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i was away but now continuing my research on music science.

"Chords are built systematically through the categorization of six chord types. Major, Minor, Diminished, Suspended, Augmented and Extended. Each individual chord type holds the exact level of consonance/dissonance, regardless of its key, which is then complimented by extensions such as 7ths,9ths,11ths and so on to add further consonance/dissonance to the chord."

I don't want to build chords in a macro form by utilizing triad chords mentioned in the above statement and then adding additional intervals. I want to build chords in a micro manner which would be approached by interval to interval relationships which is what triads are built from themselves.

I have access to an interval ranking (levels of consonance/dissonance of intervals across an octave) and i would like to be demonstrated how i can utilize that interval ranking to build chords. Maybe a more useful approach would be demonstrating how triads are composed through intervals and that would possibly provide me the basis on how to create chords through intervals.

To add further context in relation to your answers, i understand that composers use triad forms (maj,min,dim,etc) as a foundation for building more complex chords but i am not looking to approach it in this manner. I want to understand how chords are built through interval relationships stemming from the interval ranking.

I want to build chords from ratios and not triad or interval names.

Many many thanks!

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    Do you know of someone who already does the sort of thing you're describing? What does the interval ranking look like, something like this one? sethares.engr.wisc.edu/images/image1.gif – piiperi Reinstate Monica Sep 10 at 15:45
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    If you are going to insist on interval rankings, you might consider looking into partial orders – ex nihilo Sep 10 at 15:55
  • I say that this format is actual music theory and the traditional way of looking at music are summarized and condensed systems which i do not care to interact with. Yes, that is the interval ranking! – Seery Sep 10 at 15:55
  • I went through the article David and for me its like a monkey trying to understand quantum physics. Thank you though! – Seery Sep 10 at 15:59
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    Why are you opposed to interval names? At some point you need a vocabulary for a language. To me you seem to be opposed to the very language of music. Please help me understand where I am wrong. Also, is your goal to use simply mathematical relationships that do not appeal to any historical music theory? – ggcg Sep 10 at 16:30
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I'll add a new answer, because before writing the first one I hadn't even bothered to find and read the article where the "interval ranking" picture was taken from. I think the OP may not have completely understood how the curve was produced and what its purpose is. Here is the article: https://sethares.engr.wisc.edu/consemi.html

and here's the picture:

original interval ranking picture

The article is based on the assumption that the sensory harmonic dissonance of a sound is based on the combined dissonances between all individual component sine waves that the sound consists of, and that the elemental sine-vs-sine dissonance (or "roughness") behaves according to the curve experimentally obtained and published in 1965 by Plomp and Levelt in their article Tonal Consonance and Critical Bandwidth (Journal of the Acoustical Society of America 38, 548-560).

That picture shows the calculated combined sensory dissonances of two notes/pitches of a timbre with 6 harmonics, played in an interval represented by the x axis. The point of that picture is to tell you "look where the dips in the curve are - they are right where people have traditionally considered the most stable intervals to be when playing actual instruments". The picture is trying to convince you that the Plomp and Levelt sine-vs-sine sensory dissonance curve, and the presented computational method for predicting the total sensory dissonance of intervals played on actual instruments with complex timbres, can be considered, if not a "truth", at least somewhat credible and not complete rubbish.

I don't think you're supposed to "use" that picture for anything. The picture is there to convince you of the computational method. If you are convinced, what you do is: you go and make your own computer programs that utilize the presented principles and methods. You do not take the picture and try to use it as some kind of a magical recipe.

I took a Python translation of Sethares's curve calculator made by GitHub user endolith: https://gist.github.com/endolith/3066664

I made some modifications to the Python snippet to parameterize things a bit and show the harmonics in the same picture, to make an animation that shows how different timbres behave regarding their calculated two-note sensory dissonance curves.

It starts with a timbre with only a pure sine wave.

sensory dissonance curve with only one sine wave

When we add a harmonic, the curve gets more complex.

sensory dissonance curve with only one harmonic above the fundamental

and with six sine waves in the timbre, we get the notorious "interval ranking" curve.

sensory dissonance curve with six partials

If we keep adding harmonics, more details are introduced

sensory dissonance curve with nine partials

If all the six partials are equally loud, we get this curve: (notice how with this timbre, the major third is more dissonant than the minor third, when with the previous timbre it's the other way around)

sensory dissonance curve with six equally loud partials

Here is the animation:

sensory dissonance curve animated with different timbres

How about chords?

Now, what happens if you have e.g. three simultaneous notes or more, so you could really talk about chords? Let's try something. A four-note chord with three fixed pitches at 12TET pitches 0, 5 and 10, plus a variable note, with the variable note's pitch presented on the X axis. The Y axis has been extended from 0..4 to 0..8, to allow for higher "dissonances", so that the curve fits in the display. Chords are way more dissonant than two-note intervals. ;)

calculated dissonance for a 4-note chord

Does that make sense? Is it useful for something? I don't know. With a pure sine wave as the instrument, the picture looks like this:

calculated dissonance for a 4-note chord, pure sine wave

With nine partials and the default 0.88 dampening factor (amplitude = 0.88 ^ harmonic index) in the timbre, the calculated sensory dissonance goes through the roof:

calculated dissonance for a 4-note chord, 9 partials

... so we have to make the partials dampen more quickly, this is with amplitude = 0.66 ^ harmonic index:

calculated dissonance for a 4-note chord, 9 partials 2

So, what to make of all this? I don't know. Maybe it's possible to use the dissonance calculation method for finding chords. At least William A. Sethares himself is using it in music-making. This stuff starts to look interesting when I get to toy around with the code - even though I haven't really listened to any of this. I haven't even checked what the example timbre sounds like.

Here are some more 4-note chord curves:

calculated dissonance for a 4-note chord, 0-3-10-x

calculated dissonance for a 4-note chord, 0-4-11-x

calculated dissonance for a 4-note chord, 0-4-10-x

If you try playing those chords, does the curve correspond to how you subjectively, intuitively feel about the sounds? I don't know. :)

And then, how to do chord progressions? Then you'd have to have some kind of a facility for memory and harmonic context that isn't continuously sounded but exists in the listener's head. Some kind of a pitch memory, where previously perceived pitches gradually dampen at some rate, and that take part in the dissonance calculation even when they're not being "heard" at the moment in question.

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    @DavidBowling you had an idea about what to try and calculate for four-note chords? – piiperi Reinstate Monica Sep 14 at 15:04
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    I deleted the comment because, after reflection, I thought that it wasn't really helping OP's project of building with intervals outside of triadic harmony along, but here it is again: instead of moving from two notes to three notes, you could move to four notes. In particular, I thought that it would be interesting to calculate the curves for shell voicings (1-3-7 of M7, m7, and dom7) with a fourth voice added as the independent variable. This might give some insight into dissonance in 7th chords with extensions or altered extensions. – ex nihilo Sep 14 at 19:00
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    I am a little skeptical of the utility of these sorts of dissonance curves, but that doesn't mean that I don't think that they are interesting. Maybe they would find the most use in algorithmic or computational music settings. – ex nihilo Sep 14 at 19:01
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    @DavidBowling I added the 0-3-10-x, 0-4-11-x and 0-4-10-x curves. Do the "consonance" dips make any sense intuitively? With this answer I'm trying to say that the curves themselves aren't really meant to be used, they're only there to reflect what's happening inside the computational model. And in a way that's the answer to the OP's question about constructing chords from intervals using the presented dissonance/consonance calculation model. The assumption here is that total dissonance is built from the dissonances between all individual sine wave pairs. Which is a lot of pairs. :) – piiperi Reinstate Monica Sep 14 at 19:58
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    Anyway, the more I tinker with this, the more it feels like the OP might be on to something with the idea of looking at individual interval pairs for building harmonies. Of course there are many further questions to answer, for example the question about chord root note, "harmonic context" memory, rhythm, etc. And the whole Plomp and Levelt basic sine-vs-sine dissonance curve - the calculated total dissonance figure depends on the basic "atomic" curve. – piiperi Reinstate Monica Sep 14 at 20:03
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The triad way of looking at chords is very valuable imo. I learned in as polychord theory and can help in making decisions on what to play but there is a more basic and fundamental way to build chords. You can think of chords as a stacking of thirds, or the maj scale played in thirds instead of steps. For example the Maj scale in thirds would be...

1, 3, 5, 7, 9, 11, 13, 1

This produces a maj13 chord. All others are deviations of this, e.g. dom7 (13) would be {1, 3, 5, b7, 9, 11, 13}.

In chord construction we usually take everything before the highest extension in theory, but in practice we eliminate notes that are either too difficult to play or not germane to the functional use of the chord. For example the classic 13th chord is as stated in a previous sentence but on guitar we usually only play {1, 3, 5, b7, 13} or even {1, 7, 3, 13} in that order. We drop the 9 and 11. If possible we play the 9th too, a classic voicing on guitar is {1, 3, b7, 9, 13}. Even the dominant 7th chord in classical harmony theory can have the 5th dropped without losing functional value.

So you can think of your chords as...

{1, 3, 5} - classic triad

{1, 3, 5, 7} - 7th chord

Here come the extensions

{1, 3, 4, 7} + {9}

{1, 3, 4, 7} + {9, 11}

{1, 3, 4, 7} + {9, 11, 13}

And that's it.

In theory, all others can be made by altering one or more notes in the list provided above. All the dom7 extensions just have b7, then you can do things like #5, b9 etc. Minor 7ths have b3 and b7, etc.

It is worth noting that Pat Martino had a way of generating all "uesfull" chords by altering the full diminished chord {1, b3, b5, bb7}.

I hope this helps somewhat.

  • 2
    You can also make chords by stacking fourths, fifths, etc. – Todd Wilcox Sep 10 at 21:58
  • Triads are valuable but i feel its not the most potent form of chord building and can almost put you in a "music theory box". How are polychords different from just adding extensions to a chord? It seems 2 of the same, no? Again, 3rds is great but im focused on building chords from interval ratios/names as opposed to triads or numeric systems. "not germane to the functional use of the chord" could you elaborate on omitting intervals for functional use? How would you say an interval dictates function, Do people omit intervals solely for practical playin and function or are there other motives? – Seery Sep 12 at 21:24
  • I will also add that you gave a stupendous writing of approaching chord building and although it is not the format i am seeking, it provides wonderful context and expansion of chord knowledge. Also, in the question i asked above, it serves purpose for questions i may have had later down the line so thank you for your contribution ggcg! – Seery Sep 12 at 21:27
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    I think I gave an example of notes that are removed from chords. One classic example is the 5th (or 5) from the V7 chord. It doesn't move anywhere. The 3 of the V (7 of the key) moves to the 1, the 7 of the V is the 4 of the key and moves to the 3, and the 1 of the V is the 5 of the key and stays there. I am not sure how you equate the standard ways of building chords as being in a box but... – ggcg Sep 12 at 22:53
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    I do see value in making the choice based on dissonance/consonance. One of my favorite examples of voicing diff is the #9th chord. The standard way to play on guitar is (1, 3, 7, #9). The #9 is enharmonic to the min 3rd. Try the voicing (1, #2, 7, 3). Very simple change and very different chord. All it is is a new voicing. Would your approach build different voicings of a chord as different chords? – ggcg Sep 12 at 22:56
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The concept of stacked intervals isn't all that uncommon. I, and many other musicians, actually prefer to think this way. Standard chords (triads, 7ths and their extensions) are all based on stacked thirds, so each note is either a major or minor (or aug or dim) third above the previous note. The same concept has been applied to fourths (occasionally in jazz), but this is very uncommon and quite limited. Stacking fifths is really just thirds, but skipping every other, and major/minor seconds essentially just make up scales (and tone clusters). Anything above a fifth, or some combination of thirds, fourths, fifths etc. can usually be shuffled around and classified via a different stack (i.e. stacking sixths, C-A-F, is just a spread out stack of thirds, F-A-C, creating a major triad; and stacking a fourth, a third and a fifth, C-F-A-E is also just a stack of thirds, F-A-C-E, a major 7th).

So, 95% of the time, thirds are all we really need. Because we only have 2 options for each interval (m3 and M3) we can determine every possible combination of stacked thirds using a binary system where m3 = 0 and M3 = 1. We will only go up to a 13th, so 7 notes, but since our "digits" are the intervals between them, we only need 6. This gives us 64 (2^6) possible permutations.

A few things to keep in mind:
1. This list does not include any chords involving diminished and/or augmented thirds, such as AugDom7: C-E-G#-Bb (sorry, I didn't feel like figuring out 4,096 permutations...)
2. For especially ridiculous voicings (AugMaj7, ##9, ##11, ##13) I used enharmonic inversions (Maj7, #9, b13), where possible, shown in parentheses
3. Doubled notes (#9 in a min7, b11 in a Maj7, bb9 bb11 bbb13 in a dim7, etc.) were omitted

m3-m3-m3-m3-m3-m3 = dim7
m3-m3-m3-m3-m3-M3 = dim7, bb13
m3-m3-m3-m3-M3-m3 = dim7, b11, bb13
m3-m3-m3-m3-M3-M3 = dim7, b11, b13
m3-m3-m3-M3-m3-m3 = dim7, b9, b11, bb13
m3-m3-m3-M3-m3-M3 = dim7, b9, b11, b13
m3-m3-m3-M3-M3-m3 = dim7, b9, 11, b13
m3-m3-m3-M3-M3-M3 = dim7, b9, 11, 13
m3-m3-M3-m3-m3-m3 = 1/2dim7, b9
m3-m3-M3-m3-m3-M3 = 1/2dim7, b9, b13
m3-m3-M3-m3-M3-m3 = 1/2dim7, b9, 11, b13
m3-m3-M3-m3-M3-M3 = 1/2dim7, b9, 11, 13
m3-m3-M3-M3-m3-m3 = 1/2dim7, 9, 11, b13
m3-m3-M3-M3-m3-M3 = 1/2dim7, 9, 11, 13
m3-m3-M3-M3-M3-m3 = 1/2dim7, 9, #11, 13
m3-m3-M3-M3-M3-M3 = 1/2dim7, 9, #11, #13
m3-M3-m3-m3-m3-m3 = min7, b9
m3-M3-m3-m3-m3-M3 = min7, b9, b13
m3-M3-m3-m3-M3-m3 = min7, b9, 11, b13
m3-M3-m3-m3-M3-M3 = min7, b9, 11, 13
m3-M3-m3-M3-m3-m3 = min7, 9, 11, b13
m3-M3-m3-M3-m3-M3 = min7, 9, 11, 13
m3-M3-m3-M3-M3-m3 = min7, 9, #11, 13
m3-M3-m3-M3-M3-M3 = min7, 9, #11, #13
m3-M3-M3-m3-m3-m3 = minMaj7, 9, 11, b13
m3-M3-M3-m3-m3-M3 = minMaj7, 9, 11, 13
m3-M3-M3-m3-M3-m3 = minMaj7, 9, #11, 13
m3-M3-M3-m3-M3-M3 = minMaj7, 9, #11, #13
m3-M3-M3-M3-m3-m3 = minMaj7, #9, #11, 13
m3-M3-M3-M3-m3-M3 = minMaj7, #9, #11, #13
m3-M3-M3-M3-M3-m3 = minMaj7, #9, #13
m3-M3-M3-M3-M3-M3 = minMaj7, #9
M3-m3-m3-m3-m3-m3 = Dom7, b9
M3-m3-m3-m3-m3-M3 = Dom7, b9, b13
M3-m3-m3-m3-M3-m3 = Dom7, b9, 11, b13
M3-m3-m3-m3-M3-M3 = Dom7, b9, 11, 13
M3-m3-m3-M3-m3-m3 = Dom7, 9, 11, b13
M3-m3-m3-M3-m3-M3 = Dom7, 9, 11, 13
M3-m3-m3-M3-M3-m3 = Dom7, 9, #11, 13
M3-m3-m3-M3-M3-M3 = Dom7, 9 #11, #13
M3-m3-M3-m3-m3-m3 = Maj7, 9, 11, b13
M3-m3-M3-m3-m3-M3 = Maj7, 9, 11, 13
M3-m3-M3-m3-M3-m3 = Maj7, 9, #11, 13
M3-m3-M3-m3-M3-M3 = Maj7, 9, #11, #13
M3-m3-M3-M3-m3-m3 = Maj7, #9, #11, 13
M3-m3-M3-M3-m3-M3 = Maj7, #9, #11, #13
M3-m3-M3-M3-M3-m3 = Maj7, #9, #13
M3-m3-M3-M3-M3-M3 = Maj7, #9, 13
M3-M3-m3-m3-m3-m3 = AugMaj7, 9, 11, b13
M3-M3-m3-m3-m3-M3 = AugMaj7, 9, 11, 13
M3-M3-m3-m3-M3-m3 = AugMaj7, 9, #11, 13
M3-M3-m3-m3-M3-M3 = AugMaj7, 9, #11, #13
M3-M3-m3-M3-m3-m3 = AugMaj7, #9, #11, 13
M3-M3-m3-M3-m3-M3 = AugMaj7, #9, #11, #13
M3-M3-m3-M3-M3-m3 = (Dom7, #9, b13)
M3-M3-m3-M3-M3-M3 = (Maj7, #9, b13)
M3-M3-M3-m3-m3-m3 = Aug, #9, #11, 13 (no7)
M3-M3-M3-m3-m3-M3 = Aug, #9, #11, 13 (no7)
M3-M3-M3-m3-M3-m3 = (Dom7, #9, b13)
M3-M3-M3-m3-M3-M3 = (Maj7, #9, b13)
M3-M3-M3-M3-m3-m3 = (Dom7, b13)
M3-M3-M3-M3-m3-M3 = (Maj7, b13)
M3-M3-M3-M3-M3-m3 = (AugMaj7)
M3-M3-M3-M3-M3-M3 = Aug

  • Thank you very much for this additional education Will. I've extracted the relevant information and i'm appreciative for your effort listing out the 3rds to chords. It proved to be insightful. – Seery Sep 17 at 23:51
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You might also look at some of the figured bass theories from the late 1500s to early 1700s. These (sometimes) consider structures over a bass (or sometimes tenor) line. One gets things like comparing a inversion C chord relative to a root position E minor: E-G-C vs E-G-B.

  • I may look into that shortly down the line if it proves to be of interest in my research. Thank you ttw! – Seery Sep 12 at 21:29

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