4:5 Ratio of Contiguous 3rds

[Robin Hufford]

RicM
 The beat rates quoted below from the last paragraph below, which takes
data from,  I think, table III in White are incorrect, if they are meant
to be the rates for the contiguous thirds you indicate.   As I don't
have the book I can't determine whether the quote is in error, whether I
misunderstand you, or whether the table itself is incorrect.  I would be
interested in determining if the table in the book itself is accurate or
not.     Additionally, possibly from rounding errors due to the
laborious calculations of the time, or perhaps other reasons, even  the
quoted rates  are not in a constant ratio, and are thus necessarily
inaccurate.   For example 11.87/9.43 is 1.2587486; 14.95/11.87 is
1.2594777675; and 18.86/14.95 is 1.261538462 .
     A quick use of the calculator suggests that the quoted rates cannot
apply to F temperaments at pitches  A440; A435  or a C-temperament.  I
wonder what the answer is, perhaps another temperament?

     The beat rates of the continguous thirds to which you refer, using
a pitch of A-440,  are:
        6.9294214:    8.7305239:     10.999777:    and   13.858842.

     I have calculated a very useful constant, among several,  which is
the ratio of the beat rate of a tempered  major third  to the frequency
of the lower note of the third.   This constant is .00396842.  Using
this constant all one has to do to arrive at the beat rate of any given
major third is to simply multiply the frequency of the lower note of the
third by this constant.  For example:  A(220) * .00396842 is
8.730523912.    Emphasizing that this constant applies to any frequency
and is most useful therefore for this reason, one can see in the use of
it,  again, for example, at  A(221), its easy utility, as it readily
produces the beat rate for a tempered major third whose lower note is
this frequency and that is  8.770208112.   The theoretical beat rate for
any frequency which is the basis for a tempered interval  can be readily
obtained thereby.  These constants can be calculated for any interval.

     While I  know the first part of the quote is not yours, I
nevertheless  agree that the 4:5 comparison is of great utility yet I
still say we should be careful in making the categorical statement that
this is the ratio of continguous major thirds since it in fact is not,
and appropriately qualify or use a phrase similiar to that which I
indicated eariler, or otherwise indicate a consciousness that the
practical value is as approximately a 4/5 ratio.   Furthermore, the
ratio of the beat rates of contiguous thirds is a rational number and
not one of the irrationals, although it surely may seem irrational to
some.
Regards, Robin Hufford

Richard Moody wrote:

> > 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