Re: What is enharmonic? (was: Microtonality repression)

Hello, there, and I'd like to offer a brief explanation of the term
"enharmonic," inviting further elucidation from people such as John
Chalmers on the topic of the Greek divisions of the tetrachord.

In ancient Greek theory, the enharmonic genus involves the use of
intervals equal to approximately half of a diatonic semitone. The
tetrachord (or division of the fourth) in this genus is based on the
intervals of a major third and of two "diesis" each equal to about
half of the remaining semitone. For example, in Pythagorean tuning,
with intervals shown in rounded cents, and an asterisk (*) used to
show the raising of a note by a diesis (about 45.11 cents, half of a
256:243 semitone such as E-F at about 90.22 cents):

E E* F A
0 45 90 498
45 45 408

On the role of the enharmonic in ancient Greek music, John Chalmers or
others might be able to comment, but the basic idea is the division of
a semitone into two precisely (or approximately?) equal parts.

Although the enharmonic division itself doesn't seem to play much of a
role in medieval European composed polyphony, Marchettus of Padua in
1318 advocates the use of an interval he terms a "diesis" equal to
"one fifth part of a tone" -- whether an equal or unequal division is
a point of much debate. However, if he is advocating an equal fivefold
division, the mathematics would be rather different than for
conventional Pythagorean tunings of the era where an interval of this
size (~41 cents) does not occur with tuning sets of 17 or fewer notes
of the kind discussed by theorists of the era.

Around the second quarter of the 15th century, interestingly, Ugolino
of Orvieto in discussing a 17-note Pythagorean organ keyboard (Gb-A#),
mentions the possibility of adding two extra notes dividing the
semitones E-F and B-C into two equal parts after the "ancient"
practice -- that is, the enharmonic genus. However, he remarks that
this is not in use by the "moderns," that is, in late medieval styles
familiar to him.

The enharmonic genus really gets revived as a central element of style
by Nicola Vicentino (1511-1576) in his treatise on _Ancient Music
Adapted to Modern Practice_ (1555).

This "adaption" is in part evidently result of musical serendipity: in
the meantone tunings favored by the 16th century for their pure or
near-pure thirds (an element of the new Renaissance styles), the
interval between such alternative accidentals as G# and Ab is equal to
size comparable to that of the ancient enharmonic diesis. In 1/4-comma
meantone with pure major thirds, for example, it is equal to 128:125
or ~41.06 cents.

What Vicentino does is to take this interval, and to use it
deliberately for its "gentle" and subtle effect in vertical and
melodic progressions, building an _archicembalo_ or "superharpsichord"
which divides the octave into 31 essentially equal diesis intervals.
Actually, in a 1/4-comma tuning, there are two slightly different
sizes of dieses or "fifthtones": the usual meantone diesis at 128:125,
and a slightly smaller fifthtone at about 34.99 cents, or ~50:49, the
difference between a meantone chromatic semitone at ~76.05 cents and
the larger 41-cent diesis.

Vicentino's enharmonic diesis or fifthtone is "microtonal" in a sense
beyond that of a conventional European tuning such as 12-note meantone
or 12-tone equal temperament (12-tET) for lutes in the same era: the
use of this interval delighted some listeners, and perplexed others
such as Vincenzo Galileo, who argued that the enharmonic genus was
ill-proportioned to the human ear.

How the term "enharmonic" came to be applied to accidentals regarded
as equivalent in a given tuning, e.g. G#/Ab in 12-tET or B#/Cb in
19-tET, I'm not sure; but in 1/4-comma meantone, G#-Ab or B#-C is
indeed equal to an "enharmonic diesis," so possibly this usage derives
from the enharmonicism of Vicentino and his followers in the 16th
century.

Most respectfully,

Margo Schulter
mschulter@value.net