I want to create chords from solely interval structure as opposed to building upon triads?

Question

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!

Answer 1

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:

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.

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

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

If we keep adding harmonics, more details are introduced

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)

Here is the animation:

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. ;)

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:

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:

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

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:

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.

Answer 2

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.

Answer 3

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

Answer 4

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.