Guitar Chord Voicings with Prolog

Generating guitar chords is my thing . Over the years I’ve written thousands of lines of code andeven more words all dedicated to the process of algorithmically generating and recommending guitar chords. In the spirit of generation, let’s throw another language and another few hundred additional words onto that stack!

Prolog is a logic-based programming language that, as I’m learning (disclaimer: I’m very new to Prolog), excels at representing logical relationships between data. The guitar fretboard is a never-ending landscape of interesting relationships ripe for exploration, so Prolog seems like a valuable tool to have at our arsenal as fretboard explorers.

Let’s see how we can use it.

The Magic of Prolog

One of the most mind blowing aspects of Prolog, from the perspective of someone new to the language, is the fluidity and ambiguity of inputs and outputs to “predicates” (think of them as functions).

For example, we can ask the built-in member/2 predicate if 1 is a member of the list [1, 2, 3] :

member(1, [1, 2, 3]).

And Prolog will tell us that yes, 1 is a member of [1, 2, 3] :

true.

We can also bind the first argument of our call to member/2 to a variable, and Prolog will happily report all possible values of that variable for which the predicate holds true:

member(X, [1, 2, 3]).

X = 1 ;
X = 2 ;
X = 3.

When the second argument of member/2 is [1, 2, 3] , the first argument can either be 1 , 2 , or 3 .

But we can take things even further. We can bind the second argument of our call to the member/2 predicate to a variable and ask Prolog for all of the lists that contain our first argument, 1 :

member(1, X).

X = [1|_5982] ;
X = [_5980, 1|_5988] ;
X = [_5980, _5986, 1|_5994] ;
X = [_5980, _5986, _5992, 1|_6000] ...

This implementation of the Prolog runtime ( SWI-Prolog 8.0.3 ) represents unbound variables with leading underscores. So the first possible value of X is 1 prepended to any other list. Another possible value of X is some value prepended to 1 , prepended to any other list. And so on, forever.

The member/2 predicate simply defines the relationship between it’s two arguments. If one of those arguments is omitted, it can be recovered by applying or reversing that relationship appropriately.

Is your mind blown yet?

Chordal Relationships

Let’s write a predicate that accepts a few arguments that describes our guitar’s fretboard in terms of tuning and number of frets , the quality of the chord we’re looking for, and the notes of a specific chord voicing given as string/fret tuples. Our predicate will either confirm or deny that the notes given live within the bounds of the fretboard and accurately depict the desired chord quality.

For example, on a normal guitar tuned to standard tuning, we could ask if fret 3 played on string 1 (starting from the lowest string), fret 2 played on string 2 and the open fret played on string 3 constitute a C major ( [0, 4, 7] ) chord voicing:

voicing([[0,40], [1,45], [2,50], [3,55], [4,59], [5,64]],
        18,
        [0, 4, 7],
        [[1, 3], [2, 2], [3, 0]]).

And the answer is yes, they do:

true.

If we assume that both our Tuning array and the final Voicing array are sorted in terms of string number we can build our predicate with a simple walk across the strings, analyzing each note in the chord along the way.

For every string on the fretboard, we first check that a note in our Voicing lives on that String . If it does, we need to make sure that the Fret being played on that String is between/3 0 and the number of Frets on the fretboard. Next, we calculate the Pitch of the fretted note and verify that it’s a member of the chord Quality we’re checking for. Lastly we remove that pitch from the set of qualities, and recurse to check the rest of the strings and remaining notes in our chord voicing:

voicing([[String,Open]|Tuning], Frets, Quality, [[String,Fret]|Voicing]) :-
  between(0,Frets,Fret),
  Pitch is (Open + Fret) mod 12,
  member(Pitch, Quality),
  subtract(Quality, [Pitch], RemainingQuality),
  voicing(Tuning, Frets, RemainingQuality, Voicing).

If a string isn’t being played as part of the given chord voicing, we can simply move on to check the next string on the fretboard:

voicing([_|Tuning], Frets, Quality, Voicing) :-
  voicing(Tuning, Frets, Quality, Voicing).

Eventually, we’ll run out of strings to check. In that case, if the remaining set of notes in the chord voicing and the remaining set of pitches in our chord quality are both empty, we can say with confidence that the given set of notes is a valid voicing of the specified chord quality:

voicing([], _, [], []).

If we run out of strings and we’re still looking for either notes in the voicing, or pitches in the quality, we know that something has gone wrong, and the chord we’re looking at isn’t a valid voicing.

Altogether, our complete voicing/4 predicate looks like this:

voicing([], _, [], []).

voicing([_|Tuning], Frets, Quality, Voicing) :-
  voicing(Tuning, Frets, Quality, Voicing).

voicing([[String,Open]|Tuning], Frets, Quality, [[String,Fret]|Voicing]) :-
  between(0,Frets,Fret),
  Pitch is (Open + Fret) mod 12,
  member(Pitch, Quality),
  subtract(Quality, [Pitch], RemainingQuality),
  voicing(Tuning, Frets, RemainingQuality, Voicing).

We can write a helper predicate that assumes an eighteen fret guitar in standard tuning:

voicing(Quality, Voicing) :-
  voicing([[0,40], [1,45], [2,50], [3,55], [4,59], [5,64]],
          18,
          Quality,
          Voicing).

We can use our new voicing/4 or voicing/2 predicates to ask whether a certain set of notes played on the fretboard are a valid C major voicing:

voicing([0, 4, 7], [[1, 3], [2, 2], [3, 0]]).

And Prolog happily tells us that it is a valid voicing!

true.

Excellent.

Reversing the Relationship

We’ve seen that we can use our voicing/4 or voicing/2 predicate to check if a given set of notes on the fretboard are a valid voicing for a given chord quality. For example, we can ask if the notes [[1, 5], [2, 5], [4, 4], [5, 6]] represent a G7 ( [5, 9, 0, 3] ) chord voicing, and our Prolog program will confirm that they do.

But what else can we do? We were promised exploration!

Our voicing/4 implementation didn’t explicitly lay out the steps for constructing a chord voicing of a given quality, but it did define the relationships between a fretboard configuration, the quality of the chord we’re looking for, and the notes in a given chord voicing. Just like we reversed the relationships in member/2 to construct all possible lists containing 1 , we can reverse the relationships defined in voicing/4 and find all possible voicings of a given chord quality!

All we have to do is leave the Voicing argument unbound when we call our voicing/2 predicate, and Prolog will reverse the relationship and spit out every possible voicing of our G7 chord spread across out fretboard:

voicing([5, 9, 0, 3], Voicing).

Voicing = [[2, 1], [3, 2], [4, 1], [5, 1]] ;
Voicing = [[2, 1], [3, 2], [4, 1], [5, 13]] ;
Voicing = [[2, 1], [3, 2], [4, 6], [5, 8]] ...

Awesome! This is basically the heart of Glorious Voice Leader compressed into ten lines of code.

Future Work

We should be able to dig deeper into these relationships. In theory, we should be able to leave the Quality off of our call to voicing/2 and Prolog should tell us all of the possible qualities a given set of notes could be interpreted as.

Similarly, we should be able to leave the Tuning argument unbound, and Prolog should give us all of the possible tunings that would give us the given type of chord with the given voicing.

Both of these types of query sound extremely useful and interesting for someone exploring their fretboard and trying to deepen their understanding of the guitar, but they’re infeasible with my current implementation of the voicing/4 predicate. If we try either of them, Prolog will think forever and never give us an answer. If we trace through the execution we’ll see an enormous amount of time being wasted on inevitably doomed partial solutions.

If I were a better Prologger, I’m sure I could implement a version of the voicing/4 predicate that could give us these answers, but I’m just not there yet. Consider it future work.