Bach and E.T. or W.T.

[Richard Moody]

>   JIM  writes:
> <<  at least be an equal opportunity 'temperament basher' and
> continue this thought by saying that "unequal" temperaments are also "out
of tune" but unequally so! :-) >>
>
>  Greetings,
>       This seems to be a slight oversight that I would like to respond to.
> It is difficult to say that a temperament is "out of tune",  without first
> defining what "in tune" actually means.  .......Or does "in tune"  refer
to realized >super particular ratios and Just intonation, such as one might
find in a >meantone? Ed Foote


I think we can agree that "In tune" (as opposted to "tempered") means no
beats in the interval being tuned.  That is usually called a Just interval
and whose coincident  partials are at the same
frequency, ie no beats.  (I think this term should be in caps so it won't be
confused with 'just' as in "just another interval".) Such an interval is
also called pure, meaning 'absence of beats'.
A superparticular ratio is one whose numerator exceeds the demoninator by
one. Thus 3/2, 4/3 and 5/4 are superparticulars.  (the 5th, the 4th, the
3rd)  There is a belief that the simplest superpaticular ratio of an
interval is what defines 'Just'.  Others include the simplest ratio as in
the 6th, (5/3).  The act or art of tempering is tuning intervals slightly
away from Just.  These tempered intervals of course not "in tune" as the
orchestra would play them but are considered in tune if they meet the
criteria of the temperament.  Thus a pure 5th would be considered a mistake
in ET, however in Pythagorean or many of the Wells it is acceptible if in
the right place.
    In the scheme of temperaments some intervals "come out" Just.  For
example in Pythagorean, the diminished 4ths sound as a pure 3rd.  The
augmented 2nds sound as a pure minor 3rd. But this is not by design. Since
Pythagorean is only a series of pure fifths, it might be debated that it is
a temperament at all.
    In Meantone theory,  the objective is for 3rds to come out pure, however
in practice it is practical to tune them pure, rathar than tuning a series
of 4 flat 5ths hoping for a Just 3rd. ---ric