source file: m1474.txt Date: Mon, 13 Jul 1998 21:58:26 -0700 (PDT) Subject: Positive tunings and adjectives From: "M. Schulter" Hello, there, everyone, and thank you all for making such a valuable forum possible. Already, some of you have offered me invaluable assistance and dialogue, and in fact this hospitality is one factor drawing me here. As a curious topic for a first post, why don't I focus on a question of language and stereotyping -- happily not of people, but of a somewhat undervalued group of intervals: Pythagorean thirds and sixths. Also, as a kind of coda, I suggest that Ludmila Ulehla's concept of "dual-purpose" sonorities as a third category somewhere between full concord and urgent discord might enrich discussions of various tunings and their characteristic intervals. -------------------------------------------- 1. Gothic and neo-Gothic interval aesthetics -------------------------------------------- While it is generally agreed that a "just" major third has a ratio of 5:4, and that a "pure" major third likewise implies this simplest ratio, how about some friendly adjectives for the Pythagorean major third of 81:64, or for that matter the major sixth of 27:16? Here my special interest is Pythagorean tuning in the context of the Gothic polyphony of the 13th and 14th centuries in Europe, where these Pythagorean intervals play an integral role in the harmonic style. Carl Dahlhaus and Mark Lindley, for example, have written eloquently on this point. In a Gothic setting an M3 of 81:64 (about 407.82 cents) or an M6 of 27:16 (about 905.87 cents) have the virtue of expanding very efficiently to the stable goal of a fifth and octave respectively. How can we express the positive qualities of these intervals in a medieval setting while making clear their differences from the 5:4 and 5:3 forms favored in tertian styles of harmony? These same issues may apply with even more force to tuning systems such as 17-tet, which offer major thirds and sixths _wider_ than Pythagorean, which expand even more efficiently to the fifth and octave with diatonic semitones even keener than the usual 90-cent limma of 256:243. Marchettus of Padua (c. 1318) advocates cadential leading tones of only around 1/5-tone or 2/9-tone (depending on one's interpretation of his unorthodox division of the whole-tone) for the M6-8 and M3-5 resolution. While a literal reading would produce a cadential M3 at around 450 cents, and M6 at around 950 cents, we might take him more freely to recommend a cadential tuning of these intervals somewhat wider than there usual Pythagorean forms, possibly comparable to those of 17-tet (about 423.53 cents and 917.64 cents). The following standard Gothic cadence, tuned in regular Pythagorean and in 17-tet, may illustrate these points, with apologies for slight rounding discrepancies. Numbers in parentheses represent vertical intervals above the lowest voice, and other numbers represent the melodic motion of each part up or down, all measured in cents. Classic Pythagorean 17-tet e'-- +90.22 -- f' e'-- +70.59 -- f' (905.87) (1200.00) (917.64) (1200.00) b -- +90.22 -- c' b -- +70.59 -- c' (407.82) (701.96) (423.53) (705.88) g -- -203.91 f g -- -211.76 f M6 8 M6 8 M3 5 M3 5 In our classic Pythagorean example, and even more dramatically in 17-tet, the active M3 and M6 expansively seek the fifth and octave, The generous whole-tone motions and keen semitonal motions of the parts contribute to the total harmonic effect of active tension very efficiently resolved. We arrive at the ideally concordant three-voice combination of the Gothic with outer octave, lower fifth, and upper fourth (string-ratio 6:4:3, frequency ratio 4:3:2). ------------------------------------------- 2. Positive adjectives for positive tunings ------------------------------------------- Here I would like to suggest that Pythagorean thirds and sixths in a Gothic context may felicitously be described as "active," "vibrant," "bright," and "dynamic." They are also, of course, "unstable," but at the same time regarded in theory and practice as _relatively_ blending. They are points of motion, or of diversion, not points of arrival. In contrast, pure thirds and sixths in a Renaissance context might be described as "restful," "optimally blending," "smooth," "tranquil." This kind of contrast applies not only to these intervals, but to others whose _musical_ uses change along with stylistic developments, and thus also, quite possibly, their optimal tuning for a given purpose. One interesting reflection of a Gothic viewpoint, possibly relevant to lines of experimentation with 17-tet and other positive tunings (my special interest), is that a major third or sixth of full Pythagorean proportions is said to be "perfected," because it approaches as closely as possible its cadential goal of the fifth or octave. This "striving" to expand to stability is an element of the Gothic musical language which maybe might be emphasized more often in discussions about the Pythagorean intervals and their aesthetic qualities. ----------------------------------------------------- 3. Dual-purpose sonorities: a xenharmonic perspective ----------------------------------------------------- Finally, an aside which might apply to many musical styles, but I suspect especially to contemporary just intonation (Pythagorean or otherwise) and xenharmonic settings. Often there is a tendency to divide intervals and combinations into two categories: consonance/dissonance. However, whether we are analysing Gothic or impending 21st-century music, a middle category might go far to increase the subtlety of the discussion. Ludmila Ulehla has proposed the term "dual-purpose" sonority to describe an interval or combination which may be somewhere between stable concord and urgent discord; the medieval concept of "imperfect concord" or "imperfect discord" ("imperfect" meaning "partial, semi-") seems to convey a similar concept. Although Ulehla focuses on the era from around 1750 to the 20th century, her approach may nicely fit earlier practice and theory as well as new xenharmonic developments. In a 13th-century setting, for example, a interval such as 81:64 or three-voice combination such as 9:6:4 seems to be regarded as relatively blending and euphonious, but unstable; and the same category might fit many of the sonorities of extended just intonation or xenharmonics in certain settings. To describe such sonorities simply as "concordant" may not convey their instability, while the adjective "discordant" may not convey their perceived level of independent euphony or "coloristic" appeal apart from any immediate resolution. Of course, both dual-purpose sonorities and stronger "discords" In these terms, the apparent transition during the 15th century from Pythagorean to meantone keyboard tunings reflects a shift in the role of M3 and M6 from active dual-purpose sonorities (81:64, 27:16) to more restful concords (5:4, 5:3). Similarly, the possible tunings of the minor seventh at 16:9, 9:5, and 7:4 may represent different levels of tension desired in divergent styles, as well the exigencies of fitting m7 into a larger scheme of things. In conclusion, I should emphasize that 17-tet, for example, is by no means limited to "neo-Gothic" applications, but might be very congenial to such applications. Also, 81:64 might have xenharmonic applications quite different from those obtaining in medieval polyphony, which suggest only one very effective use of this interval. Most respectively, Margo Schulter mschulter@value.net