hello,
I wrote:
> >consecutive fifths wich are about just intonation and divide the
> >divergence between 12 just intonated fifths and the octave between the
> >other fifths. As I remember, this specific system also includes a
> >correction for the thirds.
> Phil Taylor wrote:
> I stand corrected. However, if the system used involved distributing
> the accumulated error from twelve perfect fifths among all the notes,
> the result will surely be an equally-tempered scale, even though it's
> mathematical basis is different?
I think I was not quite clear in my writing. what I wanted to describe
is a intonation system based on 12 fifths which are from two (or more)
different sizes. All 12 together have the same size as in an equal
temperament but inside there are bigger and smaller fifths what makes it
possible to have perfect fifths (and thirds) in favorised keys and such
that are not so good at the far end of the circle of fifths. In case of
Werckmeister the circle of the fifths is "closed" not only because
thelve fifths equal the octave but also because all fifths are of
musically usable size. For example there are no fifths which are to big,
what would sound very bad (there are tonal systems wich include such
fifths, meaning that one cannot play in all keys, but in the range of
four or five keys these tuning systems sound very brilliant - these
systems are very usefull for music from the sixtenth and sevententh
century europe). So the kind of tunings people like Werckmeister worked
out made it possible to play music like the "welltempered piano" because
all the scales starting from all 12 (piano) keys are well sounding and
even better, each has an individual coulor because of their different
distance to the more brilliant center of tonalities arround these four
or five perfect fifths.
for music which does not use a 12 key (piano) keyboard there is no real
need to use those intonation compromises. The color (intonation) of
every interval, step or harmony can be choosen more freely and the A
bevore the modulation must not be the same as after . Typical example:
The A wich is the sixth of C is lower in many just intonation systems
than the A as secund of G if C and G are common to and a perfect fifth
in both keys. For computer generated music it should be possible to give
up the equal tempered intonation and to calculate the intonation
outgoing from the tonic center, maybe even when using a twelve key piano
keyboard for input.
Back to abc and traditional music this would for example mean that the
K: signature should influence the intonation.
Simon Wascher - Vienna, Austria
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