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You Can Never Escape Math Class: The Fretboard as a Grid

Another excerpt from “How Do I Bass?”. See also this post about how the fretboard is not a grid. In the actual book this chapter will come before that one.

Before I understood anything about music theory, I learned the bass by finding visual patterns. And even now that I understand a fair amount, I still find myself resorting to and using these patterns, because they’re useful. Unlike a guitar, which has a major 3rd between the G and B strings, all the intervals on the bass are perfect fourths. So if you find a sequence of pitches you like, which could represent a scale, an arpeggio, or a melody, you can transpose it easily by moving the starting note and reproducing the same sequence.

graph - blank

This is a grid with x and y axes. You may remember it from school. For right now we’ll be concentrating on the upper right portion, the positive integers, so we can relate it to the fretboard. Of course, a certain amount of abstraction is necessary here, because frets are never evenly spaced. They look something like this:

even fretboard

Frets are spaced closer together as you go up the string to compensate for physical properties of the string. But right now we just care about one property: pitch. So for the purposes of today’s discussion, we’ll be treating the fretboard like this:

fret abstraction with axes

Just like a point on a graph can be described by its coordinates, a pitch on the fretboard can be described by fret and string position. So (A,0) is the open A string. (E,3) is the third fret of the E string, which is a G. Of course, movement along the Y axis has a greater effect on pitch than movement along the X axis. If I go one fret up on the same string, I am only traveling up a semitone (half step), the smallest unit in equal temperament. If I go one string up on the same fret, I am traveling five semitones, or a perfect fourth. So a really proportional graph would look like the following diagram, in which I have spaced out the lines on the Y axis to reflect the amount of musical distance covered. Going up or down one string is the same as going up or down five frets. I like to think of a 5×5 square, in which each unit of 1 is a semitone. The upper left and lower right corners of the square are the same pitch:

the square of equivalence

Here’s an example of this thinking in action, showing every position of G2 on the fretboard:

fretboard with G2

The red square represents the position of your finger. Fret dots are for your reference. Each vertical line is a fret, and the grey area represents the nut (so a red square in the grey area is an open string). You can see that the placement of these identical pitches suggests a nice, neat diagonal line. Of course, because this diagram represents the actual physical object of the bass neck, there are no points you can land on between the strings. Or rather, you can put your finger anywhere, I guess, but you’re not going to be happy with the result.

this is not actually possible


What I like about visualizing the fretboard in this way is it makes you conscious of the shapes your hands are making as you play, and how they can be moved. Here are three ways (far from the only three, but you get the idea) to play a 1-octave G major scale starting on G1:

G Major animation

You’ll see that I’ve deliberately avoided open strings here. There’s a good reason: fingerings relying on open strings are only possible in a few places on the neck, whereas fingerings in which you fret every note are movable to any place that will accommodate the entire shape. Of course, once you’ve settled on a particular set of coordinates for a given part, you can start to incorporate them again. And it can go the other way too: you can build a line around open strings that wouldn’t otherwise be possible.

Unlike a lot of other instruments, transposing a particular pattern is as simple as moving it around, preserving the visual relationship between the notes. So here’s our major scale moving around a bit. As long as the shape stays the same, the sound stays the same.

major scale in motion

If one side of your shape extends to the left of the nut, to the right of your highest fret, above your highest string, or below your lowest string, you need to adjust it accordingly – either by moving the entire shape so it fits, or by moving only the notes that are out of bounds. So if I’ve got a pattern in the key of Eb which I’m playing starting on Eb2 (the sixth fret on the A string, and I need to transpose it to Ab, I have the following options.

  1. I can move the whole thing up on the A string

  2. I can adjust it to make it fit starting on the D string

  3. Or I can move the whole thing an octave down and start it on Ab1 on the E string

Ab Major

Which of these options makes the most sense depends on the most efficient fingering, the other parts that precede and follow, and of course the timbre of each potential placement on the neck. And it goes without saying that these three options are just what I consider to be the most efficient and sensible of an almost limitless number.

Depending on how your mind works, this way of thinking is either profound or profoundly useless. But stay tuned, we’ll be building on this concept in subsequent posts, and talking about how it relates to some cool musical concepts. For right now, the takeaway is that whatever fingerings you know for a scale, an arpeggio, or an entire bass part are far from the only fingerings possible, and thinking about the grid is a good way to arrive at new and possibly better ones.

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