Alternative Temperaments

[Richard Brekne]

Hi list.

Paul Erlich wrote me this and asked me to pass it on to all interested.
It is in reply to Richard Moodys last post.

Grin... Caught in the middle again..

Richard Brekne
I.C.P.T.G.   N.P.T.F.
Bergen, Norway

Please post the following reply:

>2/7 comma (Zarlano) doesn't tell you
>much of anything.  It gives a Third of LESS than pure, but what is the
objective??

The objective is to optimize the tuning of both the major third and the
minor third. In Zarlino's tuning both are only 1/7 comma (about 3 cents)
off
just.

>Today computation of
>this and beat rates is rather straight forward with a calculator, but
from
the 14th
>century to the 19th the theory of tuning and the practice of tuning
probably had
>little in common.

I would say they had much in common, even if the correspondence was not
nearly as exact as today. Jorgenson gives some plausible schemes for
tuning
the historical temperaments according to principles that were known at
the
time. Many, using "equal-beating" schemes, yield results different from
the
theoretical ideal, but are close enough that one can meaningfully
distinguish between different tuning systems used in different periods
of
time, in both theory and practice:

900-1450 Pythagorean
1450-1500 Schismatic
1500-1700 Meantone, around 1/4-comma
1700-1800 1/6-comma meantone yielding to circulating well-temperaments
1800-present mildly unequal well-temperaments yielding to equal
temperament

These are not sharp dividing lines -- organs in England were tuned to
1/4-comma meantone until 1850, and one still finds Pythagorean tuning in

many cultures on the periphery of the West.

>If the octave has 1200 cents, then the 12 note octave should have 100
cents
per
>note.  The 19 note octave (equally divided) then should have 1200/19,
or
63.16 cents
>per note. The 22 equal note octave would be 54.5454 cents.    But
sometimes
the "12
>note octave" is called a thirteen note octave with 12 intervals.   Do
we
know how
>the 19 or 22 note octave is set up?

1200/19 and 1200/22 are correct.