The music for "Happy Birthday to You" lends itself well to the initial stages of teaching Schenkerian analysis, as can be seen in the fine Reddit post by "Xenoceratops" here, and its accompanying analysis here. A couple of years ago in this blog, I said that Schenkerian theory "distills tonal structures to their linear-harmonic essence, where prominent starting notes and normative cadential notes are stitched together into a well-formed contrapuntal design." It is a complex theory to learn and teach, but it can be as rewarding as it is complex. Here I offer a method to (somewhat) automate the application of the theory, a way that some might find crude, but it nonetheless matches competent analyses in some, perhaps even many, cases. It also reveals a hidden symmetry in this music's linear-harmonic structure that, consistent with my last three posts, involves a combination of retrograde and inversion, as shown at the end of this post.
The method begins with a triadic reduction of the music. Here is the melody for "Happy Birthday to You" in F major, along with a simple and common bass line and a standard triadic harmonization of this bass line. The number of chords in this harmonization, which I will abbreviate as #c, is seven. (I have considered the cadential 6/4 chord as a separate chord; another approach might elect not to do so.)
The next step is, within certain constraints, to write out (or at least imagine) all the possible realizations of this harmonic progression using three upper voices -- soprano (always highest), alto (always second highest), and tenor (always second lowest) -- each making a four-part chorale-style composition. (A simple keyboard accompaniment that one is likely to play when supporting the singing of this tune is likely to be one of these realizations.) The constraints are as follows:
1) For every triad, the three voices should collectively cover its three members.
2) With each motion from from triad to triad, the three upper voices should all either upshift or downshift. Xshifting (X = up or down) means that a voice either stays on the same note or moves X either by second or third.
There are exactly 6*(2^(#c-1)) realizations, starting octave position aside, because there are six possible assignments of the first triad's three notes to the three voices of soprano, alto, and tenor, and there are #c-1 instances when a triad voice-leads to the next triad, and each voice-leading may be an upshift or a downshift. In the case of "Happy Birthday to You," #c is 7; therefore, the number of realizations is 6*64, or 384. However, for the purposes of emulating Schenkerian analysis, it only matters to which triadic member the first soprano pitch is assigned; therefore, we need only concern ourselves with half this number, or 192 realizations.
For example, the first realization below starts the soprano on the third, and alternates back and forth between upshifting (u) and downshifting (d), and the second realization starts on the root and constantly upshifts. This first realization could be abbreviated as 3-ududud, or 3-42: 3 is for third, and, if u is 1 and d is 0 in ududud, then the resulting base 2 number written in base 10 is 42. The second realization could be abbreviated as R-uuuuuu, or R-63.
These are two of the 192 realizations of interest. These 192 realizations provide 192 different soprano lines. One way to emulate a Schenkerian analysis of "Happy Birthday to You" is to choose one of these 192 soprano lines that is most like, or best represents, the melody. This soprano line then becomes what is called the Urlinie, or fundamental treble line, in Schenkerian theory. One way to quantify the degree of this resemblance or representation is to choose a soprano line that maximally overlaps with the melody, regardless of the octave placement of the notes. For example, the two figures below show that the soprano line of realization 3-42 overlaps with the melody of "Happy Birthday to You" for a total time of five quarter-note durations, and the soprano line of realization R-63 overlaps with the melody of "Happy Birthday to You" for a total time of six quarter-note durations (four quarter notes plus a half note at the end).
3-42 plus overlap with melody
R-63 plus overlap with melody
Surely we can do better than five or six. The two realizations with the most overlap of ten quarter notes are 5-duuudu, or 5-29, and 5-dudddd, or 5-16, as shown below. Among adaptations of Schenkerian theory where the Urlinie is allowed to ascend, or skip, or change direction, the Urlinie of 5-29 might find quarter. However, in orthodox Schenkerian theory, an Urlinie may only descend by step. If, in this case, we further restrict ourselves to realizations with soprano lines that only descend, or only move by step, or never change direction, then the soprano line of 5-16 is the only one with maximal overlap.
5-29 plus overlap with melody
This emulator could be refined by, for example, avoiding realizations with parallel perfect fifths or octaves (for example, toward the end of R-63 above), or weighting the overlap through the consideration of accented notes, non-chord tones, cadential notes, etc. With these refinements (but perhaps even without them), I suspect that this emulator would be able to demonstrate an important theoretical component of Schenker's project. If one runs this emulator on thousands of musical phrases that end on tonic harmony written by Western composers from 1720 to 1890, I speculate that a plurality of chosen soprano lines will descend by step to ^1, and the majority of chosen soprano lines will at least end with ^3-^2-^1. Thus the preference for 5-16 over 5-29 exercised earlier reflects a norm-based bias.
Lastly, a hidden symmetry of "Happy Birthday" shows itself when all of the chord tones are labeled, as shown below, by their assignment to soprano, alto, and tenor according to the realization of 5-16. This makes more evident the Schenkerian notion that, for example, the descents in mm. 5-6 and m. 7 are descents "into an inner voice." Inner-voice notes actually higher than adjacent soprano-voice notes due to octave equivalence—like the "tenor" note in m. 1 or "alto" note in m. 3—are what Schenker calls "superpositions" or "cover tones."
This labeling also reveals an invariance of structure. Imagine an inversion, defined on three upper voices, whereby alto (A) and tenor (T) switch places within triadic harmony, but the soprano (S) maintains the same triadic position. The sequence of voice assignments of the melody from "Happy Birthday to You"—STSATSATSAS—is thus a retrograde and this inversion of itself, just like the passages from Four Last Songs and The Aristocats mentioned in earlier posts are retrogrades and pitch-(or pitch-class-)inversions -- that is 180° flips -- of themselves: they are unflippable in the same manner than Messiaen's palindromic rhythms are non-retrogradable.
STSATSATSAS --> SASTASTASTS --> STSATSATSAS
(inversion: A and T switch places) (retrograde: sequence is flipped in time)
The fulcrum of the retrograde is also in the center of the melody.