Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 1 Measuring Time in Morton Feldman’s Late Music MTSMA 2020 Jeremy Tatar 10 Minutes Hello everybody, and thanks for tuning in, wherever in the world you may be! My name is Jeremy Tatar, and I ’ m a doctoral student in music theory at Montreal’s McGill University. In this presentation, I’ll be examining Morton Feldman’s rhythmic materials fr om a set - theory perspective, and draw on various resources from this analytical toolkit to explore his piece “For Christian Wol f f.” I hope that you enjoy, and don’t forget to get in touch if you want to talk further about any of this. In the description of this video you’ll find links to both a brief handout and a script, so you can follow along with either of those if you so wish. Surveying the range of analytical challenges presented by the late music of Morton Feldman, Dora Hanninen argues that the main problem is not the sometimes gargantuan length [ SLIDE] of these pieces , which range from around 80 minutes to 6 hours (give or take some change). Rather, “ t he real problem ,” she writes, [SLIDE] “ is not quantitative but qualitative: not the duration or the number of notes, but the identification of salient features that support memory and conceptualisation” ( 2004, 227). In these pieces, t he listener is faced with what Hanninen calls [SLIDE] “ a superabundance of nuance that eludes conceptualisation” (ibid.), as patterns of material are continually modified by minute adjustments of timbre, duration, and pitch. The opening of Feldman’s 1986 piece For Christian Wolff is particularly illustrative of this process, where three pitches and three timbres — flute, piano, and celeste — w eave amongst themselves in a flat landscape of triple - p. Let’s listen to the first few measures, for now without a score. , notated as part of Figure 1 in your handout. PLAY So, t aking Hanninen’s comments as my springboard, this presentation propose s an adaptation of beat - class set theory to investigat e Feldman’s rhythmic language , an area which has thus far received scant attention in analytical studies of his music . I argue that this beat - class set - theory Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 2 approach is helpful both for securing those [SLIDE] elusive nuances described by Hanninen , and for tracing the ir possible development s over the course of a piece For purposes of time, I will be focusing solely on For Christian Wolff , though there are many shared characteristics between this work and other s by Feldman from the 1980s. But f irst, [SLIDE] what do I mean by beat - class set theory? Simply speaking, I mean “regular” old set theory [SLIDE] , except using attack points with respect to a downbeat in place of pitch - classes with respect to C. We could begin by imagining a measure of 12/8 [SLIDE ] and assigning every eighth note an integer from 0 to 11 , beginning from the downbeat. A beat - class, then, is the metric location of this integer within a measure , in the same way that we understand pitch - classes as “occupying” a position in the chromatic scale ; a beat - class set is a group of these metric locations within a segmentational unit This rhythm [SLIDE] , for example, gives us the beat - class set {0 , 5, 6, 8}, [ SLIDE] since those are the integer positions where attacks fall. I mark this set with curly braces, [ SLIDE] and indicate its more abstract set - class in square brackets We are then positioned to “do” the same sort of music theory [SLIDE] that we are accustomed to in pitch - space; this rhythm [SLIDE] , {2,7,8,10] for example, is t 2 of the original pattern, while this one [SLIDE] , {0467} relates to the second rhythm by i2. M usic , of course, is written in meters other than 12/8, meaning that we often need to use modulo systems other than our familiar mod 12 arithmetic when dealing with beat - class space Two big aspects of this theory that I am leaving aside for today are quest ions of SEGMENTATION and the status of INVERSIONAL EQUIVALENCE , but I encourage you to ask about th ese in the question period Or, you know, send me an email or something. Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 3 Returning now to For Christian Wolff , Figure 1 [SLIDE] annotates each of the opening eight measures with its cumulative beat - class set in curly braces, along with the set - class of this rhythm in square brackets In measure 1, for example, attacks at positions 1, click 3, click 5, click and 6 click , reduce down to the set - class [SL IDE ] [0135] N ote that in these analyses I use the measure as a segmentational boundary, yield ing a consistent [ SLIDE] mod 9 beat - class universe due to the 9/8 meter ; Nota Bene, because this will of course affect what our math looks like. Such a beat - class set perspective allows us to describe , for example, that despite the differing surface rhythms of measures 4 and 5 [SLIDE] — attacks at 0356 vs 0368 respectively — they are connected by a shared rhythmic set class — 0136 — and related by t 3 [SLIDE] Let’s listen once again to this section , with the audio excerpt ending in the middle of measure 6 [SLIDE] The instrumental distribution of pitches in this example, where the flute oscillates between A - flat - 4 and G - flat - 5 with keyboards both on G5 , remains unchanged for the first 50 measures — about 4 minutes in clock - time . In m. 51, however [SLIDE] , the flute breaks this stasis by shifting to G5 and A - flat - 5, while the keyboard s both move to F - sharp - 5 Figure 2 shows this change in registration, which is accompanied by the first appearance of b eat - class set [0124] [SLIDE] Although the pitches G ♭ (or f♯) , G , and A ♭ still persist from the trichord established in the opening, after 50 measures the change of voicing here is almost seismic , and its coincidence with a previously un - encountered rhythmic set - class further articulates this moment of difference Measure 52 [ SLIDE] , also shares this new [0124] set - class , which is related to that of measure 51 by i 5 M easure 54 [SLIDE] likewise contains the first appearance of [0134] , elements that all work together in tandem to consolidate the “new - ness” of this musical moment. Let’s listen to this excerpt, beginning exactly where Figure 2 picks up [SLIDE] Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 4 Drawing now on a different analytical resource of set theory, this diagram [SLIDE] offers an abstr act map of voice - leading for tetrachords in mod 9 space . A line joining any two set - classes in this graph means that they are related by some single stepwise displacement : to move from [0123] to [0124] [SLIDE] is a displacement of one step , while to shift from an [0123] to an [0145] [SLIDE] requires more voice - leading “work ” With the aid of this map , we c an further flesh out our story of For Christian Wolff’s rhythmic development After beginning with the set [0135 ] in measure 1 , [SLIDE] , we soon encounter sets [0136] [SLIDE] , [0146] and [0246] , which together form a relatively tight rhythmic network over the first 50 measures The change of voicing at m easure 51 [SLIDE] , however , brings with it the new set [0124], which is significantly separated from the extant network . The markedness of this moment , which we just observed , is thus also accentuated by the “different - ness” [ slide] of its rhythm from anything that had come before i t Staying with this idea of voice - leading space, examining the moment - to - moment progression of these beat - class sets using the “transformational voice - leading” [slide] technology of Joseph Straus illuminates further aspects of the piece’s rhythmic organisati on. In Straus’s terminology, a “crisp” transformation [SLIDE] occurs when all voices move by the same amount , while a “fuzzy” transformation, marked by an asterisk, means that one or more voices deviate slightly from this amount. Straus calls this deviatio n an “offset,” which appears beneath the transformation in brackets. Figure 3 [SLIDE] presents the b eat - class sets of measures 1 – 8 from this voice - leading perspective, where a fuzzy t 0 with a small offset is the prevailing transformation between measures. If it helps, compare this against Figure 1, which you can find in the attached handout. Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 5 And while at first glance a whole bunch of t0’s might seem inconsequential, attending to their o ffsets reveals a certain progression: from measure 1 to measure 4 [SLIDE] , the “earliest” voice in the pattern , which is at the bottom of each rectangle, shifts twice from beat - class “1” to “0”, producing a total offset of “ minus 1” after three moves [SLIDE] . Following a crisp t3 between measures 4 and 5 [SLIDE] , which we noted before , the “earliest” voice now wanders from beat - class 0 to 2 in measure 7 [SLIDE] and then back to 1 in measure 8 , eventually producing an offset of “plus 1” [SLIDE] As these arrows indicate, a cumulative displacement of “one beat class earlier” in the first 4 measures , which results from the sum of the [SLIDE] three offsets - 1, 1 and - 1, is counterbalanced by the shift “one beat class later” [SLIDE] in the subsequent 4 ; t his second “phrase , ” from measures 5 to 8, could thus be understood as some form of compensation for the first. I wish to emphasise here that I am as interested in the ways that these individual motions trace a sort of journey with its own twists, turns, and switchbacks, as I am in ultimate destination that they reach. I hope that such an attitude might bring us closer to understanding the series of [ SLIDE ] “c haracteristic gestures , ” in David Lewin’s words (1987, 159) that build up this excerpt , as well as how its rhythmic surface is shaped by a relatively restrained compositional palette. So, to summarise. [SLIDE] Despite an ongoing focus on aspects of temporality in Morton Feldman’s music, the specifics of his rhythmic language remain underexplored. As I have argued in this video [SLIDE] , interpreting his rhythm s as beat - class sets helps us track larger - scale shifts and correspondences between musical events. Jeremy Tatar ( jeremy.tatar@mail.mcgill.ca ) Measuring Time; MTSMA 2020 6 Furthermore [SLIDE] , tracking the “voice leading” between these rhythmic sets helps us better understand certain consistencies in transformation at a local scale. Ultimately, such a theoretical apparatus works toward overcoming what Hanninen calls the “ qualitative difficult y ” of his music that I referenced earlier . I hope that, in the rhythmic domain at least, Feldman’s “superabundance of nuance , ” (Hanninen 2004, 227) need no longer remain so elusive. [ SLIDE] Thank you so much for listening, and have a look at the handout attached for my bibliography. Big thanks to the Music Theory Society of the Mid Atlantic, and to everyone else who helped make this research possible. Remember to send me an email, and I hope to see some of your faces sometime i n the not - too - distant future. Bye for now!