Re: LMY (was Re: [tuning] Reply to William Annis)

Pagano and Beardsley are missing Erlich's point that a duodene or
periodicity block of 12 tones fits a matrix of threes and fives optimally.
Young was able to put the WTP tuning on a 12-tone keyboard by giving up
consistant mappings of intervals (7/4s are mapped onto major sixths and
major sevenths) as well as scalar order -- going up a chromatic scale on the
WTP yields a sequence with a few descending intervals thrown in.

That said, Beardsley is exactly right that the tuning of the WTP was neither
intended nor suited for 5-limit triadic harmonies, but for a entirely
different project.

DJW

I wrote,

>>about the duodene:
>
>> (a) it seems particularly geared toward the 12-tone periodicity block
that
>> results on the 3-5 plane, so some other number of notes other than 12
might
>> be more appropriate when using other constructing intervals (I'd be happy
to
>> pursue this if you're interested); and
>> (b) one should remember that it fails to allow even some of the simplest
of
>> Western triadic music to be performed.

>It seems like you're refering to the tuning
>of The Well Tuned Piano.

Nope, I'm referring to the 12 notes on the 3-5 plane called the "duodene" by
Ellis (Helmholtz's English translator), also known as the just system of De
Caus. The Well Tuned Piano uses pitches from the 3-7 plane.