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<font size="-1"><font face="Georgia">The line of thinking is somewhat
suspect, indeed. <br>
<br>
It is true that an augmented 6th in QCM meantone is a close
approximation of the interval 7:4 ratio, being only 3 cents wider. The
QCM 3d is by definition pure, equal to 5:4, and the QCM 5th is a
reasonable approximation of a pure fifth (narrowed by 5,5 cents). So,
yes, a German augmented 6th in QCM is an approximation of overtones 4
through 7. It is however difficult to evaluate the extent to which the
better "natural" 7th</font></font><font size="-1"><font face="Georgia">
compensates for</font></font><font size="-1"><font face="Georgia"> the
tempering of the 5th</font></font><font size="-1"><font face="Georgia">.
Some tunings in just intonation probably produce better results – eg
Marpurg's monochord n° 4 (Murray Barbour, <i>Temperament and Tuning</i>,
p. 100), with an augmented 6th Eb-C# only 12 cents wider than 7:4 and
the other intervals (Eb-G and Eb-Bb) pure. I all cases, however, the
number of playable augmented 6ths was limited to two in the octave for
an instrument of 12 pitch classes.<br>
<br>
In any case, nobody could have associated these intervals with the
overtones 4 through 7 before Sauveur's early 18th-century experiments
at the earliest. The first mention of the possibility of a "consonant
seventh" is, unless I am mistaken, Euler's <i>Conjecture sur la raison
de quelques dissonnances</i>, 1766. There seems little reason to
stumble on the interval 7:4 as particularly consonant before the modern
fascination for the overtone series.<br>
<br>
The perception of the sound quality of a "natural" 7th as compared to a
diatonic dominant 7th supposes a very vertical, harmonic way of hearing
sevenths. Singers and instruments of variable pitches may at times have
sung or played rather flat sevenths, but for reasons more probably
deriving from contrapuntal constraints than from attempts to produce
"natural" (?), "consonant" (???) 7ths.<br>
<br>
But the major difficulty with all this is that we will never know...<br>
<br>
Nicolas Meeùs<br>
<a class="moz-txt-link-abbreviated" href="mailto:nicolas.meeus@paris-sorbonne.fr">nicolas.meeus@paris-sorbonne.fr</a><br>
<i><br>
<br>
<br>
<br>
</i></font></font>Eric Knechtges a écrit :
<blockquote
cite="mid:AANLkTikmE31BQpJOd8bzcRMGXbNhDR8hRe+LYb5pH99a@mail.gmail.com"
type="cite">Thank you for those of you who have responded already.
Just to clarify why I was asking:<br>
<br>
A number of just intonation zealots have remarked that a QCM tuning of
a German augmented sixth chord is a fairly close approximation of
overtones 4 through 7 in the harmonic series... and some sources go so
far as to say that composers actually *intended* that quality of sound
for the Ger+6, distinct from the enharmonic dominant seventh.<br>
<br>
I find this line of thinking suspect, since I'm not sure that there was
much historical overlap between QCM and the German sixth. I can
believe it might have been intended in Gesualdo, given his
improvisation and experimentation on Vicentino's microtonal
harpsichords... but I'm not sure about other composers.<br>
<br>
Eric Knechtges, DM<br>
Assistant Professor, Coordinator of Composition/Theory<br>
Northern Kentucky University<br>
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