Fretboard Memorisation with Modular Arithmetic | Ó hAoḋa<br>Fretboard Memorisation with Modular Arithmetic<br>2024-03-01<br>Table of Contents Introduction§<br>Memorising (or “learning”) the fretboard refers to when an individual learns which musical notes correspond to what<br>positions on the fretboard of their instrument, a skill that is useful for playing just about any instrument that<br>has frets.<br>In particular, “learning the fretboard” is a topic frequently discussed in guitar circles for its importance in<br>improvisation & soloing.<br>In general, the common approaches to learning the fretboard seem to consist of a combination of just committing it to<br>memory and using some tricks to help with that memorisation; however, I have a number of problems with this approach:<br>It’s too much effort: I am far too lazy to memorise the note that corresponds to up to $24 \times 6 = 144$<br>positions on a 24-fret guitar (or $24 \times 5 = 120$ positions if you take account of the two $E$ strings<br>being identical except for their pitch).<br>It doesn’t translate to other tunings: it’s all well & good memorising the note that corresponds to each<br>position in standard tuning, but your effort is rendered useless when you switch to a different tuning or a<br>different instrument.<br>Aside: What is Modular Arithmetic?§<br>While I would encourage anyone not already familiar with modular arithmetic to read up on it in detail themselves, I’ll<br>give a brief explanation here so that anyone can understand what I’m talking about (a good overview can be found in the<br>book Discrete Mathematics – An Open Introduction by Oscar Levin<br>here).<br>Modular arithmetic refers to number systems in which the values don’t get higher and higher towards infinity, but<br>rather wrap around and start again from 0 once a certain maximum number called the modulus is reached.<br>Perhaps the most famous example of modular arithmetic is the 24-hour clock.<br>The 24-hour clock starts at 0, and goes up to 23 before starting at 0 again.<br>There is no way to exceed 23 on a 24-hour clock.<br>In this sense, the 24-hour clock is an arithmetic system modulo 24 .<br>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<br>01:00 as $23 + 2 = 25 \equiv 1 \text{ mod } 24$ (25 is congruent to 1 modulo 24).<br>Modular arithmetic can also be thought of as dividing any given number by the modulus and getting the remainder.<br>For example, $30 \text{ mod } 24 \equiv 6 $ because $30 \div 24 = 0 \text{ remainder } 6.$<br>Incidentally, many resources refer to the 12-hour clock as a classic example of a system of modular arithmetic, but I think<br>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<br>(which starts at 0 and ends at 11) and also not quite modulo 13 (which start at 0 and ends at 12).<br>My System§<br>I realised when I was teaching myself guitar that the chromatic scale can be thought of as a system of modular<br>arithmetic modulo 12: there are 12 notes ($A$ through $G^\#$, including the sharps/flats).<br>If you assign each note a number starting with $A$ at 0 and ending with $G^\#$ at 11, you get a system of<br>modular arithmetic that corresponds perfectly to the frets on the $A$-string.<br>If you don’t fret the string and pluck it, you get $A$, if you fret the string at the first fret and play it, you<br>get $A^\#$ (or $B^\flat$), et cetera.<br>If you fret the string at the 12th fret, you get $A$ again, and if you fret the string at the<br>13th fret, you get $A^\#$ again, in accordance with $12 \text{ mod } 12 \equiv 0$ and $13 \text{<br>mod } 12 \equiv 1.$<br>Hence, we already have a system for memorising all the notes on the $A$-string: it’s just arithmetic modulo 12.<br>All we need to do is memorise a value for each note in the chromatic scale.<br>Then, to find out what note a certain fret corresponds to, we can just find its value modulo 12, and to find a certain<br>note on the string, we can just look at the frets at multiples of the value of the note.<br>$A$$A^\#$ / $B^\flat$$B$$C$$C^\#$ / $D^\flat$$D$$D^\#$ / $E^\flat$$E$$F$$F^\#$ / $G^\flat$$G$$G^\#$ / $A^\flat$$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$11$This same system can actually be generalised to all the other strings on the guitar with little adaptation.<br>The only difference is that we need to remember at what number on the scale each string starts and add this to the<br>number of each fret on that string to get the integer value corresponding to the note at that fret.<br>Take for example one of the $E$ strings: this string is tuned to the note $E$ (which has value 7).<br>To find out what note is at the 5th fret of the $E$ string, we use the same approach as with the $A$<br>string, but add the value that the string starts at to the fret number before getting its value modulo 12:<br>$7 + 5 = 12 \text{ modulo } 12 \equiv 0$, indicating that the note at the 5th fret of the $E$ string<br>is an $A.$<br>$$<br>v \equiv r + f \text{...