I wonder how many people simply accept our keyboards as they appear to be and look at pitch as the obvious octave divided into 12 semitonal notes of which each will be an equal step along the way?
The origin and construction of our musicscape is different.
Why 12 notes?
It's really based upon the simplest complexity accessible to us which provides an adequate infinity of permutations for the purposes of making music.
If we take a note, that note is the result of a certain number of vibrations per second. The octave is perhaps the simplest interval, as it's twice the number, and then the fifth is three times the number.
We can derive notes from harmonics - as intervals the fourth, the third and minor third and second derive from the next notes on this spectrum of the series of harmonics obtained by multiplying the fundamental frequency by successive amounts.
But these harmonically derived notes, in combinations sound beautifully pure but don't always fit in exactly with harmonic series derived from any of those resultant notes when taken as a fundamental note.
I referred above to the "simplest complexity". The simplest relationship is the octave and the next simplest is the fifth, being a ratio of 1 1/2 times the fundamental frequency, 3/2. When we take 12 fifths, they coincide with 7 octaves and we get 12 notes as a result.
But the problem is that 7 octaves, being 2x2x2x2x2x2x2 = 128 and 12 fifths, being (3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2)x(3/2) = 129.7 . . . just recognisably being the same note but being that much slightly different.
Perhaps the natural thing to do is to distribute the error equally between all of the fifths, but in terms of the development of music, it doesn't really work. When we sing or play in brass bands, we like to produce pure intervals and certainly the origin of polyphony in the 12th century, I understand, was in the experience of the sweetness of singing in pure thirds. It was for this reason that Meantone developed giving 8 pure thirds giving unparalled sweetness to the music but making 4 keys, G# B C# and F# unplayable.
If one wants to start to play in all keys, one has to start juggling.
I start from a premise of allowing good thirds to be pure or beating slowly enough to enjoy the beats rather than losing count of them per second.
If we want to be able to play in keys denied to us by Meantone, we need to be a little pragmatic in the extent to which we allow modest shifts to our thirds and fifths . . .
For the purpose of forming a rationale, let's start with the three thirds centred on C - F-A C-E G-B as being substantially pure.
With 3rds and 5ths, this means that we are starting with defining C-F-A-E-B-G and then if we want Bb-D-F#.
Can anyone start to look at how the implications might be of shifting these intervals as against their equal temperament counterparts so as to achieve better sounding 3rds?