Crying "wolf".

Joe Downing wrote:
> By the way: In Just Intonation, how do you avoid a wolf in a
> progression like the following: I V I vi ii V I? I think you have
> to choose between a wolf and a comma slip, and I generally prefer
> the former.

and Gary Morrison replied:
> Orrrr... Let the tonic wander. (Some of you probably guessed that
> was coming, eh?)

I'd like to deal with two points arising from this exchange: in this
message the application of the term "wolf", and then, when I have time
to write another message, the issue of wandering tonics in JI.

The term "wolf" has long been applied to diminished sixths which
appear on a 12-note-per-octave keyboard as fifths. The usage goes back
at least as far as Schlick's *Spiegel* (1511): if the keyboard was tuned
from Eb to G# (irrespective of whether the tuning chosen was
Pythagorean, meantone or Schlick's own irregular temperament), then the
interval G# to Eb, when the G# was unhappily pressed into service as an
Ab, would constitute the wolf "fifth"; it was up to the keyboard player
to avoid a sustained wolf by means of decoration, permitting tritones
that singers would avoid, or ad hoc ficta adjustments. The number of
notes per octave is not the cause of the problem, which will always
reappear in any tuning or temperament that is not closed (closed being
12TET or the various 18th-century irregular well-tempered systems: to
"well-temper" was simply to remove the wolf). In much music c.1500 the
wolf, even on a 12-note keyboard didn't necessarily constitute a great
problem, at least for the clavichord: if the piece to be played included
a G#, all was well; if it included an Ab, then the G#s could all be
retuned within a minute or two by conscientious players. The need for
both G# and Ab did not arise so often as to defeat the measures I have
mentioned above (the same problems and solutions were present for C#
versus Db etc.).

Now here is the point: a JI keyboard, such as Zarlino envisaged, would
also contain a wolf: i.e. a diminished sixth which the unwary player
might call upon to serve as a perfect fifth. But in addition to this, it
would contain a 40/27 -- a perfect fifth less a syntonic comma. Without
dividing the lower rank of keys -- i.e. our white keys today -- this
would appear between D and A if the Ptolemy sequence was chosen as the
basis of the tuning (as Zarlino wished). Leaving aside the question of
whether a JI keyboard is practicable (I would certainly say it isn't,
and that 1/4-comma meantone is the practical keyboard equivalent of
5-limit JI), we are left with a terminology problem: we have two
varieties of interval which are not what they seem from the layout of
the keyboard, namely the 40/27 and the diminished sixth (192/125).

Do we really want to use the same term "wolf" for these two very
different intervals? The 40/27 is a syntonic comma flat of a 3/2 perfect
fifth, while a 5-limit diminished sixth is 41 cents _sharp_ of a 3/2. To
complicate matters further, the established usage of wolf for a
diminished sixth in any open tuning/temperament allows it to be applied
legitimately to a range of intervals: e.g. the 3-limit diminished sixth
is 262144/177147, a Pythagorean comma _flat_ of a 3/2. So even leaving
aside meantone and its relatives, and various open irregular
temperaments used in the 16th and 17th centuries, we have a 5-limit wolf
41 cents sharp of 3/2 and a 3-limit wolf 21.5 cents flat of 3/2.

Dressing up a 40/27 in wolves' clothing, however many recent precedents
can be adduced, therefore seems a recipe for irretrievable theoretical
confusion.

--
Jonathan Walker
Queen's University Belfast
mailto:kollos@cavehill.dnet.co.uk
http://www.music.qub.ac.uk/~walker/

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