Radiohead - Pyramid Song, the analysis

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

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.

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.

Radiohead - Everything In Its Right Place

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.

Muse - Knights of Cydonia, the analysis

I was following some of Trevor de Clercq's analyses on pop and rock tunes and felt inspired enough to see if I could contribute to that area as well.  This would be mostly for my own amusement but also to chronicle the things that come into my mind, all within the convenience of this blog format.

Knights of Cydonia is a song from Muse's 2006 album Black Holes and Revelations.  I first heard it played incessantly on the radio while going to and from work around that time as well.  I found it to be a catchy and genuinely interesting rock hit, but it wasn't until I started taking a closer look at it that I noticed it had some neat compositional features.

The work doesn't have a typical song structure for starters.  It's not your average AABA or ABABCB song form which so many pop tunes are built on.  It can be broken down into this general form:

Intro - A - A' - A'' - Intro - B - Coda

I've omitted any indication of how many times the Intro, B, or Coda is repeated, instead just choosing to show how the basic formal divisions are mapped out.  It may seem a little peculiar to have the recurring "Intro" lodged within the middle of the piece, but I didn't want to call it a letter name, which I thought might imply it being more structural (it functions like a slow intro, and is merely repeated after A'' concludes).  You can see from what I've organized above that the song is therefore a kind of binary form.

The three instances of A is where I find the most interesting part of the song takes place.  It's composed around a key modulation scheme which is derived from a classical tradition rather than learned from any pop school.  Most pop/rock songs don't even modulate; this one traverses paths which are seriously atypical of a mainstream work!  I can think of a few pop songs like The Police's "Every Breath You Take" which modulate to [bVI], but it's brief, in a major key, and bounces right back up to [I], rather than boldly modulating even further like Knights does.

I've placed to the right of here a visual representation of the way the keys change within the three instances of the A section (circle of fifths graphic shamelessly stolen from a GIS).

A starts in Em and modulates to C minor
A' starts in C minor and modulates to G#m
A'' starts in G#m and modulates to Em

The keys divide the octave into major thirds, something which you may be familiar with in a piece like Beethoven's "Waldstein" Sonata, where the second theme modulates to [III] instead of a more typical [V].  It wasn't until composers like Schubert and Liszt came around that there would be more extensive 3rd related key modulations which divide the octave up totally.  Knights of Cydonia mimics this latter method.  Using roman numerals, here's a way to show the chords progress within each A section:

               [i]:  i - III - VI - III - V - VI - 
                                        [vi]:  I - III - V - i - V - VI - III - V* - VI - III - V - i

I don't think I can totally convince myself where the exact pivot chord is, but I'm sure this is comprehensive enough.  Where I've placed the asterisk is a point I found noteworthy; while the chord is a dominant V, the guitar melody on top plays the flat-third of the V chord, creating an interesting false relation.  I figure this is some kind of subtle blues derived idiom; either way, it's not too jarring because the background chords only have the root and 5th notes of that chord primarily perceptible.  It's only at 1:53 do we hear the brass play the raised 3rd (F-double sharp) right before the guitar plays the regular unaltered-3rd (F-sharp) of the V-chord.

Since I've just mentioned a point in the song relating to a specific time, here's an embedded version of the song to follow along with:

When that elaborate chord progression of the A section outlined above is applied three times, as occurs in the piece, the keys come full circle (or in the case of the graphic, full triangle) and return to E minor, where it all began.  One thing I particularly enjoy about the modulation is how seamlessly Muse is able to go from [i] to [vi] within each A section.  The lead guitar provides a motion from VI5-6 to #VII (reinterpreted as III in [vi]), and it is performed without my ears hearing it as abrupt or foreign.  I would wager to say that your average listener is unaware that the keys cycle through this anachronistic scheme.


The piece also harbors some harmonic motivic unity -- a chromatic line that outlines a major second.  For example, in 1:03 a III - V - VI progression occurs with the guitar outlining D-D#-E (decorated with an F# neighbour), and once we reach the modulation to [vi] at 1:07, we hear the guitar play A-A#-B.  How convenient that they are a perfect-5th apart.  

I hope this analysis was informative.  I feel compelled at the end of it to say that  it's not an attempt to elucidate the value of the piece.   I have not included many things which contribute to the quality of this song, but I think showing the objective harmonic features give an impression that this is different from the more common pop/rock songs one hears on the local radio station, certainly more interesting from an analytical perspective, in my opinion.