonant. Depending on the intervals used within a chord, the desired level of consonance, dissonance or possibly a mix of the two within a chord, is produced.

Correct? If not could you kindly correct this statement?

Follow up question..

If the statement is correct, how exactly would you specifically measure the consonance or dissonance of a chord based off the intervals used?

To give added context, im seeking this information to produce the specific desired amount of tension and resolution along an entire chord progression, lets say a 4 bar progression.

Thank you!

Answer 1

The tension from a chord depends on context. The section on Ludmila Uhlela's book at thereelscore.com has a nice outline about the perceived tensions in various styles. Trivial example is the major-minor seventh which is a medium dissonance when used as a dominant seventh: V7-I or G7-C whereas as s German Sixth Ge6-I64-I or Ab7 (with the Gb enharmonically interpreted as F#)-C64-G7-C.

Even the 64 chord exhibits a different feeling of tension as a passing chord C-F64-C6 (C-E-G, C-F-G, E-G-C) or as part of a cadence ii65-I64-V7-or F-A-C-D, G-C-E-G, G-B-D-F, C-E-G-C (if parallels are taken care of).

Answer 2

Those intervals [of chords] hold a position in a 12TET interval ranking from most consonant to most dissonant.

No.

The problem is making a general statement like that and then trying to make it apply to your new composition method.

If the premise about chord interval tension where true, then the chords C major and G major would have exactly the same tension, as they are both major triads.

But just like dissonance in the other question tension and resolution are style dependent.

In common practice style C major and G major aren't equal within a key. In C major C is the tonic and a resolution point, but the G chord is the dominant and either resolves to C or can be a partial resolution point. In this way, in the key of C major, the G chord has more tension than the C chord. If the key is G then G major is the tonic and C major is the subdominant which yet again have different levels of resolution. Your could use C major and G major in the key F major and change the resolution roles yet again!

Also, in common practice notions of resolution aren't very dependent on chord dissonance. The viio6 chord is diminished and the V chord is major, both commonly resolve to the I chord. viio6 I isn't heard as a stronger resolution than V I. If anything, it's the other way around. V I would be the stronger resolution.

Common practice resolution is not a mathematical property. It's also not a matter purely of dissonance. In common practice chords operate more like a grammar. V I is the strong resolution, because that's the idiom. You could even say it isn't logical. The more dissonant viio6 should provide a stronger resolution in a logical system. But that isn't the idiom of common practice.

Of course you don't need to follow common practice conventions. You can make up your own mathematical system of chords and resolution, but you don't need to look for existing musical premises to do so. You don't need a universal definition of tension and resolution to make your system.

Answer 3

In a sense this question rests on the same core set of assumptions as your previous question, which I tried in my answer to address, but it seems that you are essentially viewing your goal through the same framework as before.

First let me say, the project of trying to understand music from theoretical, logical (even mathematical) principles is one I admire, and a passion I share. And it seems like you have some interesting thoughts on the matter too. However, I would caution you that when trying to approach this problem, "how to look at music through an essentially scientific standpoint", you look at the music that is in front of you, and try to work out theoretical principles that fit to the music, not the other way around.

When Newton looked at the motion of the planets, he didn't think up a neat and beautiful theory, and then work out how the planets motion could be explained by it; he looked at the planets and tried to work out what theoretical principle best explained their motion.

And when Einstein later approached the same problem, he didn't insist on making the planets conform to Newton's (or any other laws), he theorized, and then extrapolated his theories to their effects in reality. If the theory matched the observation, it was correct, and if not then it was discarded.

You propose the following ideas in your question (which fundamentally are the same ideas embedded in this previous question:

1. that all simultaneous combinations of pitches (or chords) have a quality of "dissonance" or "consonance"

2. that this quality is not absolute, but granular. A chord can be said to be "more consonant", "more dissonant" or some combination of the two. If you like, that there is a sliding scale between "consonance" and "dissonance" for a given chord, and every chord falls somewhere along that scale.

3. that this position on the scale between dissonance and consonance can be described in scientifically objective terms. One chord can said to be "more consonant"/"less dissonant" than another, and vice verse. Let us call this property "harmonic stability" so that we have an unambiguous word to describe it (call it anything you want of course, to (mis)quote Feynman, "you can call it lemon meringue pie if you like, it doesn't matter as long as it works"

4. that the "harmonic stability" of a chord can be measured or predicted in a way that is derived from a ranking of the stability of intervals which constitute that chord, perhaps by an addition of the ranking of the intervals within a chord for example.

5. that the tension or resolution of a series of chords (known is western music as a "chord progression") can be described based on this "harmonic stability", that a "more harmonically stable" chord will resolve to a "less harmonically stable" chord, and so on.

to quote you:

"Correct? If not could you kindly correct this statement?" and, if correct: "How exactly would you specifically measure the consonance or dissonance of a chord based off the intervals used?" with the qualifier "I'm seeking this information to produce the specific desired amount of tension and resolution along an entire chord progression"

So in order to answer these questions, let us go through the assumptions as I have outlined them (the form is different from your question, but I believe I have accurately characterised your standpoint).

1) all simultaneous combinations of pitches (or chords) have a quality of "dissonance" or "consonance"

This one ought to be relatively uncontroversial. I have in my time here occasionally come across people who have argued "it's all relative" or "all cultural", but, while that is at least in part true, a look at the science (psychoacoustics), philosophy (a logical analysis of the our own subjective experience in relation to music from other cultures for example) or anthropology (more specifically, from comparative musicology/ethnomusicology) tells us that there is some fundamental human response to "euphony" and "cacophony" that is in some sense universal between different cultures.

2) This quality is not absolute, but granular

Again, let us for the purpose of this answer take this as read, for similar reasons.

3) This position on the scale between dissonance and consonance can be described in scientifically objective terms, (which I will call "harmonic stability")

Now we're starting to get to the interesting part, the meat of the problem. Whether this is true is indeed a very interesting question, and some theorists have done some very interesting work on this subject (which I admittedly, have only glanced at). Whether this has been established as possible I couldn't say for certain, but it's certainly a worthy goal, and that's good enough for me (and I expect for you too).

4) The "harmonic stability" of a chord can be measured or predicted in a way that is derived from a ranking of the stability of the intervals which constitute that chord

Now this is certainly an interesting hypothesis. I've not heard this suggested in such terms before, but it's certainly food for thought. I don't know that it's necessarily original (it seems like a relatively logical first step), but, on the other hand, it might be.

Your question seems to be "how is this done?", but in a sense, that's the wrong question, because it isn't yet something that's "done", to my mind, you're proposing a novel approach.

Other theorists have proposed ways of measuring the "harmonic stability" of individual chords (some of which I linked to in my previous answer), but I am not aware of any who used this approach. My advice would be this; develop your approach, and see if it results in sensible answers that match your subjective observations of the "harmonic stability" (or euphony/cacophony) of individual chords. If it works, then you might want to compare it to the results of other theorists, see whether the answers allign, or whether they differ.

I suppose a way you might do it, for example, would be to assign each interval a number based on the euphony/cacophony of that interval, and then look at all of the intervals within a chord, add them up, and assign a value based upon that. Just an example, I'm not saying this would necessarily give good results, I'm just saying that if you want to pursue this "interval ranking" based approach, you yourself (or perhaps others too on a more "discussion based" forum rather than a Q&A site) need to come up with a way to realise this. I can't answer the question "how is this done" because to my knowledge so far it never has been done. And as for how it would be done, that's still an open question (but one I invite you to by all means pursue further)

5) The tension and resolution of a series of a "chord progression" can be described by looking at the "harmonic stability" of the individual chords within it.

No. It cannot.

At least, not in western music, and not exclusively.

One of the reasons I coined "harmonic stability" as a descriptor, and used the greek "euphony and cacophony" rather than the latin "consonance and dissonance" is because the terms "consonance" and "dissonance" are associated with a western musical tradition which looks as harmony not just in terms of the individual chord itself, but as it appears in a context. When looking at tension and resolution in western terms, there are multiple factors to take into account. Yes, one of them is the stability of an individual chord, devoid of its context, but also:

• the functional relationship between the current chord and its preceding and following chords (e.g. G chord following an F chord)

• the voice leading between these previous and subsequent voicings of each chord (which is different from the functional relationship, and looks at the flow of individual voiced of the chord.) As Jacob Collier points out, sometimes good voice leading is even more important than functional harmony, and nonsense chords can become part of a pleasant musical "sentence" even if there isn't much of a functional relationship, just because of the good voice leading.

• the relationship of the chord to the "key" of the piece (e.g. an F chord in the key of C vs an F chord in the key of E are very different beasts)

The terms "consonance" and "dissonance" are often used in ways that include all of these concepts, not just the stability of a chord in its own right, but also in a music context (the key, the voice leading, and the functional relationships). The Greek terms "euphony" and "cacophony" more unambiguously refer to just the chord it its own right; how "stable" or "unstable" a chord is, and the term I coined "harmonic stability", can mean whatever I want it to mean, so I decided to use it to refer to the property of a chord on its own.

Now, if your goal is to create a new way of writing music, then it would be very interesting to hear what music that completely eschews tonality and functional harmony, and instead focuses exclusively on the consonance and dissonance of individual chords, then that would be an interesting project, and I would be really interested to hear what music composed along those principles sounded like (it would certainly be unique!).

If your goal is to describe western music, as it exists, then it is insufficient for this purpose.

However, it is certainly one aspect of western harmony, and one of the many tools in the palette of a musician. I think think of examples where using chords and voicings that are "less stable" at moments of tension, and chords and voicings that are "more stable" at moments of resolution has been used to great effect by musicians. But it's by no means the only consideration.

I'll give an example here, I know what piece of music I want to use, and how this will show how dissonance and consonance of individual chords can be used to great effect, but how it also isn't sufficient to explain the song, but right now I don't have the time to write all the transcription (this has taken long enough as it is), so I'm just going to leave this answer here, as-is, in abstract terms without an example for now, and come back when I have some more time.