Wolf with a difference tone

--------------------------------
A Wolf with a Difference (Tone):
Essay in Honor of Ervin Wilson
--------------------------------

The work of Ervin Wilson, to which I have only recently been
introduced as part of my ongoing initiation into the world of
xenharmonics, has a richness inviting enthusiastic if imperfect
emulation. At the outset, it would be well to emphasize that while
Ervin has indeed provided both the conceptual basis and inspiration
for the analysis which follows, its imperfections are my own.

One of the most intriguing chapters in the history of Western European
music and tuning, with many potential byways yet open to explore, is
the epoch around 1400. Here I would like to focus on a cadence from
this era, which, in a prevalent keyboard tuning of the time, would
have a striking Pythagorean "Wolf" fourth.

Let us consider first a conventional analysis of this cadence based
primarily on late Gothic theory, and then a difference tone analysis,
concluding with a bit of philosophical musing about harmonic
"implications" (acoustical or otherwise) and stylistic context.


---------------------------------
1. Anatomy of a 15th-century Wolf
---------------------------------

In Pythagorean tuning, all tones and intervals are generated by a
series of pure 3:2 fifths (about 701.955 cents each). Around 1400, a
popular arrangement of the chain of 11 fifths for a full chromatic
keyboard scale was as follows:

Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B
F# C# G#

In this tuning, all accidentals are positioned as flats, with the
three notes at the flat end of the chain used as substitutes for true
Pythagorean sharps. One effect of this substitution is to make major
thirds and sixths involving these "quasi-sharps" a Pythagorean comma
smaller than usual, and minor thirds and sixths a Pythagorean comma
larger, resulting in nearly pure forms.[1]

Another result is to make the fifth B-Gb, or the fourth Gb-B, a
Pythagorean "Wolf": the fifth is a full Pythagorean comma narrower
than just (262144:177147, around 678.495 cents), and the fourth a
comma wider than just (177147:131072, around 521.505 cents).

A keyboard octave chart may illustrate these points:

256:243 32:27 1024:729 128:81 16:9
90.22 270.67 588.27 792.18 996.09
db' eb' gb' ab' bb'
_90.2|113.7|90.2|113.7_ _90.2|113.7_90.2|113.7_90.2|113.7_
c' d' e' f' g' a' b' c''
1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1
0 203.91 407.82 498.04 701.96 905.87 1109.78 1200
203.91 203.91 90.22 203.91 203.91 203.91 90.22

To solve the problem of the Wolf, some musicians of the time advocate
keyboards with 13 or more notes per octave including a true F#
key. However, while this solution may have been implemented on some
keyboards, documents of the period suggest that these instruments were
the exception rather than the rule. [2]

In fact, much keyboard music of the period (e.g. the earlier pieces
from the Buxheimer Organ Book) seems to fit the G-Bb tuning very
nicely. An especially charming resource is the contrast in color
between active Pythagorean thirds and sixths involving diatonic notes
or flats, and almost-pure schisma thirds and sixths involving sharps, the
latter being only a schisma (about 1.954 cents) from the simplest
5-based ratios (5:4 for M3, 6:5 for m3, 5:3 for M6, 8:5 for m6):

Regular Pythagorean intervals Schisma intervals
interval example ratio cents interval example ratio cents
M3 g-b 81:64 407.82 M3 d-f# 8192:6561 384.36
m3 d-f 32:27 294.13 m3 c#-e 19863:16384 317.60
M6 d-b 27:16 905.87 M6 a-f#' 32768:19683 882.40
m6 e-c' 128:81 792.18 m6 f#-d' 6561:4096 815.64

Some of these pieces, however, include a popular cadence of the 14th
and early 15th centuries which takes on a very special quality in this
tuning. In the notation which follows, numbers in parentheses identify
vertical intervals in cents between a given note and lower notes in a
sonority; signed numbers show the melodic motion of each part up or
down.

b' -- +90.2 -- c''
(905.9, 521.5) (1200, 498.0)
f#' -- +113.7 -- g'
(384.4) (702.0)
d' -- -203.9 -- c'

In this standard Gothic cadence, the unstable major third expands to a
fifth and the major sixth to an octave, arriving at the complete
stable harmony of an outer octave, lower fifth, and upper fourth. This
stable combination has a frequency ratio of 2:3:4, and plays a role
analogous to that of the 4:5:6 triad in Renaissance-Romantic harmony
and of various more complex sonorities in 20th-century music. It
represents what I call the point of _stable saturation_, the richest
possible sonority which may yet be perceived in a given stylistic
setting as a restful point of arrival.

What is less standard about this cadence in the Gb-B tuning is the
combination in the unstable sonority d'-f#'-b' of a usual Pythagorean
major sixth (d-b) with a schisma third d'-f#'. This produces a Wolf
rather than pure fourth between the two upper voices, a Pythagorean
comma wider than just.

While conventional wisdom tells us that a Pythagorean Wolf is the very
epitome of an "unplayable" interval in contexts where a pure fifth or
fourth is expected, Mark Lindley interestingly has a more tolerant
verdict to render regarding this cadence.

Finding such a progression "less noxious to the ear than one might
expect," he suggests that "[s]omehow the cadential context and the
purity of D-F# alleviate the effect of the wolf fourth."[3]

One possible factor at work here is the tendency for "discordant"
intervals of various kinds between two upper voices to be somewhat
mitigated in their impact if both affected voices form "concords" with
the lowest voice. For example, in a 16th-17th century meantone
setting, Easley Blackwood has observed that an augmented fifth such as
Bb-F# may be acceptable where these notes form a minor sixth and major
tenth above the lowest tone. Here I assume 1/4-comma meantone:

f#'
(1586.3, 772.6)
bb
(813.7)
d

Although the augmented fifth between the upper voices is a full diesis
(128:125, about 41.058 cents) narrower than a just minor sixth, the
concordant intervals of both voices with the lowest part somewhat
"cushion" this tension.

As Lindley also notes, the "cadential context" plays a role in making
our cadence with the Pythagorean Wolf fourth more acceptable: an
unstable cadential sonority, after all, is often expected to be a
point of harmonic tension rather than of restful euphony. Yet another
factor is that in the examples he cites [4], these sonorities are not
especially prolonged.


-------------------------------------------
2. Can a difference tone make a difference?
-------------------------------------------

Having presented a more or less conventional analysis of this cadence
with its striking Wolf fourth, I would like to consider an analysis in
terms of difference tones.

One complication of such an analysis might be that while the outer
Pythagorean major sixth of d'-f#'-b' (or in this tuning d'-gb'-b') has
a rather simple ratio of 27:16, the schisma third d'-f#' (8192:6561)
and the Wolf fourth f#'-b' (177147:131072) are much more complex.

A possible solution is to treat the latter two intervals as if they
were defined by simple 5-based ratios, disregarding the difference in
each case of a schisma between acoustical reality and simplified
model. In such a model, we treat d'-f#' as if it were 5:4, and f#'-b'
(actually gb'-b') as if it were 27:20, a Wolf fourth differing from just
by a syntonic comma (81:80, about 21.506 cents) rather than a
Pythagorean comma:

b' 27
7
f#' 20
4
d' 16

If we accept this simplification, then the difference tone between the
lower two voices (20:16) is equal to 4; the difference tone between
the upper voices forming the Wolf is 7. These tones would imply a
sonority like the following:

b' 27
f#' 20
d' 16
(c~) 7
(D) 4

>From this point of view, we might caption this cadence when realized
in the Gb-B tuning as "Pythagoras meets Partch." The difference tone
of 4 implies a doubling of the fundamental of our d'-f#'-b' sonority
at two octaves below, while the difference tone of 7 implies a pitch
notated above as c~ -- that is, c lowered by a septimal comma (64:63).

Curiously, a Pythagorean tuning system having a highest prime of 3 and
a highest odd factor in multiplex (9:1) or superparticular ((9:8)
ratios of 9 is producing not only schisma approximations of 5-based
intervals, but difference tones approximating 7-based intervals. Might
these intervals, including those implied by difference tones,
contribute to the harmonic color or force of the cadence?

While these tones might be perceived to add color, I find it more
difficult (at least at first blush) to interpret them as playing a
directed cadential role in early 15th-century terms. The cadential
paradigm for this epoch involves progression from mildly unstable
intervals to stable ones (e.g. m3-1, M3-5, M6-8) by conjunct contrary
motion. Here the progression d'-f#'-b' to c'-g'-c'' is already
complete: no further difference tones are required.


----------------------------------------------
2.1. Xenharmonic time travel: which direction?
----------------------------------------------

As it happens, the period around 1400 marks the point where seconds
and sevenths come to be treated more and more cautiously in practice
as well as theory, one of the changes often taken to mark the
transition from the late Gothic to the early Renaissance.

To have our difference tones make a difference in terms of _directed_
harmony, we must apparently travel to an era where vertical seconds
and sevenths play a more "essential" role in cadential action than in
the early 15th century. We might journey in either direction: either
back to the 13th century, when such intervals regularly participate in
directed resolutions, or ahead to the 17th and 18th centuries, when
they again come to play a prominent role in bold cadences.

Let us consider the latter alternative first, since it is simpler from
the viewpoint of the difference tone levels required.

b' 27
f#' 20
d' 16
(c~) 7
(D) 4

>From an 18th-century viewpoint, our c~ is a "harmonic seventh," and
from a viewpoint of 7-limit just intonation, the interval c~-f#'
(20:7) is an octave transposition of 7:5, the 7-based diminished
fifth. These intervals, in such viewpoints, form an ideally euphonious
form of tertian dominant seventh chord, inviting a resolution to a
sonority based on G. While the actual cadence is to a stable Gothic
combination on C -- not a triadic sonority on G -- possibly a listener
oriented to 18th-century style might argue that the f#'-g' progression
of the actual middle voice fulfills some of these expectations.

>From a 13th-century viewpoint, we can derive an interesting result by
considering not only primary difference tones, but what I would call
"secondary difference tones" arising these primary tones:

b' 27

f#' 20

d' 16

(c~) 7
3
(D) 4

The difference tone of 7:4 is 3, and we might also take the sum tone
of 20:4 or 24: these procedures imply an additional note at A or a'.
If one is willing to consider such procedures as musically relevant,
then we can happily derive a 13th-century progression where the new
tone indeed reinforces a compelling directed cadence. Note that the
schisma thirds and Wolf fourth are 15th-century anachronisms in this
earlier Gothic context, but the other intervals fit both the
expectations of a more classic Pythagorean tuning and the patterns of
harmonic action:


b' -- +90.2 -- c''
(905.9, 521.5, 203.9) (1200, 498.0, 498.0)
a' -- -203.9 -- g'
(702.0, 317.6) (702.0, 0)
f#' -- +113.7 -- g'
(384.4) (702.0)
d' -- -203.9 -- c'

In a 13th-century context, the "implied" tone a' takes part in two
directed resolutions by contrary motion: an m3-1 resolution with the
other middle voice, and a very dynamic M2-4 resolution with the
highest voice. Most typically, this cadence occurs in the 13th century
as g-b-d'-e' to f-c'-f' rather than d-f#-a-b to c-g-c'.

Having derived a pure Pythagorean fifth from a schisma third
associated with a Wolf fourth -- a curious line of descent, one is
tempted to quip -- we can factor out these 15th-century elements to
arrive at a very common 13th-century cadence for three voices with all
intervals in their classic Gothic sizes:

b' -- +90.2 -- c''
(905.9, 203.9) (1200, 498.0)
a' -- -203.9 -- g'
(702.0) (702.0)
d' -- -203.9 -- c'

Here a major sixth (27:16) expands to an octave, and a major second
(9:8) to a fourth.

>From the intriguing transitional world of keyboard tunings around
1400, we have managed through a curious musical logic to return to the
more traditional but likewise colorful harmonic universe of the Ars
Antiqua. It would seem that difference tones can be a refreshing
mechanism for stylistic time travel.


---------------------------
3. In search of conclusions
---------------------------

The concept of difference tones has been used in many ways, from very
tangible applications in FM synthesis to the drawing of various
conclusions for composition and analysis. For example, it has been
argued that parallel fifths (3:2) imply parallel octaves, because the
difference of 3:2 is 1, generating octaves below the lower note of the
fifth. Possibly this phenomenon might be used to explain why parallel
fifths and octaves alike are accepted in 13th-century harmony but
excluded from usual 16th-century harmony, for example.

"Wolves" are one area where difference tones may be significant, at
least for some intervals, periods, and styles. Here I am indebted to
the enthusiastic wit and wisdom of Ervin Wilson, and open to
suspicions of being overly imaginative at the least in applying such
concepts to what may be quite unlikely stylistic contexts.[5]

>From one point of view, difference tones may provide a kind of musical
ink blot test: one can take a 15th-century cadence and come up with
either an 18th-century or a 13th-century cadence. Whether such
associations have much connection with what a listener actually hears,
they offer in any case a new way to move around the centuries of music
history and alternative tunings.


-------------------------
Notes
-------------------------

1. 1. Lindley, Mark, 1980a. "Pythagorean Intonation and the Rise of the
Triad," _Royal Musical Association Research Chronicle_ 16:4-61. ISSN
0080-4460.

2. Thus see ibid., at pp. 10-11, for a diagram of a keyboard showing
true F# (a pure fifth above B), as a useful note not included on
actual keyboards; see also pp. 13, 15-16, and 30 for statements
indicating that a true F# key was a desired rather than standard
feature of 15th-century instruments. On musical evidence for the use
of actual 13-note instruments in this period, see pp. 44-45.

3. Ibid. p. 43. Lindley follows traditional 20th-century terminology
in describing this progression as a "double-leading-note" cadence, and
indeed one of its distinguishing marks is the presence of two
ascending semitonal progressions (f#'-g', b'-c''). While I am not
aware of any 15th-century term for the cadence as a whole, the term
"double-expansion cadence" might not be inapposite also: the major
third expands to the fifth, and the major sixth to the octave.
Theorists of the period 1300-1450 often emphasize the "perfection" and
force of these expansions, and the cadence combines them in a mutually
reinforcing manner.

4. Ibid. p. 50, Examples 29a and 29b.

5. Possibly if one compares Ervin Wilson's teachings to Greek theories
concerning tunings and modes, and this paper to medieval European
interpretations of those theories, the analogy might not be
inapposite.

Margo Schulter
mschulter@value.net
26 July 1998