4:5 Ratio of Contiguous 3rds

[Richard Moody]

> the contiguous 3rds
> test is the most valuable and useful tool there could ever be
for aural
> tuning of ET.  It does not matter one iota what all those
irrational numbers
> are that say it is not exactly a ratio of 4:5.  The FACT is that
it a a
> relationship of "a *little* slower" to "a *little* faster".
> William Braide White's instructions did not provide this
diagnostic tool.


"Thus F#--A# beats faster than F--A, but slower than G--B; G#--C
beats faster than G--B but slower than A--C# and so on
throughout."

    William Braid White, _Piano Tuning and Allied Arts_.  Chapt IV
"The Art of Tuning in Equal Temperament". p 92.  (5th edition
1946).

See also Table III  Beats per second in Equal tempered Intervals.
p 68.

The first edition of PTAA was 1917.   It would be interesting to
see if the instructions and beat tables appeared then.    From the
above there might be some  confusion between successive 3rds and
contiguous 3rds.   Examples of contiguous 3rds are F--A, A--C#,
C#(Db)--F,  F--A.   The beat rates between these are  (from table
III) 9.43 --11.87 -- 14.95 -- 18.86.
White gives examples of two "7-8-9-10" series involving minor
3rds, major 6ths and  major 3rds in succession which pretty much
locks in ET.  (p. 90--93)   ---ricm