Volume 19, Number 1, March 2013
Copyright © 2013 Society for Music Theory
Radiohead’s “Pyramid Song”: Ambiguity, Rhythm, and Participation
KEYWORDS: rhythm, meter, ambiguity, audience participation, internet, Radiohead, “Pyramid Song”
ABSTRACT: This article demonstrates how the confluence of ambiguity and rhythm in a pop/rock song creates a powerful force for audience participation. Focusing on Radiohead’s “Pyramid Song” (2001) as a case study, I document in detail the myriad ways listening audiences have made sense of its meter and how this knowledge has informed their interpretations of compositional intent. I conclude with further thoughts on the roles and possibilities of ambiguity and the directions it points towards mass participation and collaborative problem solving in the realms of aesthetics and music theory.
Ambiguity begets participation. (Levitin 2009, 26)
 I first heard Radiohead perform live on August 19, 2008 at the University of British Columbia’s Thunderbird Arena. I hadn’t purchased a ticket; unfamiliar with their music at the time (minus the ubiquitous “Creep” due to heavy radio play) but encouraged to attend by numerous friends, students, and professional colleagues, I wandered out into the rain from my university apartment only three blocks away and stood, umbrella in hand, for the full duration of the show just outside of the northern wing of the arena. I was in good company—a large group of students, families with children, and tourists had similarly amassed for the free musical strains—and in spite of the bad weather and bass-heavy sound making its way over the stands, I was intrigued enough to purchase their entire oeuvre on CD and begin to work through their songs, one at a time.(1)
 I made the decision early on to tackle their albums in chronological order, listening to each one a minimum of three times (and at different times of the day), without any reference to guides or record reviews, before moving on to the next. And so it took me a while to make it to their fifth full-length studio release, Amnesiac (2001). I had already noticed up to that point that Radiohead was fond of metrical shifts and other interesting rhythmic constructions, but when I came to the second track on the CD, titled “Pyramid Song,” I was completely baffled as to what the song’s underlying meter could be. With an opening that featured just a series of sustained piano chords with no drum beat or discernable regular pulse, it would take repeated listenings to begin to come up with my own rhythmic understanding.
 Feeling energized by such activity, and full of curiosity as to how others might interpret the meter, I went to the web to search out potential fellow lovers of rhythmic ambiguity, albeit with only modest expectations for what I would find. What I stumbled upon was simply astonishing: a large and rich body of material directly addressing rhythm in “Pyramid Song” that spanned dozens of web sites and hundreds of individual entries over roughly a five-year period. As an ethnomusicologist and reader of the cultural studies and sociology literature, it didn’t surprise me to find a fan base that was actively engaged in listening and constructing musical-social meaning—including online criticism and the sharing of playlists, photographs, and videos—rather than a body of passive consumers (Crafts et al. 1993; DeNora 2000; Hubbs 2008; Kot 2009). What I wasn’t quite prepared for, however, was the role that creative listening with attention paid specifically to metrical complexity played in the formation of such meaning, even among listeners with little formal music training.
 The central task of this article will be to demonstrate how the confluence of ambiguity and rhythm in a pop/rock song creates a powerful force for audience participation. As is shown in the opening epigraph, music is predicated on ambiguity in its lack of one-to-one correspondence to language, emotions, and tangible objects of human experience. Numerous writers have shown how ambiguity—understood in its full sense of encompassing uncertainty/opacity, two or more possible meanings and/or interpretations, and the blurring of boundaries—feeds our imaginative and interpretive engagement (Bernstein 1976; Copland 1980, 7; Thomson 1983; Cook 1990, 14; Cross and Morley 2009, 69).(2) Pop/rock music more specifically has often played upon vagueness and/or multiple possibilities to great critical and commercial success; it is easy to recall examples of such ambiguity with regard to race and vocal timbre (Elvis, Chuck Berry), gender (Michael Jackson, Boy George, Prince, kd lang), and even species (David Bowie/Ziggy Stardust, Lady Gaga/various human-alien hybrids). In all cases, such lack of concrete references—visual, lyrical, musical—allows a fertile space for active listeners to personalize the experience in ways that are deeply and complexly emotional, intellectual, and spiritual.(3)
 Rhythm, as has been amply documented across numerous disciplinary fields, is a fundamental aspect of music and its appreciation.(4) Meter in particular plays an essential role in our ongoing and real-time perceptual organization of music, with research demonstrating how the average listener attempts to impose a meter on whatever he or she hears (Povel and Essens 1985; Temperley 2001, 24, 205–6). Knowledge of this foundational structure—including what the beat of a song is and how to predict its recurrence—is a key ingredient in creating (positive) musical emotions, with our human concern or preoccupation with seeking out a regular pulse in musical sound most likely based on an evolutionary adaptation (Levitin 2006, 168–9, 171–80; Patel 2006; Sacks 2007, 240). In the context of the typical pop/rock song, it is the beginning or intro that creates that pregnant moment when listeners attempt to engage in the groove, to be rewarded by the “promise of entrainment” (London n.d., 10). Through an understood meter glimpses of forthcoming timbres, textures, chord progressions, and melodic fragments unfold, with the more “successful” (i.e., innovative or creative) intros “conceal[ing] enough to stimulate the appetite without blunting it” (Hennion 1983, 165). As this article will show, in the case of “Pyramid Song” where the meter itself is withheld, many listeners went the extra mile to search it out, adding additional layers of meaning and personal investment.
 Rhythmic ambiguity draws its power from connections between emotions and expectations buried deep in human evolutionary history. As David Huron observes:
Huron goes on to note that our capacity (and desire) to predict confers very specific biological advantages, including our preparedness for advantageous opportunities and the avoidance of potentially dangerous situations; accurate expectations also optimize our expenditure of energy and help focus our attention in economical ways (2006, 3, 176). We see such factors at play in the writings of music theorists who have long noted that musically ambiguous or even suspenseful events are often pleasant for a listener, depending on how long it takes for clarification or resolution to occur (e.g., Meyer 1956, 27–28). Similarly, Daniel Levitin has made the analogy between such resolutions and joke telling, since in both cases the listener enjoys being led (temporarily) astray and realizing that there is more than one way to proceed through a piece/passage/story (2009, 110).
 The analysis and insights in this article complement and develop ideas from Mark Butler’s Unlocking the Groove (Butler 2006), particularly in his focus on metrical ambiguity, audience participation, and the specific role that underdetermination (metrical vagueness)—the form of ambiguity featured in “Pyramid Song”—plays in fostering interpretive multiplicity.(5) Butler engaged in ethnographic fieldwork, an approach he openly credits to his studies in ethnomusicology (2006, 26–29); this article similarly draws upon ethnography (as well as fan reception studies) and its application to music theory and analysis. My research diverges from Butler’s, however, in the following ways: (1) the responses of my studied subjects are drawn almost exclusively from the internet; (2) these responses form the central text in my analysis (whereas in Butler’s work such remarks serve more as occasional commentary); and (3) my focus is on active listeners, who may or may not be dancing as they experience the music. Our research parameters nevertheless encompass the same domains, as stated in the opening to Butler’s second chapter: “I will be concerned with rhythm and meter in a broad sense: with the ways that those who listen to the music have described and categorized its rhythmic and metrical attributes, and with music-theoretical concepts that can clarify these phenomena as they occur in this repertory” (2006, 77).
 Ethnographic research focused on internet communities has particular value for the expansion of existing music-theoretical and cognitive science models. As the responses in my sample will clearly illustrate, audiences are no longer homogenous (if they ever were) due to the eclectic nature of modern listening, continued travel and immigration, and exposure to and knowledge of non-Western musical genres and rhythmic constructions. Categories such as “Western listener” or “youth audience” are thus problematic on a number of fronts, calling into question many assumptions and protocols in (Western) analyses of expectation and entrainment. Internet research engages audiences on their own terms—where they provide their own analytical-interpretive language, instead of having to choose from pre-established categories (a norm in academic research questionnaires)—and in their natural places of experiencing the music (often within the home from their computer, stereo, or mp3 player, rather than in a university lab). Re-presenting these views in their entirety, preserving the “grain” of the individual voice, reflects as well a long-held ethical imperative within ethnomusicology and anthropology not to speak for others.
 As Butler also notes, the use and integration of field research can serve as an alternative to music theory’s tendency to emphasize individual analytical engagements (2006, 27). Including audiences’ views from dedicated web sites thus signals the move away from typical author/lone-analyst centered music analysis toward a more interactive and collaborative research model. The potential benefits of such dialogic encounters are at least twofold: (1) concrete examples of audience responses expose the richness and complexity of fans’ listening experiences; and (2) broader theoretical perspectives or possibilities are allowed and encouraged through a decentralized approach to analysis and interpretation. In the context of the present research, a music-theoretical sophistication or savvy on the part of audiences emerges that has as yet gone unacknowledged in the literature on fan cultures and the media,(6) at the same time calling out for the tools of music theory to standardize and thus compare these views.
 I begin by providing a general context for understanding the place of “Pyramid Song” within the broader Radiohead repertoire—historically, textually, visually, and musically. The main body of the article then outlines in some detail the myriad ways listening audiences have made sense of its meter—or lack thereof—in the absence of reliable published scores and expressed intent by the band.(7) The primary goal is to document the activities of a community that coalesced around a riddle of metrical ambiguity. I will also draw attention to the fans’ expressions of their interpretation of meter as a key to unlocking semantic content, as well as the sources of authority on which they drew. I conclude with thoughts about additional research on musical ambiguity and further possibilities for mass participation and collaborative problem solving in music theory and aesthetics.
 Radiohead is Thom Yorke (lead vocals, guitars, piano), Ed O’Brien (guitars, backing vocals), Phil Selway (drums/percussion), and brothers Jonny Greenwood (guitars, keyboards) and Colin Greenwood (bass). Formed in Abingdon, Oxfordshire in 1985, the band has remarkably kept the same personnel over its roughly 27 years of existence, though members’ roles have changed over the decades, as has their arsenal of musical instruments (including electronics). As of 2013 Radiohead has released eight full-length studio albums, six under contract with Parlophone/EMI and the last two independently via the label tbd records (see Example 1).(8) Generally classified as “alternative rock” (a designation now almost meaningless), they are acknowledged and admired for their eclectic influences ranging from classic rock, jazz, post-punk, and the avant-garde to electronic music (including techno and post-serialist studio work). Radiohead is regularly featured on fans’ and critics’ top choice lists, and they have won numerous prizes on both sides of the Atlantic, including three Grammys for Best Alternative Music Album.(9)
 As is apparent from Example 1, a number of consistencies appear across the albums when viewed as a composite whole. For many critics, fans, and even the band members themselves, the “true” Radiohead sound and approach began to emerge with their second release, The Bends (1995). Sonically it introduced the work of Nigel Godrich as a recording engineer, brought on under the producer John Leckie. Godrich’s ear and work habits were such that Radiohead would ask him to be their producer on the subsequent OK Computer (1997), a position he has held for the remainder of their recorded work (Godrich even joined Thom Yorke as a keyboardist on his solo tour in 2009–10 with the band Atoms for Peace). The Bends also introduced to the world the artistic vision of Yorke’s personal friend Stanley Donwood (b. Dan Rickwood), a fellow art student from University of Exeter days, who has been responsible for all of Radiohead’s cover and liner art since (see Donwood and Tchok 2007).(10) And while it is clearly reductionist to subsume the band’s entire lyrical content under a single emotional rubric, one researcher has noted the prevalence of anxiety spanning the output from the The Bends through to Hail to the Thief (2003), a catchall that includes the recurring themes of broken relationships, paranoia, powerlessness, fear of technology, mistrust of government, physical and emotional brutality, suicide, death, hell, and alien abduction (Letts 2005, 85–87).(11)
 The Bends also critically marks a tacit agreement between Radiohead and its serious fans that ambiguity would form a central pillar on which they would build their collective and ongoing working relationship. Intentional vagueness and/or multiplicity of meanings in the artwork, lyrics, musical references, web sites, and videos created a fan culture (and significant academic base) that routinely pores over the imagery, sounds, and words looking for clues and hidden meanings: “such deliberate ambiguity has become a hallmark of Radiohead’s presentation and has created a community of fans joined by their shared interpretations of the band’s projects” (Letts 2005, vi; see also Forbes and Reisch 2009). The academic and popular press now routinely speak of Radiohead’s reliance on ambiguity (Tate 2005a, 1–4), obscurity (Paytress 2005, 4; Clarke 2010, 65, 95–96), and the enigmatic (Fricke 2001a, 44), with “the band’s reflexive esthetic effectively disrupt[ing] naïve consumption, confronting the listener with music and art that adheres to opacity versus authenticity as a guiding principle” (Tate 2005b, 115). In an interview with the music critic Alex Ross, Radiohead—in an uncharacteristically straightforward manner—likened their work to the solving of musical puzzles:
It is also a culture built on respect and trust: as has now been amply documented, in late 2007 with perhaps their greatest gesture of invitation to the fans, Radiohead released In Rainbows digitally such that fans could choose what to pay for the download (including nothing at all), could create and donate music videos, were given access to some of the music files to create their own unique pieces, and were allowed to write their own reviews before large outlets and the press were sent the standard media packet (see Lawson 2009, 62–79 and Randall 2011, 235–40, 235–40, 254–56).
 Ambiguity was a special hallmark of the pair of albums Kid A and Amnesiac, recorded during the same sessions but released eight months apart as separate projects. In both cases the musical roles of the band members became newly obscured or obliterated, with Radiohead choosing to leave their guitar-based sound behind in favor of electronic experimentation and non-traditional lineups such that it was often impossible to match a particular sound with a specific individual (band members also switched instruments, or were absent on some tracks). Even Thom Yorke’s distinctive voice was manipulated or masked by studio techniques so that the comprehension of words and at times entire phrases was made extremely problematic (Hainge 2005, 63). Such opacity was enhanced by Radiohead’s choice not to include lyrics with Kid A (a first for the group), and to provide only snippets of text—some of it from sung lyrics, others not, but all out of order—on Amnesiac; Yorke admitted in an interview with The Wire that it was his desire that listeners not focus too carefully on the words with this pair of releases (Reynolds 2001, 26). As former Guardian music critic Garry Mulholland acknowledged, “No one else on Radiohead’s creative and commercial level makes music like Amnesiac and Kid A that felt so open to individual interpretation [and] gave so much credit to their fans for an adventurous ear” (2006, 348; see Collins 2004 and Letts 2010b for further discussion of similarities and differences between these two albums [Collins also compares their relationship to the album that followed]).
 Amnesiac was initially released in three formats: a standard CD, a deluxe box set (standard CD, CD with B-sides, DVD, and three postcards), and a Special Edition Book (CD with bound book, winner of the 2001 Grammy Award for Best Album Package). Where Kid A was essentially sprung on the world without the usual fanfare of singles, videos, or concerts to promote the album (it nevertheless shot to Number 1 on the charts, and won the Grammy for Best Alternative Music Album), with Amnesiac Radiohead decided to return to more conventional practices by releasing three singles with B-sides (“Pyramid Song,” “I Might Be Wrong,” and “Knives Out”; refer to Example 2), producing a number of videos, and embarking on a world tour.
 Before even taking the standard CD out of the case, audiences are confronted with—and ultimately challenged by, if so choosing—the packaging of Amnesiac. On the cover of the booklet is the picture of an orange/red-colored book with a damaged spine; the only design is a cartoon figure with its hands over its eyes, crying, superimposed on what appear to be star charts.(13) Once the booklet is opened, the viewer is presented with 26 separate artistic works that include drawings, sketches, cartoons, photographs, computer-generated art, x-rays, and hand-written and typed excerpts of texts, with no apparent connection or identification provided. The cartoon character from the cover does reappear on some of the pages (one only discovers that this is a “weeping Minotaur” if one goes to Radiohead web sites or reads print interviews),(14) but overall its presence seems only loosely related as a kind of general commentary on a collective ethos of sadness, confusion, destruction, and despair reflected in images of gravestones, skeletal faces, fires, otherworldly figures, and nuclear explosions. Such mysteries are compounded for the fan who also buys the box set and special edition book: apparently some of the art work matches specific songs (the postcards in the box set make connections to “Pyramid Song” and “Knives Out”), and the standard CD booklet is missing six of the special edition book’s prints (begging the obvious question, Why?).(15) And then there are those text excerpts, requiring the enthusiast with additional energy to first seek out the lyrics on an unofficial web site, then try to match them against what is found.(16)
 While a later analyst argued against thinking of Amnesiac in terms of a concept album, because of “material that requires effort on the part of the listener to make sense of, yet offering little to no plot, characters, or dramatic action to clarify the concept” (Letts 2005, 167), both Yorke and artist Donwood offered enticing, if not characteristically ambiguous, commentary on their project:
 It is against this backdrop of richly and complexly textured imagery, themes, and texts that I now turn to the track “Pyramid Song,” and to the myriad detailed audience responses elicited by its metrical challenges.
 Known alternatively as “Egyptian Song” and “Nothing to Fear” during its gestation period, what came to be called “Pyramid Song”—debuted in Amsterdam at the 1999 Tibetan Freedom Concert—is now considered by the band and its audiences to be one of their finest works (Paytress 2005, 59). The choice of the final title has been scrutinized by fan and critic alike—nothing new for Radiohead—as there are no direct references to Egypt or pyramids in the lyrics (there is, however, a sizable community that believes it has found pyramid-like dimensions in the metrical structure of the song, to be addressed below). Many theories abound: (1) Yorke immersed himself in Tibetan and Egyptian books of the dead in preparation for the Amsterdam concert, which could account for lyrical references to rivers, boats, angels, and the heavens (Hale 1999, 116–17); (2) musically the song was inspired by Charles Mingus’s song “Freedom” and its subject matter relating to the flight of Moses and the Israelites, hence the earlier reference to Egypt and rivers (Letts 2005, 136); (3) the last line of the verse is nearly identical to content found in Tom Waits’ 1985 song “Clap Hands,” which itself is lifted from “The Clapping Song” by Shirley Ellis (1965), which refers back to the handclaps in Mingus’s “Freedom” of 1963 (http://www.greenplastic.com/radiohead-lyrics/amnesiac/pyramid-song/); and (4) the lyrics suggest an Egyptian funeral (Pareles 2001, 74; Dimery 2006, 883). Yorke’s own musings on the album title are also suggestive: “I read that the Gnostics believe when we are born we are forced to forget where we have come from in order to deal with the trauma of arriving in this life. I thought this was really fascinating. It’s like the river of forgetfulness” (quoted in Fricke 2001b, 25).
 “Pyramid Song” is composed of a single verse with refrain (“there was nothing to fear and nothing to doubt”) played twice.(17) For the first statement of the verse there is only Yorke singing, accompanying himself on acoustic piano. An orchestral string section enters with the intro to the second verse, then drums followed by upright bass provide a context for interpreting the song’s meter (the overall texture later enhanced by the sounds of the ondes Martenot).(18) What first attracted me and many others to this track, however, was the opening of the song where little is provided for the listener in terms of rhythmic orientation. There is no “beat” or regular pulse, no percussion or text, just a series of piano chords played at an even dynamic level with (slight) pauses occurring after the third and eighth chords (the pauses are repeated structurally at the same points throughout the remainder of the song). Most listeners identify this series as a kind of cycle divided into two phrases of five chords each because of the upward and downward movement of the bass line and inner voices and the parallel placement of pauses after the third chord in each group of five. And most focus on the third or fourth iteration of the cycle as a kind of model on which to base their analyses, most likely because these versions are played more than any other.(19) (Notice that the holdover or suspension of the G natural in the top line of the piano from the end of bar 2 into bar 3 creates a kind of temporary structural/harmonic ambiguity.) In Example 3 I have provided a piano reduction of the first four iterations of the piano cycle; for readers unfamiliar with the song, I have looped the fourth iteration in the audio. It should also be noted that many respondents commented on the cycle’s “gypsy” feel due to its adherence to a standard I–
 The opening of “Pyramid Song” is a clear example of ambiguity of metrical type featuring underdetermination, as defined by Butler: “Underdetermination usually occurs when one or more layers of motion needed to make a decisive metrical interpretation are absent; in such a case, the meter is ‘not clearly defined’ as in the first sense of ambiguity described by the OED” (2006, 129–30).(20) The Radiohead introduction also fits especially well within Justin London’s category of “vague metric context” under his taxonomy of metric ambiguity: “[I]n metrically vague situations there is a discernable sense of regularity, but the listener is stymied when he or she tries to construe any particular metrical organization” (2004, 85–86). It is this lack of metrical clarity, taken in conjunction with its relationship to the later entrance of the drum set, that has provided the impetus behind the outpouring of listeners’ interpretations, not only of the meter but also of Radiohead’s underlying semantic intent. What is important for me here is the wide range and inventiveness of perspectives and meanings generated by these audiences as a potent indicator of ambiguity inviting participation. Looking to maintain a decentralized approach to analysis and interpretation, it is not in my present interests to advocate for any particular viewpoint. What is central to this process (for reasons outlined in the Introduction) is to let respondents speak in their own voices, so that the original feeling and texture of the analytical and emotional content remains intact.
 To this end, I have provided below the unedited responses of participants drawn from a pool of six print publications, 13 websites, and 261 individual web entries logged between 2007–11 related to rhythm and meter in “Pyramid Song” (this number runs to well over a thousand if all threads addressing this song and/or Amnesiac are included).(21) These responses—chosen as representative examples of a much larger base—are organized into nine categories of metrical interpretation to facilitate comparison: firstly by the existence/absence of a meter, and secondly by the quality of the meter. To further cut down on redundancy, I have used the following method for marking web entries: “FB Stephen” identifies an entry by “Stephen” (in some cases these are real names, in others an assumed one) from the Facebook (FB) page titled “Time Signature in Pyramid Song???” (refer to the Web Sites section at the end of this article for the full list of abbreviations and corresponding page titles).(22) It is worth noting that while “participation” for many of the respondents represented “just” listening and commenting, there were also a number of instances of fans attempting to play the music themselves, demonstrating various instrumental lines on YouTube, producing original-content video, or running the song through sound-analysis software.
Opening Philosophical Musings
 The stakes, at least for one philosophically minded listener, are high—extending beyond this individual song to the sources of individual temporal experience. In his contribution to the tome Radiohead and Philosophy, Michael Thompson begins with an observation that perfectly encapsulates my own initial encounter with the song:
From here Thompson goes on to provide a more ambitious explanation for the song’s ability to challenge a listener’s sense of self, time, the world, and even the afterlife:
Two Separate Songs
 Most listeners made attempts at exacting a rhythmic structure and/or meter—either right from the beginning or based on the interaction of drums and piano in the second iteration of the verse—and then assumed that this structure was applicable to the song in its entirety. While this was my own strategy (and, generally, the way I listen to this song), the following respondents were happy to think of the two verses as inhabiting different realms, almost as if they were two separate songs linked together by a common text (without going any further to establish a unifying, underlying rhythm):
After having spent numerous listenings perfecting my tapping abilities to an underlying but hidden pulse (see below), I now also enjoy hearing the first time through the verse as a kind of rhythmic dream state without a beat, allowing myself to be surprised by the entrance of the drums the second time through.
A Regular Meter
 As mentioned above, the majority of respondents felt that there was some kind of regular meter that applied to “Pyramid Song” from beginning to end. Radiohead on the whole has been silent on this issue,(23) though a quote attributed to Thom Yorke suggests a circular orientation that could be construed as a reference to a regular meter:
What is more significant about a unifying metrical structure in the context of this song is that it requires the listener to abstract out a silent or “hidden” beat (at least at the beginning) which is not provided by the piano chords or singing—a very difficult task, even for professional musicians. Parallels here with African drumming and its rhythmic orientation are striking: John Blacking speaks of “the rhythm of an invisible conductor” (1973, 30), Simha Arom conceives of a “mould” into which surface structures would fit (2004, 18), and Richard Waterman describes an additional (unheard) rhythm against which an obvious surface structure pulled (1952, 211–12;12; see also Chernoff 1979, 49–50). Justin London further identifies this phenomenon in “Pyramid Song” as a kind of “metric fakeout” which “starts with [a] complex beat pattern that then becomes [the] rhythmic figure against [a] ‘correct’ meter” (n.d.). This explanation is similar to David Huron’s application of the “garden path” phenomenon to music in which someone might feel s/he understands an utterance, but must then reanalyze it in light of a later happening (in this case, the entrance of the drums; 2006, 279–81).
No Specific Meter Identified, or Respondents Unsure
 One of the first responses I encountered on the web essentially matched my bare-bones description of the piano chords at the beginning of this article, a view that seemed to assume that all the chords were the same length (no syncopation) with a kind of artistic loosening of the tempo (rubato?) applied to every third and eighth chord:(24)
A number of listeners, however, were content to just “feel” the piece, regardless of what the meter might be:
 Numerical ratios also captured the imagination of a number of individuals; while certain meters are implied by such discussions, I have included the following ruminations more for their relationship to the title and deeper meaning of the song (these ratios will appear again, and will be fully explicated, in categories that follow):
 Other listeners had more philosophical and/or creative ways of accounting for the feeling and structure of the song:
And my sympathy goes out to the following individual:
Non-Isochronous Meters: Alone and in Combination with Simple Meters
 Before moving on to more common simple and compound meter constructions, I begin with examples of non-isochronous meter interpretations, both alone and in combination with simple meter interpretations. A non-isochronous meter, as defined by London (2004, 100–15), refers to a collection of asymmetrical pulse lengths where the integrity of the subtactus is maintained (such as in the example ). My assumption with the responses that follow, therefore, is that each number in a non-isochronous grouping refers to a single piano chord placed metrically (i.e., aligned with a pulse, not syncopated against some other underlying meter).
 Non-isochronous meter discussions centered on the possibility of 11/8. In each case the 11/8 interpretation remained consistent, each phrase of the piano cycle (five chords) envisioned as composed of five pulses with the first, second, fourth, and fifth chords being equal in length ( ):
On the surface the core interpretation of combined non-isochronous and simple meters created by listeners—7/8, 3/4, 5/8, 2/4—bears little resemblance to the 11/8 examples. However, if one doubles the 11/8 phrase to cover an entire piano cycle (22 eighth notes in length)—maintaining the same pulse length relationships—then what one discovers is an astounding yet logical way of dividing the cycle that breaks across the symmetrical model of two bars of 11/8 but is still composed of ten pulses arranged in a way that mirrors the harmonic rhythm of the first iteration of the piano cycle (especially the changes at Aadd6, Gmaj7, and finally Gmaj). Written in eighth note groupings, the full cycle is represented by 2+2+3, 2+2+2, 2+3, 2+2; written using time signatures, we are presented with , 3/4, , 2/4 (see Example 4):
 For the majority of respondents “Pyramid Song” was in a more common compound or simple meter. Within the compound meter camp, many individuals cited a “jazz” or “swing” feel to the entrance of the drums at the second verse to support their interpretation of metrical intent (and many web sites and blogs made references to Yorke’s inspiration in Charles Mingus’s jazz classic “Freedom”):(26)
Example 5 provides a simplified “compound meter” realization of the entrance of the drums without any grouping (or “measure”) indications within each set of five chords (listen to the audio with Example 5). Two important features distinguish all compound meter interpretations structurally from those of the non-isochronous type: (1) the first two piano chords (as well as a number of subsequent chords) are viewed as unequal in duration; and (2) a number of chords are felt as syncopated against a stream of symmetrical pulses.
 A number of individuals were happy with a straight 9/8 or 12/8 organization:
A larger group, however, had fun employing a cycle with alternating compound meters:
 Because of the recurrence of certain numeric ratios in the previous discussion (3–2–3, 3–3–2, 4–5–6), but also to be able to compare these ratios with time signatures that follow, I have provided in Example 6 a 9/8, 9/8, 6/8 realization of the first phrase (opening five piano chords) of the third iteration of the cycle. The top row of boxes represents the smallest rhythmic unit implied by this meter (the eighth note), bracketed off into groups of 9, 9, and 6 (9/8, 9/8, 6/8); immediately below this I have added slurs to indicate possible groupings of the piano chords. I say “possible” in that only one of the responses provides specific piano chord lengths (4, 5, or 6 eighth notes), and that this only makes sense when corroborated with another entry that dictates that one should “syncopate” the chords (even with this knowledge, I had to exercise some interpretive license). Below this I then provide the chord qualities, with the remaining two rows of boxes showing groupings by beat (according to chord changes) and eighth note lengths of the piano chords. Two items of note: (1) beat groupings by chord change (3–2–3) do not coincide with the bar lengths (3–3–2), unless one chooses a 9/8, 6/8, 9/8 cycle; and (2) 4–5–6 only makes sense when looking at the eighth note groupings for the piano chords but written in the order 5–4–6–5–4. A listening challenge in the Appendix will help readers explore the metric interpretation of Example 6, as well as that of Example 7 (presented below).
4/4 with Swung Eighth Notes
 In keeping with common practice in jazz notation (and, to a certain extent, pop), a number of respondents chose to describe the meter in “Pyramid Song” as 4/4 with the instructions to “swing” the eighth notes. While a number of writers and theorists refer to “swing” in such a context to describe expressive timing (including Butler 2006, 22), most of the respondents below used such terminology to indicate what they felt was essentially a 12/8 compound meter (and, in fact, a number of scores referenced gave direct instructions to play each eighth note pair as a quarter-eighth triplet(28)):
This last response identifies clearly what I set out previously as distinguishing features of the compound meter interpretations: (1) the first two piano chords are unequal in duration (the first is longer than the second), and (2) the chords are felt as syncopated against a stream of symmetrical pulses (see Example 7, a rendition of the first phrase written out in 12/8 time to illustrate visually what happens when swinging the eighth notes). In such a “swung” (12/8) interpretation there is further syncopation with the metrically consonant third and fourth chords landing on weak beats (beat 4 and beat 2, respectively), as well as beat groupings by chord change that do not coincide with the bar lengths (3–2–3 versus 4–4).
 When comments begin to address specifics, however, the language and numerical ratios become more obscure. For the following entry, the note lengths of the piano chords are correct when writing out the swung eighths in full (here the sixteenth note is considered the smallest rhythmic unit, but with the same numeric results):
For the remaining entries and published scores one must look at a strictly 4/4 rendition (no swung eighth notes) to understand the numbers, which I have provided in Example 8. This then accounts for the sequence “33433” (refer to the row of boxes showing eighth note groupings for the piano chords), and its related “1.5 beats” (3 eighths) and “ 2 beats” (4 eighths), and/or dotted quarter note (3 eighths) and two tied quarter notes (4 eighths):
4/4, 1 Complete Cycle over 4 Bars (16 Beats)
 If any conflict or rivalry exists within the “Pyramid Song” metrical world, it is felt most strongly by those aligning themselves with the “straight” (simple meter) 4/4 camp versus those who adhere to some kind of compound meter. While I have no way of evaluating the view from the opening philosophical musings that those who “see life as a sequence of [a] small cluster of events with strange emphases” will recognize “a simpler time signature [4/4]” (Thompson 2009, 223), music theorists and cognitive scientists have documented how Western listeners carry with them a default binary expectation—because of a predominance of binary meters in Western music (especially popular genres)—which they will generally apply when confronted with an unknown rhythm (Brochard et al. 2003; Huron 2006, 194–95).(29) Precisely whether or not this applies to the listening perceptions of the 4/4 contingent is beyond the scope of this article; in most cases, the stated reasons for the simple meter choice was one of pragmatism: it was easiest to think of it this way, and the beats added up most directly to bars of four beats. In order for this interpretation to work, however, one of two viewpoints must be held: (1) the first verse (without drums) is played in its own (simple) meter in a different way from the second verse (an acknowledgement of a change to compound meter); or (2) the drum part is heard as binary (ternary divisions of the beat are not recognized). Either way, significantly the opening two piano chords are seen as equal in duration.
 A number of respondents identified a 4/4 meter, but without specifying the length of the entire cycle:
Most web entries, however, described or notated the full cycle—ten piano chords, returning to the
The strongest argument for 4/4 was made by an unnamed listener who created a YouTube video with the first iteration of the piano cycle looped over a click track to “prove” his 4/4 explanation (see Example 9).
 It is also interesting to note that a number of individuals heard the cycle beginning on the third (longer) piano chord, such that the song then began on the last two beats of the previous cycle:
4/4, 1 Complete Cycle over 2 Bars (8 Beats)
 Identifying a pulse or beat to tap along with is of course a subjective enterprise, although there are some parameters by which most of us seem to adhere (see Huron 2006, 175–76; Martens 2011). While a minority view within the 4/4 group, a significant number of listeners nevertheless heard “Pyramid Song” at exactly half the speed of the above category, so that a complete cycle took only 2 bars to complete:
Mentally it is easy to just halve all of the values found in Example 8; but for ease of comparison—and for a clear visual representation—I have provided a realization of the entire cycle in Example 10.
 In this final section I have documented various mixed meter interpretations, mixed meter understood as a stream of perceptually isochronous beats with asymmetrical groupings by bar (such as in the example 3/4, 5/4, 2/4). Such responses often resulted in unusual and/or challenging breakdowns, though in every instance the larger integrity of the phrase (8 beats) or entire cycle (16 beats) was maintained, as was a feeling that the first two chords are equal in length (the second chord syncopated against the pulse stream).
 The following respondents felt a significant break at the Aadd6 chord (sixth beat in 4/4 meter), similar to the non-isochronous grouping illustrated in Example 4 (b):
Others felt the first significant break to occur with the Gmaj7 chord (fourth beat in 4/4 meter):
And numerology and the pyramids also found their way back into the fray:
 Taken as a whole, the community that coalesced around the rhythm/meter of “Pyramid Song” represents a diverse and energetic body that demonstrated considerable patience, sophistication, and interpretive insight. In looking to solve a musical riddle, their arsenal of tools drew on music theory, structural formulae, notation, intuition, numerology, mathematics, cosmology, history, philosophy, technology, classical training, help from friends and teachers, and perspectives from non-Western music. Their responses reflect a blend of curiosity, adventure, posturing, passion, deference, and humor. And while this article only focused on those who were self-consciously reflective and vocal about the metrical aspects of this song, there are of course many others who didn’t or couldn’t identify the rhythm but who nevertheless felt something special about this composition, a kind of deep affinity for something textually, visually, and (crucially) musically that in combination created a work that was deeply moving and otherworldly:
 Ambiguity in the form of underdetermination created a rich forum for participation as listeners grappled with the song’s rhythmic organization and developed strategies for entrainment—satisfying biological and cognitive needs. Ethnographic research focused on internet communities brought these perspectives to light, though the tools of music theory were required to make sense of many of these interpretations. Radiohead’s use of ambiguity, however, extends much further into their compositional and performative strategies, encompassing the realms of lyrics, song forms, chord progressions, vocal timbres, imagery, and human-technology interfaces. A full account of the ways these various components intersect, diverge, and play off one another provides many fruitful avenues of research yet to be explored.
 Whether one appeals to the “wisdom of the crowds” (Sorowiecki 2004), mass collaboration (Tapscott and Williams 2006), or “crowdsourcing” (Howe 2008), in structure and function the increasing reliance on and participation through the internet by large communities of like-minded individuals signals an era of mass participation and collaborative problem solving that has come to characterize our age. This is as much a political as it is a pragmatic reality: by reading through and mulling over entries by listeners with highly varied training, education, and life experiences, I was presented with a much broader view of the possibilities—in fact, listener realities—than I would have found on my own. The open-ended, interactive, and inclusive nature of many of these web sites has created public forums for debate that to my mind indicates a healthy new direction and long overdue reinvigoration of musical criticism and aesthetics, at the same time suggesting alternative, decentralized analytical approaches to musical works.
 Musings on ambiguity remind us that often the best songs are those worth revisiting. These are the melodies, structures, and words that continue to challenge and surprise us over the years, the ones that reward our efforts and personal imprint. They are also a mark of our imagination, of our need for ambiguity and the space it affords our feelings and personal stories:
In Examples 11–13 below I have provided mp3 files with the following content:
Readers can listen to any one of these tracks in isolation or combination to challenge themselves to (1) hear the placement of the other parts, or (2) develop their own interpretation of the meter. Conversely, one can listen to Example 13 while viewing Example 6 (9/8, 9/8, 6/8) or Example 7 (12/8) to see if such realizations match or diverge from one’s own metrical understanding.
Nathan D. Hesselink
Brochard, Renaud, Donna Abecasis, Doug Potter, Richard Ragot, and Carolyn Drake. 2003. “The ‘Ticktock’ of Our Internal Clock: Direct Brain Evidence of Subjective Accents in Isochronous Sequences.” Psychological Science 14, no. 4: 362–66.
Brown, Steven, Björn Merker, and Nils L. Wallin. 2000. “An Introduction to Evolutionary Musicology.” In The Origins of Music, ed. Nils L. Wallin, Björn Merker, and Steven Brown, 3–24. Cambridge: MIT Press.
Cross, Ian and Iain Morley. 2009. “The Evolution of Music: Theories, Definitions and the Nature of the Evidence.” In Communicative Musicality: Exploring the Basis of Human Companionship, ed. Stephen Malloch and Colwyn Trevarthen, 61–81. New York: Oxford University Press.
Lawson, Selena Michelle. 2009. “Radiohead: The Guitar Wielding, Dancing, Singing Commodity.” Communication Theses. Paper 47. http://digitalarchive.gsu.edu/communication_theses/47
—————. 2010a. “ ‘I’m Not Here, This Isn’t Happening’: The Vanishing Subject in Radiohead’s Kid A.” In Sounding Out Pop: Analytical Essays in Popular Music, edited by Mark Spicer and John Covach, 214–44. Anna Arbor: University of Michigan Press.
Lewis, Justin. 1994. “The Meaning of Things: Audiences, Ambiguity, and Power.” In Viewing, Reading, Listening: Audiences and Cultural Reception, ed. Jon Cruz and Justin Lewis, 19–32. Boulder: Westview Press.
Linder, Brian. 2009. Online IGN Review of Radiohead’s work (March 24). http://music.ign.com/articles/965/965634p1.html
—————. n.d.(b). “Metric Fake Outs” (Excel spreadsheet). http://www.people.carleton.edu/~jlondon/
Mazokopaki, Katerina and Giannis Kugiumutzakis. 2009. “Infant Rhythms: Expressions of Musical Companionship.” In Communicative Musicality: Exploring the Basis of Human Companionship, ed. Stephen Malloch and Colwyn Trevarthen, 185–208. New York: Oxford University Press.
Osborne, Nigel. 2009. “Towards a Chronobiology of Musical Rhythm.” In Communicative Musicality: Exploring the Basis of Human Companionship, ed. Stephen Malloch and Colwyn Trevarthen, 545–64. New York: Oxford University Press.
Thompson, Michael. 2009. “The Signature of Time in ‘Pyramid Song’.” In Radiohead and Philosophy: Fitter Happier More Deductive, ed. Brandon W. Forbes and George A. Reisch, 221–28. Chicago: Open Court.
Waterman, Richard Alan. 1952. “African Influence on the Music of the Americas.” In Acculturation in the Americas (Proceedings of the Nineteenth International Congress of Americanists), ed. S. Tax, 207–18. Chicago: University of Chicago Press.
Web Sites(all web sites last accessed December 2012)
Web Sites(all web sites last accessed December 2012)
8N (http://www.8notes.com/school/riffs/piano/radiohead_pyramid.asp), “Pyramid Song Piano Tab”
FB (http://www.facebook.com/topic.php?uid=6979332244&topic=5330), “Time Signature in Pyramid Song???” [web site no longer active]
FH (http://www.freehandmusic.com/sheet-music/pyramid-song-290128), “Pyramid Song (Digital Sheet Music)”
GPL (http://www.greenplastic.com/lyrics/pyramidsong.php), “Radiohead Song Lyrics and Info” [web site no longer active]
GPT (http://www.greenplastic.com/lyrics/tabs/pyramidsong.txt), “Radiohead Pyramid Song (Piano Tab)”
MG (http://mortigitempo.com/really_bored/index.php?/topic/78491-what-do-you-count-pyramid-song-in/page__st__40), “What Do You Count Pyramid Song In?” [web site no longer active]
PS (http://www.youtube.com/watch?v=Ydc2HYjhO20), “Pyramid Song 4/4 Explained.” [web site no longer active]
PS2 (http://www.youtube.com/watch?v=zbKQPqs-cqc), “Pyramid Song”
RM (http://www.rockmagic.net/guitar-tabs/radiohead/pyramid_song.btab), “Pyramid Song Bass Tab”
SF (http://www.songfacts.com/detail.php?id=1524), “Pyramid Song by Radiohead”
WP (http://en.wikipedia.org/wiki/Pyramid_Song), “Pyramid Song”
WPS (http://en.wikipedia.org/wiki/File:Pyramid_Song_for_Wikipedia.jpg), “Pyramid Song for Wikipedia”
YT (http://www.youtube.com/comment_servlet?all_comments=1&v=cqP0WNpojFM), “Radiohead — Pyramid Song”
1. Research for this article was completed during a sabbatical year spent at the University of Oxford. Gratitude is firstly offered to Martin Stokes, Noel Lobley, and the graduate students in ethnomusicology at St John’s College for providing such a warm and stimulating environment. While in Oxford I also benefited from the institutional support of St John’s Research Centre and its director, Linda McDowell, as well as numerous conversations with Dai Griffiths at Oxford Brookes University. I would also like to thank Yonatan Malin and two anonymous readers at Music Theory Online for their collective careful eye and guidance, and my colleague John Roeder for further help.
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2. For an overview and synthesis of the ways ambiguity has been discussed in the music theory literature, see Butler 2006, 121–23. Other realms of ambiguity that combine with music in the creation of the commodified pop/rock CD (a product that relies on liner notes, packaging, and advertising) include photography, especially of men and women (Sontag 2001), language (Pinker 2008, 175–85), and symbolic imagery (Jung 1997 , 203–84).
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3. David Huron in a work on anticipation makes it clear that such engagement requires the listener “to be challenged, not simply pampered” (2006, ix; italics in original); in a similar vein, a large part of the Beatles’s success has been credited to the “vagueness” of their lyrics (MacDonald 2005, xii).
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4. These perspectives include human evolution (Brown et al. 2000, 12), Western rock music analysis (Moore 2001, 37), cross-cultural or “world music” studies (Tenzer 2006, 22–5), early childhood development (Mazokopaki and Kugiumutzakis 2009, 189), and human biology (Osborne 2009).
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5. Two passages from Butler’s work that clearly illustrate this viewpoint are as follows: (1) “Much of the significance of ambiguous structuring in electronic dance music, however, lies in its potential for drawing the listener in. Rather than demanding a particular way of hearing for the listener, passages such as the one heard in example 3.2 encourage each of us to seek out our own preferred interpretation — to actively participate in the construal of our musical experience” (2006, 127); and (2) “[M]etrically ambiguous sections encourage the listener to construe the meter actively rather than absorb metrical information passively” (2006, 137; italics in original). While not directly acknowledged in this work, Butler’s theory is bolstered by Justin London’s idea of “meter as a kind of attentional behavior” (2004, 9–26). The idea of meter as embodied experience has also been explored more recently in connection with other forms of dance/popular music (Attas 2011; this research is openly indebted to Hasty 1997).
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7. With degree of accuracy dependent on the artist, publisher, and transcriber, Allan Moore has noted (at least in the past) that published scores are “notoriously inaccurate as a coded version of the sounds heard” (2001, 61). In the case of the “official” notation of “Pyramid Song” sanctioned by Radiohead’s record label, the work was carried out by an editorial team, not by members of the group.
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8. Radiohead also released I Might Be Wrong: Live Recordings (2001), a collection of live tracks recorded during a European and North American tour taken primarily from Kid A and Amnesiac, and a collaborative remix album of The King of Limbs titled Radiohead TKOL RMX 1234567 (2011).
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9. Further biographical information is readily available on the web (various fan and Wikipedia sites) and in print sources (see, as examples, Hale 1999, Paytress 2005, Clarke 2010, and Randall 2011).
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12. Such practices have continued on through their recent release, The King of Limbs (2011). In the Rolling Stone review by Jon Dolan, he comments (in reference to the track “Codex”): “Maybe it’s about a drowning, maybe it’s about a swimming lesson. The fun is not knowing. Taking the plunge into this band’s mysteries is one of rock’s true pleasures” (2011, 62).
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14. The context for the genesis and use of the “weeping Minotaur,” as well as its accompanying cartoon character the “toothy Bear” and various interpretations as to their meaning, is found in Hainge 2005, 78–84, Leblanc 2005, 93–96, and Donwood and Tchok 2007 (second essay).
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15. For the truly curious reader, I have correlated the artwork of the special edition book and the standard CD as follows. The first number in each pair is the special edition book pagination, followed by the standard CD booklet page number (an empty space represents a “missing” page in the booklet): 1/1, 2/2, 3/3, 4/4, 5/5, 6/6, 7/7, 8/8, 9/9, 10/10, 11/11 (both pages modified so that subtle differences exist), 12/12 (both pages modified with subtle differences), 13/ , 14/13 (crying Minotaur added in special edition), 15/14, 16/15, 17/16, 18/17, 19/18, 20/19, 21/20, 22/21, 23/22 (booklet image altered), 24/23, 25/24, 26/25, 27/26 (instruction sheet added to booklet), 28/ , 29/ , 30/ , 31/ , 32/ .
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16. Here I have also done the legwork for those who might be interested. The following pairings identify the page of the standard CD booklet and the track from which a text excerpt is provided (remembering that no titles are given anywhere in the booklet): 10/ “Morning Bell/Amnesiac,” 12/ “Packt Like Sardines in a Crushd Tin Box” and “Like Spinning Plates,” 16/ “I Might Be Wrong” and “Like Spinning Plates,” 18/ “Packt Like Sardines in a Crushd Tin Box,” 21/ “Life in a Glasshouse,” 22/ “Knives Out.”
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17. The refrain is one of two “slogans” printed on the back cover of the CD booklet (the other being “spine damaged,” a reference to the condition of the book printed on the front cover of the booklet).
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18. The ondes Martenot, an early electronic instrument named after its inventor Maurice Martenot in 1928, was first made popular by the composer Olivier Messiaen in his 1937 composition “Fête des belles eaux” (Celebration of the Beautiful Waters). The instrument was added to Radiohead’s sonic palette by Jonny Greenwood.
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20. This kind of rhythmic ambiguity has been similarly described by the psychoacoustician Ernst Terhardt in terms of “insufficiency of structural information included in the stimulus” (1991, 229).
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22. While it would be of considerable interest to know who these audiences and/or fans might be in terms of gender, age, place of residence, occupation, and socio-economic class, there is unfortunately not enough information revealed in the entries to warrant such speculation (and personal contact information is not as a rule provided).
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23. In an interview with Alex Ross, Radiohead’s drummer Phil Selway was quoted as saying “there is no time signature [in “Pyramid Song”],” though Ross (and many others) recognize Selway’s shuffling rhythm implying a compound meter standard in jazz (2010, 95 [to be discussed in more detail below]). It is also difficult to imagine how Radiohead could perform a piece with such intricacy and required coordination without any metrical organization.
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25. This analysis matches a listening experience described by Longuet-Higgins and Lee: “In choosing a rhythmic interpretation for a given note sequence the listener seems to be guided by a strong assumption: if the sequence can be interpreted as the realization of an unsyncopated passage, then that is how he will interpret it” (1984, 424).
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29. While I should reiterate that I am not advocating for any particular single metric interpretation of “Pyramid Song,” if one accepts the eighth note groupings of the piano chords in Examples 6 and 7 (5–4–6–5–4) we are presented with a series of note durations with highly complex ratios that are difficult to hear: “[W]e understand notes as being in one rhythmic category or another, rather than merely perceiving them as continually varying. This process of sorting or ‘quantizing’ notes has been demonstrated experimentally as well; when played patterns of alternating notes whose durations are related by complex ratios (such as 1.5:1 or 2.5:1) and asked to reproduce them, subjects tend to adjust the durations toward simple ratios (such as 2:1)” (Temperley 2001, 25; see also Povel 1981). The note durations 5–4–6–5–4 become 1.25:1, 1:1.5, 1.2:1, and 1.25:1; without any other outside stimuli, it could account for why many listeners hear the first two chords (and the fourth and fifth) as being even in length.
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