Temperament Question

[Richard Moody]

> Hello list
>
> Is there any location where I can find the exact definition of
> all these temperaments, including the more recent ones ?
> Pietro Aaron temperament, ... and others ?
> Thank you
>

For definitions the best place I know of is New Groves .  In particular the
article "Temperament" by Mark Lindley.  IMHO it is as standard and
definitive as you can get.
    For tuners a temperament is defined by the cents off-set so that it can
be compared to other temperaments and also tuned electronicaly.    These off
sets are of use to aural tuners with spread sheets that convert then to
beats.  But in in beats or cents, temperament might not mean much to the lay
person or musican.  For one of the first to "define" temperaments in terms
of cents, see J.M. Barbour  _Tuning and Temperament, a Historical Suvey_
1951.
    Traditonally tempermants were conceived by theorists and defined in
terms of "commas".   Again unless you know about commas, terms like 1/4
comma Meantone  might not mean much.   But this is all explained in
LIndley's article.
    For Pietro Aaron, most writers ascribe 1/4 comma Meantone to him, not as
the inventor but as the earliest description of it.   Lindley though
suggests there may be other interpretations.  Aaron's writings are now
published in English and  I am preparing a draft using the complete text
(all of a page and half) of his instructions, with interpretations and
analysis sentence by sentence from a piano tuner's point of view.    For
anyone interested I will gladly email it.

    If indeed Pietro Aaron was describing 1/4 tone Meantone, then briefly
this consists of  a temperament of pure thirds gotten by flattening each 5th
in the Circle of Fifths by 1/4 comma.   Since four fifths in that circle
give a 3rd (from the starting note) that is sharp from pure, this sharpness
is called a comma which is the difference between a pure 3rd and the sharp
or Pythagorean 3rd.  Tuning by pure 5ths is called Pythagorean.  Since it
takes four 5ths to get this sharp (wide) 3rd then by logic flattening each
5th by 1/4 of the comma should render a pure 3rd.  BUT, not every 3rd can be
pure in an octave so having two pure 3rds leaves the last one very sharp, so
disagreable as to be called a "wolf".
To complete the description of Meantone temperament(s) would take 2 to 20
more pages depending how far you want to go.  Then one should play and or
listen.    If you had a midi tunable  synth you could listen to .mid files
in any temp you wanted.  Just enter the off-sets there you are. ---ric