π Sounds Classically Evil From the Start

Happy Pi Day! It can be fairly straightforward to turn π into a melody, and there are many ways to do so. For example, in π's decimal representation (3.14159...), one could assign 0 to middle C (C4) and the other nine digits to the nine white notes above C4: 1 is D4, 2 is E4, and so on, as shown below.

Or one could assign the ten digits to some other group of ten notes. Or, since the common scales of pentatonic, diatonic, and chromatic have five, seven, and twelve notes respectively, one could represent π in base 5, 7, or 12, so that each digit would correspond to a unique note in the corresponding scale, octave differences aside. The example below shows a diatonic rendition of π in base 7 (3.0663651...) in which register is freely chosen. (In base 7, .066 is quite close to .1, which is 1/7 in base 10. This is another way to see that π is very close to 3 1/7, a well-known rational approximation.)

You can find multiple examples of such representations around the internet, such as here. The resulting music sounds as one might expect: even though the digits of π are not random, the melody sounds more or less as if it were randomly generated.

However, as with so many random or apparently random phenomena, the appearance of randomness in this sequence does not preclude identifications of design. For example, one could find multiple digits in a row, such as the series of six 9's in a row that starts with the 762nd digit in the decimal representation of π. Or one could find an incremental series: 0123456789 occurs first at the 17387594880th digit. Or one could find one's birthday (mm/dd/yy, or dd/mm/yy, or otherwise) within the sequence. Any such series would be even more remarkable if π began with them, which, in the last case above, it would for someone born on this day four years ago (if one allows 3 to substitute for 03).

Aspects of design could also be identified through musical conventions. 999999 would create a distinctive sound if played as a melody -- repeated notes -- as would 0123456789 -- straight through the scale -- if adjacency of digit corresponded to adjacency within the scale. However, other aspects of design are more particular to music. For example, one distinctive design of Western classical music -- found especially in keyboard accompaniment patterns -- is a succession of evenly spaced notes whereby pairs of notes separated by a fixed time length (labeled as n notes below) are no more than a step apart in the prevailing scale, simulating smooth voice leading in multiple virtual parts. Below are some diatonic examples. The numbers below each note show the number of diatonic steps the note is away from the note that occurred n notes before it. The series of +2, +3, +3 below the excerpt from Schumann's music shows exceptions to the stepwise relations.

As shown with the first example above, π base 10, when realized as diatonic steps on and above middle C, starts with such a design with two notes in between: the next-adjacent notes F-G, D-D, and G-A are no more than a second apart from one another. This design is more infrequent with more notes in between: the first such design with three notes apart starts at digit 24 and the first with five notes apart starts at digit 28 and overlaps with the previous, as shown below. The first such design with four notes apart does not happen until digit 502, assuming the notes are in fixed registers.

Another distinctive musical design is a progression of triads, one of Western classical music's most privileged harmonies. In the second example above, which is in base 7, the first triad between successive notes is a B triad representing the digits 3, 6, 1, and 3, which begin at order position 13. However, there is no different triad immediately before or after this B triad, so there is no triadic progression. The first such triadic progression in π base 7, shown below, starts at the 696th note.

The preceding exposition provides the context for what makes the following so remarkable. Here are the opening 17 notes of π base 12, realized as notes in the chromatic scale in which 0 = C, 1 = C# or Db, and so forth. I use the digit B for 11 in base 12; it so happens that this number also stands for the note B when C is assigned to 0.

If one considers the whole-number portion of this number -- the 3 -- as a "before-the-beginning" pickup note, then π base 12, represented by chromatic notes, begins with both a triadic progression and an arpeggiated design that simulates smooth voice leading. The next such series of notes derived from the π base 12 sequence that has both of this properties does not start until the 5763rd note.

Moreover, this triadic progression is between two minor triads -- C-sharp minor and A minor -- whose roots are a major third apart and in which the "higher" triad in the minor-third relation is more like tonic -- in this case, because it comes first. As I discuss here with regard to its use in motion pictures, such a progression has been associated with villainy and the shadowy in a lot of Western music.

The four-sharp diatony of what follows, and how standard the implied harmonic progression is among the first seventeen notes, is also quite remarkable. For a Western classical musician, the opening triadic progression and its immediate continuation might as well be the equivalent of starting the fractional portion of π with 999999.

In Common Time, Ain't No Onset on a Strong Beat When Threes Unfold, Unless There Is

A couple years ago, Richard Cohn wrote an article exploring how music in "pure duple" meter — that is, music that divides time into units of powers of 2 — interacts with a rhythmic pattern that evenly divides time into a succession of 3s. There are no integers x and y such that 2^x = 3y. This means that, if a pulse with successive onsets separated by 3 unit durations -- say, sixteenth notes -- starts on the first downbeat of 4/4 music, no onset will fall on metrically relatively important moments such as beat 2 of m. 1 (4 sixteenth notes later), beat 3 of m. 1 (8 sixteenth notes later) downbeat of m. 2 (16 sixteenth notes later), downbeat of m. 3 (32 sixteenth notes later), downbeat of m. 5 (64 sixteenth notes later), and so forth. Relatively metrically important can mean, among other things, that a change of some musical aspect, such as harmony or form, is more likely to occur at these moments. The notation below demonstrates this initial stages of this pervasive non-coincidence: never is an onset from the bottom part synchronized with an onset from the top part.

The relationship between these two parts can be inverted: if in the top part, sixteenth rests and sixteenth notes are converted into one another -- producing the complement, or negative image, of the original rhythm -- then always is an onset from the bottom part synchronized with an onset from the top part, as shown below.

In his article, Cohn spends some time with Bill Withers's song "Ain't No Sunshine." The first two verses of this song each unfold over an eight-bar span, which I hear as a shortened form of the twelve-bar blues structure, with each bar in 4/4. Instead of a third verse, Withers chains together twenty-six instances of "I know" in a single breath, each sung to what would be notated as an sixteenth-eighth rhythm to match my proposed eight-bar-verse notation. The notation below shows two possible metrical readings of this music.

Cohn puts forward Reading #1. This works out quite well for many reasons:

• the "I know" chain begins a quarter of the way during the eighth measure of the second verse's span, exactly as each of the two previous verses begin a quarter way during the measure that precedes each verse's span
• the strings fade out at this reading's beginning of the third-verse substitute
• the last "know" falls on a power of 2
• the meter falls right in line with the fourth verse to come

However, I can also hear Reading #2, which corresponds to the second example above. In fact, I find that sometimes I have to work against Reading #2 in order to hear Reading #1. Something about the new musical idea prompts a resetting of the meter for me. Or perhaps Reading #2 compels, especially in retrospect, because it continuously reinforces the pure duple's junctures with onsets. In Reading #2, the word that falls on a power of 2 toggles back and forth between "I" and "know." This means one cannot stop the pattern on any power of 2, unlike Reading #1, where the word that falls on a power of 2 -- if there is a coincident onset at all -- is always "know."

One can generalize this phenomenon beyond 2s and 3s. There are no integers x,y, and z such that 2^x = zy, and z is not a power of 2. Cohn's article, and the discussion above, concern the situation when z = 3. The next largest z would be 5. I have in mind a 23-second passage in a well-known song by a progressive rock band for which z = 5 would be appropriate. However, it neither continuously avoids pure-duple moments (like my first example) nor continuously articulates them (like my second example). Rather, it inhabits a happy medium between these two extremes, creating both a pulling away from stability and a push toward resolution, all within a single perpetual process. I will blog about this music next February.

Another Protuberant 3 in Some Mozart

In January posts on this blog from two and three years ago, I recognized two late eighteenth-century sonata movements in which the recapitulation of the second theme was altered to give scale degree 3 more salience that it had during its exposition.

Here's another example, from very well-known music of Mozart: the first movement of his piano sonata in C major, K. 545. These examples show the end of the second theme in the exposition and recapitulation, respectively. The descending-fifth transposition in the recapitulation avails more room for the treble melody to stretch out in the highest register, of which Mozart takes advantage in m. 65, which is an octave higher than what one would expect given its correspondence with m. 20. It so happens that the highest note during this stretch is E6, which is scale degree 3 in C major. Likewise, Mozart stretches out in m. 69 from what would be down a fifth from m. 24, up via a new scalar arch to the high E6 (which is connected to the beam in red, and followed by another scale degree 2 beamed in red, albeit back down in the treble-clef staff).

I will leave it there. If one wonders how this might intersect with a Schenkerian reading of this entire movement (of which my reading is neither necessarily of the entire movement, nor entirely Schenkerian), I recommend consulting John Synder's intriguing approach to this movement (cited here) at some point.

Hanson's Second Symphony "Romantic" Inverts Wagner's Tristan

On this day sixty years ago, the American composer Howard Hanson recorded his Symphony No. 2, nicknamed "Romantic," with the Eastman-Rochester Symphony Orchestra. This symphony refers to nineteenth-century music in several ways. One way is the resemblance between the symphony's opening and the famous opening of Wagner's opera Tristan und Isolde. The beginning of each work is provided below in a grand-staff reduction.

These two beginnings are similar in several respects:

• As shown with an enclosure, each uses a progression of four chords that are grouped together by repetition and/or silence.
• In each four-chord progression, one treble-clef voice changes to a different note from Chord 1 to Chord 2 and Chord 3 to Chord 4, and all notes change to different notes from Chord 2 to Chord 3. In accord with this -- more voices typically change to different notes at moments of greater metrical accent -- Chords 1 and 3 are more metrically accented than Chords 2 and 4.
• The top voice of each begins on G#4/Ab4 and rises by step to end on B4: Wagner entirely by half step, and Hanson with a whole step then a half step.
• The second-to-lowest voice of each sounds B3 for Chords 1 and 2, and then G#3/Ab3 for Chords 3 and 4, creating a voice exchange with the top voice. 
• In contrary motion to the top voice's rise, the bottom voice of each descends by step from its note of Chords 1 and 2 to its note of Chords 3 and 4.
• Each progression is soft, slow, and -- not shown in the reduction -- features the woodwinds of the orchestra. 

Wagner's progression uses four voices, while Hanson's uses five. However, in the eleventh measure of the symphony, Hanson removes his second-to-lowest voice: the one with B3 and Ab3, marked with little blue dots in the notation above. This slimmed-down progression reveals other connections to Wagner's music.

These two four-voice progressions are shown below, more abstractly. Each voice, along with each chord, has been numbered: the highest voice is Voice 1, the second highest voice is Voice 2, and so so forth. In Wagner's music, Chords 1 and 4 are conventionally tertian: specifically, Chord 1 is a half-diminished seventh chord and Chord 4 is a major-minor seventh chord. In Hanson's music, Chords 2 and 3 are conventionally tertian: specifically, Chord 2 is a half-diminished seventh chord and Chord 3 is a major-minor seventh chord. Although the two remaining chords in Wagner's music do not similarly match the two remaining chords in Hanson's music, this difference could nonetheless be described with a permutation: Chords 1 and 2 switch places, and Chords 3 and 4 switch places. This can be represented with the notation (12)(34).

The graphic below takes a closer look at the half-diminished (ø7) and major-minor seventh (Mm7) chords from each progression. Each colored arrow measures the number of semitones between the two notes spanning the arrow as if these two notes were transported by octaves to put them as close as possible. For example, in Wagner's Chord 4, the top voice's B and the bottom voice's E are separated by a perfect twelfth, which is seventeen semitones. However, if the top note was lowered by two octaves (or the bottom note was raised by two octaves), they would span a mere five semitones. This five-semitone span is represented by the color white. The correspondence between each arrow's color and the number of semitones of its span is also shown in the graphic below.

The double lines in the graphic above single out the red (three-semitone) arrows between Voice 1 and Voice 3 in all four chords, and the blue (two-semitone) arrows between Voice 2 and Voice 4 in all four chords. In Wagner's chords, the notes in Voices 2 and 4 (D# and F) move in parallel motion to new notes (D and E), while the notes in Voices 1 and 3 (G# and B) switch places (disregarding octaves), as shown with the crossed diagonal arrows. This switch in Wagner's progression could be labeled as a (13)(2)(4) permutation. In Hanson's chords, the notes in Voices 1 and 3 (Bb and Db) move in parallel motion to new notes (B and D), while the notes in Voices 2 and 4 (F and G) switch places (disregarding octaves), as shown with the crossed diagonal arrows. This switch in Hanson's progression could be labeled as a (1)(24)(3) permutation.

These two scenarios flip when we consider each note as labeled by its intervallic environment within its chord. For example, the G# in Wagner's Chord 1 (first chord) and Voice 1 (top voice) is three semitones, three semitones, and five semitones away (disregarding octaves) from the other three notes in Chord 1, as represented by the two red arrowheads and one white arrowhead in the G# cell of Wagner's Chord 1. Therefore, in Wagner's Chord 1, G# can be labeled as two parts red and one part white, which is the distribution of colors on the Austrian flag. In the graphic below, the G# in Wagner's Chord 1 is positioned on the Austrian flag.

I have chosen the colors so that the other notes can be positioned on other country's flags -- Germany (black, red, yellow), Estonia (blue, black, white), and Armenia (red, blue, yellow) -- although, with apologies especially to Armenia, the colors have been standardized to the same primary or secondary hues. The four intervallic environments of the four notes of a major-minor seventh are the same as the four intervallic environments of the four notes of a half-diminished seventh, as shown by the same four flags in each vertical column. However, although Wagner's Chord 1 and Hanson's Chord 2 assign the same flags to the same voices, the registral (vertical) ordering of the flags for Wagner's Chord 4 and Hanson's Chord 3 are different from this and each other. The arrows of the graphic below show how the assignment of each flag to each voice permutes in the progression from ø7 to Mm7 for each composer's work. From this vantage point, Wagner's permutation is (1)(24)(3), because the flags of Voices 2 and 4 switch places, while those of Voices 1 and 3 do not. Hanson's permutation is (13)(2)(4), because the flags of Voices 1 and 3 switch places, while those of Voices 2 and 4 do not. These permutations are swapped from those shown earlier.

These two voice permutations also share a relationship with the aforementioned (12)(34) chord permutation, as shown below: the latter -- called an automorphism -- transforms one voice permutation to the other.

Henry Klumpenhouwer's 1991 dissertation from Harvard inspired this analysis.

A Gutsy Prelude by Maria Szymanowska Turns a Chord Inside Out

Just over three years ago in Paris, a scholarly conference called "Maria Szymanowska and Her Times" was wrapping up its focus on the talented Polish pianist-composer who flourished during the first three decades of the nineteenth century. If I had been on the program, I might have talked about the innovative aspects of the seventeenth prelude of her Twenty Exercises and Preludes, which were published in Leipzig almost two centuries ago in 1819. Below is a summary of the music's tonal, harmonic, and melodic materials. In my reduction, some of the chords have been modified from this edition to achieve more formal-harmonic consistency.

The lower-case letters a, b, c refer to distinctive melodic ideas, all of which move in more or less continuous sixteenth notes. Idea "a" is a initially vaulting stepwise rise of parallel thirds that later float downwards; variants on this are more wave-like. This idea serves as the ritornello that articulates the tonic chord of each new key, and the return to the main key of B-flat at the end. Its four statements divide the music into four different rotations.

Idea "b" uses chromatic half-stepping neighbors harmonized in parallel sixths. Idea "c" alternates between harmonic third and sixths as each interval descends by step. The reuse of idea "b" divides the prelude into two parts, each with two rotations.

The notated key-signature changes are Szymanowska's, which partition the prelude into four parts as shown with the blue brackets on the left. The proximity of systems to one another in my layout reflects the rotational form, which begin in alignment with the tetrapartite key-signature form, but then cut across it toward the end.

While the first and last rotations unsurprisingly start in B-flat major, the main key of the prelude, the second is in C major and the third is in E major. These are unusual, even audacious, subordinate keys to be thematically and formally articulated in such a small-scale piece like a prelude, perhaps even the two most unusual of such among classical tonal works if we categorize relations to the main key regardless of mode and direction but simply by one of the six interval classes: minor second, major second (B flat to C), minor third, major third, perfect fourth, and tritone (B flat to E). (Root motion by major second and tritone is also the largest distance between the roots of consonant triads in this space.)

Also, m. 29 has a curious pivot chord. Its curiosity can be understood by first recognizing that the interpretation of tonal materials is often hierarchical, sometimes deeply hierarchical, empowered by the preposition "of," and that this hierarchy has the potential for rearrangement. For example, the highest note in the first chord of Beethoven's First Symphony can be interpreted as the third of (the chord built on) the fifth (scale degree) of the (scale whose tonic is a) fourth of (the scale whose tonic is) C. Sometimes the ordering of the intervals in the hierarchy is permuted, creating a different interpretation of the same note, chord, or key. For example, the fourth note of the famous G-G-G-Eb motive at the beginning of Beethoven's Fifth Symphony in C Minor is typically heard as ^3 in i ("three of one"), but, when conductors repeat the exposition, which ends in E-flat major, one can hear this same Eb as ^1 in III ("one of three"). The chord at the end of a typical sonata-form exposition is I of V (V:I), but the same chord at the end of a typical sonata-form development is V of I (I:V).

In Szymanowska's prelude, an F-sharp minor chord sounds in m. 29, which is very close to exactly halfway through the 56-measure prelude. In a larger formal context, this chord's root is the tritone above the formally articulated tonic of C, which is in turn a whole step above the prelude's tonic of B-flat. But Szymanowska recycles a technique from mm. 18-20, reinterpreting a triad as a supertonic in the following key. This puts m. 29's root a whole step above the next key of E, which is in turn a tritone above the prelude's tonic of B-flat. (Szymanowska achieves the modulation from E to B flat in m. 41: the German sixth in E becomes a V7/V in B flat. This enharmonic pivot is quite common to modulate up or down by semitone, but it's much less common to use it to modulate by tritone.) The tonal hierarchy of the F-sharp minor chord in m. 29 has been turned "inside out," as the roman numerals and the pivot-chord analysis in red show above.

Amy Beach & Her "Old World" Symphony

American composer Amy Beach began composing her second symphony near the end of 1894, around a year after she had heard the premiere of Czech composer Antonin Dvořák's ninth symphony, subtitled "From the New World." Her symphony, subtitled "Gaelic," was premiered on this day in 1896.

In her book Amy Beach, Passionate Victorian: The Life and Work of an American Composer, which was published twenty years ago, Adrienne Fried Block recognized that Beach's "Gaelic" Symphony was "both inimitably her own and at the same time influenced by the 'New World' Symphony's use of folk idioms."

The influence may have also involved key choice as summarized in the table below, which provides some information about the first movement of each symphony. The numbers indicate measures and are distributed proportionally in the figure, unless measure markers are too close, as in the case of Dvořák's mm. 396 and 400. The counting of Dvořák's measures begins after his twenty-three-measure slow introduction. The vertical/diagonal lines indicate the degree of discrepancy between the time -- proportional to each work's sonata-form span -- that the keys (along with their accompanying formal sections, in some instances) arrive in each movement.

Both symphonies are in E minor. Moreover, the first movements of both symphonies, each cast in sonata form, use a three-key exposition (Expo.): E minor, G minor, and G major. Both composers place their primary theme (P) in E minor. Dvořák assigns the his two secondary themes (S1 and S2) to G minor and G major, respectively, while Beach uses G minor as part of the transition (Tr) to G major, in which both of her secondary themes reside. This similarity of expositional key design is not that distinctive, as the relative major is a very common secondary key for minor-mode movements, and preceding this relative-major key with its parallel-minor key is also fairly common. Their development (Dev.) sections diverge with respect to their key areas (X).

However, in their recapitulations (Recap.), both first movements proceed through the keys of E minor, G-sharp minor, and A-flat major before returning to E major and then finally E minor. These choices of keys for a recapitulation's secondary themes (#iii and #III enharmonically) are extremely unusual for a nineteenth-century sonata-form work, which suggests that Beach's keys were modeled after Dvořák's. However, while Dvořák's keys and two secondary themes align (S1 in Gm and G#m, S2 in GM and AbM), Beach restores her S2 to E major, which is the traditional tonal adjustment of secondary themes in a sonata-form movement. Dvořák never provides a theme or even a cadence in E during the secondary-theme portion of his recapitulation, upending a time-honored convention. For this reason (and other reasons, which I may revisit in a later post), Beach's movement is more conservative, more "old world."

Beach took issue in the mid-1890s with Dvořák's recommendation that American composers should use "negro melodies" in cultivating a distinctively American music. In her biography, Block relates how Beach opposed this view, "believing that blacks were no more 'native American' than 'Italians, Swedes or Russians,'...rather, she believed that composers should look to their own heritage." Beach's incorporation of Gaelic folk songs in her symphony makes it clear that she is looking back over the pond for inspiration, but her use of more traditional European relationships among key, theme, and form than what occur in Dvořák's symphony suggests a more subtle way to achieve her posture, which is at once critical and tradition-oriented.

Scalar Transformations in Some of Chen Yi's Music

Below are the first fifteen measures of the solo cello part of the last movement of Chen Yi's Ballad, Dance, and Fantasy for cello and orchestra (2003). The entire published score can be found here. I have added an editorial marking—a flat over the G in m. 5, as this would be more consistent with mm. 8 and 11—and a performance suggestion in mm. 4-5 and in subsequent parallel measures marked simile.

The first five notes in m. 1 can be transformed into the first five notes in m. 5 through the process laid out below. First, the notes are retrograded and inverted around C2/G2. Second, since the intervals between adjacent notes are all either one or two steps within the chromatic scale, these sizes can be exchanged with one another. Lastly, the scalar context for the succession of step sizes is changed from the chromatic scale to a pentatonic scale. Just as moving from C2 to D2 in my example skips over exactly one note in the chromatic scale (hence, +2 or up two steps), moving from Eb2 to Ab2 in my example skips over exactly one note in a pentatonic scale. As Dr. Yi's music is billed as blending "Chinese and Western traditions, transcending cultural and musical boundaries," this last transformation seems especially apropos.

But perhaps this connection is a coincidence: too far to go transformationally for too indistinctive of a design. However, as shown below, if one applies the same series of transformations to the first nine notes of the cello part, then the resulting nine notes appear in mm. 4-5, albeit not all immediately adjacent. A cellist could adumbrate this connection by bringing out these notes, as I have suggested with my added articulations.

While echoing interests in R and I from earlier this year, this post also kicks off a three-post series featuring music written by women.

Thanks to Xiaolai Zhou for introducing me to this music.

And that's five years of monthly posts.

Music Provides a Personality Analysis using Successive Octaves

After being closed for approximately nine months for renovations, the General George Patton Museum of Leadership in Fort Knox, Kentucky reopens today. Many movies and television programs have explored the WWII general as a subject, but the most well-known is the 1970 film Patton with George C. Scott in the title role. Jerry Goldsmith's musical score for this film earned him an Academy Award nomination for Best Score. Goldsmith's score incorporates three distinctive musical ideas. One features a recording of two trumpets playing parallel perfect fourths in a triplet rhythm, which is then fed into an Echoplex, creating continuous and fading repetitions. While this idea is essentially a fanfaric motive, extended through reiteration, the other two ideas are bona fide themes: a spirited march and a slower-moving chorale. As Goldsmith explained in a 2002 interview, these three ideas represent different aspects of Patton's personality. The fanfare represents “the archaic part of [Patton], the historical, the intellectual part of him,” and the echo effect in particular reflects Patton's belief in reincarnation. More obviously, the march represents Patton's military side, and the chorale his religious faith.

In the interview, Goldsmith shared that he fashioned these three ideas “so that all three could be played simultaneously or individually or one or two at a time.” Given the fanfare's harmonic simplicity, its combination with either the march or the chorale is relatively straightforward. However, the merger of the march and chorale themes, which occurs both during the movie's main title and during the movie proper, requires more craft. The score below shows the two themes, how Goldsmith combines them in counterpoint, and the harmonic intervals that frame their contrapuntal interaction. (This music is transcribed in 6/8 (instead of, say, 12/8, or in 4/4 with triplets) to aid in its comparison to another famous simultaneity of two themes I will discuss on this blog a year from now.)

The fact that both themes basically arpeggiate tonic harmony for the first six of each eight-measure phrase (and dominant harmony for the last two) facilitates well-formed counterpoint. However, something about the combination contravenes classical tonal practice: measures 3 and 4 include successive octaves, as highlighted. This would be quite unusual to find in music from Bach to Brahms. One could interpret this succession as either faulty counterpoint, or counterpoint that has shaken off the shackles of classical rules. Or one's interpretation could instead espouse the idea that successive octaves undermine the autonomy of lines; therefore, the successive octaves in measures 3 and 4 subtly bring the march and chorale themes into a closer affinity with one another. This is consistent with Patton's biography, according to historyonthenet.com: "To Patton, prayer was a 'force multiplier'—when combined with or employed by a combat force, it substantially increases the effectiveness of human efforts and enhances the odds of victory. In this sense, prayer was no different from training, leadership, technology, or firepower."

Tables Turn in an Atypical Place in Bach's Music: R and I (Part V)

295 years ago today, Johann Sebastian Bach premiered his cantata Herz und Mund und Tat und Leben (BWV 147) in Leipzig. Each of the two parts of the cantata ends with a movement based on the Lutheran chorale tune "Werde munter, mein Gemüte," shown below.

The music for these two movements is well known today in instrumental settings with the name "Jesu, Joy of Man's Desiring." Here are the first five measures for these two movements.

One does not need to analyze deeply the famous stream of eighth notes to find within it the chorale tune's opening three notes (B-C-D), as shown below: they occur on beats 2, 3, and 1 in mm. 1-2.

The B-C-D progression—also on beats 2, 3, and 1—also occurs in the bass in mm. 3-4, as shown on the bottom of the full score provided above. In fact, these two pairs of measures enjoy a closer relationship. Below are just the soprano and bass onsets for these six beats. The first six notes, if retrograded and diatonically inverted around middle C, becomes the next six notes. The inverted clef and signatures at the end of the system also indicate this: if you turn this figure 180°, it looks exactly the same.

A "table canon"—whereby a line of music is placed on a table in between two performers, who read it from either side—also contains this kind of symmetry. This reduction of Bach's music could be performed by placing either of the two lines above on a table between a treble performer and a bass performer.

A Hidden Unflippable Pattern, via Schenker, in “Happy Birthday To You”: R and I (Part IV)

On this day 22 years ago (1996), "Happy Birthday to You" was inducted as a "Towering Song" into the Songwriters Hall of Fame, whose website reports that, on this day 125 years ago (1893), this song was first published, although it was only the music that was published in this year: the words probably came later.

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.

3-42

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

5-16 plus overlap with melody

The last step is to choose a registral realization of 5-16 that maximizes registral overlap with the melody. The examples below show that starting the soprano on the C above middle C creates an overlap with a total time of six quarter-note durations, instead of the four quarter-note durations if the soprano started on middle C. This soprano line, in both its content and pacing, matches the Urlinie of the aforementioned analysis by "Xenoceratops" here, and the Urlinie that I suspect most Schenkerian analysts would choose.

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.