Two simple things

[ted]

I have never understood why books discussing equal temperament spend hundreds of pages of meandering historical reasoning on something which seems to me very simple. Cannot the whole thing be summarized as follows?

We desire to find divisions of the octave which contain as many close approximations of the type 2 to the power of (a/b) = c/d, where a,b,c,d, are integers and c and d are integers as small as we can make them.

In fact, a moment's thought tells us that it suffices to find a close approximation for 3/2, the next simplest ratio after ½. A few minutes with a hand calculator soon reveals that 12, 19, 24 and (bit of a surprise) 29 are the lowest divisions of the octave containing a very close approximation to 3/2.

12 is the obvious, indeed probably the only, choice for a practical manual instrument.
24 gives the quarter tone scale. Why is any more analysis needed ?

As an exercise I experimented with 19 and 29 using an Amiga and found that the first embeds the diatonic scale as the partition 3,3,2,3,3,3,2 and the second embeds it as
5,5,2,5,5,5,2. I didn't try what 19 sounded like, but the division of 29 sounds great. It also yields almost symmetric scales of 4,7,5 and 6 (hence 2 and 3 ) notes. I recorded a simple sequence modulating around the circle of 29 keys and tried it on musicians. It surprised me that even professional musicians couldn't tell they were actually hearing 29 keys instead of the usual 12 !

The second topic which seems to me to be burdened with needless complexity is analysing how many chord types there are ignoring pitch and position (i.e. major, minor, diminished etc). Seeing the chromatic scale is a cyclic group of order 12 and a chord type is therefore just a partition of 12, a quick application of the Polya Burnside theorem (or by just counting) tells us that including a silence there are 352 chord types (or scales, regarding a scale as a chord) playable on the piano. Further, a second application of the theorem gives an obvious way of looking at the chords within each partition. For example the partition 4,3,3,2 has three permutations corresponding to the seventh chord (4,3,3,2), the sixth chord (4,3,2,3) and the minor sixth chord (4,2,3,3)

The whole thing has a pleasing unity to it which I used to imagine would appeal to serialists. I couldn't find it in any books so I published it myself many years ago in the New Zealand Mathematics Magazine (Vol 16, No 2) because no musical publication I could find would accept it.

[rachfan]

Despite the implication for precision of the term "equal temperament", if you discount  eletronic tuning machines, it is a theoretical concept residing mainly in the ear of the beholder (and tuner), despite the fact that it can probably be reduced to a simple and rigorous mathematical representation as you correctly assert.  

Moreover, equal temperament in practice is uneven temperament.  That is, if one is attempting to equalize or temper the different tones within the octave to imbue each with an equal degree of purity of interval with any resulting imperfection being divided among the whole of the octave, it begs subjectivity, not objectivity.  You'll find subtle differences in the way different tuners tune to the ideal of 12 semitones.  The ideal pitches in their minds' ears  for equalizing intervals is often a different ideal than that of another tuner, nevermind the theorem.  And as they each implement and "test" their tunings throughout the octave with unisons, thirds and  fifths, it remains ever subjective, never objective.  In fact, the way a tuner sets the scale is a primary reason why one would want to continue with that tuner or favor another.  Thus, I believe that while the most simple and straightforward mathematical representation might indeed be elegant and accurate, it will remain only a construct.  A particular tuner, biased toward apportioning much of the imperfection to the fourth rather than dividing it equally across the octave, proceeds in reality to do exactly that--oblivious in the moment to the proof of the theorem.