As discussed by Kyle Gann, there are subtly different tunings that have some other advantages. The fundamental problem with equal temperament, or really with any tuning system, is that the mathematical ratios underlying harmony can be combined in an infinite variety which would require densely packed notes infinitesimally spaced, if the mathematical ratios were to be represented precisely by the tuning system. For a tuning system to provide only a manageable set of notes to a performer, some or all of the intervals will have to be adjusted, or tempered.
Any tuning system that is not equal tempered will not be symmetrical, by that very inequality. A sequence of notes will sound somewhat different if it is shifted up or down the keyboard. For example, the fifth from C to G might be slightly different than the fifth from E to B. A musical piece will have a slightly different character when transposed to a different key.
This variety of character opens up an interesting musical possibility. Given modern keyboard technology, the pitch of any note on a keyboard can be shifted slightly in an instant. To change the character of a piece of music, one could leave the music in the same key, but just tweak the tuning of the instrument on the fly, during a performance.
This is analogous to an orchestral harp. A harp has only seven strings. One can use the pedals of the harp to sharpen or flatten strings, one pitch class at a time, so the harp can be played in the different keys. My proposal here is to take a keyboard that can has twelve pitch classes available, and to use a pedal, or perhaps hand operated controls, to shift one or more of those twelve pitch classes slightly sharper or flatter, to make available multiple non-equal temperaments during a performance.
Here is one concrete proposal for a set of operations to shift tunings. Studying Kyle Gann's presentation of Young's Well Temperament from 1799, one can see that the deviation of the pitches of the various notes from equal temperament follows a simple pattern when plotted along the sequence of fifths:
Typically one will want to shift the meanings of the notes, the harmonic structure, by a fifth. One could just shift the tuning pattern directly:
This shift leaves four pitch classes unaltered, sharpens four, and flattens four. One problem with this shift is that the four unaltered pitch classes are in two separate pairs. Musically, probably only two are significant at any time.
One can also provide two alternatives: shifting up
and shifting down
These two shifts also leave four pitch classes unaltered, but the four pitch classes are all together in the circle of fifths, and so would generally work better as a tonal center during a shift.
A complete system could provide these three alternative shifts when moving up a fifth, and the corresponding three when moving down a fifth, for a total of six single step possibilities. Each step could be repeated indefinitely. Shifting over repeatedly would bring one back to the same tuning after twelve shifts. Shifting up or down repeatedly would keep sharpening or flattening notes, so the whole tuning would be drifting up or down in pitch. Of course, a more typical usage would likely be to shift up when moving a fifth in one direction, then to shift down when moving a fifth in the other direction, which would return the keyboard to the starting tuning.