It is probably fair to say that the single greatest advantage of the major thirds tuning is that intervals between pairs of notes are instantly recongnizable by the patterns they form on the fretboard. The diagrams below should be interpreted in the following way.
Suppose you have a six-string guitar tuned in major thirds, from low E to high C, and we draw the fretboard from fret 0 (open strings) to 8, both included. There is a C note in fourth fret on the 2nd and 5th string. Those two notes, as well as other C notes, are indicated by hollow dots. The purpose of the diagrams is to show how notes separated by intervals from zero to twelve semitones are connected across strings, using the two C notes in fret 4 as references.
Since adjacent strings are four semitones apart, you get the same note if you move one string up and four frets down, or you move one string down and four frets up. Consequently, you can connect identical notes by a line sloping steeply from high frets and low strings towards low frets and high strings as illustrated in the diagram at the top left hand corner of the table. If you move three strings up or down while staying in the same fret, you also get the same note but now it is one octave away (because 3 times 4 is 12).
| Identical notes | 1 semitone | 2 semitones | 3 semitones | 4 semitones | 5 semitones | 6 semitones | ||||||
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| Octaves | 11 semitones | 10 semitones | 9 semitones | 8 semitones | 7 semitones |
An advanced interpretation using the note indices can help you to understand the properties of the straight lines. When a note is moved up or down a certain number of semitones, the note index always changes by a fixed amount. For example, an interval of 5 semitones changes the note index by 1, just as an interval of 1 semitone or 9 semitones (because 1, 5, and 9 modulo 4 is 1). Similarly, for intervals of 2, 6, and 10 semitones the note index always changes by 2 (because 2, 6, and 10 modulo 4 is 2), and for intervals of 3, 7, and 11 semitones the note index always changes by 3 (because 3, 7, and 11 modulo 4 is 3). The procedure might sound complicated but once you get used to it is very powerful. Say you want to find out where the fifth is relative to the root note of a chord. The fifth is seven semitones above the chord's root note, and since the index of seven is three you know the fifth is found in a fret three higher, or one lower (-1 is equivalent to 3 modulo 4). You probably don't want to do that sort of analysis on the fly but there are only a few combinations you need to learn. If you pay attention to how the frets numbers are related to the intervals while you are playing you will most likely come up with your own system anyway.