GoKawiilFretboard Memorisation with Modular Arithmetic
2024-03-01
Memorising (or “learning”) the fretboard refers to when an individual learns which musical notes correspond to what positions on the fretboard of their instrument, a skill that is useful for playing just about any instrument that has frets. In particular, “learning the fretboard” is a topic frequently discussed in guitar circles for its importance in improvisation & soloing. In general, the common approaches to learning the fretboard seem to consist of a combination of just committing it to memory and using some tricks to help with that memorisation; however, I have a number of problems with this approach:
It’s too much effort: I am far too lazy to memorise the note that corresponds to up to $24 \times 6 = 144$ positions on a 24-fret guitar (or $24 \times 5 = 120$ positions if you take account of the two $E$ strings being identical except for their pitch). It doesn’t translate to other tunings: it’s all well & good memorising the note that corresponds to each position in standard tuning, but your effort is rendered useless when you switch to a different tuning or a different instrument.
Aside: What is Modular Arithmetic?§
While I would encourage anyone not already familiar with modular arithmetic to read up on it in detail themselves, I’ll give a brief explanation here so that anyone can understand what I’m talking about (a good overview can be found in the book Discrete Mathematics – An Open Introduction by Oscar Levin here). Modular arithmetic refers to number systems in which the values don’t get higher and higher towards infinity, but rather wrap around and start again from 0 once a certain maximum number called the modulus is reached.
Perhaps the most famous example of modular arithmetic is the 24-hour clock. The 24-hour clock starts at 0, and goes up to 23 before starting at 0 again. There is no way to exceed 23 on a 24-hour clock. In this sense, the 24-hour clock is an arithmetic system modulo 24. If an event start at 23:00 and takes 2 hours, we don’t say that it finishes at “25:00”: instead, we say it finishes at 01:00 as $23 + 2 = 25 \equiv 1 \text{ mod } 24$ (25 is congruent to 1 modulo 24).
Modular arithmetic can also be thought of as dividing any given number by the modulus and getting the remainder. For example, $30 \text{ mod } 24 \equiv 6 $ because $30 \div 24 = 1 \text{ remainder } 6.$
Incidentally, many resources refer to the 12-hour clock as a classic example of a system of modular arithmetic, but I think that this is a poor example as the 12-hour clock starts at 1 and ends at 12, meaning that it’s not quite modulo 12 (which starts at 0 and ends at 11) and also not quite modulo 13 (which start at 0 and ends at 12).
My System§
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