equal temperament

A fundamental property of an equal tempered scale is symmetry. The intervals
available at any note are the same as those from any other note in the
scale. This property really defines equal temperament.

This property impacts the design of multi-stringed fretted instruments. As
long as each string is tuned to a note in the scale, the frets will need to
be in essentially the same place for each of the strings. (Ignoring effects
due to string diameter etc.)

But the symmetry property has far vaster impact than instrument design. I
think I picked this up from Danielou. Mainstream European music is built on
modulation, key changes. The tonal center moves around. So the function of a
note is always shifting, it depends on the current key context. Since every
note can play every role as the context shifts, then the uniform availability
of intervals at each note is required. So modulation pushes one to equal
temperament.

If one gives up modulation, if each note has a constant and specific
function, then the intervals available at that note can reflect that specific
function, hence equal temperament becomes actually inappropriate. Danielou
argues that by holding the context fixed and building it up over the course
of a whole piece of music, then subtle differences of meaning can be given to
each note - a single note can express rich meaning - whereas the constant
context shifting of modulated music implies that the meaning of a note is
ambiguous. One needs several notes to express a clear meaning.

Of course, that very ambiguity can be used artfully, creating puns etc. See
the analysis in Blackwood's THE STRUCTURE OF REGULAR DIATONIC TUNINGS.

Jim

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