David Finnamore's tunings

Hello, there, and I'd like to reply briefly to two questions of David
Finnamore about my example of one possible arrangement for his
"neo-medieval/Renaissance" tuning based on Pythagorean 9:8 whole-tones
and 256:243 semitones combined with semitones ingeniously dividing 9:8
into 21:20 and 15:14.

First, in my example, I followed the usual Gothic rule that small
semitones should be placed at diatonic locations, that is, where mi-fa
or fa-mi would be sung: e.g. a-bb, eb-d, f#-g, c#-d, g#-a. Here I take
the small 21:20 semitone as the counterpart of the usual limma
(256:243, which occurs at e-f and b-c'), and accordingly place it at
these locations. Note that "diatonic" semitones in this sense may be
defined by the rule that a flat tends to descend, and a sharp to
ascend.

Chromatic semitones (c-c#, eb-e, f-f#, g-g#, bb-b) get a large semitone of
15:14 (about 119.4 cents), a counterpart of the Pythagorean large semitone
or apotome (2187:2048). These semitones are located immediately below a
sharp or above a flat, and singing such semitones would go "against the
grain" from a usual medieval/Renaissance perspective.

This preference has both melodic and harmonic aspects somewhat
accentuated by this tuning. Melodically, the diatonic semitones of
21:20 (about 84.5 cents) are even more incisive than the usually keen
limma of about 90.2 cents. Harmonically, for example, major thirds and
sixths take on an even more active and "expansive" quality than in
classic Pythagorean tuning, making cadences where these intervals
expand to stable fifths and octaves yet more dynamic.

Thus if one shares the medieval preferences for small melodic
semitones and dynamic Gothic-style vertical cadences, and is
performing or composing in a usual medieval or "medievalesque" style
where c#-d or bb-a, for example, are much more likely to occur as
melodic intervals than c-c# or bb-b (often regarded as "unsingable" in
medieval and even Renaissance theory, although theorists as early as
Marchettus of Padua in 1318 approved of such "chromatic"
progressions), following the "small diatonic semitone" custom might be
an apt choice.

However, you are very right to emphasize that this _is_ a choice, and
that your scale permits many alternatives which might be just as usual
or more so, depending on the style.

For example, around 1400, it appears that there was a very popular
movement in favor of consistently dividing whole-tones so that the
small semitone would always be in the _lower_ part of the division. In
other words, from a Pythagorean perspective, all accidentals were
tuned as flats -- or, in your tuning, would have 21:20 below and
15:14 above.

As Mark Lindley has shown (see "Pythagorean Intonation" in the _New
Grove_ for a brief and informative summary), a major motivation for
this modification was a desire in the early 15th century to place
schisma thirds in prominent places -- that is, the almost pure thirds
(and sixths) which result when sharps are tuned as flats.

In your tuning, if we make a similar modification, we get major thirds
and sixths at locations like d-f# and e-c#' which are about 379 cents
and 877 cents respectively, if I calculate correctly. This narrow M3
is reminiscent of 1/3-comma meantone or 19-tet, but the m3 won't be
the same as in these tunings at such points, because the fifths remain
a pure 3:2, requiring an m3 of around 323 cents to fill the remaining
space. While not as close to pure as regular Pythagorean schisma
thirds and sixths, these intervals nevertheless will have a nice
contrast with their active Pythagorean counterparts (e.g. c-e, d-b)
and even more active counterparts involving flats (e.g. eb-g, bb-g').

One choice, of course, if you go in for a "quasi-Pythagorean"
arrangement of your scale, is to do what certain early 15th-century
theorists proposed: have 17 notes per octave, with ten accidentals
splitting the five whole-tones both ways (c#/db, eb/d#, f#/gb, g#/ab,
bb/a#). Where this is possible, it could give a performer or composer
great freedom to explore many potentials of your scale.

In conclusion, I should emphasize that these tunings are just a subset
of the possibilities for your scale.

Most respectfully,

Margo Schulter
mschulter@value.net

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End of TUNING Digest 1523
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