I appreciate that answer Mdwz.
Here is something I worked out a few years ago regarding Equal Temperament tuning, post Bach.
He decided that keyboards were so 'open' to multi key uses that there was no way that a modal single key tuning would suffice any longer.
So, he ripped all the beautiful harmonical majesty out of music to create THIS:
THE FACTOR 1.0594630943592
(the semitone gap between 'equal temperament' musical notes)
1.0 ^ 0
1.0594630943592 ^ 1 << * This is the basic factor
1.12246204831 ^ 2
1.189207115 ^ 3
1.25992104989 ^ 4
1.33483985417 ^ 5
1.41421356237 ^ 6
1.49830707688 ^ 7
1.58740105197 ^ 8
1.68179283051 ^ 9
1.78179743628 ^10
1.88774862536 ^11
2.0 ^12
( inverse of factor = 0.943874312682 )
I regard this mathematical adulteration as a far more serious assault on harmonical beauty that any other factor. At least until I can figure out why concert pitch has been dragged up from somewhere around 432A . Even then, I may still regard it as the most serious adulteration.
edit:
Modal tuning does NOT stick to these numbers. With modal tuning, the 3rd, or whatever the favor, are exact fractions of the root. This is only possible if the tuning is restricted to one single key.