:: Kind Of Chords

There are two kinds of chords that get lots of attention in microtonal music theory. The first kind is intervallic, and these chords have names like "Bbm6" or "G7b9#11" . The other kind of chord is defined in frequency space by ratios like "4/4, 5/4, 6/4". This post is all about the first kind - intervallic chords - and how to use them. Frequency-space chords will get their own post.

:: Rank-2 Chords

We'll start with the normal chords of grade school music theory, defined over rank-2 interval space.  There's a lot to do here. We'll start by learning 

Once we've done all that, we'll look a little bit at how to do those seven things with rank-3 and higher chords.

: Intervallic Chord Names

Musicians notate chords in slightly different ways. Some jazz musicians do it especially differently - with a shorthand that has like triangles and circles and plus signs. I've tried to come up with a systematic naming scheme that I think would satisfy most musicians, except perhaps for the jazz musicians with triangles. Here's how it works:

First up, while most western scales increase by second intervals (i.e. the letter names increase one step, like [F, G, A, B, C, D, E]), most chords instead increase by third intervals (i.e. the letter names increase by two steps, like [F, A, C,  E, G, B, D]). Chords made by stacking thirds are called "tertian" in general music theory, and we'll try to analyze all rank-2 chords as though they were tertian. Lots of musicians like using non-tertian spellings of chords, and I'm not opposed to those names exactly, but I wanted a system that would give a one-to-one mapping between chord names and interval sets, and the tertian restriction made that work. In non tertian chord names, you can have 2nd, 4th, 6th scale degrees, while in a tertian spelling, we interpret these as 9th, 11th, and 13th scale degrees. 

It will be useful for us to memorize now that a 9th interval is a 2nd plus an octave, so it comes in major and minor varieties, and an 11th is like a 4th, so it can be perfect, and a 13th is like a 6th, so it has major and minor varieties.

The most famous rank-2 chords are 

major: [P1, M3, P5]

minor: [P1, m3, P5])

diminished: [P1, m3, d5]

augmented: [P1, M3, A5]

Chords like those with three notes are called triads, but you can have more notes.

 Here's a chord with more notes:

minor thirteenth: [P1, m3, P5, m7, M9, P11, M13]

The intervals of the chord go through all the odd number up to 13, and so it's called a 13th chord of some kind, in this case a minor 13th or ".m13" for short.

One rule in my naming system is that if you have an odd scale degree in a name, like the "13" in ".m13", and it is natural, rather than sharpened or flattened, that implies that every other odd scale degree below it is also present (as a natural degree, or a sharpened degree, or a flattened degree, but it has to be there in some form). For chords, "natural" means that the interval comes from the major scale spelled by thirds, i.e. it's a perfect or a major interval. So the M13 in the chord above is natural, and thus we must also have some kind of 1st, 3rd, 5th, 7th, 9th, and 11th present below that. 

Let's look at another chord. Here's the "major thirteenth chord with a flat ninth", or ".maj13b9" for short:

.maj13b9: [P1, M3, P5, M7, m9, P11, M13]

If you spell the .maj13b9 chord by how it deviates from a major scale, then it happens to look like this in "scale degree" notation:

(1, 3, 5, 7, b9, 11, 13).

Suppose we got rid of the 11th scale degree so that there is a gap in our stack of thirds.

[P1, M3, P5, M7, m9, ..., M13]

 What would the chord be called then? Let's review our naming principle: "if you have an odd scale degree in a name, and it is natural, that implies that every other odd scale degree below it is also present". Our chord with the 11th removed can't be a kind of 13th chord because we don't have all the odd degrees below 13. And the chord can't be a 9th chord, because the "m9" interval isn't natural. It can be a 7th chord though, and it is. After we've identified what kind of chord it is, we step through the scale degrees that haven't been accounted for by knowing that it's a 7th chord in increasing numerical order, and the result looks like this:

.maj7b9(add13): [P1, M3, P5, M7, m9, M13]

Well done, us.

: Complications To The Chord Naming Rule

Actually, the rule as we will use it is a little more complicated than "write the highest odd natural scale degree for which every lower odd scale degree also appears".

One quibble is that in standard practice, the "dominant seventh" chord, ".7", has a b7 scale degree, so we could call it "5b7", but we'll stick with standard practice here and just use a bare "7". In the same way, if you start with a dominant seventh chord, ".7", and add on a major ninth interval, then your chord becomes a dominant 9th chord, ".9", with an implied m7.

.9: [P1, M3, P5, m7, M9]

This is how musicians do it, I swear. If we had a "G.13b9" chord, that would also have an implied m7 interval above the root.

Here's another concession to the musicians: the minor seventh chord of standard practice, ".m7", has a flattened seventh scale degree, so we could call it minor chord with a flat seventh, ".mb7" or".5b3b7", but again we'll stick with common practice naming. The minor ".m7" chord is [P1, m3, P5, m7], and so the m11 chord is similarly "P1, m3, P5, m7, M9, P11".

Another quibble is that the part of the chord name before the chord type, like "m", "maj", "dim", "aug", tells us about the third and the fifth scale degrees of the chord, and we list these before the chord type, instead of working our way up through the scale degrees after mentioning the 7. There's also a "minor-major seventh" chord that has a minor-third a major-seventh,

m-maj7: [P1, m3, P5, M7]

For  completeness and symmetry, my naming system also includes augmented-major seventh chords, 

aug-maj7: [P1, M3, A5, M7]

and diminished major seventh chords, 

dim-maj7: [P1, m3, d5, M7]

But I don't think I've ever seen these used. 

We'll make one last concession to standard names of music theory: the so called "diminished seventh chord" is spelled 

".dim7": [P1, m3, d5, d7],

And similarly a ".dim9" chord adds on a M9 interval and retains the implied d7 interval.

If that's a dim7 chord, then what's the diatonic 4-note chord that you make by thirds, starting on the 7th scale degree of a major scale, for example [B, D, F, A] in the key of C? Standard practice calls this a "minor seventh chord with a flat fifth".

".m7b5": [P1, m3, d5, m7]

even though the triad at the bottom is diminished rather than minor.

Those are all the ways that you can specify the third, fifth, and seventh at the start of a chord, before you get on to the altered upper chord tones. If those options don't adequately specify the lower tones of your chord, like if you have a minor third and an augmented fifth, then you do the regular thing up walking your way up through the scale degrees in numerical order, e.g. 


.m9#5: [P1, m3, A5, m7, M9]

In total, my code recognizes 720 unique chords. Those are the chords you get when the scale degrees (^1, ^3, ^5, ^7, ^9, ^11, ^13) are allowed to range over these options:

^1: [P1]

^3: [m3, M3]

^5: [d5, P5, A5]

^7: [d7, m7, M7]

^9: [m9, M9, A9]

^11: [P11, A11]

^13: [m13, M13]


and also scale degrees ^7, ^9, ^11, and ^13 are optional; they don't have to make an appearance. Also, d7 is only an option for scale degree ^7 when scale degree ^5 is a d5. I suppose I could extend my code to include #3, #7, b11, and #13, but I'm also fine with the thing that I have.


This system can give you some pretty crazy names like:


.m-maj9#5#11b13: [P1, m3, A5, M7, M9, A11, m13]


if you feed it crazy chords. Here are a few random chords with fairly short names that you can look over to cement your understanding of the naming scheme:


.dim#11: [P1, m3, d5, A11]

.9b5#11: [P1, M3, d5, m7, M9, A11]

.11b9b13: [P1, M3, P5, m7, m9, P11, m13]

.mb13: [P1, m3, P5, m13]

.maj7b13: [P1, M3, P5, M7, m13]

.m7b13: [P1, m3, P5, m7, m13]

.dim11b9: [P1, m3, d5, d7, m9, P11]

.11b5: [P1, M3, d5, m7, M9, P11]

.m#5b13: [P1, m3, A5, m13]

.majb13: [P1, M3, P5, m13]

.7b9b13: [P1, M3, P5, m7, m9, m13]

.m13b9: [P1, m3, P5, m7, m9, P11, M13]

.m7b9: [P1, m3, P5, m7, m9]

.maj13b5: [P1, M3, d5, M7, M9, P11, M13]

.m#5: [P1, m3, A5]

.9b5: [P1, M3, d5, m7, M9]

.m13: [P1, m3, P5, m7, M9, P11, M13]

.11: [P1, M3, P5, m7, M9, P11]


: Root Motion Cadential Progressions

If you play a bunch of chords, one after another, you might not have a chord progression. A chord progression, in contrast to a chord sequences, makes progress - it goes somewhere. The most basic way to go somewhere with chords is to cadence. The term "cadence" comes from the Latin verb meaning "to fall". You find yourself in some crazy place musically, and you fall back down in a simple structured way.

I used to think cadences were pretty boring and I could write more interesting music without them. Cadences are indeed boring, but music without them is actually less interesting. A cadence is the way that listeners expect a chordal phrase to end, and the way that know that a chord phrase has ended, and without cadences, the chords sequences don't feel structured, so people lose interest. You can and should do weird beautiful amazing things with chord sequences - but if you don't end them in boring ways, they can become too complex to be appreciated.

This was a really hard pill for me to swallow: I'm a staunch and devout xenophile, I'm always trying to expand my aesthetic horizons, I want the advanced shit, I want to "be the aliens you wish would contact the world". But cadences are really important, and music is worse without them. And it's not just important to have them; you need to cadence obviously, conspicuously, sometimes ostentatiously. You have to  make a great flamboyant show of being boring or people will miss what happened.

Okay, that was a lot of justification for cadences. Now, what are they?

There are dozens of cadences. We'll have to learn a little bit about voice leading and chord voicing before we can talk about all of them in depth. But the most basic categories of cadence talk about root motion: how the root of one chord relates to the root for the next chord.

The most basic, boring, powerful and compelling cadence defined by root motion is moving from one chord to second chord that has its root note a perfect fifth down from the root of the first chord. This motion is the so called the "perfect" or "authentic" cadence. It's often written "V -> I", suggesting a tertian chord based on the fifth degree of a 7 note octave spanning scale moving to a tertian chord built on the first degree of a such a scale. But you should still use perfect cadences even if you're writing music that doesn't have a normal diatonic scale, or any scale at all. Perfect cadences often use some kind of dominant 7th chord quality for the "V" chord. Western classical music has more conventions to make perfect cadences more obvious, and we'll learn about these in time, but the root movement is the core.

Another basic cadence is called the "plagal" cadence. In a plagal cadence, the root moves down by a perfect fourth instead of a perfect fifth. A plagal cadence is often notated in roman numerals as "IV -> I". Plagal cadences are not as powerful as perfect cadences - they can almost be thought of as retrogressions instead of progressions, but they're still a recognizable structure for chordal phrases, and they're used idiomatically in some musical genres, and they go somewhere. Maybe they fall up instead of down, of they fall off the stage onto the audience instead of down the trap door on the set, but they go somewhere, and you can work with that. Chords built on the fourth scale degree in tonal music are often called "subdominant" because they're below the dominant seventh chords built on the fifth scale degree.

If you play a dominant chord - a "V" chord - especially one with a major or dominant 7th quality built on the fifth scale degree in tonal pieces,  and you don't follow it with a chord that has a root a perfect fifth down, that's a "deceptive cadence". They usually go to a VI chord, but they don' t have to. So we've seen that it's a cadence to go from "V -> I". And it's also a cadence to go from "V -> anything else". So anything with V chord is a cadence? Kind of! The V chord has to be the second to the last chord in your phrase, and it has to be clearly there - you have to outline its intervals or drone on it or blast it and slide into it an an unusual way, you have to make it noticeable. And then yes, the last chord can be anything. But it's much stronger if you move down by a perfect fifth. If you're not doing that, then there are some other options that are fairly stong while being deceptive rather than unstructured. In C major, for example, you can progress [G.7 -> A.m] or sometimes [G.7 -> Ab.maj] if you're into modal mixture. If you're in A minor, some fairly strong but deceptive options are to progress [E.m7 -> F.maj] or [E.7 -> F.maj].

One more root motion cadence is the "half cadence". In a half cadence, you land on a V chord, usually from I, II, IV or VI. It's great to precede a perfect cadence by a half cadence, but you can also just end a phrase on a half cadence sometimes. If you want to do a call-and-response song structure, then half-cadences are great way to end the calling phrases.

It should already be clear that there are lots of types of cadences, and it shouldn't be particularly constraining to use them. The most important things to do are to make your chord transitions very clear at the end of phrases, and to often move your chord roots by a perfect fourth or perfect fifth between your last two chords, unless you're being deceptive, but then you should keep in mind that dramatically violating your audience's expectations can sometimes requiring building up those expectations in the first place.

Don't have perfect fourths and fifths in your music? Perhaps you can use a similarly sized interval similarly. Don't have major chords or dominant chords in your music? I bet you can find pairs of chords that have a similar feeling of tension and resolution, and you can highlight them in a similar way. Don't have chords at all? Okay, you got me. Monophonic music doesn't have to have cadences. This chapter on chords might not be for you.

: Chord Voicing

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: Passing Chords And Voice Leading

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: Advanced Cadences And Modulations

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: Chord Substitution and Reharmonization

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: Upper Chord Tones and Harmonic Fields

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: Modal Mixture Grammar

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:: Rank-3 Chords And Higher 

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