Possible Playable Chords on a Guitar

Fingerstyle Guitar Chord Diversity Check

Considering a $20$-fret $6$-string acoustic guitar and supposing that the fretting range (inclusive of the fingered notes) for an average hand is $4$ frets in the first $8$ frets, $5$ frets in the $9-14$ fret region and $6$ frets on the remaining of the fretboard, how many combinations of at least $3$ different notes can be strummed if we ignore barring and use at most $4$ fingers to fret?

We would have the worst fretting range if our fingered frets involve any of the $3$ prescribed regions down the fretboard (i.e. $8$ fret region, $9−14$ fret region and the remaining of the fretboard). Suppose you decide to fret $12$ on the first string, then, the lowest other fret you can finger is $9$ since fretting $8$ would require a fret range of $5$ while playing $8$ reduces your fret range to $4$ since we consider the worst fret range on overlapping fret regions.

In our case, a chord is defined as a collection of $3$ to $6$ distinct notes strummed together. Each chord is unique. For example, fretting $3$ on sixth string, $2$ on the fifth string, $3$ on the first string and strumming all 6 strings is the same as fretting $3$ on sixth string, $2$ on the fifth string, $3$ on the second string and $3$ on the first string and strumming all 6 strings as both would produce a $\text{G}$ chord.

Solution 1:

The following is not really specific to guitar music, but a general approach that could be specialized to guitar music with further restrictions.

Dmitri Tymoczko has tried to develop a kind of geometrical theory of chord spaces. I found it interesting insofar as it managed to get some structure, I didn't find it at all useful when it comes to making music though. But I'm sure this is something you want to look into. His is not the only approach though.

Anyway, while trying to understand his papers, I started to do my own computations to determine how many different chords of 3, 4, 5,... notes there exist. The key here is Burnside's lemma from group theory. Let me illustrate for the case of 3 note chords.

First, we have to establish the space we are working on. We'll start from the conventional Western 12 tones in equal temperament, but if you want to make quarter tone or microtonal music or use different temperaments, you can easily adapt my story. Each tone will get a number from 0 to 11, we'll work modulo 12 since the twelfth tone is just the 0th an octave higher. So that's a first symmetry that is already implicit in the definition of our space. In modern harmony, this is called a pitch class. Now, since we consider 3-tone chords, we are going to present them as follows

$$\text{CMaj}=\{\text{C,E,G}\}=\{0,4,7\} \\ \text{Cmin}=\{\text{C,Eb,G}\}=\{0,3,7\} \\ \text{FMaj 2nd inversion}=\{\text{C,F,A}\}=\{0,5,9\} \\$$

Then we are going to introduce additional symmetries. We consider all chords that can be transposed into each other to be the same chord. For instance

$$\text{FMaj}=\{\text{F,A,C}\}=\{5,9,0\}\overset{T}{=}\{5-5,9-5,0-5\}=\{0,4,7\}=\text{CMaj} \; .$$

Next, we are also going to identify inversions, which mathematically amounts to permutations of the elements

$$\text{FMaj 2nd inversion}=\{\text{C,F,A}\}=\{0,5,9\}\overset{P}{=}\{5,9,0\}\overset{T}{=}\{0,4,7\}=\text{CMaj} \; .$$

One could go further and add additional symmetries, for instance

$$\text{Cmin}=\{\text{C,Eb,G}\}=\{0,3,7\}\overset{I}{=}\{7-0,7-3,7-7\}\overset{P}{=}\{0,4,7\}=\text{CMaj} \; .$$

But since this turns major chords into minor chords and visa versa and these are usually considered distinct in conventional harmony, we're not going that way. One might argue that even inversions of chords should be considered separately because of their different functions, but for the sake of the present argument, I'm gonna stick to two symmetries $P$ and $T$.

Now, let's see how we can apply Burnside's lemma. Our space of 3-tone chords is $X$ and is acted upon by a group $G$ generated by transpositions $T$ on one hand and permutations $P$ on the other. Burnside's lemma tells us that

$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$

or in human words: the number of 3-tone chords up to transpositions and permutations is found by adding for each symmetry element the number of chords fixed by that certain symmetry and dividing by the total number of symmetries acting upon the space. $X/G$ is called in jargon the set of orbits. Whereas $X^g$ is the set of fixed elements under the action of $g$.

Applying this to our 3-tone chords set, we really only have to look at what happens under permutations since transpositions can never fix elements. So, we have 6 possible permutations:

$$ \{\text{Id},(12),(13),(23),(123),(132)\} \; .$$

The identity fixes all $12^3$ elements of our space. The three following permutations fix elements of the form $\{a,a,b\},\{a,b,a\}$ and $\{b,a,a\}$ resp. thus accounting each for $12^2$ fixed elements. The last two are more tricky. Obviously elements of the form $\{a,a,a\}$ are fixed. But also elements of the form $\{a,a+4,a+8\}$ and $\{a,a+8,a+16\}$ are. This accounts for a total of $6 \times 12$ elements. Applying Burnside's lemma, the amount of distinct 3-tone chords is

$$\frac{12^3+3\times 12^2+6\times 12}{12\times 6}=31 \; .$$

Among these 31 chords are however a bunch of chords containing repeated tones, 12 to be precise. If we leave those 12 out, we are left with 19 chords. Leaving out 9 more that contain a minor second interval leaves us with the following 10 chords

$$\{\text{C,E,G#}\},\{\text{C,E,G}\},\{\text{C,Eb,G}\},\{\text{C,D,F#}\},\{\text{C,E,Gb}\},\{\text{C,D,G}\},\{\text{C,D,E}\},\{\text{C,Eb,F}\},\{\text{C,Eb,Gb}\},\{\text{C,D,F}\}$$

The first three chords are resp. the augmented triad, the major triad and the minor triad. The second to last triad is the diminished triad. Those are very common in classical harmony. Chords like $\{\text{C,D,G}\}$ are used in quartal and quintal harmony. It is however also understood as a suspended fourth in classical and jazz harmony.

Note that it is possible in my analysis to exclude repeated notes from the onset. The sequence of integers for the number of chords consisting of several distinct notes is listed on OEIS with number A035495 and continues as follows:

$$1, 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1$$

Finally, a small "dirty" picture I made to illustrate the 10 chords and how you can move by stepwise motions between them, i.e. by altering just one tone a half step up or down.