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

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

These two beginnings are similar in several respects:

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

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

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

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

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

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

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

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

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

A Gutsy Prelude by Maria Szymanowska Turns a Chord Inside Out

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

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

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

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

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

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

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

Amy Beach & Her "Old World" Symphony

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

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

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

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

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

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

Scalar Transformations in Some of Chen Yi's Music

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

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

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

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

Thanks to Xiaolai Zhou for introducing me to this music.

And that's five years of monthly posts.

Music Provides a Personality Analysis using Successive Octaves

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

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

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

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

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

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

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

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

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

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

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

The music for "Happy Birthday to You" lends itself well to the initial stages of teaching Schenkerian analysis, as can be seen in the fine Reddit post by "Xenoceratops" here, and its accompanying analysis here. A couple of years ago in this blog, I said that Schenkerian theory "distills tonal structures to their linear-harmonic essence, where prominent starting notes and normative cadential notes are stitched together into a well-formed contrapuntal design." It is a complex theory to learn and teach, but it can be as rewarding as it is complex. Here I offer a method to (somewhat) automate the application of the theory, a way that some might find crude, but it nonetheless matches competent analyses in some, perhaps even many, cases. It also reveals a hidden symmetry in this music's linear-harmonic structure that, consistent with my last three posts, involves a combination of retrograde and inversion, as shown at the end of this post.

The method begins with a triadic reduction of the music. Here is the melody for "Happy Birthday to You" in F major, along with a simple and common bass line and a standard triadic harmonization of this bass line. The number of chords in this harmonization, which I will abbreviate as #c, is seven. (I have considered the cadential 6/4 chord as a separate chord; another approach might elect not to do so.)

The next step is, within certain constraints, to write out (or at least imagine) all the possible realizations of this harmonic progression using three upper voices -- soprano (always highest), alto (always second highest), and tenor (always second lowest) -- each making a four-part chorale-style composition. (A simple keyboard accompaniment that one is likely to play when supporting the singing of this tune is likely to be one of these realizations.) The constraints are as follows:

1) For every triad, the three voices should collectively cover its three members.

2) With each motion from from triad to triad, the three upper voices should all either upshift or downshift. Xshifting (X = up or down) means that a voice either stays on the same note or moves X either by second or third.

There are exactly 6*(2^(#c-1)) realizations, starting octave position aside, because there are six possible assignments of the first triad's three notes to the three voices of soprano, alto, and tenor, and there are #c-1 instances when a triad voice-leads to the next triad, and each voice-leading may be an upshift or a downshift. In the case of "Happy Birthday to You," #c is 7; therefore, the number of realizations is 6*64, or 384. However, for the purposes of emulating Schenkerian analysis, it only matters to which triadic member the first soprano pitch is assigned; therefore, we need only concern ourselves with half this number, or 192 realizations.

For example, the first realization below starts the soprano on the third, and alternates back and forth between upshifting (u) and downshifting (d), and the second realization starts on the root and constantly upshifts. This first realization could be abbreviated as 3-ududud, or 3-42: 3 is for third, and, if u is 1 and d is 0 in ududud, then the resulting base 2 number written in base 10 is 42. The second realization could be abbreviated as R-uuuuuu, or R-63.

3-42

R-63

These are two of the 192 realizations of interest. These 192 realizations provide 192 different soprano lines. One way to emulate a Schenkerian analysis of "Happy Birthday to You" is to choose one of these 192 soprano lines that is most like, or best represents, the melody. This soprano line then becomes what is called the Urlinie, or fundamental treble line, in Schenkerian theory. One way to quantify the degree of this resemblance or representation is to choose a soprano line that maximally overlaps with the melody, regardless of the octave placement of the notes. For example, the two figures below show that the soprano line of realization 3-42 overlaps with the melody of "Happy Birthday to You" for a total time of five quarter-note durations, and the soprano line of realization R-63 overlaps with the melody of "Happy Birthday to You" for a total time of six quarter-note durations (four quarter notes plus a half note at the end).

3-42 plus overlap with melody

R-63 plus overlap with melody


Surely we can do better than five or six. The two realizations with the most overlap of ten quarter notes are 5-duuudu, or 5-29, and 5-dudddd, or 5-16, as shown below. Among adaptations of Schenkerian theory where the Urlinie is allowed to ascend, or skip, or change direction, the Urlinie of 5-29 might find quarter. However, in orthodox Schenkerian theory, an Urlinie may only descend by step. If, in this case, we further restrict ourselves to realizations with soprano lines that only descend, or only move by step, or never change direction, then the soprano line of 5-16 is the only one with maximal overlap.

5-29 plus overlap with melody

5-16 plus overlap with melody

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

This emulator could be refined by, for example, avoiding realizations with parallel perfect fifths or octaves (for example, toward the end of R-63 above), or weighting the overlap through the consideration of accented notes, non-chord tones, cadential notes, etc. With these refinements (but perhaps even without them), I suspect that this emulator would be able to demonstrate an important theoretical component of Schenker's project. If one runs this emulator on thousands of musical phrases that end on tonic harmony written by Western composers from 1720 to 1890, I speculate that a plurality of chosen soprano lines will descend by step to ^1, and the majority of chosen soprano lines will at least end with ^3-^2-^1. Thus the preference for 5-16 over 5-29 exercised earlier reflects a norm-based bias.

Lastly, a hidden symmetry of "Happy Birthday" shows itself when all of the chord tones are labeled, as shown below, by their assignment to soprano, alto, and tenor according to the realization of 5-16. This makes more evident the Schenkerian notion that, for example, the descents in mm. 5-6 and m. 7 are descents "into an inner voice." Inner-voice notes actually higher than adjacent soprano-voice notes due to octave equivalence—like the "tenor" note in m. 1 or "alto" note in m. 3—are what Schenker calls "superpositions" or "cover tones."

This labeling also reveals an invariance of structure. Imagine an inversion, defined on three upper voices, whereby alto (A) and tenor (T) switch places within triadic harmony, but the soprano (S) maintains the same triadic position. The sequence of voice assignments of the melody from "Happy Birthday to You"—STSATSATSAS—is thus a retrograde and this inversion of itself, just like the passages from Four Last Songs and The Aristocats mentioned in earlier posts are retrogrades and pitch-(or pitch-class-)inversions -- that is 180° flips -- of themselves: they are unflippable in the same manner than Messiaen's palindromic rhythms are non-retrogradable.

STSATSATSAS              -->                     SASTASTASTS               -->                    STSATSATSAS

                (inversion: A and T switch places)             (retrograde: sequence is flipped in time)

The fulcrum of the retrograde is also in the center of the melody.

Two Measures of Strauss’s "Spring" Twice Reflected: R and I (Part III)

The Four Last Songs of Richard Strauss (1864–1949) were composed 70 years ago this year, and were posthumously premiered on this day 68 years ago. One of the songs, entitled "Frühling" ["Spring"], repeatedly uses a distinctive four-chord, four-voice progression at different transpositional levels. The figure below includes a transcription of the first instance of this progression in the song. Some of the notes have been enharmonically respelled from Strauss's original score so that the each chord's notation clearly reflects its harmonic quality: Chord 1 is a minor triad, Chords 2 and 3 are minor seventh chords, and Chord 4 is a major triad. The vocal line follows the top notes, but the words have been omitted. I am considering Chord 4 as essentially a B-major triad in second inversion with the F-sharp eventually on top; therefore, I am treating the G sharp, presented below using a smaller font, as a non-chord appoggiatura.

In the picture above each chord, the chord's four notes -- register ignored -- are arranged onto a circle of half steps with C# at the top and G at the bottom, and then a polygon is inscribed within the circle using the notes as points. The notes that are doubled -- the A flat in Chord 1, and the F# in Chord 4 -- are indicated with the double curves.

In short, the progression of Chords 1 and 2 is both inverted around the axis C#/G (shown with the red arrows) and retrograded (shown with the green arrows) to produce the progression of Chords 3 and 4. Even the doubling of notes is preserved in this double transformation. The figure demonstrates this visually by inserting a mirror in between Chords 2 and 3. Whereas my two visualizations of both retrograde and inversion used earlier in this blog took place in two dimensions — the vertical depiction of registral pitch and the horizontal depiction of time — here the use of circular pitch-class space accommodates both transformations as reflections in a single dimension. The double-tipped black arrows, each labeled with a duration, shows that the mirroring of the onsets of the chords is also exact in time.

Lastly, Chords 2 and 3 are related not only by inversion but also by a 60° rotation; that is, a whole-step transposition (again, with register ignored). This is the same composite relation I showed three months ago in Schoenberg's op. 19 no. 4.

Chords Flip in Pfitzner’s String Quartet in C-Sharp Minor: R and I (Part II)

The German composer Hans Pfitzner met Adolf Hitler early in 1923, and according to Michael Kater's account in Composers in the Nazi Era: Eight Portraits (Oxford, 1999), Hitler was not impressed. When Pfitzner learned of Hitler's internment in Landsberg later that year, Pfitzner hoped to reconnect with Hitler. He bought Hitler a book that he hoped would inspire him, and inscribed in it the dedication "To Adolf Hitler, the great German" and dated it April 1, 1923: 95 years ago today. Pfitzner never sent the book.

Among his chamber works, Pfitzner's Second String Quartet, op. 36, written in 1925, received special admiration. According to Kater, Paul Hindemith sought to premiere the work with his Amar Quartet. Arnold Schoenberg’s composition student Winfried Zillig was particularly impressed by the chromatic boldness of the work. The modernist journal Melos, when recognizing Pfitzner as a contributor to New Music, singled out this work and its similarity to recent works of Schoenberg, who was a couple years into his serialist period.

Pfitzner would have chafed at this comparison: he disdained the twelve-tone method. Nonetheless, two harmonic-progression pillars in the first movement of his op. 36 demonstrate -- with a little manipulation -- a retrograde-inversion (RI) relationship, a kind of musical transformation that musicians most readily associate with Schoenberg. The grand-staff reduction below shows both the movement's opening thematized chord progression in C-sharp minor, and the exposition's final cadential arrival in the relative key of E major. The four lines match the expected four quartet performers from top to bottom.

The graphic below shifts these pitches from the diatonic staff to a chromatic grid: the small numbers show when these notes occur using the measure.beat form. It simplifies and moves an octave higher the cello part of the cadential music. It also aligns the cello's C#-E resolution with the resolutions in the other parts. With these rather modest transformations, this cadential progression is a retrograde and inversion (a 180° flip) of the opening chord progression: the two stacks of notes and intervals on the left -- when fused together and turned 180° -- match the two stacks of notes and intervals on the right, allowing for enharmonic equivalence. The instrumental assignments do not follow the flip, but the flip does preserve registral arrangement, which is a more precise kind of RI relationship than the typical Schoenbergian flip.



Here is the answer to last month's crossword:

The Aristocats Perform a Half Turn: R and I (Part I)

In 1970, Disney released its twentieth animated movie The Aristocats. (The children's book based on the movie was released thirty years ago this month.) Besides the title song, the only other song of the many songs written by the Sherman brothers that made it to the final cut of the movie was "Scales and Arpeggios." In this Mozartean tune, the cute kittens Marie and Berlioz practice their solfège ("do-mi-sol-do...") and piano playing together in their upscale maison in Paris. (No cat on cornet at this point.) In the third verse, their mother, Duchess, joins Marie in song. On the third-verse line, "like a tree, ability will bloom and grow," Marie and Duchess sing the first vocal harmony in the song. The graphic below shows the two lines so that the syllables of the lyrics match up with the pitches of the tune. (C4 is middle C.) Duchess sings the orange notes, while Marie sings the green notes. The two-tone squares represent the two moments when Marie and Duchess sing the same pitch at the same time.

The notes inside the 7x7 square in the upper right display a special property. If one takes the Duchess's orange notes inside this square, plays them both backwards (retrograde, or R) and upside down (inversion, or I), fits the result as close to the same range while still achieving harmonic intervals that are all consonant, and uses the same syllables in the same order, one obtains Marie's green notes inside this square. The graphic below shows how reversing the Duchess's notes in both temporal space and pitch space, in either order, results in Marie's notes.

Another way to consider this, especially given the standard Western notation's assignment of a score's two spatial dimensions as temporal space and pitch space, is that the combination of a retrograde and inversion is the equivalent to a 180° rotation. Turn any score 180° and you will have performed a combination of a retrograde and an inversion on this score. Let me put this in drawing-tools terms:

This means that the two lines together, irrespective of who is singing which note, makes a pattern -- shown below in black squares -- that does not change when it is rotated 180° or, equivalently, when it is both retrograded and inverted.

Patterns like this one cry out for comparison to a crossword puzzle grid. Indeed, as varied as crossword puzzle grids can be, most use this standard form: each remains the same after being rotated 180°.

Not counting one-letter words, the grid above can be completed in exactly one way using only words from this month's blog post: four six-letter words, two four-letter words, et cetera. I'll get you started by including the word "Oy," which was the hardest to weave into the text above, although some of the other words didn't seamlessly blend in either. I'll show the answer next month.

This post begins a series of posts on using a combination of R and I to reveal structures in music.

T and I: Twice-Related Chords in Some Music of Schoenberg

Arnold Schoenberg started and finished the first five of his Six Little Piano Pieces, op. 19 on this day 107 years ago. Here is a score of the fourth piece in which I have selected ten groups of notes, each identified by a colored enclosure and a roman numeral.

Some of the groups of contiguous notes relate to one another by 1. disregarding both the octave in which each note appears and when it appears relative to the others in the group, 2. placing each note of each group on a circle of half steps, forming a polygon, and 3. relating two polygons as either differing by a certain rotational distance or through a certain reflection.

For example, as shown below, the groups II and V have the same triangular shape, and one shape is a reflection of the other shape around an axis that falls between the notes B and C on the circle of half steps, shown with a dashed line. Music theorists call this relation inversion (I).

As another example, as shown below, the groups III and VIII have the same pentagonal shape, and one shape is, at minimum, a 150° rotation away from the other shape (allowing for either clockwise or counterclockwise rotation). When this rotation is directed, music theorists call this transposition (T).

This second example does not surprise, as the notes in group VIII are, in chronological order, all transposed down a perfect fifth from the notes in group III. The inversion of a perfect fifth is a perfect fourth, which is five-twelfths of an octave; likewise, 150° is five-twelfths of 360°.

However, as shown below, the same 150° rotational difference also relates the seven-note groups IX and X, each of which fill a full measure and have a similar single-line-then-big-chord presentation. This suggests a covert and distinctive employment of the perfect-fifth or perfect-fourth transposition, the most important transpositions in Western classical tonal music.

It stands to reason that a certain relation would be even more distinctive if it were a composite of both a certain rotational difference (one of six possible) and a reflection around a certain axis (one of twelve possible). One such composite relation occurs at least twice in this piece. As shown below, the five-note groups I and IV are related both by a 60° rotational difference and a reflection around the axis that falls on the note C, and so are the three-note groups VII and VIII.

I will leave it to the reader to find, and comment upon, two two-contiguous-note groups -- melodic intervals -- that are related by this same composite relation of 60° rotation and reflection around C. These two intervals also have the same articulation and same melodic-contour context.

Schubert and the Colors of Hierarchy

Franz Schubert was born 221 years ago today.

This video uses different colors to demonstrate the hierarchical embedding of pitch centers -- roots of chords and tonics of scales -- in Schubert's song "Mein!" from Die schöne Müllerin. Relationships between such centers by perfect fifths are indicated by the near-complement relation between colors, and those relationships by step are indicated by colors very similar to one another.

The small dark blue circle toward the right of the thumbnail below is immediately within a purple oval, which is within a red oval, which is within the pastel green oval. This indicates a chord that is a F#°7 chord, which is vii°7 in G minor, a key which is vi in B-flat major, a key which is bVI in the song's key of D major: thus, the chord may be understood as vii°7 / vi / bVI.

After a introduction to the graph, the video synchronizes a scrolling presentation of the graph with a performance by Genaro Mendez, to whom I am very grateful for lending his voice to this little project.