This article looks at applying basic music theory to the guitar using Python in order to derive chords and scales for alternate tunings.
This is a follow-up to my previous article on learning basic music theory in Python. Some of this stuff will make more sense if you’ve read the previous article, so I highly recommend doing so. (Given the various fretboard diagrams in this article, you might want to read on a desktop or in landscape mode if on a mobile.)
You might ask yourself: why do this? My primary reason is purely practical: I’ve been playing in an alternate tuning (DADGAD; a Dsus4 tuning) for a while now and wanted to dig deeper into chords and scales in it. While there are lots of resources available for standard tuning, I’ve found few when it comes to alternate tunings like DADGAD. Thus my attempt to encode this information programmatically. Another reason is…well…fun!
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Music theory algorithmically before memorizing and practicing it (which one has to do to actually apply it in practice). Your mileage might vary of course. For those interested in learning more music theory, I can’t recommend Michael New’s playlist highly enough.
The guitar is a beast. It might look innocent enough, but due to the two-dimensional nature of the guitar fretboard, there’s a tendency for things to get really hairy. In this first section, we’ll dissect the fretboard a bit and encode our knowledge of it into a Python class.
The image shows the notes on a 21-fret fretboard of a six string guitar tuned to standard tuning. That is, there are six strings (the horizontal lines) tuned to the notes E, A, D, G, B, and E from bottom to top (or from the thickest to thinnest strings). The pitch also increases from bottom to top; therefore the E at the top left (the orange circle) turns out to be two octaves higher than the E at the bottom left (the blueish-green circle).
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Each fret (the vertical lines) is where the string makes contact when you press down on it (at the spot where the circles are in the above image). This shortens the length of the string and raises the frequency of the note. Frets are carefully spaced out on the guitar such that each successive fret is a half-note increment from the previous one. You can see this in the note labels on each fret in the image above (using only sharps, for simplicity). You’ll also notice that although there are 21 frets, you can have 22 notes per string if you include the open string (
Finally, we can note some basic relationships of notes on the fretboard: for every string, frets zero (the left-most circles) and twelve (the right-most orange circles) are identical notes but an octave apart. This is always true due to the design of the guitar where the twelfth fret sits exactly at exactly halfway between the nut and the bridge.
Second, and this is where things get whacky with the guitar, each note in the middle is identical to the same color note on fret zero of the next successive string. So for example, the two blue A’s are
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In fact, the diagram below shows how many ways the same B can be played on a 21-fret guitar. These are not B’s one octave after another, but rather
, so to speak, to the next successive one, depends entirely on the specific tuning being used. Alternate tunings, where some strings have a higher or lower pitch, will have these relationship similarly altered. You can see this below in a diagram similar to the previous one, but for DADGAD tuning:
Let’s now encode some of this knowledge programmatically in Python. We start by labeling notes and intervals based on a user-specified key. We then label notes with extra pitch information to detect octaves or equivalent notes like the B’s above.
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If you’ve read my previous article, I showed how we can pick out notes from enharmonic equivalents based on a given key. We do this by using
, relative relationships between notes, to represent the notes we want in a key-agnostic way. So we already know how to map from intervals to notes in a key. Let’s also build a reverse map for this so we can map from notes to intervals.
Above, we use the make_intervals_major() function from my previous post to generate mappings from intervals (relative to the major scale) to notes in a key. Using this data, the function further generates a reverse mapping. For each note, it calculates what interval it can be.
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Are just the same note one octave apart). This is why the reverse mapping has an array for each note. Here is the reverse mapping for the key of C showing how notes map to intervals:
Generating the notes on the fretboard is surprisingly easy. Remember the chromatic scale that consists of all the notes, each spaced apart by a half-note? Each string is just a looping chromatic scale starting with the note the string is tuned to.
Function above takes as argument the starting note (which here would be the tuning of a particular string), and the number of values to generate. We call
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Each note above is represented by all the enharmonic equivalents possible. However, since we’re really only interested in notes of the specified key, we can filter these out with a simple function as shown below:
Information we derived in the previous section already pays off. We get the following filtered notes; only the ones relevant to the key of C.
Ok, great! We now have the ability to encode the notes of a fretboard in a user-specified key. We can retrieve the note for any fret by looking up the string (from
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We can do this trivially for arbitrary tunings as well. Here is an example for a D modal tuning where strings are tuned to DADGAD (instead of the EADGBE of standard tuning):
Similar to how we labeled notes for each fret, let’s label each fret with intervals as well. This is easy now that we generated our notes. We can look up the notes for a given fret, then use our
The notes above correspond to an E in standard tuning, which in the key of C, does indeed correspond to a (major) third (a
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The final piece of information we’re going to generate is pitch information. Remember that each note can be in different octaves and we want to be able to derive this information for the guitar. Of course, this is a common problem, so there’s already a system in place for this: Scientific Pitch Notation (SPN).
In SPN, each musical note name is combined with a number to uniquely identify the pitch and the octave it sits in. The system begins at C0 and at every successive C note, the octave number goes up. So for example a C3 is a C note in the third octave, and is one octave lower than a C4. If you look at an instrument like the piano, it’s quite easy to visualize.
The guitar typically sits in the colored range of the keyboard above. A guitar in standard tuning has the layout shown below. The colored notes match the colors of the keyboard.
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The rules are fairly easy to encode though. Basically, when we see a C note, we increase the octave by one. And while traversing a string from left to right, when we observe a note that is the same as that on the zero’th fret of the next string, we record its octave number. We then process the next string starting with that saved value.
. It specifies the pitch of the lowest note on the lowest string and defaults to 2. We then process the rest of the strings using that.
Everytime it encounters a note that is identical to that of the zero’th fret of the next string, it saves it in
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What we end up with is a 2D array showing the pitches of each note. Here is the output we get if we pretty-print the
Remember that the above output is a vertical flip of the image shown earlier: string 0 corresponds to the first line of output above, but represents the lowest string in the diagram (
Chords are a complex topic. In particular, full chords that use all six strings can get complex to search for. However, we can start with the most basic chords: triads.
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Triads consist of a stack of three notes that are each a third interval apart. This third interval can be either a major 3rd (
Notes in a key played together. For the key of C these would be the notes C, E, and G, respectively. Since we have three notes, we can derive triads in any key for groupings of three strings at a time:
In the above examples, we use the string number as the key and the interval name as the value. In the next two sections we will first find a particular interval on a given string, and once we can do that, use a recursive algorithm to derive all possible
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