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Owen Jorgensen, _Tuning_, "Section 84: The Distinction between Equal-Beating
Temperament and Theoretically Correct Temperament Clarified in 1807" p310
<P>"...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..."
<P>So I stand corrected on my interpretation of the glossary entry.
<P>Ric, I don't follow
<BLOCKQUOTE TYPE=CITE>
<PRE>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?</PRE>
</BLOCKQUOTE>
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.
<P>C 1/1
<BR>G 643/430 (=[(3/2)^(81/80)]*1/1), quarter comma fifth
<BR>D 341/305 (=643/430*643/430*1/2)
<BR>A 1075/643 (=341/305*643/430)
<P>and here are two E's
<BR>E 242/193 (=1075/643*3/2*1/2) perfect fifth from A
<BR>E 5/4 (=1075/643*643/430*1/2) quarter comma fifth from A
<P>In fact the latter is the _raison d'etre_ for 1/4 comma syntonic meantone.
<P>Try a this table for 12tET fifths and then calculate beat ratios for
A and E.
<P>(I'll try not to post this one four times)
<P>Clark</HTML>