Equal Temperament, history of, judgement of

[Richard Moody]

"the likelihood that some kind of cumulative error would produce unintended
and unrecognized effects is very high.  This goes for Braide White's
instructions and for many others too.  WIthout the understanding of the 4:5
ratio of contiguous 3rds, ET is next to impossible to get exactly right
every time. "

Even a causual reading of Braid White, both in "Tuning and Allied Arts"
(1946) and "Theory and practice of Piano Construction" (1906) reveal more
than just an understanding the theoritical rates to acheive ET.  In "Theory
and Practice....  he states, "If we could secure an absolute
standardization of pitch { ie A440 and not A442}  it would be possible to
construct tables that would show the exact number of beats that ought to
occur between ann the equally tempered sounds within the whole compass."
(p126).   Now this table was published at least as early as the 5th edition
(1945) of "Tuning and Allied Arts".     He goes on to say, "In default of
such a method, {beat table} it is necessary to resort to a variety of tests
and to prove the correctness of the tempering of each interval...." (p
126).   He describes 3rds and 6ths as examples.
    Regarding "Tuning...."  5th edition ET can be tuned correctly from the
informantion he gives there.   I don't now how far back the editions carry
the beat tables, but  with beat tables there is no more accurate way to
tune ET unless you want to argue that a tuning machine is better.
    Braid White, I suppose, did not explain the 4:5 ratio of contiguous ET
3rds because it does not exist. The ratio is actually 1.25992105, Which is
sharp of the ratio of 5/4 by 13.686 cents.  (1.25992105/1.25)