source file: mills2.txt Date: Fri, 6 Dec 1996 06:31:59 -0800 Subject: Crying "wolf". From: kollos@cavehill.dnet.co.uk (Jonathan Walker) 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/ Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Fri, 6 Dec 1996 15:48 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01830; Fri, 6 Dec 1996 15:50:48 +0100 Received: from eartha.mills.edu by ns (smtpxd); id XA01833 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id GAA17779; Fri, 6 Dec 1996 06:50:45 -0800 Date: Fri, 6 Dec 1996 06:50:45 -0800 Message-Id: <199612061449.XAA00821@judy.sfc.keio.ac.jp> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu