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.

63 Tripled Units in a 64 Span, Tune by 65daysofstatic

65daysofstatic is a twenty-first-century English experimental band. Their techno-infused track "The Distant and Mechanised Glow of Eastern European Dance Parties" appears on their third album, The Destruction of Small Ideas, which was released in the United States on the first of this month twelve years ago. Below is a YouTube recording of the track, and below that is an annotated transcription of the snare drum, kick drum, and synthesized bass from 2:09-2:55.

Before the drums recuperate at 2:16 the 4/4 time signature that has governed the track since its beginning, the synth lays down a repeating pattern of sixteenth-then-eighth, creating a three-sixteenth pulse that cuts first against the implied continuation of 4/4 and then against an explicit 4/4 when the drums re-enter. At first, it seems as if the synth bass's triple pulse will stubbornly continue its transversality. But at 2:27 it resets as it drops the octave, starting again with the sixteenth-then-eighth rhythm on the downbeat as it did when it first entered. It then resets like this every four measures, simultaneous with a change of register. Therefore, during of these four-measure spans from 2:27 to 2:55, the synth bass delivers 21 of the sixteenth-then-eighth successions, shown with the blue brackets. These total to 63 sixteenths (21 successions times 3 sixteenths each), which is one sixteenth shy of 64, the number of sixteenths in four measures of 4/4. Each of the two yellow brackets indicate this single-sixteenth difference. 

In an article where he investigates the general phenomenon of a string of threes unfolding over, but then giving way to, pure duple meter, Richard Cohn shares an awareness of examples that do so over a 64-unit pure-duple span: Bill Withers's "Ain't No Sunshine" and some music of Brazilian jazz guitarist Baden Powell de Aquino. However, in none of these examples does the string immediately repeat. At the least, this passage from 65daysofstatic provides an example that does.

But, moreover, this passage provides a compromise between the two extremes I put forth in my earlier blog post expanding upon Cohn's article. In that post, I offered two abstract examples in which, in the first, an onset in the triple pattern never falls on the start of a duple span, and, in the second — the complement of the first — an onset in the triple pattern always falls on the start of a duple span.

This passage from 65daysofstatic falls in between. Sometimes a synth-bass onset does fall on the beginning of a duple span, as shown with the 4 and 16 in green in my annotations. Sometimes a synth-bass onset does not fall on the beginning of a duple span, as shown with the 2, 8, and 32 in red in my annotations. One could hear this as a cycle, undulating between working with and working against a meter that is duple on all levels. In his discussion of a similar phenomenon in Duke Ellington's "It Don't Mean a Thing," Cohn refer to this cycle as "a wave of release and relock."

However, as cycles go, an oscillating cycle is arguably less interesting than a cycle with more than two members. To get a cycle of four members, one can use a string of five units. To restate the challenge at the end of my February 2019 post, there is a 23-second passage in a well-known song by a progressive rock band that does exactly this. I plan to blog about this music at some point, but I would much prefer it if, before then, someone else found it, revealed it, and maybe even analyzed it in a comment below. Here's a hint: the band is Yes.

Approximating π Using Lower Pi-artials

Last month I offered a post for Pi Day. The idea of intoning the number π as a melody that matches the opening of its infinite decimal (or septimal, duodecimal, etc.) representation remains dependent upon this choice of base. An intonation less dependent on such is simply π as the frequency ratio between two numbers: it sounds like a slightly flat minor thirteenth.

Three years ago my April post demonstrated how to approximate the natural logarithm (e) using musical ratios. This time around I suggest a method to do so for π, using the Wallis product.

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.