Over here! (long)

[Clark]

Hi again,

Ed wrote:

> it was my understanding that by the time you divide an octave into 
> more than 31 notes, there will be enough pitches in proximity to 
> virtually anything that 12 ET provides.

That's not always a great reason for increasing the number of available
notes, but 72tET, etc. provide those pitches _exactly_. Examining 32tET
(evenly divisible by 4, so it includes the pitches of a 12tET diminished
chord), its smallest step is 37.5 cents: best 5/4 is 375 cents - 10
steps, best 6/5 is 300 cents - 8 steps, and best 3/2 is 712.5 cents - 19
steps.  The 5/4 improves upon 12tET being 11.3 cents flat instead of
13.7 cents sharp, but the 3/2 is over 5 times worse (and sharp). Still,
with unranked errors no greater that 12tET it might contain 5 (odd)
limit sonorities, but a 3:4:5 major chord won't have the best 6/5 minor
third between the 3/2 and 5/4, so it's not _consistent_ through 5-limit.
12tET is.

> It must be considered that the piano, tuned and used as we know it, 
> will not be a forever thing.  After several generations come of age 
> in an environment that places the piano along side so many other 
> instruments,  it may come to be seen as an anachronistic, expensive,
> limited instrument.

I think a century of sameness easily can sustain the instrument for the
foreseeable future. As Bill pointed out, extended keyboards span the
history of more normal ones (by which process five accidentals were
added to form the familiar one), but the best known examples are museum
pieces: condition and taste may exclude antique pianos from popular use,
but not any immediate harmonic limitation (pitch centers, maybe). Your
reports from the frontier underscore that the somewhat small tonal
palette of 12 available pitches isn't so limited at all.

> Ain't life grand?

At the moment, it's kind of square out here.


Clark