Re: Is the ditone (or semiditone) a third?

Hello, there, and Johnny Reinhard has proposed an interesting thesis for
which I might not be the most unlikely person to offer what I hope be a
friendly and spirited dissent:

> The ditone (or the 81/64 or 408 cents) interval is not a "third" in any
> functional sense. It is not a holistic part of a harmony, but rather
> the sum of 2 whole tones (of 9/8 or 204 cents). And it was never called
> a third during its use in Europe.

Here we really have two questions: first, was the Pythagorean ditone at
81:64 ever referred to as a "third" (Latin _tertia_) during the era of its
characteristic usage, say 1200-1420? Also, was it sometimes describeed as
a "practical third" later in the history of European music and tunings?

The answer here is a simple _yes_, since Jacobus of Liege, in his
_Speculum musicae_ (c. 1325), possibly the most encyclopedic exposition
available both of medieval Pythagorean tuning and of 13th-century musical
practice, refers to the two regular varieties of the third as _una tertia
in ditono_ ("a third in the ditone [ratio]," i.e. 81:64) and _una
tertia in semiditono_ (i.e. "a third in the semiditone [ratio]," 32:27). A
fifth may be "split" (_fissa_) into two such _tertiae_ by a third
simultanous voice, producing the mildly unstable and pleasing _quinta
fissa_ or "split fifth" sonority, e.g. c-e-g or d-f-a.

Of course, "functionality" is a matter of definition, and also a point of
debate. I would say that both 13th-century and 18th-century harmony have
their own "functional" progressions -- maybe "directed progressions" is
just as good and less controversy-provoking (grin).

Anyway, I say that medieval Pythagorean thirds in the era 1200-1400, and
some time before that, are indeed "thirdlike" in their own way: they are
less concordant in theory and practice than fifths, and more concordant
than M2 and m7, say. Note that one could debate whether a diminished
fourth in 1/4-comma meantone or 5-limit JI (32:25, c. 427 cents), or an
interval such as 6/17 octave (c. 423.5 cents), is a "third" in the same
sense; various listeners might find it _more_ tense than M2 or m7, for
example.

Of course, as Ed Foote might remind me, the ditone or near-ditone _was_
accepted as a "third" (albeit a tense one) in various well-temperaments of
the late 17th-19th century, and Charles, Earl of Stanhope (hope I got the
first name right), declared around 1800 that the "ditonic" thirds in
remote keys of such tunings had a "pathetic" power which 12-tone equal
temperament lacked, just as it also sacrificed the smoothness of the most
common keys. Thus Stanhope indeed regarded the ditone as a third, and as a
memorable one worth keeping in a keyboard tuning system.

> Playing the musicology game of "psychological differences regarding thirds
> through history" is to help disguise ignorance. This is an important
> distinction because there are many different sizes of tempered thirds.
> However, the ditone is not even a tempered interval.

In a Gothic setting, I'd agree on the last point -- I consider the ditone
as a "true" major third in this context, not a tempered interval. I'd say
"a just 81:64," but that might prompt both controversy and needless
confusion. One could also follow Prosdocimus and Ugolino in the early 15th
century, and speak of a "fully perfected" major third ideally poised to
expand into a fifth. Incidentally, Ugolino uses _tertia_ ("third") to
describe both major and minor thirds, also referred to as respectively
"perfect" and "imperfect."

From an 18th-19th century perspective, of course, the ditone _is_ in
effect a highly tempered major third, although generated from pure fifths.

Again, this is one of my favorite intervals, so people know my biases
<grin>.

Most respectfully,

Margo Schulter
mschulter@value.net