C |
o |
D |
o |
E |
F |
o |
G |
o |
A |
o |
B |
C |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0* |
1* |
2* |
3* |
4* |
5* |
6* |
7* |
8* |
9* |
10* |
11* |
0* |

An interval between two numbers is what you have to add to the first to get to the second, in other words:

I(X, Y) = Y - X

To compute that in the most efficient manner we can reduce the amount of distance we move by using the small representation:

I(X, Y) = Y - s(X)

To define an anchor interval you need an anchor note, and an other note, the anchor interval between them is simply the interval between the anchor note and the other note.

Thus given an anchor note A* and an other note N*, to efficiently determine the interval between them we compute:

I(A*, N*) = N - s(A)

The relative interval collection of a major scale is:

0 2 4 5 7 9 11

and of the minor:

0 2 3 5 7 8 10

The open strings of the guitar are:

4* 9* 2* 7* 0* 5*

Note that the next note is computed by taking the one on it's left and adding 5, a fourth is an interval of 5 steps so this is why it's called fourths tuning.

Given that you are on string S*, and you are looking for the note N*, the fret at which it can be found is the I(S*, N*) which is equal to N - S. This is correct because moving up frets increases your note by one, thus the fret number is the number which you are adding to S*.

Play scales on the guitar combines all of the previous skills we've built up, by remembering the tuning of the instrument, we know that by staying on the same fret but moving horizontally over by one string (towards thinner strings) will increase our note by 5, so as we move through the relative interval collection we can also move horizontally across the fretboard instead of just staying on one string.