Equal Beating...(Jorgensen quote, table - long)

[Clark]

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Owen Jorgensen, _Tuning_, "Section 84: The Distinction between Equal-Beating Temperament and
Theoretically Correct Temperament Clarified in 1807" p310

"...musicians, tuners, and even theoreticians of past centuries were not aware that when two
adjacent intervals of a kind seemed to have the same color-quality or their beat frequencies
were equal that the intervals were not the same size having the same ratio. Inequalizing a D
between a low G and upper A a ninth higher, the intuitive, instinctive and easiest manner of
tempering was to average out D until the qualities of the two fifths GD and DA seemed
identical. Equal qualities also included equal beat frequencies. ..The tuners of the past did
not know that in the latter example the fifth GD was narrower than DA when both fifths
sounded the same. This method is now known as the equal-beating technique of tempering...Some
organ tuners were the exception. They applied the progressive beat frequencies published by
Robert Smith in 1749..."

So I stand corrected on my interpretation of the glossary entry.

Ric, I don't follow

> Yet in pure fifths, a 3rd
> and a 6th from the same root have the same rate. And these are gotten by a constant
> ratio 1.5 (pure fifth)  Now why would 1.498307 (ET 5th) change that symmetry? In
> Werckmiester III with three 1/4 comma fifths and one pure fifth that gives a third
> that equals its sixth. Wonder why?
>
Personally, I don't find it at all handy to use decimal for intervals. The following table is
of Werkmeister chain as you describe above, with octave reduction.

C 1/1
G 643/430 (=[(3/2)^(81/80)]*1/1), quarter comma fifth
D 341/305 (=643/430*643/430*1/2)
A 1075/643 (=341/305*643/430)

and here are two E's
E 242/193 (=1075/643*3/2*1/2) perfect fifth from A
E 5/4 (=1075/643*643/430*1/2) quarter comma fifth from A

In fact the latter is the _raison d'etre_ for 1/4 comma syntonic meantone.

Try a this table for 12tET fifths and then calculate beat ratios for A and E.

(I'll try not to post this one four times)

Clark

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