I wonder why they called notes by names like A..G. Wouldn't it be better to give them numbers? Could this be troublesome or better?
E.g., I have numbered notes on my guitar neck in standard tuning as below, with 0 being the very lowest note (6th string open) and up from that. E.g., open 5th string gets number 5, same as the note on the 6th string, 5th fret.
24|25|26|27|28|29|... 19|20|21|22|23|24|... 15|16|17|18|19|20|... 10|11|12|13|14|15|... 5| 6| 7| 8| 9|10|... 0| 1| 2| 3| 4| 5|...
Now a chord is a set of numbers and by subtracting them I can immediately see note intervals in terms of half-steps, because a difference of 1 equates to 1 half-step. E.g., the C-major chord would be [8,12,15,20,24]. Between the lowest C (8) and following E (12) there are 12-8=4 semitones, that is 2 tones.
Moreover I can easily found barre positions for open chords, because it is just a matter of finding out the same chord numbers. E.g., I immediately see C-major can be also played by fully barring fret 5 with a capo and playing note 8 on the 6th string, note 12 on the 5th and note 32 on the 1st string (this is a C with a G-shape). So [3,7,27] is G and by adding 5 to each number of this vector we get [8,12,32] which is a high-pitched C. Indeed C and G are 5 frets apart, because 3+5=8.
Inversions of triads are also easy to work out. Eg in [8,12,15] (C-E-G) I do not play the 8, add 12 to it and stick the result to the end. I get [12,15,20], which is E-G-C.
If for convenience we do not want to represent each note as a frequency number and work only with a finite number of frequencies in a geometric progression (with the common ration being the 12th square root of 2), it makes sense to me to locate each progression element with its sequence number, which is a natural number.
Maybe I am not the first to have thought of an integer notation for music. If this is so, why didn't it take?