source file: mills2.txt Date: Wed, 8 Nov 1995 16:18:10 -0800 From: "John H. Chalmers" From: mclaren Subject: Rhythm, tuning & the extension of Charles Seeger's 1930 article into n-j n-e-t scales --- "Look within. Let neither the peculiar quality of anything nor its value escape you." [Marcus Aurelius, "Meditations"] With typical insight, Ivor Darreg once commented that the rhythms of Baroque music arose from the 18th-century love of clockwork and chronometry. And in fact one of the greatest achievements of Enlightment technology was the development of a ship's chronometer capable of keeping sufficiently accurate time to allow Royal Navy ship to navigate along lines of longitude. The inventor won a 10,000-pound prize--at the time a huge fortune. In a larger sense, the infrastructure of music (the tuning) has probably always exerted a potent influence over its overt superstructure--specifically, over the rhythms used. It's only in the last century, however, that the relationship appears to have been restored to the approximate level of complexity characteristic of the music of the late 14th century. Just intonation music naturally lends itself to a complementary arrangement of ratios of small integer numbers of beats. Ben Johnston pioneered this idea in the rhythms of his quartets. Thus a 4:3 will often be mirrored with 4 beats against 3, a 6:5 with 6 beats against five, and so on. Toby Twining picked this procedure up from Johnston, and employs it in his own just intonation choral compositions: the polyrythms consistently mirror on a larger time-scale the vibrational ratios produced at the waveform level. This is not a new idea. The masters of ars subtilitas in the 1380s played with this idea extensively and with great subtlety: and in the 1920s Leon Theremin implemented it with his now-lost instrument "the rhythmicon." Theremin's instrument responded to the presence of dancers in a space (sensed by changes in capacitance as detected by 3 sensors) and produced polyrhythms accompanied by just intonation ratios. As it turns out, just intonation intervals are very much easier to generate with analog electronic circuits than equal-tempered intervals. So Theremin's rhythmicon was an early JI synthesizer as well as an interesting example of very early integration between rhythm and tuning. In this century the massive strip-mining and subsequent exhaustion of the 12-tone equal- tempered tuning appears to have led to an increasing dissatisfaction with chronometric 18th-century rhythms. Thus the history of avant garde music in the 20th century is a history of ever less regular rhythm. First 3 against 4... most famously in the section of "Rite of Spring" where the conductor is supposed to conduct 3 beats with one hand and 4 beats in the other. Then, onward and upward to other beat-ratios. "As tonally in 900, so rhythmically in 1900, the relations 2:3 and 3:4 represented the ultimate in harmonic comprehensibility." [Seeger, Charles, "On Dissonant Counterpoint," Modern Music, Vol. 7, No. 4, 1930, pp. 25-31] Copland's 1924 piano concerto and Gershwin's Rhapsody in Blue from the same period both use extensive syncopation and hemiolas. Elliott Carter's metric modulation procedures extended the irregularity of the basic pulse, as did Bartok's essentially barline- less compositions, ditto Varese's "Density 21.5," "Arcana," "Ionisation," etc.--in many cases using a new key signature every barline. The real break came when Khaikosru Shapurji Sorabji introduced multiple embedded n-tuplets during the 30s and 40s, and when Nancarrow started stretching the tempi of multiple polyphonic lines by different rates simultaneously. You get multiple simultaneous tempo-shift curves going on that deform time in a completely plastic way: rhythmic regularity has completely disappeared. The basic pulse is continuously changing even within the individual note. This has led to rhythmic complexities like those of Michael Gordon's "Yo, Shakespeare!", or Trimpin's, computer-controlled vorsetzer works, or Warren Burt's "I Have My Standards" and "Notes From the Jungle of Intonational Complexity." It seems likely that the reason for this increasing irregularity and complexity in rhythm is that composers simply beat the 12-tone equal tempered scale to death. By the 9th century A.D. they had introduced the third as a consonance, by the 18th century they'd started using sixths as consonant intervals, by the 1920s and 1930s the major and minor 2nd and major and minor 7ths as part of the spectrum of consonance. Schoenberg's "emancipation of the dissonance" effectively placed all intervals on a continuous scale--there were no longer any "forbidden" intervals. Everything could be a consonance, depending on context--and the "rules" of consonance and dissonance could be turned upside-down, if desired. Charles Seeger's "dissonant counterpoint," introduced in 1916, was a typical example: "Dissonant Counterpoint...is essentially an inverted species counterpoint, the species of the older discipline remaining intact but *dissonance* (seconds, tritones, sevenths, ninths) becoming the norm and *consonance* (thirds, fourths, fifths, sixths, octaves) requiring preparation and resolution." [Nelson, Mark, "In Pursuit of Charles Seeger's Heterophonic Ideal: Three Palindromic Works by Ruth Crawford," Musical Quarterly, Vol. 72, No. 4, 1986, pg. 459.] The above prescription is a blueprint for post-Webern modernism up to the late 1970s: and indeed, the exhortations of Kyle Gann's music professors to "use more good solid 20th century intervals--tritones, minor seconds, major sevenths," is of course nothing but standard Palestrina species counterpoint turned inside out: the "good" intervals of the 1500s have become the "bad" intervals of the 1920s-1970s, and vice versa. Modernism did not expand the language of music, of course: there were no new intervals introduced. The list of preferred intervals had merely been swapped for the list of intervals formerly proscribed. Thus, by the 1930s there was nothing left to do with harmony or melody in the 12-tone equal tempered system. The harmonic resources had been played out. The 12-tone tuning had been strip-mined, leaving a hole in the ground and an enormous amount of bad wannabe-Webern. That left rhythm. So, starting circa 1948, composers began to explore ever more complex, ever more irregular divisions of the beat. More than one commentator has suggested that Nancarrow represents some kind of "ultima thule" for rhythmic complexity in this progression toward ever more complex time-relationships. However, this is obviously incorrect. One of the most interesting frontiers in xenharmonic composition is, in fact, the extension of rhythm in accord with the tuning of non-just non-equal- tempered scales. This is nothing more than a self- evident expansion of Charles Seeger's 1930 suggestion of "a recognition of rhythmic harmony as a category on a par with tonal harmony." [Seeger, Charles, "On Dissonant Counterpoint," Modern Music, Vol. 7, No. 2, 1930, pp. 25-31.] (Since this is an obvious extension of Seeger's classic modernist insight, naturally no academic has yet suggested it. Score another one for the same no-talent PhDs who barred the greatest tape music composer of the 20th century from the Columbia-Princeton Electronic Music Center because of his "lack of credentials"-- the composer being, of course, Tod Dockstader.) As we've seen, just intonation compositions naturally lend themselves to small integer ratios of beats (as Johnston, Twining, Partch, et al., have skilfully shown). Indeed, Kenneth Gaburo produced a composition, "Lemon Drops," using a bank of sine wave oscillators at the U. of Illinois, which uses the same principle of moving micro-ratios on the waveform level into macro-ratios on the level of time of the individual measure. (Circa 1972?) And, as we've seen, equal tempered compositions appear to lend themselves to much more complex divisions of the beat: embedded n-tuplets, metric modulation, and so on. On the macro-level of the individual measure this is very similar to approximating an irrational number with two large rationals. If you've heard a complex rhythm like 7 in the time of 4 inside 11 in the time of 9 inside 3 in the time of 2 inside 17 in the time of 13, you realize that the end result is a set of timings that sound nearly like ratios of irrational numbers--above a sufficient level of embedded n-tuplets, no underlying pulse is audible at all. This is obviously akin (on the macro-level) to the ratio of irrational numbers on the micro-level of the individual waveform which defines an equal-tempered scale, in which all pitches are some Nth root of K. By analogy, the next step in rhythmic complexity is self-evident: move to ratios of transcendental numbers on the macro-level of the beat, mirroring the non-just non-equal-tempered pitches of n-j n-e-t scales. Our present system of notating music has no way of dealing with such divisions of the beat. They are really impossible to notate with anything like conventional musical notation. 5 11 17 While regular pulses like cut time or 8 or 8 + 8 can be easily notated, and even very complex ratios made up of embedded n-tuplets *can* be written down in conventional notation, the kind of rhythmic pulsations I'm talking about here lie completely outside the range of Western notation. The system just breaks down. Conventional notation can't handle these rhythms *at all.* Let me give an example, so you can get an idea of what I'm talking about here: At a tempo of 60 each quarter note lasts exactly one second. So common time (4/4) produce measures lasting 4 seconds, with each quarter note lasting one second, each eighth note last 1/2 second, each 16th note lasting 1/4 second...and so on. A triplet 8th note would use 3 8th in the time of 2, so each 8th note would last 1/3 second. This is a simple extension of micro-ratios at the waveform level into macro-ratios at the level of the beat. A more complex embedded tuplet might require, say, a measure in 4/4 to have 11 eighths in the time of 8 eighths, with 5 in the time of 4 inside it, with 3 in the time of 2 inside that. If we had a measure like this: 4 |-------- 11:8------------------------------------------| 4 |----5 : 4---------------------| 8th 8th 8th 8th 8th 8th 8th 8th 8th 8th |----3:2--------| 8th 8th 8th Working from the outside in, the divisions of the beat are: the last 7 8th notes have a duration of 8/11 of 1/2 second = 8/22 of a second; the first 3 8th notes have a duration of 4/5 of that, or 4/5*8/22 = 40/110 or 20/55 of a second, and the 3 eight note sof hte inenrmost n-tuplet have a duration of 2/3 that, or 40/165 of a second. This is number complex enough that any underlying pulse (if audible) is quite obscure and irergular-sounding. Again, a reasonable analogy to the irrational Nth root of K ratios of equal-tempered pitches. However, moving on to non-just non-equal-tempered tuning produces a new level of rhythmic complexity, when the individual scale pitches are projected upwards into the macro-level of the individual measure. A typical n-j n-e-t scale is one of the subset of tunings produced by taking ratios of inifnite continued fractions. The fraction N1 + N2 ___ N3 + __ N4 + __ N5 + ... in general produces numbers which are neither simple integers nor Nth roots of K. For example, if N1...NJ = 1, the result is 1.61803399, or phi (the Golden Ratio). By using a very simple computer program (3 lines) to evaluate such a continued fraction out to, say, N20, it's easy to calculate the frequencies of such scale-steps to an accuracy of 7 figures. The first 5 infinite continued fractions for N1...NJ = 1 through 5 are: f1 = 1.618034 f2 = 2.414213 f3 = 3.302775 f4 = 4.236067 f5 = 5.192582 f6 = 6.162277 If we let f0 = 1.0, this gives 6 scale-steps. Now, notice that these ratios when expressed as rhythms *cannot be notated in any conventional way.* There is just no reasonable method of writing down a rhythmic system in which the longest note lasts 1.0, the next longest note last 1/1.618034, the next longest note lasts 1/2.414213, the next longest note lasts 1/3.302775, and so on. The concept is totally alien to anything in our notational convention. These kinds of rhythms just blow Western notation right out of the water. In fact, not only can we not *write music* that *notates* such divisions of the measure, at present we cannot even *talk* about such divisions of the measure with a comprehensible vocabulary. We do not even have the *words* to begin a discussion of such temporal divisions. We are, literally, mute. Why does this matter? It matters because one of the clearest and simplest compositional strategies involving non-just non-equal- tempered scales is to work with such rhythms on the level of the individual measure, then in blocks of phrases which last for times proportional to these ratios, then in sections of the work which last for times also proportional to these ratios on a larger time-scale. This continues traditional musical practice in a sensible and straightforward way: namely, by systematically extending the micro-level of frequencies up into the macro-level of the beat. In such a non-just non-equal-tempered composition, all timbres could (using Csound) easily be made up of additive sets of freuqnecies described by a non-just non-equal-tempered tuning, and all the durations of the notes *also* described by the same ratios, along with durations of sections, movements, etc. However! Trying to keep track of the rhythms is mind-bending and maddening. Because of the total inadequacy of notational systems or even vocabulary, I have been forced to notate these rhythms as delta start times: that is, note durations--which must then be added to the absolute run time (as demanded by Csound). It's *incredibly* meticulous, and requires a *great deal* of bothersome calculation. Interestingly, the rhythms sound jazzy and almost improvisational. They are not conventional. And in particular, when two or more strata of such rhythms are going at once, made of notes broken down into subvalues of these infinite continued fraction convergents, with each note-stream at a tempo also described by the an infinite continued fraction, the results are truly exotic. Next post, some suggestions for a generalized rhythmic vocabulary that would at least allow an approach to coherent manipulation of time- streams and durations derived from non-just non-equal-tempered tunings. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 9 Nov 1995 06:50 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id UAA26952; Wed, 8 Nov 1995 20:50:30 -0800 Date: Wed, 8 Nov 1995 20:50:30 -0800 Message-Id: <951109043956_71670.2576_HHB32-6@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu