Bizarre Beethoven

Beethoven -- or whom I occasionally like to call "the male Emilie Mayer" -- was born 250 years ago this month. His music, and discussions about his music, have been recently getting even more airtime than usual as a consequence of this sestercentennial, like this piece in the New York Times, where various luminaries find passages in the composer's music that are "resigned," "wondrous," "euphoric," 'furious," "eruptive," "rampant," "spiritual," "life-affirming," "passionate," "disruptive," "experimental," "daring," "heroic," "incredible," "extraordinary," "ferocious," "heart-stopping," "playful," "combative," "maniacal," "lyrical," "modest," "heartfelt," "reflective," "unpredictable," "transcending," "gloomy," "thrilling," "sophisticated," and "sublime."

I have another passage and another adjective to add to the list. The passage is from his Diabelli Variations, one of the works cited in the New York Times article. Below is the score for the twentieth variation.

This is strange music. For one nit-picky example of this strangeness, as enclosed in red below, this music as notated contains ascending thirds of both diminished and augmented varieties, each starting on the note two semitones below C (notated as A sharp in the diminished third and B flat in the augmented third).

As shown with diagonal lines below, a fairly strict but intermittent outer-voice canon with a delay of two measures -- stretching to three measures in its last appearance, indicated in green -- does offer some organization to this music's structure, but not all of it.

The one moment I find most bizarre in this already strange music, a moment that lies outside of the canon shown above but insinuated with the diminished third shown above that, is in m. 11. The stack of notes in the second half of this measure, enclosed below, is deliciously flabbergasting, for multiple reasons.

First, this is a root-position tonic triad in the key of the theme and the entire variation set: C major. Outside of its local context and ignoring its top note for now, this kind of chord should be the most stable sonority in the entire work's tonal fabric. And yet not the theme, nor any variation before or after this one, places a root-position C-major triad at this point in the two-reprise form: if anything, at this point in the form, the music is pulling away from the orbit of C major as a key, and certainly not landing on it as a stable chord.

Second, a fully-diminished chord occurs immediately before this C-major chord. Such chords, outside of other contextual patterns, are most likely to resolve to a consonant triad whose root is one semitone above some pitch -- octave aside -- in the fully-diminished seventh (as a vii°7) or to a consonant triad whose root is the same as some pitch in the fully-diminished seventh (as a "common-tone chord"). The two recompositions of the eleventh measure below show one example of each -- note how soprano and bass lines maintain the same contour of those in Beethoven's original, which matches the same of the prior two measures, which -- at least in the soprano -- matches the theme's sequential continuation in mm. 9-12.

Beethoven's resolution does neither of these. Rather, it resolves in a third, and the only remaining, way: to a consonant triad whose root is one semitone below some pitch  -- octave aside -- in the fully-diminished seventh. This third way is not unheard of in classical tonal practice. However, in this third way, two voices typically remain as common tones (which Beethoven does), and each of the two moving voices typically uses a melodic interval that is typically no larger than two semitones (as heard) and a step (as seen) (which Beethoven does not do, on both the hearing and seeing fronts). Furthermore, in this third way, the music is typically preparing a subsequent tonic that -- or at least a subsequent chord whose root -- is a perfect fifth below the root of the consonant triad to which the fully-diminished seventh resolves. In other, odder words, the consonant triad is "dominantized." In this case, with C as the "dominantized" chord's root, that would mean that the eleventh measure's strangeness might be mitigated with an F root or tonic soon thereafter, but no such emphasis on F materializes; rather, as in the theme, the music modulates to G. (If C's function is to be "-ized" for this purpose, it should be "subdominantized," not "dominantized."). The recomposition below shows a typical third-way  continuation, reversing the motivic soprano ascent to achieve even higher atypicality.

The C-major chord does make some sense when m. 12 repeats m. 11 verbatim (capitalizing on the clichéd maxim better known in jazz that it's not a wrong note if you repeat it), and, in retrospect, the C-major chord is "subdominantized" or "predominantized" as the music moves to a D7, the dominant in G. But, for me, the bizarreness remains. Perhaps Beethoven wanted the performer to prepare the listener for this bizarreness when he prescribed a terraced dynamic change (the pianissimo, from piano) in m. 11, in the middle of the four-measure sequential continuation -- something he does neither in the theme nor in any other variation.

Turning Some Music for Stargate: Atlantis

A number of my 2018 posts to this blog demonstrate how, by flipping or turning a representation of some music, you get either the representation of the same music or that of music nearby. Examples include a piano piece by Schoenberg, a song from a Disney movie musical, a string quartet of Pfitzner, a song by Richard Strauss, a popular celebratory melody (and one part-writing assignment of its notes in a typical harmonization), and a well-known cantata movement by J.S. Bach.

This video provides another example of this. It is most like the Pfitzner analysis, in that a particular rotation not only leaves the notes unchanged, but it also preserves the tonicizations as well.

Chika's "High Rises" and the Transformation of a Popular Form of Part Writing

"High Rises," the 2019 single by hip-hop artist Chika, enlists a vamp throughout the song. Here is my transcription of only the top line of this vamp.

This line receives consistent homorhythmic triadic accompaniment. In my first hearing of this music, I defaulted to an incorrect assumption about its accompaniment. I will introduce this incorrect assumption with a relevant but somewhat lengthy excursus. This line has six different pitches: D-E-F#-G-A-B. Removing either the F sharp or the G yields some pentatonic scale. Adding a C or a C sharp yields some diatonic scale. It is well understood that both the pentatonic and the diatonic scales play an important role in popular music. However, the same might be said for the kind of six-note scale -- what is sometimes called the Guidonean hexachord -- to which the six pitches in this line belong, or at least one particular employment of this kind of scale.

The number of pitches in a Guidoniean scale -- six -- is not a prime number, unlike the pentatonic and diatonic scales. A scale with a non-prime number of pitches -- let us say it has XY pitches, where X and Y are integers -- permits a certain kind of procedure: one can 1) find a smaller pitch set of X pitches within this scale that is evenly distributed within the scale, 2) transpose this smaller set up or down by step through the scale, and, thus, 3) generate a cycle of Y elements. It might be helpful to think of this cycle as analogous to a ring of blinking holiday lights; in the 96-light example below, X = 16 and Y = 6.

For example, in the four-pitch (XY = 4) scale on the left below, this procedure can generate a two-member (Y = 2) cycle of consonant dyads (X = 2) on the right below, a progression, or parts of it, commonly referred to as "horn fifths."

One place where this cycle occurs in the repertoire is from near the end of Smetana's Die Moldau.

For another example, in the two-pitch (XY = 2) scale on the left below, this procedure can generate a one-member (Y = 1) cycle of tritones (X = 2) on the right below.

For a third example, in the more familiar and scale-like twelve-pitch (XY = 12) scale on the left below, this procedure can generate, among many cycles, a three-member (Y = 3) cycle of fully-diminished four-note tetrads (X = 4) on the right below.

During a portion of Chopin's E-major Etude, these two previous cycles are woven together: mostly just  the right hand unfolds the smaller cycle mostly with every other sixteenth note, and the two hands together unfold the larger cycle mostly with every sixteenth note, although downwards instead of upwards as shown above.

For a fourth example, in the six-pitch (XY = 6) Guidonean hexachord on the left below, this procedure can generate a two-member (Y = 2) cycle of consonant triads (X = 3) on the right below. The half-note noteheads reveal how the '''horn fifths" cycle mentioned earlier is embedded within this cycle.

This cycle alternates between two of the four consonant triads available in the Guidonean hexachord: the only pair of triads whose roots are a step apart, one major, one minor. Above, those triads are D major and E minor. In most cases, one of these two triads is clearly treated as tonic, and the other triad serves as either supertonic or subtonic harmony.

This cycle plays a role in various forms of popular music that is frequent enough to warrant a name. It probably has already been given a name, but I do not know it. I speculate that one of its first uses was in gospel music, so I will provisionally call the employment of this cycle "gospel part writing."

Here are three examples of gospel part writing from popular music where the major triad is treated as tonic:

Here are three examples of gospel part writing from popular music where the minor triad is treated as tonic:

If the "High Rises" melody were harmonized throughout with gospel part writing, it would sound like this, with D-major tonic triads alternating with E-minor supertonic triads:

This is close, but not completely right; rather, this is the aforementioned incorrect assumption I made. Instead, this is how the line is harmonized:

Although the top line can be said to move stepwise through a six-note Guidonean hexachord, the bottom voices initially step through a seven-note D-major scale, generating the new triads of B minor and A major, foregoing the use of an E-minor and D-major triads for the lowest two notes of the melody. However, the gospel part writing cycle does apply at the end of the vamp to the highest two notes of the melody.

This interpretation of the harmonization of this melody as having some, but also lacking some, of the components of a gospel part writing cycle can make interesting what happens in the bass, which I have yet to show but do so now:

The bass primarily arpeggiates an E-minor triad, which is the "same" triad that the upper parts first entertained but then sidestepped when the harmonization slipped out of the Guidonean scale and into the D-major scale. This choice of bass makes more simultaneous and tonally balanced the two triads of gospel part writing that are typically only successive and tonally imbalanced. Through this mash-up of D-centered music up top and E-centered music down low, I relish the sound of not one but two different "soul dominant" chords (to use Mark Spicer's term): DM above E (dominant of A?) and AM above B (dominant of E!). But I also hear a remarkable struggle between two triads whose pecking order in the hierarchy of tonal status in gospel part writing is usually unequivocal -- I think this struggle resonates quite well with the autobiographical "origin story" Chika tells in the song. 

Does a Le Beau Sonata Exposition Succeed or Fail? You Decide

The exposition of a sonata-form movement is the first of the movement's three big parts. Especially in eighteenth- and earlier-nineteenth-century music, this part of the three is usually the easiest to recognize, because it is often surrounded by repeat signs. In modern-day practice, sometimes performers take these repeats and sometimes they do not. Occasionally, the composer will provide first and second endings at the end of the exposition.

A sonata-form exposition typically modulates to a secondary key and articulates a perfect authentic cadence in this key toward the end of the exposition. In their 2006 book Elements of Sonata Theory, James Hepokoski and Warren Darcy define a "failed exposition" as an exposition in which the secondary material does not end with a perfect authentic cadence in the secondary key, what the authors call the "essential expositional closure," or EEC. The last movement of Beethoven's Fifth Symphony is a good case in point. After the music modulates from the primary key of C major into the secondary key of G major, the music never articulates a perfect authentic cadence in this key before the end of the exposition, which is marked clearly by first and second endings.

Below are the string parts, which are sufficiently representative of the music's content to recognize formal aspects (and their absence), with some annotations. An earlier opportunity to cadence is avoided; instead, another theme begins at the point of cadential effacement. This theme is set up in a antecedent-consequent periodic design. Toward the end of the antecedent, a G sharp steers the music toward an A-minor triad, which functions as a predominant chord for the half cadence appropriately ending this antecedent phrase. As the louder consequent phrase begins, Classical practice suggests that this music will head toward a perfect authentic cadence. However, when the melody arrives again at the G sharp, Beethoven respells it as A flat and locks the bass first on C, then F, turning this harmony into a minor-mode predominant in C major. Both this theme and this exposition fail to achieve the customary cadential closure.

The first movement of Luise Adolpha Le Beau's Violin Sonata, op. 10, from the second half of the nineteenth century, begins with an exposition that, like that of the first movement of Beethoven's Fifth Symphony, begins in C minor and modulates to E-flat major for the second theme. The entire exposition is shown below, with a couple of annotations. Le Beau does write one perfect authentic cadence in E-flat major at the end of the second theme, but it is within the first ending. So if a performance takes the first ending and the repeat, like this one, the exposition succeeds, or at least, the first pass through it does. If a performance does not take the repeat and opts for the second ending after a first time through, like this one, the exposition fails. I do not know of another sonata-form exposition that puts the only possible EEC in the first ending and not in the second ending, but I would not be surprised to find out that there are others: please let me know with a comment below. (If Erwin Schrödinger had written a sonata...)

A Reminiscence of Beethoven in Some Music of Florence Price

In the 2007 book Black Women and Music: More than the Blues, edited by Eileen M. Hayes and Linda F. Williams (University of Illinois Press), Teresa L. Reed contributed a chapter called “Black Women and Art Music.” On page 191, while Reed is discussing the life and music of Florence Price (1887–1953), she writes "[t]he first movement of [Price's] piano Sonata in E Minor, for example, is a conservative rendering of sonata-allegro form. Its introductory bars are even mildly reminiscent of the opening measures to Beethoven's Pathetíque [sic] Sonata [in C Minor]."

Although both opening movements begin with a slow introduction (10 measures in Beethoven's movement, 12 measures in Price's movement) and the opening measures of these introductions enjoy a little mutual resemblance, the beginning of the two movements' Allegro sections are more alike and are probably the measures to which Reed refers.

I have provided the first seventeen measures of both below, with some color-coded annotations that point out similarities beyond the most obvious (tempo, cut time, etc.). I modified the Breitkopf und Härtel edition of Beethoven's movement so that the number of measures in each system is the same as in the Price, to allow for easier comparison. Most annotations, perhaps all, are self-explanatory. One that is perhaps not is the area shaded in blue toward the end of each of the first two systems. This highlights when the second system departs from an exact repetition of the first by staying on a half-note harmony for twice the duration. This extension pushes the predominant (PD) --> dominant (D) progression later in time by a half measure, altering the metrical position in which the music arrives on the dominant, which sets up two contrasting phrase-ending experiences.

I will leave it to the reader to decide how mild the reminiscence is.

Westworld Complements Game of Thrones

Ramin Djawadi turns 46 years old today.

He wrote the main theme for the HBO series Game of Thrones (2011–2019). Below is a transcription of just the opening and the basic harmonic and melodic components. A time signature of 6/8 or 12/8 might be more appropriate than 3/4, but there is a rationale for this choice.

There are six consonant triads, or consonant fifths, in the diatonic scale, which can be identified by their Greek-mode names. For example, the Game of Thrones main theme uses the three-flat diatonic scale, and the six consonant triads, or fifths, in this scale are rooted on E-flat (Ionian), F (Dorian), G (Phrygian), A-flat (Lydian), B-flat (Mixolydian), and C (Aeolian). Two of the six consonances in the diatonic scale are the common tonic consonances: Ionian and Aeolian. Of the four remaining consonances, Game of Thrones uses three of the four: Phrygian, Mixolydian, and Dorian. Only Lydian remains unused.

There are three quarter-note spans that begin on each of the beats in 3/4: beat 1, beat 2, and beat 3. Any of these three spans could be subdivided into two eighths notes. In the Game of Thrones main theme, the accompanimental motive subdivides the span starting on beat 3. The primary melody subdivides the span starting on beat 1. Only beat 2 remains undivided.

Djawadi also wrote the main theme for the later HBO series Westworld (2016–). Below is a transcription of just the opening and the basic harmonic and melodic components. The headless stems indicate pitches hard to hear. A time signature of 6/8 or 12/8 might be more appropriate than 3/4, but, once again, there is a rationale for this choice.

The Lydian consonance unused in the Game of Thrones main theme is the first non-common-tonic consonance to be used in the Westworld main theme, starting in m. 5. The beat-2 eighth-note subdivision unused in the Game of Thrones main theme is the first eighth-note subdivision to be used in the Westworld main theme, probably starting in m. 41 but clearest starting in m. 44. 

More can be said of how these main themes relate, such as a focus on transformations of the [0234] diatonic set. But I will stop there for now.

In Psycho, Herrmann Stabs Before Mother Does

On this day 60 years ago, Alfred Hitchcock's movie Psycho was premiered at the DeMille Theater in New York City.

Among other things, this movie is famous for its chilling music by Bernard Herrmann and its shower scene, where an unrecognizable assailant stabs Marion Crane to death. Although Hitchcock did not want music for the shower scene, Herrmann wrote some anyway, which has since become one of the most recognizable musical passages of film music and even beyond.

In an interview published in this book, Herrmann suggested that the horrific acts in Hitchcock's film are anticipated by his opening music: "The point, however, is that after the main title nothing much happens in the picture, apparently, for 20 minutes or so. Appearances, of course, are deceiving, for in fact the drama starts immediately with the titles! The climax of Psycho is given to you by the music right at the moment the film begins. I am firmly convinced, and so is Hitchcock, that after the main titles you know that something terrible must happen. The main title sequence tells you so, and that is its function: to set the drama."

Back in 2009, I wrote an (fairly serious (?)) essay for a book on horror-movie music that supported Herrmann's point about the main title using some technical analysis of the prelude's voice leading. Here I provide a (less serious (?)) analysis of the voice leading of the music immediately after the main titles to do the same thing.

The most common way to label four voices ordered from high to low in register is to assign the highest as soprano, the second highest as alto, the third highest as tenor, and the lowest as bass. These four voices are often abbreviated using their first letter: s, a, t, and b. Four-voice music is often called SATB music. However, these voice-based designations can also be used for music that is not specifically for voices.

When the composer twice assigns some set of four distinct musical elements to these four voices, one can describe that which "transforms" one of these two assignments to the other. For example, below is shown measures 7-11 of Anton Bruckner's motet "Locus iste." The first halves of measures 7, 9, 10, and 11 each feature a G7 chord, which contains the notes G, B, D, and F. These notes are passed around from voice to voice. For example, from the G7 in m. 7 to the G7 in m. 10, the soprano's D goes to the tenor, the tenor's B goes to the bass, the bass's F goes to the soprano, and the alto's G stays in the alto. Mathematicians summarize this "passing around" of notes with parenthetical notation. In this case, (stb)(a) is shorthand for s --> t --> b --> s and a --> a.

The use of the word "stab" in the title of this blog post probably hints at where my analysis is going. There are twenty-four permutations for four elements. As Wolfram Mathworld reminds us: "There is a great deal of freedom in picking the representation of a cyclic decomposition since (1) the cycles are disjoint and can therefore be specified in any order, and (2) any rotation of a given cycle specifies the same cycle." For example, (stb)(a) could also be written as (a)(stb) or (tbs)(a). That being said, there is nonetheless only one of the twenty-four permutations that can be written as (stab).

Immediately after the main title (at 1:54 in the video below), the film proper begins with a panoramic shot of Phoenix, Arizona, followed by a long multi-shot zoom through the window of a hotel room where Marion Crane and her boyfriend are trysting.

Herrmann accompanies this footage with a cue called "The City," which begins with the two measures provided below. Much of this cue features four-note chords: in mm. 1-2, the downbeats present B°7 -- a fully-diminished seventh chord with the notes F, Ab, B and D -- while the other three beats present Fø7 -- a half-diminished seventh chord with the notes F, Ab, B, and Eb. The latter chord is especially appropriate for the amorous scene to follow, as it matches the so-called "Tristan chord," which is associated with desire in Richard Wagner's opera Tristan und Isolde and in Herrmann's score for Vertigo, released two years before Psycho.

Each string section is divided into two parts. Violins 1 and 2 play the complete four-part harmony, and violas and cellos play the same down an octave. Although instruments rather than voices perform this music, it is still reasonable to consider the four lines in violins 1 and 2 (or the four lines in violas and cellos) as -- arranged from highest to lowest -- soprano, alto, tenor, and bass. Although the chords have the same (or almost the same) pitch content, each voice does not play the same pitch throughout: rather, they descend through these chords. As they do, the notes in each chord are passed from voice to voice. The permutation for all four progressions from one "Tristan chord" to the next is indeed (stab). If one allows the Eb as a substitute for D -- or vice versa -- this permutation describes all seven progressions in these two measures.

And, by at least one person's count, Marion is stabbed (around) seven times in the shower scene.

Compared to a Classical Example, The Threes in Porcupine Tree's "Even Less" Trickle Up Even More

Recent articles here and here and my blog posts here, here, here, and here have considered what composers do in the situation when a pulse's even division of time unfolds over a underlying meter that does not include this pulse's frequency or any multiple of it. In all of these considerations, it is assumed that the underlying meter does not yield; rather, the pulse yields by 1. dissipating, 2. shifting to another nearby frequency, or 3. trading its beginning-synchrony for end-synchrony. Examples of these three solutions are given below for the specific case of three(s) unfolding over 4/4. An underlying meter that does not yield in this situation is especially true in more popular music.

These situations do not arise as often in pre-1900 Western music as they do in post-1900 Western music, but, when they do, classical composers more often offer another solution: make the underlying meter yield to the pulse, rather than the other way around. I call this temporal process "trickling up": a pulse at a relatively quick frequency that is incommensurate with the prevailing meter begins on a relatively strong part of the meter (beginning-synchrony), and then the meter stretches to include some multiple of this pulse's frequency, setting up an end synchrony within this new meter. Here's one example of trickling up from the last movement of Haydn's String Quartet op. 76 no. 4, nicknamed "Sunrise." Below are this movement's first 21 and 3/4 measures.

The opening establishes a pure-duple meter on multiple levels, such as common time (4/4), two-measure hypermeasures, and four-measure phrases. A one-measure extension in m. 19, which uses an ascending-third melodic sequence to lift the top voice to the precarious ninth of the dominant harmony, breaks from the two-measure pulse. However, it appears the two-measure pulse returns with the repeat of m. 20 as m. 21 and the new idea in m. 22. Yet, this new idea begins a repeating three-eighth-note motive (G-F-Eb in Violin 1) that initiates a three-eighth-note pulse, although the bowing playfully continues the quarter-note pulse of the pure-duple meter. A post-1900 pop-music response to this situation might be something like the following:

But Haydn does not do this. Instead, he stretches the two-measure unit to a three-measure unit. This allows the accent scheme of the three-eighth pulse to be both beginning- and end-synchronized, while also cleverly dovetailing the end of the G-F-Eb motive with the pick-up of the recapitulated main theme (marked as A).

In this example, the threes trickle up by a factor of 8, which is 2 to the power of 3 (2^3): the repeated motive is three eighths long, and the three-measure unit that accommodates this repeated motive is eight times as long as this motive.

"Even Less" is the opening song on Stupid Dream, the fifth album by the progressive rock group Porcupine Tree, released in 1999.

Below is a diagram -- click it to zoom in -- that maps out the proportions, metric organization, and form of the seven-minute version of the entire song; Porcupine Tree later released longer versions of the song. The colors represent similarity and difference of material. (Curiously, what I've labeled as the chorus has no singing, but otherwise it functions a lot like a chorus.) I am considering 1 as the beat, around 124 bpm. The asterisk indicates the ordering of the metrical hierarchy: X*Y means that there are Y instances in a row of a X span.

Much can be said about temporal proportion in this song. For this post, I will focus on the 192-beat solo section from 4:33-6:06. This solo section divides into two equal spans of 96 beats. The first span features a loud guitar-based three-beat pulse, a more distorted guitar solo, and an accompaniment with fewer layers in its texture. The second span features a switch to a softer bass on the three-beat pulse, cleaner guitar solo, and an accompaniment with more layers in its texture. Unlike my analysis of all other spans of the song, each of these two solo spans is labeled in two different ways: 2^5*3 and 3*2^5. The 2^5*3 means that the 96-beat span is divided into three equal parts by the repetition of some pitch pattern. The 3*2^5 means that the 96-beat span is divided into 32 equal parts by the three-beat pulse. Rather than continue the pure-duple meter of the chorus into the solo section, and amend the three-beat pulse using one of the first two methods shown in the first two examples above, the 64-beat span is stretched into a 96-beat span. This allows the accent scheme of the three-beat pulse to be both beginning- and end-synchronized, creating satisfying metric resolutions at both 5:20 and 6:06.

In this example, the threes trickle up by a factor of 32 (2^5), four times that of the example from Haydn. This is the highest trickle-up factor I have come across; please post a comment if you know of one that is greater.

Neatly Near Self-Similarity in Survivor's "Eye of the Tiger"

A pop quiz about 80s pop: Below is a transcription of music played by guitar of the first part of the instrumental opening of "Eye of the Tiger" by Survivor, written for Rocky III, the fourth highest-grossing movie of 1982. Two contiguous measures are not correct, particularly in the timing of events. Which two measures? (If you think you know 80s pop well, answer before listening to the beginning of the video below.) My answer is below the video in both text and notation.

The three-chord (I-VII-I) motive in what I am notating as measure 20 is shifted later by two eighth notes (or two of the intro's shortest durational units played as repeated Cs, however they are notated), breaking the clear pattern established beforehand. (According to songfacts.com, Jim Peterik, one of the song's creators, was matching these motives to visuals from the movie; he "started slashing those chords to the punches we saw on the screen." I suppose one way to gain the upper hand in a boxing match is to surprise an opponent by breaking a pattern.) This shifted rhythm results in another, but much shorter, example of the kind of "end synchrony" I recognized toward the end of Yes's "Our Song": starting a series of 3-durations (in this case, dotted quarters) after the beginning of a pure-duple span (in this case, a 4/4 measure) can set up the synchronization of an onset in this series with the end of this span. This lends the downbeat of m. 21 and its tonic harmony a substantial phenomenal accent.

This phenomenal accent comes three-fourths of the way through the big sixteen-measure section in measures 9 through 24. Self-similarly, the recurring, non-shifted three-chord motive also articulates the moment three-fourths of the way through its measure. The sixteen-measure section would achieve more self-similarity with the three-chord motive if there was also a substantial phenomenal accent three-eighths of the way through the sixteen-measure section. Three-eighths of the way through measures 9 through 24 is the downbeat of measure 15. The onset of the first of two big sustained chords at the end of measure 14 is near this downbeat, but not quite. Moreover, the chord is a VI, resulting in long-range I-VI-I progression, which is not quite I-VII-I. However, as shown below (click for detail), each of these two states of nearness—one in time-space, one in pitch-space—neatly differ from their exact self-similar counterparts by exactly two of the smallest units in those respective spaces used earlier in the song: the semitone (G-Ab in measure 22) and the eighth notes of the omnipresent pedal point.

For those looking for more self-similarity in this song, consider how the analysis above compares to the rhythm and hypermetric location for "It's the..." and the big downbeat that starts the chorus.

Earlier, in Yes's Our Song...

At the end of last month's* post, I suggested that, earlier in Yes's "Our Song," there is another n-against-powers-of-2 cycle that does not quite reach its completion, and this interruption could make the completion of the later 5-against-powers-of-2 cycle all the more satisfying.

Below I have sketched out that earlier almost-cycle, which starts at 0:41. In this case, n = 3. I have chosen to retrospectively notate this music in 4/4. Although there is little to nothing to support this meter during the instrumental intro to the verse (0:41-0:53) and the first verse (0:53-1:13), its material repeats every 16 quarter notes, and the music of both the pre-chorus (1:13-1:28) and chorus (1:28-1:48) are much more clearly in 4/4, grouping these measures clearly into twos and fours. In short, my 4/4 notation of 0:41-0:53 gets a head start on what follows, for better or worse.

A triadic progression in bright synthesizer unfolds onsets three quarter notes (a dotted half note) apart, and a later bass-guitar addition subdivides this 3-quarter pattern into a 3-eighth pattern. Since each of these patterns begins on beat 2 of the first 4/4 measure, if it were to continue, one of its onsets would land on the big downbeat, shown in green, at the start of a span of 16 quarter notes (2-to-the-power-of-4). But it does not: rather, it peters out and a unison riff in guitar and bass, with assorted drum hits joining the notes marked with accents, knocks the implied continuation of each 3-pattern off its downbeat-targeted course by displacing it backwards an eighth-note duration, shown with red arrows. This happens repeatedly underneath the entire first verse.

What I have withheld thus far (because, in full disclosure, I did not think of this until after I wrote the end of last month's blog), is that the first instance of this riff actually precedes the first instance of the triadic progression, and the 3-pattern of the former leads right into the 3-pattern of the latter, as shown below. This obviously changes the narrative of "targeting" and "knocking off."

This being said, the riff could have both initiated the 3-pattern and, with an adjustment an eighth note later, ushered this pattern to its big-downbeat cyclic completion, as notated below.

It does not do this -- which, again, sets up the idea that the later completion is more satisfying -- but it could have. For the skeptic who thinks that such big-downbeat-finding displacements of a riff have no precedent, I will next post a discussion of one such well-known displacement from a pop song released during the year before "Our Song."

* (actually, three months ago, as COVID-19 set back this blog a bit, so I will be backdating the next couple of posts)

The 5-Against-Powers-of-2 Cycle in Yes's Our Song

A year ago, I blogged about a 23-second passage in a progressive rock song that does something rather special, but did not reveal the passage, instead promising to reveal it this month.

That blog post a year ago investigated how powers of 2s and multiples of 3s can interact in different ways in music, so, to complement that presentation, I will explore in this post how powers of 2s and multiples of 5s can interact in different ways in music.

If an even division of time with inter-onset intervals of 5 units aligns its first onset with the beginning of pure duple music (straight eighths, 4/4, 4-measure groups, etc.), a subsequent onset will never coincide with a power-of-2 beat, as shown below with the eighth note as the unit. Such a coincidence could be produced if the music broke from the quintuple regularity, like 5+5+6 = 16 or 5x12+4 = 64. This adjustment, what Richard Cohn calls a comma, converges the otherwise divergent quintuple and pure duple divisions of time, much like a leap year day—like today—helps to reconcile the otherwise incommensurate daily and annual divisions of time.

A corollary of this observation is that a rhythm that is the unit complement of the quintuple rhythm above will always place an onset on a power-of-2 beat. The spoken-word music below—simulating a group of folks both sporting and evaluating neckwear—demonstrates this corollary by stringing together a series of four-unit four-syllable phrases with a unit rest in between. The phrase "I like your tie" is used when at least one of its syllables coincides with a power of 2. Notice that the coincidences cycle through these four syllables in the following order: LIKE on 2, TIE on 4, YOUR on 8, and I on 16. The four-element cycle begins to repeat with LIKE on 32; the reader can verify that, if this pattern continues, the next power-of-2 coincidence will be TIE on 64, YOUR on 128, and so forth. (I'm also using this cycle to showcase, as many others have, how emphasizing different words in a sentence can change its meaning, as is sometimes done with the phrase "I never said he stole your money.") This four-cycle is analogous to the two-cycle that results when a complement of a 3-unit rhythm interacts with pure duple moments, as in "I know" from Bill Withers's "Ain't No Sunshine."

The simultaneity of multiples of 5 and powers of 2 are not as common as that of multiples of 3 and powers of 2, but, of those I have heard, most align the beginning of each pattern; therefore, these two ways of dividing time subsequently never align, unless adjustments are made later. One example of this is from the first track entitled "Shofukan" from the 2014 album We Like It Here by the American jazz fusion group Snarky Puppy. The simultaneity starts at 4:48 in the video below. Listen for the 5-note ostinato in the keyboard and guitar parts, with the highest note (a B) marking its beginning.

Now for the reveal: the aforementioned 23-second passage starts at 3:25 in the song "Our Song" by the English progressive rock group Yes from their 1983 album 90125.

Vocals, bass line, and a keyboard ostinato are transcribed below, with some annotations. This passage is pure duple: 128 eighth-note units sandwiched in between the last statement of the chorus (which ends with the line "Music has magic / It's good clear syncopation") and a return to the opening 7/4 instrumental introduction.

The ostinato iterates a three-note motive (B-A-D) that is five eighths long: both its registral contour—like the Snarky Puppy ostinato (which also has the same starting note and close to the same tempo)—and durational content—long, long, short—clearly emphasize the first of its three notes. However, the ostinato does not begin until three eighth notes into the passage. This sets up a cyclic, rather than divergent, relationship between the ostinato's beginning and the 6 powers of 2 in this passage that remain after timepoint 2. Although it starts with a misalignment (the +1 in red between timepoints 3 and 4), it realigns right away (the 0 in green at timepoint 8). But then it continues on the four-element cycle, passing through differentials of +3, +4, and +1, before returning to realignment at timepoint 128. For me, this realignment creates a powerful arrival, very much akin to a strong attainment of the G-major tonic harmony after so much subdominant and dominant, and the conjunction of the ostinato's B with the intro's first-note B. The lyrics appear to reference this well-planned coordination as well.

Next month I'll blog about an earlier n-against-powers-of-2 cycle in the same song that becomes dislodged soon before a moment of convergence, prohibiting a realignment. I believe that this suppression makes the later cycle even more satisfying.

A Textbook Omnibus for All to Use

The omnibus (Latin, "for all") is a class of progressions that prototypically involves minor triads, major-minor seventh chords, and contrary semitonal motion in two voices that not only connects chord to chord but also perpetuates past just a two-chord progression.* In 1998, Victor Fell Yellin wrote a book and Paula Telesco an article about omnibus progressions. If the contrary semitonal motion perpetuates over five chords, in which the first and last have the same root and quality, Telesco calls this the "classic omnibus." If it goes so long that another another pair of voices needs to takes over the contrary semitonal motion, and then another pair, and then another, returning to a chord with the same root and quality, Telesco calls this an "omnibus cycle."

I say "prototypical," because the term "omnibus" has also been applied to progressions with occasional fully-diminished seventh chords and whole-step voice leading, but I get the sense from Yellin's and Telesco's writings that an omnibus progression with only minor triads and major-minor seventh chords is, if not more common, nonetheless a more idealized default definition of the progression.

Unsurprisingly, the classic omnibus is more common than the omnibus cycle, regardless of what repertoire or time period you consider. Between them, the 1998 publications of Yellin and Telesco have two examples of a prototypical ommibus cycle: Hummel's Piano Sonata in F-Sharp Minor, first movement, mm. 118–23, and Tchaikovsky's Sixth Symphony, first movement, mm. 259–63.

I have another, from nineteenth-century Norwegian composer Johan Svendsen. The first movement of his first symphony fills some of its coda with a big prolongation of A7, the dominant of D major, the key of the movement and symphony. For this big prolongation, Svendsen puts four iterations of a classic omnibus (well, two iterations, but both forwards and backwards) and a complete prototypical omnibus cycle back-to-back. It is the most "textbook" display of the omnibus idea in a single excerpt I've ever run across. Here it is in my short-score reduction. You can listen to it below: the transcription starts at 8:50.

* I nonetheless still find a little value in calling, for example, a A7-->C7 progression with contrary semitonal motion in two voices something like "an omnibus-component progression." Telesco labels a three-chord omnibus progression a "small omnibus."