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
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