source file: m1486.txt Date: Sun, 26 Jul 1998 23:10:44 -0700 (PDT) Subject: Re: Wolf with a difference tone From: "M. Schulter" -------------------------------- 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