Preface
From 1998 to 2006, I worked extensively on Minimalist music as a secondary field to my main research on fourteenth-century music. Under the caring guidance of Reinhold Brinkmann, I gave several papers on the topic, considering a dissertation on Glass's Einstein on the Beach and the analysis of minimalist music.
By the end of my Ph.D., I had three mostly written articles which needed the sort of fleshing out to turn conference paper into publication. Good events, such as having many obligations at MIT, discovering computational musicology/music21, and having more to say on medieval music, conspired to make it so these articles never got polished nor made it into print in any way beyond the few people who still had handouts from random conferences where I presented the work.
It is nearly 2016 and I have not worked in minimalist circles for almost ten years now. During this time, minimalist studies have exploded: the Society for Minimalist Music has been founded, numerous conferences have taken place, and whole monographs on significant works such as Nixon in China, De Staat, and so on have been published. Minimalism has gone from being a research area to sneer at to one of the foundational parts of modern music studies. Thus, my work was becoming more and more dated with each year that I did not keep up with new bibliography, new terminology, and new discoveries.
It has become time to admit that it's extremely unlikely that I will ever work up these thoughts into a format that could be published in a significant journal. I can already imagine the "revise and resubmit" requests to cite so many people whose work is relevant to my own, but which I don't know now and was not written when I wrote the words below. I have tenure now, so formal publication is less important to my career than it was a couple of years ago. Yet I do think that there are probably some tidbits of theories here that might still be useful to someone. What I present below are unrevised (except in the case of typos or sentences that trailed off or references to video clips etc.) versions of talks given between 1999 and 2005. I would welcome comments on places where I can add bibliography and cite others who published this work first (which I will update with a note) and I apologize in advance for all the already published work that is not included. If there is interest (by a journal that does not mind that this has appeared on the web), I could revise later, but none of this information was doing anyone any good sitting on my hard drive, so might as well get it up where maybe it could help someone.
Given the best traditions of blogging, I will try to break these posts into approximately 1,000 word chunks. The label "minimalist publication project" will help find other contributions to this series as they are uploaded.
Ambiguity and Certainty in Minimalist Processes
Dublin Conference on Music Analysis, June 2005
A useful way of looking at repeated processes in minimalist music is to consider the amount of ambiguity or certainty they introduce. This view moves beyond description of the mechanisms of processes (additive, divisive, cyclic) and focuses on their effects on the form of a work or a section of a work, and from there on the expectations of listeners. I begin by discussing how some processes have the potential to create ambiguities in perception. We will then observe the opposite case: pieces where we are able to perceive order in the midst of a highly complex or seemingly somewhat homogenous texture.
(2015): The paper began with several definitions of process in minimalism which were given in expanded form in other articles. They will be put in a separate post. It ended with discussions of process in Lucier and Beethoven, which will also be put in a separate post. What remains here are the elements of the paper that fit into the narrow niche of Ambiguity and Certainty. For this reason, Figure numbers do not begin at 1.
I’m mostly going to confine myself to “top 40” minimalist pieces; the hits, in order to keep things in more familiar territory. Let us consider a passage from Glass and Wilson’s Einstein on the Beach, given in a modified score as Figure 4. The music is taken from the connecting passage between the first and second acts, Knee Play 2.
(2015): The paper began with several definitions of process in minimalism which were given in expanded form in other articles. They will be put in a separate post. It ended with discussions of process in Lucier and Beethoven, which will also be put in a separate post. What remains here are the elements of the paper that fit into the narrow niche of Ambiguity and Certainty. For this reason, Figure numbers do not begin at 1.
I’m mostly going to confine myself to “top 40” minimalist pieces; the hits, in order to keep things in more familiar territory. Let us consider a passage from Glass and Wilson’s Einstein on the Beach, given in a modified score as Figure 4. The music is taken from the connecting passage between the first and second acts, Knee Play 2.
In this section, a series of additive and subtractive processes augment and diminish the lengths of the arpeggios. Let me focus on one place where I believe a clearly defined process can produce ambiguous results.
I believe no matter what, we have to hear the passage as a gradual change in tempo. But we are given the opportunity to choose among two or more different tempo progressions. In one of these ways of hearing, the beat of the passage is tied to the repetition of contour and its emphasis on the repeated bass note. In this mode of listening, lines two and three have more but faster beats than the preceding lines. This process is described by the line marked "contour" in Figure 5 What we hear is a subtractive process—a quickening of the tempo from six eighth notes per five beat section to four eighth notes per ten beat section. The process continues now by removing a single eighth note and playing fifteen three-note beats per repetition. Then one further eighth note is removed and we are left with two-note beats.4
4 After the two eighth-note contour, I have chosen six eighth notes to be the fundamental motivic beat for the final line of Figure 5's contour analysis rather than the one eighth-note beat. This choice was based on research that showed that tempos near to or father than 300 beats per minute are usually unable to be perceived as beats. See, for instance, Simon Dixon, "Automatic Extraction of Tempo and Beat from Expressive Performances," Journal of New Music Research 30 (2001).
Intriguingly, many listeners hear the passage in the opposite way, emphasizing the chord changes as primary over the repetition of contour. This hearing is given in the line marked "harmony" in Figure 5. In this way of listening, the passage is primarily a large-scale ritardando except for the motion between passage 3 and passage 4 which is unambiguously an acceleration of beat no matter which mode of listening is chosen.*
* (2007/2015) In seven talks and classroom presentations I have conducted an experiment where I asked listeners to tap silently along with the changing beat of Knee 2 before giving this section of the paper. The results were always nearly evenly divided between those who chose the contour interpretation of the beat and those who chose the harmonic interpretation (a few listeners could not find a beat at all). In later talks I asked listeners to identify whether or not they had absolute pitch after conducting the experiment. The minority of listeners with absolute pitch always chose the harmonic interpretation over the contour interpretation, perhaps suggesting that the relationship between two motives with similar contours but different harmonies is much weaker for absolute pitch possessors than for the general population. I have been wanting to reproduce this experiment in a more formalized setting, but I admit now that this is not going to happen any time soon. In an April 13, 2005 interview with Philip Glass I conducted in Rome, I asked him about the changing beat and he said, "You mean where it goes 6, 4, 3, 2." I mentioned that some people hear it as slowing down because of the chords and he replied with some surprise, "Really?" but then returned to calmness saying that it was not too surprising because he tried to put ambiguous interpretation into his works and told the Beckett story with which this paper continues. I want to publicly thank Philip Glass for being so generous with his time and for JoAnne Akalaitis for facilitating the interview.
Ambiguity was a fundamental part of Glass’s musical philosophy in writing Einstein—he has frequently stated that he admired the quality of Beckett’s drama that each listener could experience the epiphany of the work in a different place; yet it is rare that a quality so fundamental and usually simple such as tempo can be experienced in opposing ways by different listeners. The balance that allows ambiguous perception is also fragile. The addition of a chorus sustaining the chords when the material returns in Knee 4, for instance, emphasizes the ritardando interpretation.
Stronger accidents at the beginning of each repetition of contour could have the opposite effect. Once the ambiguity in the passage is noticed, there is also the possibility of consciously or unconsciously switching-gears at any moment from the contour to the harmonic interpretation or back to create other paths of acceleration and deceleration.
Although the work uses only two rhythmic values (quarter and eighth notes) and simple diatonic intervals favoring stepwise motion and motion by thirds--in a word, simple--the melodic line is constructed with practically no repetitions among various sections. The line has enough distinct material that even very short melodic sections played by any musician and which jump out of the texture give enough information to identify where each instrument is in the melody. Figure 7 gives a list of the places where hearing two or more notes is insufficient to precisely locate a player within the sixty-five-note score.
There are fifteen two-note segments which do not identify the player's location. These represent thirty-three of the sixty-four possible two-note starting places, or about half. If three contiguous notes can be heard from a single instrument in the texture, then there are only five segments that do not identify the location of the player. In fifty-five of the sixty-three possible places, the musician's position in the score will be known to the listener. With four contiguous notes, there are only two places of ambiguity, and with five notes, the listener always can have complete certainty of a musician's location.
I am not saying that Rzewski has purposely arranged his material to create maximum distinctiveness—in fact, I had a computer program generate 1000 random melodies, sharing only Rzewski’s notes and rhythms and his predilection for stepwise and 3rd motion, and the results were similar.5 The distinctiveness of short motives is not a result of his ordering, but rather of his choice of melodic material (non-tonally oriented skips and no apparent reason behind the choice of longer notes) and the lack of any distinctive ordering. We can contrast the 65 notes of Moutons with the first 65 notes of the clarinet entrance of Mozart’s concerto. Hearing any isolated note in Mozart’s work will give you a better idea of your location in the work, but there are many more locations where hearing three, four, or even five or six notes will not pinpoint your location. A movement from a Bach solo ’cello suite would almost certainly have a lower level of distinctiveness by this metric.
Ambiguities of perception and certainty within chaotic or hard to perceive processes are essential but overlooked components of minimalist music. Considering them in the light of previous characterizations of minimalist processes can bring out the many complexities hidden within seemingly simple pieces.
In Einstein, we have an ordered and predictable process which can create ambiguities in hearing a fundamental part of the music, the beat. Conversely, some composers have paired minimalist processes which create surface disorder with compositional decisions which limit ambiguity. Frederic Rzewski’s Les Moutons de Panurge is a piece for any number of players on any instruments consisting of a single melodic line. First only one note of the line is played, then two, three, and so on until all sixty-five notes are played, after which notes are removed. See the score and the realization of the opening in Figure 6.
As normally performed, due to rhythmic errors in playing, the musicians will make mistakes that cause them to temporarily be off from one other. Unlike standard performance situations, where the musicians would then get back in sync, the score tells the musicians to remain off from the others, eventually creating a jumble of different layers.
Although the work uses only two rhythmic values (quarter and eighth notes) and simple diatonic intervals favoring stepwise motion and motion by thirds--in a word, simple--the melodic line is constructed with practically no repetitions among various sections. The line has enough distinct material that even very short melodic sections played by any musician and which jump out of the texture give enough information to identify where each instrument is in the melody. Figure 7 gives a list of the places where hearing two or more notes is insufficient to precisely locate a player within the sixty-five-note score.
There are fifteen two-note segments which do not identify the player's location. These represent thirty-three of the sixty-four possible two-note starting places, or about half. If three contiguous notes can be heard from a single instrument in the texture, then there are only five segments that do not identify the location of the player. In fifty-five of the sixty-three possible places, the musician's position in the score will be known to the listener. With four contiguous notes, there are only two places of ambiguity, and with five notes, the listener always can have complete certainty of a musician's location.
I am not saying that Rzewski has purposely arranged his material to create maximum distinctiveness—in fact, I had a computer program generate 1000 random melodies, sharing only Rzewski’s notes and rhythms and his predilection for stepwise and 3rd motion, and the results were similar.5 The distinctiveness of short motives is not a result of his ordering, but rather of his choice of melodic material (non-tonally oriented skips and no apparent reason behind the choice of longer notes) and the lack of any distinctive ordering. We can contrast the 65 notes of Moutons with the first 65 notes of the clarinet entrance of Mozart’s concerto. Hearing any isolated note in Mozart’s work will give you a better idea of your location in the work, but there are many more locations where hearing three, four, or even five or six notes will not pinpoint your location. A movement from a Bach solo ’cello suite would almost certainly have a lower level of distinctiveness by this metric.
5 A random distribution of the notes used by Rzewski was shuffled in a way that favored motion by seconds and third by having a 50% chance of reshuffling the notes if motion larger than a third was created, and a 75% chance of reshuffling if it was larger than a fourth. On average, this process resulted in 14 two-note matches per piece, 1 or 2 three-note matches per piece, a four-note match every six pieces, and a five-note match every fifty pieces
A similar effect can be heard in Satie’s oft-cited “proto-minimalist” work, Vexations, whose bass line is given in Figure 8.
Here Satie has composed the line out of extremely distinctive intervals, approaching the construction of an all-interval set. Any two consecutive pitches or intervals in the bass will uniquely identify where the performer is within the line. The piece is thus constantly sending signals about where the performer is in within this short line. The larger form of the piece is completely different, consisting of 360 repetitions of this bass line organized into 840 larger repetitions played over 12 to 24 hours. Maddeningly, these interval-based signals constantly give localized information about the position in the line but give absolutely no information about where we are in the overall form of the work.
A similar effect can be heard in Satie’s oft-cited “proto-minimalist” work, Vexations, whose bass line is given in Figure 8.
Here Satie has composed the line out of extremely distinctive intervals, approaching the construction of an all-interval set. Any two consecutive pitches or intervals in the bass will uniquely identify where the performer is within the line. The piece is thus constantly sending signals about where the performer is in within this short line. The larger form of the piece is completely different, consisting of 360 repetitions of this bass line organized into 840 larger repetitions played over 12 to 24 hours. Maddeningly, these interval-based signals constantly give localized information about the position in the line but give absolutely no information about where we are in the overall form of the work.
