Figuring Out How Many Partial Capo Configurations There Are
Anyone out there who is a real mathematician feel free to double-check this...
To figure how many ways you can put a partial capo on a guitar, we don't t count 0 0 0 0 0 0 which is no strings clamped, but we do count clamping all 6 strings like a full capo.
With one capo it is pretty easy, just 2 to the s power minus 1, since each string can be either up or down. This means a 6-string guitar yields 2 to the 6th power, minus one= 64-1 or 63 configurations
So one universal partial capo can clamp on a 12 fret neck 63*12 = 756 configurations
It is a lot harder when you use multiple capos, since if more than one capo clamps the same string, only one of them does anything.
where f over c is "f things taken c at a time" which from combination math is this formula
So with 2 capos I get 66 * 728= 48,048 configurations
So with 3 capos I get = 900,900 configurations
12 capos I get = 4,826,808 configurations. You can also get this same number more easily by thinking that the 1st string has 13 different fretting choices on a 12 fret neck (you count the open string)-- so a 2-string instrument would have (13x13)-1 combinations, and thus a 6-string guitar would have (13x13x13x13x13x13)-1= 4,826,808 configurations. So a 14-fret guitar would yield (15x15x15x15x15x15)-1= 11,390,624 configurations
This means there are over 11 million ways to put partial capos on a guitar in every single tuning!