Remembering The Notes Of On A Fretboard Using Math

[NC11-AM] In math, there is something call congruence modulo n. Congruence is used for sets of numbers that are in some way, related to each other. For example, a clock is the best example of congruence: A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12 hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00.

Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic modulo 12. 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 12 ß 0 mod 12. In 24 hour time, one uses 0:00 for midnight. 24-hour time uses arithmetic modulo 24.

Hope, that didn't bore you too much but it's for a reason. Now, like a clock, the "notes" on the fretboard repeat after 12 notes. That is the first fret of the 6th string is F so, if you were to go up 12 frets, you'd be at the 13th fret and at F again but an octave higher. So, when reading tabs, this might be helpful.(Since there are 12 "notes" in the chromatic scale.)

|---------------|
|---------------|
|---15----------|
|------14-12----|
|-12------------|
|---------------|

Say you wanted to find out what these notes were but you really don't know the fretboard well. Since 12 ß 0 mod 12, that means that 12 is simply the name of the note that is played open on that string,(e.g. 12 on the a string is A, 13 on the a string is A#/Bb, 14 on the a string is B, 15 on the a string is C, etc.) So our first note played in this lick is A. The next note is 15 on the g string. This means that 15 mod 12 = 3 thus, 15 is A#/Bb since this is just three frets higher than the open G, just an octave higher. This may explain it better:

0  1  2  3  4  5  6  7  8  9  10 11 (First octave)
12 13 14 15 16 17 18 19 20 21 21 22 (Second octave)

The number underneath would be the note equivalent, just an octave higher. For example, the E string.

0  1  2  3  4  5  6  7  8  9  10 11 (First octave)
E  F  F# G  G# A  A# B  C  C# D  D# (Chromatic scale)
12 13 14 15 16 17 18 19 20 21 21 22 (Second octave)

so the notes of the riff would be:

|---------------|
|---------------|
|---15----------|
|------14-12----|                
|-12------------|
|---------------|
 
|---------------|
|---------------|
|---A#----------|
|------E-D------|
|-A-------------|
|---------------|