A Geometric Analysis Of The Harmonic Structure of "In My Life"
AA GEOMETRIC ANALYSIS OF THE HARMONIC STRUCTURE OF
IN MY LIFE
JAMES S. WALKER AND GARY W. DONA
BSTRACT . After our book [1] was published, we found a striking example of the impor-tance of the
Tonnetz for analyzing the harmonic structure of The Beatles’ song,
In My Life .Our
Tonnetz analysis will illustrate the highly structured geometric logic underlying thenumerous chord progressions in the song. Spectrograms provide a way for us to visualizechordal harmonics and their connection with voice leading. We shall also describe the in-teresting harmonic rhythms of the song’s chord progressions. A lot of this harmonic rhythmlends itself well to a geometric description.
1. I
NTRODUCTION
The Beatles, especially in some classic songs of John Lennon and Paul McCartney, used muchmore elaborate chord progressions than one finds in typical popular songs. A fine example of thisoccurs with their song,
In My Life . For whatever reason, the chord progressions and instrumentalmelodies found in various sheet musics for Lennon and McCartney songs, such as
In My Life ,are aural transcriptions made by others. This has led to some differences in the various sheetmusics, and chord listings, that are available. Our approach to this difficulty is to do an initialanalysis on a basic template for the song found in the lead sheet version in [2]. After analyzing itsharmonic structure, consisting of a
Tonnetz analysis of its chord progressions and an examinationof its harmonic rhythm, we then look at a computer analysis of the recorded performance of
In MyLife from The Beatles’ album
Rubber Soul . The harmonic structure of this recording involves justa few enhancements of the lead sheet version.2. H
ARMONIC S TRUCTURE OF THE L EAD S HEET V ERSION OF
In My Life
The lead sheet lays out the song form for
In My Life , in the key of A . We show this song formhere: Verse Bridge Verse Bridge Interlude Bridge Coda (1)In this section, we shall analyze the harmonic structure of each distinct part—Verse, Bridge,Interlude, and Coda—of this song form.2.1. Harmonic Structure of Verse.
The lead sheet in [2] gives the chord progressions for Verse.These progressions, in terms of
Tonnetz transformations, are A T −→ E T −→ A T −→ E T −→ A T −→ f (cid:93) T −→ A T −→ D T −→ d T −→ A T −→ f (cid:93) T −→ A T −→ D T −→ d T −→ A (2)In the second instance of Verse, the progression A T −→ E only occurs once. It is worth notingthat all of these progressions are single Tonnetz transformations. They follow an interesting paththrough chords on the
Tonnetz .In Figure 1, we have plotted the motion of these progressions through the chords on the
Tonnetz .In these plots, any seventh chords are treated as embellishments of underlying triadic chords. ForVerse, the motions of the chord progressions are shown on the left side of Figure 1. We will not speculate whether this is due to the common assertion that Lennon and McCartney couldnot read music, or due to them simply leaving musical transcriptions of their songs to others. a r X i v : . [ m a t h . HO ] A ug JAMES S. WALKER AND GARY W. DON
Verse Bridge Interlude Coda
Figure 1.
Red arrows show chord progressions on the
Tonnetz for each of the parts in thesong. In Verse, Interlude, and Coda, the circled chord A is the starting and ending chord.In Bridge, the circled chord A is the ending chord of Verse or Interlude, hence A T −→ f (cid:93) isthe transition to the beginning of Bridge. The chord A is also the ending chord of Bridge.The blue rectangles, and connecting arrows, in Interlude illustrate the movement of pitchesthat undergirds the double Tonnetz transformation E T T −−→ f (cid:93) . The blue dot next to the pitchclass B indicates its double position in the Tonnetz . The blue rectangle in Coda marks thenote D found in both the E chord and the d chord. The ending progression A T −→ D T −→ d T −→ A , corresponding to the triangular path around thepitch class hexagon A is particularly interesting. Here A is functioning as a secondary dominant for the chord D , i.e., a dominant for the non-tonic chord D . In roman numerals, this is denotedas V / IV . Hence, V / IV T −→ IV is the roman numeral version of A T −→ D in the key of A . The transformation D T −→ d is a neo-Riemannian transformation D P −→ d which introducesa modal mixture . It is well-known that The Beatles were experimenting with songs in variousmodes and modal mixtures on the Rubber Soul album, and the
Revolver album that followed it.For example, the Mixolydian mode is used in the song
Norwegian Wood on Rubber Soul . There isalso a mode mixture used in
Eleanor Rigby on Revolver , which we discussed in some detail in [1,Example 7.4.6]. For
In My Life , using the minor chord d —which belongs to the key of a -minorparallel to the key of A -major—provides a mode mixture. The particular progression of IV P −→ iv is often used in music to provide a nice contrast in “color” of these parallel chord types. Withinthe song In My Life , D P −→ d harmonizes well with the chromatic descent in the lyrics from an A note to a natural F note. See Figure 2. Moreover, when the ending chord A is included to formthe cadence D P −→ d N −→ A , there is a smooth voice leading between the pitches F (cid:93) → F (cid:92) → E inthe respective chords. See again, Figure 2. In roman numerals, this cadence is IV → iv → I , i.e., GEOMETRIC ANALYSIS OF THE HARMONIC STRUCTURE OF
IN MY LIFE a plagal cadence with modal mixture. We have focused here on the ending cadence, but the otherprogressions and their motions on the Tonnetz are also worth considering, especially in connectionwith the harmonic rhythm in Verse.
Figure 2.
The quadruple arrow points to harmonics for pitches of type D , A , B , F (cid:93) in as-cending order. This is why computer analysis classifies the chord as D (more on thiscomputer analysis in Section 3). In the Tonnetz we have plotted this as a D chord, as indi-cated in the lead sheet. The triple arrow at the bottom points to fundamentals for a d -chord( D , F , A ). The double arrow points to fundamentals for the chromatic descent from an A note to a natural F note, and the fundamental of F aligns with a previous second har-monic for the F note in the d -chord. The triple arrow at the top points to harmonics for thevoice leading F (cid:93) → F (cid:92) → E . The harmonic rhythm in Verse closely relates to the structure of the path followed by the chordchanges on the
Tonnetz . This path breaks down into three substructures: A T −→ E A R −→ f (cid:93) R −→ A D P −→ d N −→ A I T −→ V I R −→ vi R −→ V / IV IV P −→ iv N −→ I (3)Each of these substructures is repeated twice in moving through the Tonnetz . The roman numer-als are for the chords in the key of A -major, including the notation V / IV that we previouslydiscussed.The substructures in (3) correspond to three distinct harmonic rhythms. In Figure 3, we showthese three distinct harmonic rhythms by plotting them on rhythm clocks of eight hours each, onehour per beat (quarter note), spanning two measures each. The first substructure in (3) has a wholenote rhythm (one chord per measure), while the second and third substructures show alternatingfaster (half-note) rhythm with slower (whole note) rhythm. Furthermore, it is interesting that thesecond and third clocks in Figure 3 are reflections of each other. The reflections being through amirror passing through hours and , which are the hours for the chord onsets in the first clock. JAMES S. WALKER AND GARY W. DON AE I T −→ V (cid:45) Af (cid:93) A I R −→ vi R −→ V / IV (cid:45)(cid:27) DA d IV P −→ iv N −→ I Figure 3.
Harmonic rhythm of Verse. The eight hour clocks mark hour per quarter notebeat, spanning two measures each. In the first instance of Verse, the clock on the left iscycled through twice. While in the second instance, it is cycled through once. In bothinstances, the second and third clocks are each cycled through twice in alternation. Thearrows mark the change from one clock to the next—one harmonic rhythm to the next—asthe music proceeds through Verse, two measures at a time. Within each clock, there is alabeling of its form in terms of roman numerals and types of transformations (either general Tonnetz transformation T , or neo-Riemannian transformation R , P , or N ). Harmonic Structure of Bridge.
The lead sheet in [2] gives the following chord progressionsfor Bridge: (A T −→ ) f (cid:93) T −→ D T −→ G T T −−→ A T −→ f (cid:93) T −→ B T T −−→ d T −→ A (4)The progression (A T −→ ) is not part of Bridge. It indicates the progression from Verse to Bridge, orInterlude to Bridge.These transformations are graphed on the Tonnetz in Figure 1. Most of them are single
Tonnetz transformations. These single
Tonnetz transformations either pass across a single hexagon from theset { D , F (cid:93) , A } , corresponding to the chord D , or along an edge between two of these pitch classescorresponding to D . In fact, the D -chord lies in the center of the network of transformationsshown for Bridge in the Tonnetz diagram. All these geometric facts provide clear evidence for the tonicization of the subdominant chord D in Bridge.The two exceptions to single Tonnetz transformations in (4) are the double
Tonnetz transforma-tions G T T −−→ A and B T T −−→ d . In the chord G , the leading tone G (cid:93) has been dropped by a halfstep to G -natural. This lowering of the leading tone by a half step in order to play a VII -chordwas commonly employed in popular music at that time. In which case, the roman numeral form of G T T −−→ A represents a return to the tonic with VII → I . The musical practice here is similar to theRenaissance practice of lowering the seventh tone in a major scale in order to avoid tritones. Onthe left of Figure 4, we show the harmonics for the G -chord. It is interesting that the notes sung byLennon during the playing of this chord are a descending scale sequence D , C (cid:93) , B , A . When theseventh tone for the A -major scale is lowered by a half-step, the resulting scale is A -Mixolydian: A , B , C (cid:93) , D , E , F (cid:93) , G , A . As shown on the right of Figure 4, Lennon’s lyrical pitch descent from A crosses the secondharmonic of the fundamental for the G -chord. If we take this as an implicit sounding of a G -note, Tonicization refers to a strong emphasis, or central role, for a scale degree or chord other than the tonicfor the key.
GEOMETRIC ANALYSIS OF THE HARMONIC STRUCTURE OF
IN MY LIFE Figure 4.
Left: The triple arrow at the bottom indicates harmonics with frequencies cor-responding to pitches G , G , B in ascending order. While the triple arrow at the topindicates harmonics with frequencies corresponding to pitches D , G , B in ascending or-der. Consequently, computer analysis classifies the chord being played as a G -chord. Thedownward pointing arrow points to a harmonic for A which is an overtone for the har-monic for D pointed to by the upward pointing arrow. Right: The quadruple arrow pointsto fundamentals for the descending sequence of notes D , C (cid:93) , B , A . The first fundamental,for the D note, matches the D harmonic in the G -chord. The single arrow points to animplicit G note occurring when Lennon’s pitch descent from A crosses the G harmonic.A video illustrating this implicit note is available at [3]. then the descent is along the A -Mixolydian scale. It is interesting to note in this connection thatthe note G (cid:93) is never used in Bridge, neither in the lyric’s notes nor in the chords. Thus, perhaps wehave at least an implicit invocation of the A -Mixolydian mode in Bridge. As for the progression, B T T −−→ d , when we analyze the recording of In My Life , we shall see that B T T −−→ d does not occur there. Instead, it is replaced by B T −→ D T −→ d . Thus, in the recording, even though the chords B and d are not in the key of A , they still play a consonant role in the music due to the Tonnetz transformation sequence f (cid:93) T −→ B T −→ D T −→ d .There is also a change in harmonic rhythm in Bridge. The harmonic rhythm in [2] is a simpleone chord per measure rhythm. This contrasts with the slightly more complex harmonic rhythm ofVerse. So we have an interesting change of complexity of the harmonies when Verse changes toBridge: the chord progressions become more elaborate, while the harmonic rhythm becomes lesselaborate. That change of complexity then reverses when Bridge changes back to Verse.2.3. Harmonic Structure of Interlude.
The lead sheet in [2] gives the following chord progres-sions for Interlude: A T −→ E T T −−→ f (cid:93) T −→ A T −→ D T −→ d T −→ A (5)All of the transformations in (5) are single Tonnetz transformations, except for E T T −−→ f (cid:93) . Aninteresting feature of this one exception is that, while the chord E is sounding, the notes being JAMES S. WALKER AND GARY W. DON played are a descending series of perfect fourths: E , B , F (cid:93) . We show the pitch classes for thosenotes in Figure 1, and it is clear that they lead nicely into the chord f (cid:93) . It is clear from the Tonnetz diagram in Figure 1 that the tonic chord for Interlude is A . This makes sense as Interlude is asubstitute for Verse in the song form shown in (1).The harmonic rhythm in Interlude is mostly half-note. The only exception is that for each of thetwo passes through Interlude in the lead sheet score, the ending chord A is held for a full measure.2.4. Harmonic Structure of Coda.
There is a brief coda at the end of the song. The lead sheetin [2] gives the following chord progressions for Coda: A T −→ E T T −−→ d T −→ A T −→ E T −→ A (6)All of these are single Tonnetz transformations, with one exception: E T T −−→ d . This is a sur-prising progression, as the dominant seventh chord E would typically anticipate the tonic chord A . However, the progression to the subdominant minor chord d , used frequently throughout thesong—along with Lennon singing a chromatic descending sequence of pitches C (cid:92) , B , A —providesa nice unexpected twist to the music in this final part.The harmonic rhythm for the Coda is whole note throughout. It thus provides a slowing downfrom the half-note harmonic rhythm at the end of the last Bridge.2.5. Summary.
The lead sheet provides a foundation for understanding the fascinating harmonicstructure of
In My Life . We have seen that the
Tonnetz provides a powerful tool for understandingthe geometric logic underlying the chord progressions in the song. A geometric analysis also shedslight on some of its harmonic rhythm.3. H
ARMONIC S TRUCTURE OF THE R ECORDING OF
In My Life
The recording of
In My Life for the most part follows the structure given in the lead sheet in [2].There are some interesting differences, however, which serve to enhance the harmonic structure inthe recorded version.To transcribe the chords in the recorded version of
In My Life , we used a computer programcalled C
HORDINO . C
HORDINO is available from the link given in [4]. We used C
HORDINO as a plug-in to the free audio software A
UDACITY . To analyze the recording of
In My Life , wefirst converted the stereo file from the
Rubber Soul
CD to mono. We used the default settingsin C
HORDINO , with one modification: Following the advice given in [4] for analyzing popularmusic, we set the spectral roll-off parameter to . . C HORDINO identifies chords by detecting theharmonics for the notes composing the chords. In Figures 2 and 4 you can gain some idea howidenfifying the sets of harmonics in the sound can be used to identify the individual notes in achord, and thus identify the chords themselves.The chordal analysis given by C
HORDINO differed from the lead sheet in just a few places. Thedifferences occurred in the Verse and Bridge parts. Here we will discuss the differences for thefirst instance of Verse and the second instance of Bridge. That will capture the essence of thedifferences between the lead sheet version and the recording. We found that the lossless *.wav form of the recording did not produce good results, as C
HORDINO could not reliably reproduce known chords from the lead sheet. However, when we analyzed *.m4a (highestquality of kbs) and *.mp3 (standard quality kbs) formats, C
HORDINO reproduced the chords foundin the lead sheet and gave the same results for additional chords, and these are the chords we report here.
GEOMETRIC ANALYSIS OF THE HARMONIC STRUCTURE OF
IN MY LIFE Figure 5. C HORDINO analysis of chords in first instance of Verse. The notation N indicatesthat no identifiable chord is played at the start, so the first chord is A . Harmonic Structure of Verse.
In Figure 5, we show the C
HORDINO analysis of the chordsin the first instance of Verse. This analysis gives us the following progressions: A T −→ E T −→ A T −→ E T −→ A T −→ E T T −−→ f (cid:93) T −→ A T −→ D T −→ d T −→ A T −→ E T T −−→ f (cid:93) T −→ A T −→ D T −→ d T −→ A (7)where we have replaced some chords by their underlying triads—viewing E , E as embellish-ments of the chord E , f (cid:93) as an embellishment of the chord f (cid:93) , D as an embellishment of the chord D , and A as an embellishment of the chord A .Unlike in the lead sheet Verse, there are a couple of double Tonnetz transformations in additionto the many single
Tonnetz transformations. The motion of these chord progressions on the
Tonnetz is shown on the left of Figure 6. The overall pattern is quite similar to the one shown for Verse inFigure 1. The one difference is the two double
Tonnetz transformations of type E T T −−→ f (cid:93) . Thesedouble Tonnetz transformations enhance the pattern from the lead sheet. To be more precise, inthe key of A , they are both instances of the deceptive progression: V T −→ vi , which adds somevariety to the harmony. Moreover, the occurrence of the notes C (cid:93) in Lennon’s lyrics immediatelypreceding the chords f (cid:93) makes the transformations E T T −−→ f (cid:93) more sonorous.The sequence in (7) breaks up into four distinct substructures: A T −→ E A T −→ E T T −−→ f (cid:93) R −→ A A T −→ D P −→ d A T −→ E T T −−→ f (cid:93) (8)which differ from the three substructures in the lead sheet Verse given in (3). Likewise, there is adifferent harmonic rhythm corresponding to these substructures. This harmonic rhythm is shownin Figure 7.Thus, in the recorded version of In My Life there is a slightly more sophisticated sequenceof chord progressions (including enhanced voice leading), as well as slightly more sophisticatedharmonic rhythm. This increased sophistication also occurs in the Bridge part.3.2.
Harmonic Structure of Bridge.
The second instance of Bridge most clearly illustrates theenhanced harmonic structure of the recorded version of
In My Life . In Figure 8, we show the
JAMES S. WALKER AND GARY W. DON
Verse 2nd Bridge
Figure 6.
Movement of chords on the
Tonnetz for the first instance of Verse and the sec-ond instance of Bridge. The blue rectangle in the Verse diagram marks the note C (cid:93) thatimmediately precedes the chord f (cid:93) in the progression E T T −−→ f (cid:93) . C HORDINO analysis of the chords in the second instance of Bridge. The chord progressions iden-tified by C
HORDINO are the following: B T −→ f (cid:93) T −→ D T −→ G T T −−→ A T T −−→ B T −→ f (cid:93) T −→ B T −→ D T −→ d T −→ A (9)where we have written the chord B / F (cid:93) as simply B and written a couple of other chords by theirunderlying triads—viewing f (cid:93) as an embellishment of the chord f (cid:93) , and D as an embellishmentof the chord D . AE (cid:45) A A Ef (cid:93) (cid:45)
D dA (cid:45)(cid:27)
A Ef (cid:93)
Figure 7.
Harmonic rhythm of first version of Verse in the recording of
In My Life . Themeaning of the hours is the same as for the clocks in Figure 3. The first clock is cycledthrough twice. The second clock is cycled through once. The third and fourth clocks arecycled through in alternation (third clock, fourth clock, third clock).
GEOMETRIC ANALYSIS OF THE HARMONIC STRUCTURE OF
IN MY LIFE Figure 8. C HORDINO analysis of chords in second version of Bridge.
As with the lead sheet version, most of these transformations are single
Tonnetz transforma-tions. The pattern of the progressions on the
Tonnetz is shown on the right of Figure 6. As withVerse, this second version of Bridge is quite similar to its lead sheet version, but with some inter-esting enhancements. For one thing, the tonicization of the subdominant D chord is more clearlyemphasized as the central chord in the diagram in Figure 6. For another, the chromatic chord B receives much more emphasis. This enhanced emphasis includes the use of the chromatic medianttransformation B T −→ D .The harmonic rhythm in the recorded version is also more interesting than in the lead sheet. Inthe recorded version, if we look at the length of separation between successive chords shown inFigure 8, we can see that this rhythm consists of interspersing half-note rhythms (such as B / F (cid:93) to f (cid:93) ) and whole note rhythms (such as D to G ) with one slightly different splitting of one measure( B / F (cid:93) to f (cid:93) to B / F (cid:93) ). This is a much more nuanced harmonic rhythm than in the lead sheet.3.3. Interlude and Coda.
The Interlude and Coda of the recorded version are identical in theirharmonic structure to the lead sheet version. This is understandable as the Interlude was composedfor solo piano (played by George Martin and then sped up to sound like a harpsichord). Conse-quently, the lead sheet is a faithful transcription of the music (undoubtedly played by Martin fromhis own sheet music). The Coda is so brief that it probably allowed no room for any elaborationsby The Beatles during the recording process.3.4.
Summary.
The recorded version of
In My Life is an enhancement of the lead sheet version.We have seen how the recorded version builds upon the underlying logic of the lead sheet version.It does this by an increased emphasis on chromaticism, slightly more complex chord progressions,and slightly more complex harmonic rhythm.4. C
ONCLUSION
In this paper we have used geometric methods to analyze the harmonic progressions and theharmonic rhythms in the song,
In My Life . The
Tonnetz provided a powerful geometric tool forunderstanding the logic of the song’s harmony. Spectrograms provided a useful tool for visualizingthe interplay of harmonics in voice leading. Clock diagrams were also quite useful for analyzing the details of the changing harmonic rhythms in the song. We have endeavored to show as well thatthese two geometric approaches were closely linked in the Verse parts of
In My Life . Our analysishas revealed many of the details of how The Beatles elaborated on basic harmonic patterns to createprogressions more elaborate that typical ones used in rock music at that time.R
EFERENCES [1] J.S. Walker and G.W. Don. (2020).
Mathematics and Music: Composition, Perception, and Perfor-mance. 2nd Edition . CRC Press.[2] Lead sheet for
In My Life G -note is available here:https://tinyurl.com/ImplicitG[4] C HORDINO
AMES
S. W
ALKER , M
ATHEMATICS , U
NIV . OF W ISCONSIN -E AU C LAIRE , WALKERJS @ UWEC . EDU G ARY
W. D ON , M USIC & T
HEATRE A RTS , U
NIV . OF W ISCONSIN -E AU C LAIRE , DONGW @ UWEC ..