For years, I've found it odd that the second phrase of Jerrold Immel's iconic two-phrase main theme for the popular TV drama Dallas (1978–1991) started exactly like how the first phrase ended. Phrases typically don't do this.
Then yesterday I thought of one possible reason:
Marvel's "What If...Doctor Strange Lost His Heart Instead of His Hands?" was released on Disney+ eighteen days ago. (Warning: spoilers soon ahead.) Laura Karpman composes the music for the series. In scoring this episode, Karpman explains that she "started with these four really soft piano chords." This four-chord progression can first be heard at 7:22, during a montage in which Strange tries over and over again to stop Christine from dying. It is also heard at the end of the episode, when Strange's desperation leads to the end of his world, and the Watcher's voice-over concludes: "One life, one choice, one moment, can destroy the entire universe." The progression is shown on the first staff below; it is juxtaposed against one transposition of what Mark Richards calls the Axis-a progression, which I have shown to be an important signifier of the heroic and epic in scoring for recent film, film trailers, and television.
These progressions differ by only a single note in a single chord: the F sharp in the last chord of the one-sharp version of Axis-a is an F natural in the last chord of Karpman's four. Not even the root changes: the D-major chord becomes a D-minor chord. And yet this smallest of changes transforms the music into something very different, like a negative image of Axis-a.
Within a seven-note diatonic scale, there is a single tritone. For example, among the seven pitches of the one-sharp collection—like a G-major scale—the tritone is between F# and C. There are six consonant triads in any diatonic collection, but only two of them—one major, one minor—are "tritone-free," that is, they do not overlap with the tritone at all. For example, in the one-sharp collection, those two triads are G major and E minor. Diatonic and "classically" tonal music tends to favor these triads, using them more often than other triads, putting them at beginnings and endings of phrases, and placing them in more prominent metric locations.
The Axis-a progression, which presents all seven pitches of a diatonic scale, puts these two privileged triads in the first and third positions, which are the most metrically weighted in the four-chord loop. But Karpman's seemingly minute alteration shifts the diatonic scale, which then passes the twin mantles of "Privileged Chord" to two other triads. The second chord in Karpman's four-chord loop is one of these triads, which is arguably in the weakest metric position of the four, and the other triad is not in this progression at all. To attempt to hear the second chord as a privileged chord feels contrived, forced, even useless, like Strange's attempts to cheat death.
This will be my 88th and final musicellanea post. The total of 88 total posts matches the total of 88 keys of a standard modern piano keyboard. Furthermore, of the 12 equal-tempered pitches within an octave, C is understood as the (mostly arbitrary) starting point. Analogously, of the 12 months in the modern Western calendar, January is understood as the (mostly arbitrary) starting point. This blog's monthly posting schedule matches this mapping from C to January, since I began on October 2013 (analogous to A0, the lowest note on the modern piano) and ended on January 2021 seven years and three months later (analogous to C8, seven octaves and three half steps higher, the highest note on the modern piano).
I will conclude this blog with a lightning round of three unrelated miscellaneous musical observations.
1. The syntonic comma is the difference between four pure 3/2 perfect fifths -- like C4 up to E6, via G4, D5, and A5 -- and two pure 2/1 octaves plus a pure 5/4 major third -- like C4 up to E6, via C5 and C6. As a fraction, and as an ascending interval, the syntonic comma is 81/80, which is (3/2)^4 / ( (2/1)^2*(5/4) ). As a decimal, this is the rather humdrum 1.0125. As a fraction, and as a descending interval, the syntonic comma is 80/81. As a decimal, this is 0.987654320 repeating, but can be rounded to 0.987654321. How fun that humans have n digits, and one of the most significant musical commas is (n-1)^2-1 / (n-1)^2.
2. In Wagner's Tannhäuser, the title character sings his "Hymn to Venus" three times in Act 1. The first time, the music is in D-flat major with a tempo indication of half note = 69. The second time, the music is in D major with a tempo indication of half note = 72. The third time, the music is in E-flat major with a tempo indication of half note = 76. Not only is this music increasing in intensity by getting slightly higher and faster with each iteration, but the ratios by which each are increasing are rather close to one another: the pitch is increasing at a ratio of around 1.059 (the equal tempered half step), and the tempo is increasing first at a ratio of 1.043 (72/69) and then around 1.056 (76/72).
3. In the one-measure piano introduction to Ives's song "Tom Sails Away," my favorite moment is near the end of the measure, the moment occupied by the sole F, encircled in the music below. If I were playing this music, I would aim to use a slight change of dynamics and/or microtiming to give this seemingly uninteresting F some special treatment.
One the one hand, this F breaks a pattern. The image below teases out some three-note motives with a melodic major third (shown with a yellow arrow) in some direction overlapping with a melodic tritone (shown with a purple arrow) -- either before or after -- in the same direction. Two of these motives occur in the first half of the measure in immediate succession, the second a major seventh lower than the first. Had the F4 of my favorite moment been an F#4, then two versions of this same motive -- one retrograded, the other inverted -- would have occurred in the second half of the measure in immediate succession.
Furthermore, by using F natural instead of F# at this moment, the first measure contains all 12 pitch classes (the 12 pitches in the equal-tempered chromatic scale without attention to register) besides F#, as shown in the graphic below. (One octave of keyboard with the standard white-black key coloration is shown on the left. The image below also replaces the major-third yellow arrow with a minor-third green arrow.) The second measure completes the set of 12 pitch classes, with F#s prominently placed as the lowest note and the first note of the treble melody. Therefore, by breaking from a pattern, attention is drawn toward something that is incomplete that later music makes complete.
On the other hand, the F continues a pattern. The top voice in the first measure plays a mi-re-do dotted-rhythm motive that overlaps with its transposition down a major third -- again, this interval is shown with a yellow arrow -- producing five of the six notes of a whole-tone scale. As shown below, the F at my favorite moment completes this whole-tone scale, albeit down the octave. A continuation of this sequential pattern would transpose the motive down another major third to G to F to E flat. The vocalist enters with something very close to this -- G# to F# to E -- which is down a minor third (plus an octave) instead of a major third. Once again, the green arrow replaces the yellow arrow.
However, G to F to E flat does occur later, to begin the last vocal line, which occurs at the end of the song, as shown below. When the first vocal entry in the second measure breaks from a pattern, once again attention is drawn toward something that is incomplete that later music makes complete.
OK, that's it for this blog. I hope that you got something out of it. Thanks to Stephen Soderberg for suggesting this idea to me in the first place.
I have one last thing to share: hidden among some of the post titles of this blog is a message.
Beethoven -- or whom I occasionally like to call "the male Emilie Mayer" -- was born 250 years ago this month. His music, and discussions about his music, have been recently getting even more airtime than usual as a consequence of this sestercentennial, like this piece in the New York Times, where various luminaries find passages in the composer's music that are "resigned," "wondrous," "euphoric," 'furious," "eruptive," "rampant," "spiritual," "life-affirming," "passionate," "disruptive," "experimental," "daring," "heroic," "incredible," "extraordinary," "ferocious," "heart-stopping," "playful," "combative," "maniacal," "lyrical," "modest," "heartfelt," "reflective," "unpredictable," "transcending," "gloomy," "thrilling," "sophisticated," and "sublime."
I have another passage and another adjective to add to the list. The passage is from his Diabelli Variations, one of the works cited in the New York Times article. Below is the score for the twentieth variation.
This is strange music. For one nit-picky example of this strangeness, as enclosed in red below, this music as notated contains ascending thirds of both diminished and augmented varieties, each starting on the note two semitones below C (notated as A sharp in the diminished third and B flat in the augmented third).
As shown with diagonal lines below, a fairly strict but intermittent outer-voice canon with a delay of two measures -- stretching to three measures in its last appearance, indicated in green -- does offer some organization to this music's structure, but not all of it.
The one moment I find most bizarre in this already strange music, a moment that lies outside of the canon shown above but insinuated with the diminished third shown above that, is in m. 11. The stack of notes in the second half of this measure, enclosed below, is deliciously flabbergasting, for multiple reasons.
First, this is a root-position tonic triad in the key of the theme and the entire variation set: C major. Outside of its local context and ignoring its top note for now, this kind of chord should be the most stable sonority in the entire work's tonal fabric. And yet not the theme, nor any variation before or after this one, places a root-position C-major triad at this point in the two-reprise form: if anything, at this point in the form, the music is pulling away from the orbit of C major as a key, and certainly not landing on it as a stable chord.
Second, a fully-diminished chord occurs immediately before this C-major chord. Such chords, outside of other contextual patterns, are most likely to resolve to a consonant triad whose root is one semitone above some pitch -- octave aside -- in the fully-diminished seventh (as a vii°7) or to a consonant triad whose root is the same as some pitch in the fully-diminished seventh (as a "common-tone chord"). The two recompositions of the eleventh measure below show one example of each -- note how soprano and bass lines maintain the same contour of those in Beethoven's original, which matches the same of the prior two measures, which -- at least in the soprano -- matches the theme's sequential continuation in mm. 9-12.
Beethoven's resolution does neither of these. Rather, it resolves in a third, and the only remaining, way: to a consonant triad whose root is one semitone below some pitch -- octave aside -- in the fully-diminished seventh. This third way is not unheard of in classical tonal practice. However, in this third way, two voices typically remain as common tones (which Beethoven does), and each of the two moving voices typically uses a melodic interval that is typically no larger than two semitones (as heard) and a step (as seen) (which Beethoven does not do, on both the hearing and seeing fronts). Furthermore, in this third way, the music is typically preparing a subsequent tonic that -- or at least a subsequent chord whose root -- is a perfect fifth below the root of the consonant triad to which the fully-diminished seventh resolves. In other, odder words, the consonant triad is "dominantized." In this case, with C as the "dominantized" chord's root, that would mean that the eleventh measure's strangeness might be mitigated with an F root or tonic soon thereafter, but no such emphasis on F materializes; rather, as in the theme, the music modulates to G. (If C's function is to be "-ized" for this purpose, it should be "subdominantized," not "dominantized."). The recomposition below shows a typical third-way continuation, reversing the motivic soprano ascent to achieve even higher atypicality.
The C-major chord does make some sense when m. 12 repeats m. 11 verbatim (capitalizing on the clichéd maxim better known in jazz that it's not a wrong note if you repeat it), and, in retrospect, the C-major chord is "subdominantized" or "predominantized" as the music moves to a D7, the dominant in G. But, for me, the bizarreness remains. Perhaps Beethoven wanted the performer to prepare the listener for this bizarreness when he prescribed a terraced dynamic change (the pianissimo, from piano) in m. 11, in the middle of the four-measure sequential continuation -- something he does neither in the theme nor in any other variation.
A number of my 2018 posts to this blog demonstrate how, by flipping or turning a representation of some music, you get either the representation of the same music or that of music nearby. Examples include a piano piece by Schoenberg, a song from a Disney movie musical, a string quartet of Pfitzner, a song by Richard Strauss, a popular celebratory melody (and one part-writing assignment of its notes in a typical harmonization), and a well-known cantata movement by J.S. Bach.
This video provides another example of this. It is most like the Pfitzner analysis, in that a particular rotation not only leaves the notes unchanged, but it also preserves the tonicizations as well.
This line receives consistent homorhythmic triadic accompaniment. In my first hearing of this music, I defaulted to an incorrect assumption about its accompaniment. I will introduce this incorrect assumption with a relevant but somewhat lengthy excursus. This line has six different pitches: D-E-F#-G-A-B. Removing either the F sharp or the G yields some pentatonic scale. Adding a C or a C sharp yields some diatonic scale. It is well understood that both the pentatonic and the diatonic scales play an important role in popular music. However, the same might be said for the kind of six-note scale -- what is sometimes called the Guidonean hexachord -- to which the six pitches in this line belong, or at least one particular employment of this kind of scale.
The number of pitches in a Guidoniean scale -- six -- is not a prime number, unlike the pentatonic and diatonic scales. A scale with a non-prime number of pitches -- let us say it has XY pitches, where X and Y are integers -- permits a certain kind of procedure: one can 1) find a smaller pitch set of X pitches within this scale that is evenly distributed within the scale, 2) transpose this smaller set up or down by step through the scale, and, thus, 3) generate a cycle of Y elements. It might be helpful to think of this cycle as analogous to a ring of blinking holiday lights; in the 96-light example below, X = 16 and Y = 6.
For example, in the four-pitch (XY = 4) scale on the left below, this procedure can generate a two-member (Y = 2) cycle of consonant dyads (X = 2) on the right below, a progression, or parts of it, commonly referred to as "horn fifths."
One place where this cycle occurs in the repertoire is from near the end of Smetana's Die Moldau.
For another example, in the two-pitch (XY = 2) scale on the left below, this procedure can generate a one-member (Y = 1) cycle of tritones (X = 2) on the right below.
This cycle alternates between two of the four consonant triads available in the Guidonean hexachord: the only pair of triads whose roots are a step apart, one major, one minor. Above, those triads are D major and E minor. In most cases, one of these two triads is clearly treated as tonic, and the other triad serves as either supertonic or subtonic harmony.
This cycle plays a role in various forms of popular music that is frequent enough to warrant a name. It probably has already been given a name, but I do not know it. I speculate that one of its first uses was in gospel music, so I will provisionally call the employment of this cycle "gospel part writing."
Here are three examples of gospel part writing from popular music where the major triad is treated as tonic:
Here are three examples of gospel part writing from popular music where the minor triad is treated as tonic:
This interpretation of the harmonization of this melody as having some, but also lacking some, of the components of a gospel part writing cycle can make interesting what happens in the bass, which I have yet to show but do so now:
The bass primarily arpeggiates an E-minor triad, which is the "same" triad that the upper parts first entertained but then sidestepped when the harmonization slipped out of the Guidonean scale and into the D-major scale. This choice of bass makes more simultaneous and tonally balanced the two triads of gospel part writing that are typically only successive and tonally imbalanced. Through this mash-up of D-centered music up top and E-centered music down low, I relish the sound of not one but two different "soul dominant" chords (to use Mark Spicer's term): DM above E (dominant of A?) and AM above B (dominant of E!). But I also hear a remarkable struggle between two triads whose pecking order in the hierarchy of tonal status in gospel part writing is usually unequivocal -- I think this struggle resonates quite well with the autobiographical "origin story" Chika tells in the song.
The exposition of a sonata-form movement is the first of the movement's three big parts. Especially in eighteenth- and earlier-nineteenth-century music, this part of the three is usually the easiest to recognize, because it is often surrounded by repeat signs. In modern-day practice, sometimes performers take these repeats and sometimes they do not. Occasionally, the composer will provide first and second endings at the end of the exposition.
A sonata-form exposition typically modulates to a secondary key and articulates a perfect authentic cadence in this key toward the end of the exposition. In their 2006 book Elements of Sonata Theory, James Hepokoski and Warren Darcy define a "failed exposition" as an exposition in which the secondary material does not end with a perfect authentic cadence in the secondary key, what the authors call the "essential expositional closure," or EEC. The last movement of Beethoven's Fifth Symphony is a good case in point. After the music modulates from the primary key of C major into the secondary key of G major, the music never articulates a perfect authentic cadence in this key before the end of the exposition, which is marked clearly by first and second endings.
Below are the string parts, which are sufficiently representative of the music's content to recognize formal aspects (and their absence), with some annotations. An earlier opportunity to cadence is avoided; instead, another theme begins at the point of cadential effacement. This theme is set up in a antecedent-consequent periodic design. Toward the end of the antecedent, a G sharp steers the music toward an A-minor triad, which functions as a predominant chord for the half cadence appropriately ending this antecedent phrase. As the louder consequent phrase begins, Classical practice suggests that this music will head toward a perfect authentic cadence. However, when the melody arrives again at the G sharp, Beethoven respells it as A flat and locks the bass first on C, then F, turning this harmony into a minor-mode predominant in C major. Both this theme and this exposition fail to achieve the customary cadential closure.
The first movement of Luise Adolpha Le Beau's Violin Sonata, op. 10, from the second half of the nineteenth century, begins with an exposition that, like that of the first movement of Beethoven's Fifth Symphony, begins in C minor and modulates to E-flat major for the second theme. The entire exposition is shown below, with a couple of annotations. Le Beau does write one perfect authentic cadence in E-flat major at the end of the second theme, but it is within the first ending. So if a performance takes the first ending and the repeat, like this one, the exposition succeeds, or at least, the first pass through it does. If a performance does not take the repeat and opts for the second ending after a first time through, like this one, the exposition fails. I do not know of another sonata-form exposition that puts the only possible EEC in the first ending and not in the second ending, but I would not be surprised to find out that there are others: please let me know with a comment below. (If Schopenhauer wrote a sonata...)
In the 2007 book Black Women and Music: More than the Blues, edited by Eileen M. Hayes and Linda F. Williams (University of Illinois Press), Teresa L. Reed contributed a chapter called “Black Women and Art Music.” On page 191, while Reed is discussing the life and music of Florence Price (1887–1953), she writes "[t]he first movement of [Price's] piano Sonata in E Minor, for example, is a conservative rendering of sonata-allegro form. Its introductory bars are even mildly reminiscent of the opening measures to Beethoven's Pathetíque [sic] Sonata [in C Minor]."
Although both opening movements begin with a slow introduction (10 measures in Beethoven's movement, 12 measures in Price's movement) and the opening measures of these introductions enjoy a little mutual resemblance, the beginning of the two movements' Allegro sections are more alike and are probably the measures to which Reed refers.
I have provided the first seventeen measures of both below, with some color-coded annotations that point out similarities beyond the most obvious (tempo, cut time, etc.). I modified the Breitkopf und Härtel edition of Beethoven's movement so that the number of measures in each system is the same as in the Price, to allow for easier comparison. Most annotations, perhaps all, are self-explanatory. One that is perhaps not is the area shaded in blue toward the end of each of the first two systems. This highlights when the second system departs from an exact repetition of the first by staying on a half-note harmony for twice the duration. This extension pushes the predominant (PD) --> dominant (D) progression later in time by a half measure, altering the metrical position in which the music arrives on the dominant, which sets up two contrasting phrase-ending experiences.
I will leave it to the reader to decide how mild the reminiscence is.
