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Pyramid song is the second track off Radiohead's album 'Amnesiac'. It seems to me that it's most conspicuous feature is the meter. The introduction of the piano in the piece isn't particularly compromising in terms of its harmony, but as soon as the third chord arrives, it is the complex rhythm which immediately becomes apparent to the listener. Each bar in the piece is essentially arranged in the manner presented below (though, note the caption). There is much discussion of what is the 'correct' time signature is on various internet pages and most of what I read were poor interpretations; I think the one I've written is fairly agreeable.

For a performance score, I imagine the middle note would be notated as a dotted quarter tied to an eighth
(easier to read).

The first thing brought to mind when I saw this was non-retrogradable rhythms, a concept which composers like Messiaen deliberately used in composition. The rhythms of each measure is the same backwards as it is forwards. I wouldn't be too surprised if imitating Messiaen was the intention of Radiohead with this song. The guitarist of the band, Jonny Greenwood, frequently draws inspiration from 20th century modernist composers, and utilizes the ondes Martenot instrument on both Kid A and Amnesiac -- a favorite of Messiaen's. I've read interpretations of this palindromic rhythm as the reason for the song's title: it is 'shaped like a pyramid.' I can see where one would get the idea; the rhythmic value in the middle is where it peaks, accompanied by adjacent smaller values. I'm not entirely convinced though; in my mind it would have to be the shortest note value in the middle of the bar, not the longest, to represent the kind of peak of a pyramid which is smaller in width to the base.
I found the harmony to be quite interesting. It makes no attempt to adhere to any consonant voice leading in the classical sense, although ironically it moves with smoothest possible harmonic shifts because of its disregard for the rules of counterpoint. Of course, while imposing a contrapuntal critique of any rock song might seem a little dubious (especially to 'new' musicologists), I feel like it's still relevant to a discussion of the piece. In Pyramid Song the left hand of the piano uses raw parallel 5ths to traverse up the chord progressions of major chords like F#-G-A, with doublings in the right hand (Example 1-A). One motivic feature is the 9-8 suspension over the tonic chord, F# which appears in first phrase of the piano, and is doubled in strings in the penultimate phrase of the song. It's a nice dissonant effect, not only because it's a minor 9 suspension, but it essentially creates delayed parallel octaves with the bass. The impression of whether the musical texture is homophonic or contrapuntal is constantly distorted because of it (Example 1-B).

Example 1

Midi Pyramid Song Radiohead Chords

The key of the piece is a little ambiguous. My overall impression is that it is in the F# phrygian mode, because of the prominent G-F# motion within the song and at its final cadence. Throughout the piece, however, there are several hints at different keys. Thom Yorke's vocal lines freely interchange between the use of the notes G-natural and G-sharp. When G-sharp is on the surface, one might be satisfied if the piece were deemed in a key of F#m or F#, rather than a modal version. The fact that all the chords essentially conform to the key of Bm also might be considered. Perhaps this is why the same chords over and over do not sound tired or monotonous even after multiple listenings: if the piece were in Bm, the avoidance of resolution to the tonic chord might make the listener constantly seek a resolution which is never achieved. But then again it could just be that it is an interesting chord progression. For instance, Radiohead's 'Creep' ends each phrase with a plagal cadence, and the entire song is saturated with just 4 chords in repetition, yet I found it enjoyable and it sustains my interest.
A nice feature of this song is the constant F#5 note in the right hand (example 1); I recall seeing a comment somewhere describing this as a 'pedal tone', but my understanding of a pedal is one which is more or less in a low register. I have read in certain theory texts (e.g. Kostka/Payne) that pedals can occur in higher registers, however. Pyramid Song would therefore fit that definition. Pyramid

Midi Pyramid Song Radiohead Youtube

One impression I couldn't shake when hearing this song is its close resemblance to the song Everything In Its Right Place from Kid A. The chord progression is identical for the most part, but the distinctive rhythms and timbres are what divide them. Both songs are also predominantly played on keyboard and have the pedal tone in the high register. They are each a tritone apart in key, interestingly enough (compare to example below). NB: Everything is scored in Fm according to this publication, which also gives credence to my description of Pyramid Song being potentially in Bm and why I thought it of being potentially (I stress this word) in that key.

Midi Pyramid Song Radiohead

Radiohead - Everything In Its Right Place

Midi Pyramid Song Radiohead 1

The wikipedia article of Pyramid Song mentions that the Charles Mingus song 'Freedom' had a heavy influence on the Radiohead track. I can hear the similarities on the opening 'Ooh's in Freedom and the same 'Ooh's Yorke uses in the vocal line, but beyond that it breaks out into jazz improv, and doesn't sound anything like the Radiohead song.