Basically, I would say, a "Wolf" interval is an "unusual" interval in a given tuning system and style which is felt not to be freely substitutable for an "expected" variety of the same interval. Sometimes it is asserted that "Wolf" intervals are "unusable" or "unplayable," but this judgment is both contextual and often partial: there are often specific sonorities and usages where these intervals are musically useful even in styles where they are generally considered to be "too out-of-tune" for most purposes.

With musical tunings, as with natural languages, sounds which are normal features of one language can seem incongruous in the setting of a different language. For example, taking two languages such as French and English, one might say that some complex vowel sounds in either language would be "Wolves" in the other -- that is, heard as "wrong" or "mispronounced" or "spoken with an obvious foreign accent."

Similarly, in a setting of 16th-century Western European keyboard music, an interval of 427 cents would be perceived as an "out-of-tune major third," and an interval of 720 cents as an "out-of-tune fifth." Yet in 14th-century or 20th-century European music, the first interval might serve as one usual tuning of a kind of major third; and in gamelan music, the second interval often serves as a normal fifth.

One approach to "Wolves," at least in the context of historical European systems of tuning, is to say that they result from attempting to achieve a pure tuning of a given interval or set of intervals while restricting the system to a limited set of pitches such as 12 notes per octave. A few examples might be helpful.

In Pythagorean or 3-based just intonation, where all regular fifths have a pure ratio of 3:2, fourths of 4:3, major seconds of 9:8, and minor seconds of 16:9, we run into the potential complication that 12 such fifths will produce an interval a bit wider than a pure 2:1 octave. As a result, if we tune pure octaves, one of our 12 fifths in a chain of Eb-G#, for example, will be 262144:177147 instead of 3:2.

This G#-Eb interval is thus a Pythagorean comma (531441:524288) narrower than 3:2, this comma being equal to ~23.46 cents, the difference between 12 pure fifths and 7 pure octaves -- or about 678.49 cents. In practice as well as theory, such an interval sounds rather different from a pure fifth: it beats rapidly, and can hardly serve as a concordant fifth in a medieval setting.

Since the interval G#-Eb rarely comes up in 14th-century music, this is no major problem. However, it's easy to "solve" this problem and come up with 12 pure fifths (or fourths) per octave if we're willing to add a 13th key to our keyboard, say an Ab. Then we can use G# as a pure fifth above C#, and Ab as a pure fifth below Eb.

However, in a modified early 15th-century tuning where the chain of fifths runs Gb-B, the "Wolf" between these notes does apparently sometimes get used when the Gb key gets use as a substitute for F# in certain cadences:

    b  --  +90 -- c'
(906,522)      (1200,498)
    gb -- +114 -- g'
  (384)         (702)
    d  -- -204 -- c

Here the interesting thing is that although we have a wide "Wolf" fourth of 177147:131072 (~521.51 cents) between the two upper voices of the d-gb-b sonority, this can sound acceptable -- maybe in part because both these voices form more concordant intervals with the lowest part, and in part because we have a cadential sonority where some tension is expected.

Note, incidentally, that in early 15th-century tuning we also have a nice example of intervals that differ considerably in size but are apparently treated as interchangeable: for example, major thirds of both 81:64 or ~407.82 cents (e.g. c-e), and 8192:6561 or ~384.36 cents (e.g. d-gb in the above examples). We might say that the first tuning is somewhat more active, and the second somewhat more restful, but neither is deemed to be a "Wolf."

In tunings of the later 15th-17th centuries, we run into a different kind of "Wolf" complication: the fact that three pure major thirds of 5:4 or ~386.31 cents each do not add up to a pure 3:2 octave, but fall about 41.06 cents short. Again, if we tune pure octaves, this means that a chain of three thirds, making two of them pure (as in 1/4-comma meantone tuning, where each regular fifth is narrowed by 1/4 of a syntonic comma of 81:80), will include a very wide third interval:

c  --  e  --  g#  --  c'
  386    386     427

Such odd intervals as g#-c' are diminished fourths at 25:16 or ~427.38 cents, and sound very different than a pure 5:4 major third, different enough that they are unlikely to be satisfactorily substituted for the Renaissance interval of a concordant major third!

However, again, context and arrangement can make a difference. Thus this "augmented fifth" sonority is found in 16th-17th century music, and can be quite pleasing within this stylistic setting:

    f#'
(1586,696)
    bb
  (890) 
    d

Again, the fact that both upper parts form satisfactory intervals with the lowest part might be one factor making this "Wolf" more palatable.

Note that with 1/4-comma meantone, as with Pythagorean tuning, it would be fair to point out that these "Wolves" occur only when we play two keys together which were not advertised in the first place as producing the desired interval. Here, if we add an Ab key we can get a pure major third ab-c'; and if we add a Gb key we can get a sonority of d-bb-gb with a pure minor sixth (8:5) rather than an augmented fifth between the upper voices. In the second case, however, we might well prefer the version with the expressive "Wolf," d-b-f#'. Of course, our enlarged keyboard gives us both these choice.

Another kind of "Wolf" occurs in some systems of 5-based just intonation when realized with only 12 notes, for example, per octave. If we tune the scale interval of c-d as a pure 9:8 major second (~203.91 cents), and the major sixth c-a as a pure 5:3 (~884.36 cents), then the fifth d-a will be equal to the difference of these intervals, ~680.45 cents -- a 40:27 "Wolf" narrower than 3:2 by a syntonic comma of 81:80 (~21.51 cents). The fourth a-d', for example, at 27:20, would likewise be a syntonic comma wider than pure, or ~520.55 cents.

Interestingly enough, our Tuning List's own pianistic pioneer Dave Hill reported that this "Wolf" fourth can be quite usable in harmonizing a melody such as My Country 'Tis of Thee, if it occurs between two upper voices:

    d'
(906,520)
    a
  (386)
    f 

Here the rounding in cents is a bit awkward, so I'm calling the "Wolf" fourth a-d' 520 cents, although it's actually a bit closer to 521 cents.

In this case, again, adding more keys would give us the option of a pure fifth d-a or fourth a-d' -- what we would need is another "D" key forming a 10:9 rather than a 9:8 major second with C. Then we could use either key as needed or desired.

Now that we've considered a few examples, it's time for a couple of cautions and extra comments .

First, as already mentioned, one tuning system's "Wolf" can be another system's ideal interval. Thus the 1/4-comma meantone Wolf major third of 427 cents is very close to the normal 6/17 octave interval of 17-tone equal temperament (17-tet), and possibly quite close also to the wide cadential major third favored by Marchettus of Padua (1318) before a fifth. In fact, I can amuse myself on a meantone keyboard, in between some usual and joyous 16th-century musicmaking, by shifting to the "far side" of the temperament, e.g.

     bb --  +76 --  b
 (931,503)      (1200,503)
     f  --  +76 --  f#
   (427)          (697)
     c# -- -193 --  B

Playing such a cadence on a meantone keyboard represents a kind of "time warp": we are switching to a stylistic context or language where thirds and sixths are no longer the restful intervals of the Renaissance, but the active intervals of some medieval or neo-medieval usage.

Another and final point is that a "Wolf" interval as defined in a certain stylistic context is only a significant disadvantage if the interval is likely to be used in actual music for a given instrument. For example, European 16th-century keyboard music generally uses only accidentals in the usual meantone range of Eb-G#, so that the theoretical difficulty of a g#-c' "Wolf" (in place of ab-c') is not too much of a problem in practice. However, with the shift from modality to major/minor tonality around 1680 or so, there is more demand for remote accidentals -- and, not surprisingly, a movement by theorists such as Werckmeister toward "well-temperaments" permitting remote transpositions without Wolves.

Most respectfully,

Margo Schulter
mschulter@value.net

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End of TUNING Digest 1597
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