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."

Canon on White Christmas

Below is a recording I made of a canon -- at the sixth above -- that I discovered and arranged using the melody for Irving Berlin's song "White Christmas." It works rather well: I made one modification in the second half of measure 30. The bass line is a nod to that of the Air from Bach's Orchestral Suite No. 3.

This combination of Bach's and Berlin's music can also be heard in Peter Breiner's mashup on Naxos's Christmas Goes Baroque II from 1993.

The Pitches of the Music for NPR's All Things Considered Are (Just About) Right on Time

This post combines the music I considered in my post of two years ago with the methodology of my post here about some of Carl Vine's music and here about some of Sergei Prokofiev's music that show correlations between the pitches and tempos of a musical work.

Below is a partial transcription of the middle of the theme music for NPR's All Things Considered. During this middle, the music transitions from 4/4 to 5/8 with the eighth note as a common pulse. At this point of transition, there is also something of an authentic cadence in B major: a top-voice A#5 harmonized by a F-sharp-major dominant triad resolves to a top-voice B5 harmonized by what will probably heard as a B rooted chord, even though an E replaces the expected D# -- making a 027 -- and the F# is in the bass. In the 5/8 section, the change of bass from this F# to C and back again occurs every three measures.

Into the 5/8 section, there is a slight increase in a particular hypermetric frequency: the two-4/4-measures pulse (16 eighth notes long) speeds up just a tad to the bass's three-5/8-measures pulse (15 eighth notes long). When spanning two pitches, this 16:15 ratio is the just diatonic semitone. Indeed, if one transposes this particular 16:15 tempo interval of .4375/.4666... Hz up a few octaves so that it sounds as the pitch interval of a semitone, and moves the two notes up to the next available equal-tempered semitone -- if A4 is 440 Hz -- then this pitch interval is A# to B, the same top-voice melodic motion at this same point of metric transition. Furthermore, the 5/8 downbeats, which come three times as frequently as the .4375 Hz change-of-bass frequency, would correspondingly map onto an F#, which accompanies the B in the 027 harmony at the metric transition. These tempo-pitch relationships are summarized below, using a color coding from above (16:15:5). The two outside three-note groups are pitches in equal temperament, and the middle three-note group expresses the three aforementioned tempos as pitches, using a ten-octave transposition.

A Maximally Varied Stretto Fugue

Two years ago, I introduced an isochronous melody that 1) has no internally recurring patterns and 2) can be consonantly combined with itself at any transpositional level or at any time delay. This is the only such melody of eight notes -- allowing for individual octave transfers, or the melody's wholesale transposition, inversion, or rotation (moving some of the notes from one end to the other) -- that has these two properties.



I doubled the durations, moved some notes up an octave, and embellished this eight-note melody to make the following subject.

I then wrote a stretto fugue based upon this subject that demonstrates its special properties. After a standard exposition, and before a final entry in the subdominant, the subject overlaps with itself at seven different time intervals to create seven different strettos, each time interval one unit smaller than the last. Since each time interval is linked to a certain pitch interval, the choice to use an incremental acceleration of the frequency of subject entries thus stipulates the fugue's succession of key areas. However, the linear pattern of the acceleration translates into a convenient arch pattern of transpositions. After a presentation in the main key (A), this arch pattern takes the fugue along a relatively standard tour through the relative major (C), then its dominant (G) and then its subdominant (F, of sorts) before reversing course through these keys, ending with two subject entries in A minor only a single unit apart. The close proximity of the acceleration's final entries yields three more secondary strettos and requires the use of four voices, up from the three with which I decided to begin.

All of this is shown in the graphic below, and can be watched and heard in the video below.

This fugue has 10 different stretto intervals among 11 subject entries, which earns a ratio of .91 different stretto intervals per entry, and all of the stretto intervals are different. This can be compared to two stretto fugues of J.S. Bach. Contrapunctus VII of The Art of Fugue has 22 different stretto intervals that are all different, as I have shown here. However, with 26 subject entries, this fugue earns a ratio of .85 different stretto intervals per entry, a little lower than mine. The C-major fugue from Book 1 of The Well-Tempered Clavier has 8 different stretto intervals among 24 subject entries (ratio of .33), and duplicates some stretto intervals.

More Perfectly Contrarian Counterpoint in Berlioz's Symphonie fantastique

Last month, I pointed out a spot in the fifth and last movement Berlioz's Symphonie fantastique that uses successive parallel octaves. As shown in Liszt's solo piano transcription below, there is another spot in the previous movement that dwells upon perfect harmony between the outer voices, but in contrary motion. It's during the moment when the march to the scaffold first bursts forth with the entire orchestra playing fortissimo:

I hear this moment as pointedly transgressive of classical norms, which seems appropriate for music meant to accompany the opium-drugged artist witnessing his own execution, as Berlioz's own program notes describe the scene. There are innumerable examples of a diatonic progression comprising at least three chords in Western classical music in which 1) one outer voice moves by step in one direction, 2) the other outer voices alternates skipping by thirds and fourths in the opposite direction, and 3) the two voices always form imperfect harmonies. Below show 56 (7 x 2 x 2 x 2) possible versions with three chords, categorized by 1) the scale degree the stepwise line starts on (7 options), 2) whether the stepwise line goes down (d) or up (u) (2 options), 3) whether the stepwise line is on the bottom (b) or top (t) (2 options), and 3) whether the first melodic skip is a third (3rd) or fourth (4th) (2 options). I like to call these imperfect wedges.

Left out of the categorization is what mode (major or minor) the music is in, how the outer voices are harmonized, any transposition of one or both voices away from the other by one or more octaves, and so forth.

At least one of these progressions has been named: what the labeling system above designates as a 3ut3rd (for "stepwise line starts on ^3, goes up, and is on top; skipwise line starts with a 3rd"; it is enclosed in blue above) has been called the "champagne progression" by music theorist Gene Biringer and promoted at Open Music Theory. There it is recommended to "[o]nly use it with mi–fa–sol (or me–fa–sol) in the melody." Below each progression I have listed the number of instances of the progression I have found in a broad survey of Western classical music. While 3ut3rd (that is, the stepwise line starts on mi or me) is by far the most common of all of the ut3rd progressions, other ut3rd progressions are also used, particularly 1ut3rd. Moreover, the "champagne progression" is not the only imperfect wedge I would recommend as a schema: in my survey, 1db3rd (99 instances; enclosed in red above) is even more common than 3ut3rd (73 instances).

Berlioz, however, uses a perfect wedge: the same design but the outer voices are a third farther part. (I suppose you could call it a 4ub3rd perfect wedge.) Perfect wedges are much rarer in Western classical music: while I found 250 imperfect wedges in my survey, I only found 13 perfect wedges, including Berlioz's. This situates it as both atypical and perhaps also impertinent, since its design so closely resembles that of an imperfect wedge. Furthermore, in the Symphonie fantastique passage cited above, Berlioz's displacement of a third interval from more normative schematic counterpoint occurs at the same time as his displacement of the onsets from the strong beats of the meter.

More Successive Octaves in Some Music of Berlioz

As I post this, Quatuor Aeolina is performing a four-accordion transcription of Berlioz's Symphonie fantastique of 1830 at the Berlioz Festival in La Côte-Saint-André in southeastern France.

Here's more audacity: toward the end of the last movement of this work, Berlioz's combination of his original Witches' Sabbath theme and the preexisting Dies irae chant contains successive octaves. Like the successive octaves in Jerry Goldsmith's score for Patton I blogged about a year ago, these octaves take place from one compound-duple (e.g. 6/8) downbeat to the next, toward the beginning of a combination of two themes -- one sacred, one not.

A small rewrite of the Sabbath Round would have avoided these successive octaves. Here are a couple of possibilities:

I am not arguing that these successive octaves are inherently good or bad—plenty of ink has been spilled judging Berlioz's counterpoint in this manner—but they are unquestionably a deviation from the manner in which melodies were combined in classical Western music from the previous century.

Some Music of Brahms Sounds Like Some Music of Alice Mary Smith

Alice Mary Smith (1839-1884) was an English composer of choral, instrumental, and chamber music. In his 2003 edition of her two symphonies, Ian Graham-Jones states that she was the first woman in Britain "to have written and to have had performed a symphony, the Symphony in C Minor of 1863," her first. The first movement of this symphony bears some resemblance to the first movement of the first symphony of Johannes Brahms (1833-1897), also in C minor. Although the symphony was not completed until 1876, Brahms sent a draft of the first movement, without the slow introduction of the final version, to Clara Schumann in 1862.

The two movements realize their sonata forms in similar ways. Both movements use a primary theme (P) in C minor and a secondary theme (S) in E-flat major. This is no surprise, as most C-minor sonata movements do this. However, there are other less common resemblances. The P theme and the transition (Tr) sections are almost exactly the same length in terms of numbers of measures, and both insert a four-measure thematic introduction between the slow introduction and the start of the P theme, as indicated with the formal diagram below.

Lastly, each movement has a four-measure portion around the middle of its E-flat-major S theme that sounds very similar to the other, as shown below. (This portion's place in the form of each movement is indicated by the enclosure in the graphic above.) Both portions are soft, and both use alternating and echoing one-measure motives that involve the wind instruments of the orchestra. These motives essentially embellish upon a Bb-F-Bb-F treble succession. The harmony and bass line below this treble succession is the same in each portion, prolonging a first-inversion tonic triad, a standard "beginning-of-the-end" chord for a secondary theme. However, in characteristic fashion, Brahms displaces the harmonic rhythm from the barline.

Some Thoughts About the Chaconne from Holst's First Suite in E-Flat

Around this time next year will be the centennial anniversary of the first performance (June 23, 1920) of Gustav Holst's First Suite in E-Flat, one of the most well-known works written for wind ensemble. The first movement is a chaconne. The repeating chaconne melody is on the first line below: octave position may vary for all the melodies in this example. After nine statements of this melody, the chaconne is diatonically inverted to start also on E flat—shown as the second melody below—and presented twice as such. Next the third melody below—a diatonic transposition of the chaconne to start on G—is presented once, followed by a restoration of the original chaconne tune. (I am ignoring the last statement, which deviates from the three-flat collection.)

Octave aside, there are six other diatonic transpositions and seven diatonic inversions of the initial chaconne melody. The second and third melodies above are only one of each set. Why choose them, of all possible? There are many ways to answer this question. Here's one. The F-Bb succession in the original chaconne melody, highlighted with blue brackets, clearly expresses dominant function at the end of its two halves. An alternate way to express dominant function using the same diatonic interval class (fourth or fifth) is with the tritone. In the key of E-flat, the tritone is between D and Ab, highlighted with green brackets. The second melody above is the only inversion that replaces F and Bb with D and Ab. (Holst deviates from the second melody's inversion by ending on G instead of Ab—hence the dashed green bracket—but nonetheless delivers ^2 and ^5 just like the original melody, but in C minor.) Likewise, the third melody above is the only transposition that replaces F and Bb with Ab and D.

The second and third melodies are related by inversion around F. (There are many ways to recognize this: F is equidistant between the starting notes of Eb and G, and it is equidistant between the notes D and Ab, a dyad that inversion around F preserves.) The three-flat collection, such as E-flat major or C minor, inverts into itself around F, as shown above. This means that a melody in a three-flat collection that is inverted around F will maintain the same major and minor qualities of intervals. The figure above places a letter that shows the quality of each melodic interval above it: m = minor, M = major, P = perfect, A = augmented, d = diminished. The purple enclosure surrounds intervals in corresponding spots in melodies that express the same major or minor quality of seconds and thirds. This L-shaped enclosure demonstrates not only that, as aforementioned, the second and third melodies use the same quality of seconds and thirds in corresponding positions, but also that the first melody inverts these qualities exclusively in its first part, and matches them exclusively in its second.