Wednesday, June 28, 2017

A Pedagogical Piece Proceeding from Paganini (Part II)

Next month's post will divulge the rare chord progression that was adumbrated in last month's post and takes place in the last two-thirds of my variation. This month's post takes a closer look at the first third of the variation, which also exemplifies another notion in Cohn's book. However, my discussion of this notion follows from an approach by music theorist Ian Quinn, one of Cohn's colleagues at Yale University.

The picture below is of a musical balance "scale" that has four "pans" represented by colored thumbtacks and distributed symmetrically around a circle, instead of the two pans of a typical balance scale. Assuming octave and enharmonic equivalence, there are only four augmented triads, which are each assigned to one of these four "pans," such that augmented triads a whole step apart are across from one another on the circle. Any group of notes can be "weighed" on this "scale." For example, the picture below evaluates an A-minor triad, the tonic triad of Paganini’s famous caprice. The single orange ring on the blue thumbtack represents the root A, and the two orange rings on the white thumbtack represents the third C and fifth E. The circle thus leans directly toward the camera, touching the table at a point along the circle’s circumference a third of the way from the C-E-G# augmented triad to the C#-F-A augmented triad, or 30° away from the C-E-G# pan.



Here's another example, using the F-major chord. Now the circle touches the table at a point 30° farther clockwise away from where the A-minor chord touched the table. Therefore, we'll say that the A-minor chord and the F-major chord differ by 30°; more precisely, we'll say that the F-major chord is 30° clockwise away from the A-minor chord.



Quinn and Cohn use the structures that "scales" like this make for different purposes:

Quinn is more interested in the magnitude by which different types of chords tip the scale, and less interested in the direction of the tip. Any augmented triad would tip this scale the most, while a chord of the fully-diminished type or four-note cluster type (among others) are perfectly balanced and do not tip this scale at all. The magnitude of the tilt can give an idea of how "major-thirdy" a certain type of chord is.

Cohn is more interested in the direction that a chord of a certain type tips the scale, and less interested in the magnitude of the tip. He has noticed that this scale can give a relatively good idea how far the notes in one chord would need to travel to move to notes in another chord. In particular, when you divide the number of degrees two three-note chords are apart by 30, this sometimes, or perhaps even often, tells you how many semitones in total the notes in one chord minimally need to move to produce the notes in the other chord. For instance, in the example above, the A-minor triad and the F-major triad differ by 30°. Divide this by 30 and you get 1, which is the sum of all of the semitones minimally needed to change the notes of an A-minor triad to the notes of an F-major triad: A-->A (0 semitones), C-->C (0 semitones), E-->F (1 semitone) (0+0+1=1). Furthermore, if the change is clockwise, the minimal voice leading goes up; if the change is counterclockwise, the minimal voice leading does down. In this case, the change is 30° clockwise; therefore, the single semitonal motion is up (E-->F).

I write "relatively good idea" and "sometimes, or perhaps often" because this method works more frequently for some chord types than others. Between augmented triads, it always works. Between major and minor triads, it almost always works. The closer the notes can be put together, irrespective of octave, the less frequently it works. Cohn focuses his attention on major and minor triads, so this approach is pretty reliable.

My Paganini variation begins, as expected, with an A-minor chord, and then follows it with eight other major or minor triads, one for each measure: EM, FM, CM, Fm, DbM, AbM, C#m, and AM. (The number 8 is useful in how it matches the number of measures in the first section of Paganini's caprice.) These triads are exactly the eight major or minor triads that are no more than 60° away from the A-minor tonic triad. Three are exactly 60° away: EM, CM, and AbM. Three are exactly 30° away: FM, DbM, and AM. Two are exactly 0° away; that is, they tilt the circular "scale" in the same direction as Am: Fm, and C#m. The order of these triads both accommodates the general shape of Paganini's tune (while, at the same time, semitonal shifts of parts of Paganini's tune in mm. 5-7 accommodate the triads) and moves consistently down by (sometimes enharmonic) major third among these three trios: Am.........Fm......C#m, ...EM...CM......AbM, and ......FM......DbM......AM.

Moreover, as one triad progresses to the next in this variation, the number of degrees the circular "scale" shifts divided by 30 matches the minimal number and direction of semitonal changes needed for the triadic progression, as the video below demonstrates. The few progressions that contradict this correlation are avoided here. (One of those progressions will be examined more closely next month.) Overall, the voice leading from one triad to the next is relatively small, because no triad in this group of nine is more than 90° away from any other.


Sunday, May 28, 2017

A Pedagogical Piece Proceeding from Paganini (Part I)

Five years ago, the music theorist Richard Cohn published a book called Audacious Euphony. At one point in the book, he expressed no knowledge of the existence of a certain kind of succession of triads progression in nineteenth-century classical music. When I wrote a review of his book, I expressed the same thing. So I decided to write some music that not only used this succession, but also demonstrated other aspects from his book. It may be twenty-first-century music, but it is based upon a well-known and oft-varied piece of nineteenth-century music: the last of the 24 Caprices of Niccolò Paganini for solo violin, whose composition was completed no later than 200 years ago this November (as shown on this manuscript). Here is my variation, with a score and my approximating performance.



I'll reveal this progression, and discuss other aspects of this piece—including why I chose Paganini's caprice to illustrate parts of Cohn's approach—in the coming months.

Sunday, April 30, 2017

A Subject's Temporal Invitation to a Countersubject's Chromaticism: Bach's Fugue in A-Flat Minor from WTC II

In this blog, I have proposed pitch-time matches in the music of Carl Vine, George Gershwin, Steve Reich, Richard Wagner, Phish, and Sergei Prokofiev. Here is an example from an earlier composer.

Below is a piano-roll depiction of the beginning of the Fugue in A-flat Major from J.S. Bach's Book II of the Well-Tempered Clavier, aligned with the usual two-dimensional orientation of Western music notation. In this grid, the units of the x-axis are notated sixteenth notes, and the units of the y-axis are semitones. Therefore, the first note, shown on the left in gray, is an eighth note long, and descends three semitones to the next note.

This jaunty subject is built upon a frame of ascending perfect fourths -- 5-semitone intervals -- arranged in stepwise descent, as shown with the notes in red and the blue lines. The second and third perfect fourths occupy twice as much time as the first, which partition this part of the subject's timespan into a quarter-half-half division, or, more generally, a 1-2-2 division, as shown with the brackets above the grid. The filling-in of the second and third fourths -- the gray notes in between the red notes -- sustains the eighth-note pulse of the first three notes, compensating for the written-out rallentando of the ascending fourths.

The 1-2-2 division can also be found in semitones, as the intervallic division of this subject's third perfect fourth -- from top to bottom -- as shown with the brackets to the right of the grid.

This match may seem rather trivial; in fact, it should seem rather trivial. The duration succession is fairly short and indistinctive, and perfect fourths can also be divided diatonically into 2-2-1 and 2-1-2 as well; the second fourth in Bach's subject uses the latter division. But this pitch-time match has important implications for the countersubject, the line that appears in counterpoint with the subject.

Part of the enjoyment of listening to a work like a fugue is made available by the textural incrementalism of its opening: it begins with a monophonic subject that immediately appears again in tandem with a countersubject. Imagining what that countersubject might be given a subject can be a exciting, high-speed challenge for a seasoned listener hearing a fugue for the first time. One goes in with certain assumptions about this countersubject's likely features: diatonic, stepwise motion (especially if the subject is dominated by leaps from beat to beat), imperfect harmonic intervals with the subject, metrical conformity, steady pacing. The only three-note beginning of a countersubject that can completely meet the first four of these five assumptions for the subject's three melodic fourths is the following melody shown in green. The numbers in the green cells indicate the intervals between the notes in the countersubject and simultaneous red notes in the subject: they are all either thirds or sixths.

A deficiency of the beginning of this countersubject is that it lacks steady pacing: it slows down according to the same 1:2 proportions outlined earlier. The subject made up for this rallentando by filling in the fourths with passing motion and octave changes. Any stepwise melody can appear to quicken its pace by filling in its major seconds with chromatic passing tones. But in this case, the countersubject is perfectly suited in doing so: by virtue of the subject's 1-2-2 temporal proportions and the placement of its fourths within the scale, a complete chromatic descent from scale degree 8 to scale degree 5 is the perfect complement for the subject, as shown below. The first of the five desiderata listed earlier is traded for the fifth. The 1-2-2 temporal rate in which the melodic fourths of the subject unfold is matched perfectly to the 1-2-2 relationship between diatonic (Ab-G-F-Eb) and chromatic (Ab-G-Gb-F-Fb-Eb) steps in the top fourth of the major scale.

And, indeed, this is what happens, albeit first in the dominant key, which is the convention for this two-voice moment in a fugue's beginning.

Scholars rightfully highlight the disparity between the exuberance of the diatonic subject and the severity of the chromatic countersubject. As part of his wonderful commentary on the music of the Well-Tempered Clavier, the pianist David Korevaar recognizes that "the descending chromatic line from scale degree 1 to scale degree 5 is the standard signal for a lament – not at all the atmosphere promised in the subject!" It seems to me that this countersubject -- however dour -- is one of the few satisfying lines, and perhaps the most satisfying of them, that the subject promised.


Monday, March 20, 2017

Star Wars's Triplets in Alignment with Holst's Mars

It’s already known by many film-music aficionados that parts of John Williams’s score to the 1977 movie Star Wars sound like parts of the 1914–15 orchestral suite The Planets by Gustav Holst. For example, as demonstrated here and here, the buildup to the Death Star explosion repeats forty-six times the same colorful chord that Holst repeats nineteen times at the end of “Mars, The Bringer of War,” which is the first movement in The Planets.

There’s another, more subtle, connection between Williams’s score and Holst’s movement. It’s not surprising that when children sing back the main melody of the “Main Title”…


… they sometimes sing the melody with this different rhythm …


… like here.

They have a good excuse for doing so. Quick triplets are often on weak beats, like pickup beats, instead of strong beats, especially downbeats. Williams’s first triplet follows this convention, but his second one defies it. Children sometimes unknowingly shift the second triplet so that it matches the first and conforms to common rhythmic practice.

Why does Williams put the second triplet on a downbeat? Here’s one among many possible answers: by placing the first two triplets five beats instead of four beats apart, this matches the distance between the two triplets in the rhythmic ostinato from “Mars,” which is in quintuple time:



Saturday, February 18, 2017

Augmented Seventh Heaven and Other Deceits in "The Good Place"

The ident music for NBC’s afterlife comedy "The Good Place" uses a cliché harmonic signifier of fantasy: a progression between two major triads a major third apart, in which the triad rooted on the “higher” note of the major third serves as tonic. (An earlier blog post also spent some time with music that uses this progression.) Here’s a typical way that this harmonic progression is realized, in The Good Place’s key of A major:


Music theorist Richard Cohn has made the case that progressions like this earn their otherworldly signification because one cannot understand both the horizontal voice-leading semitones as diatonic minor seconds—a common default—and the vertical harmonic intervals as diatonic consonances. In the notation above, the triads are spelled diatonically, but the alto’s voice leading (C#-C-C#) is not, as highlighted with the dashed lines. This line could be spelled diatonically as C#-B#-C#, which would make the melodic intervals diatonic, but this would prevent the F-major triad from being spelled diatonically, as shown below; again, the dashed lines indicate non-diatonic intervals. To quote the Ninth Doctor Who: “You push one problem under the carpet and another one pops out on the other side.” This progression is simply too chromatic to hide all of it—both its chords and its lines—under diatonic carpets.


This sense that understanding a harmonic interval as commonplace keeps one from understanding a melodic interval as commonplace, or vice versa, jibes nicely with the deceits in The Good Place. However, the deceits of the first season’s last episode took the very notion of deception to another level. Likewise, The Good Place’s ident music is not your typical diatonic-interval-frustrating chromatic-mediant progression, for two reasons.

First of all, at least in nineteenth-century art music and mainstream screen music, many, and perhaps most, chromatic-mediant progressions are purely diatonic in both their vertical and horizontal intervals in their outer voices. For example, in the realizations above, the outer voices only use As, Es, and Fs, which form perfect fifths, octaves, major thirds, and minor seconds. (Same with the “Panem National Anthem” linked to above, and countless others.) Therefore, the deceit usually involves only one or more inside voices, like the alto voice above. Unlike the more conspicuous outer voices, inner voices more easily shutter the irreconcilability between horizontal and vertical away from prying ears. In other words, in the realization above, a little of the progression may not fit under a diatonic carpet, but the exposure is slight, pushed toward a dimly lit corner of the room.

A permutation of the upper voices that puts the C#-C-C# in the top voice, as shown below, would call more attention to this exposure. Perhaps because of this attention, this melodic line is not used as often for this harmonic progression.


Second of all, while it is reasonable to assume a default of a minor-second interpretation for a semitone, it does not take much to override this default. Take a purely ascending or purely descending motion through a chromatic scale, like the one below. Most composers and listeners would notate and understand nearly half of its intervals—around 42%, to be more exact—as augmented unisons (the 5 out of 12 slurs below the line) instead of minor seconds (the 7 out of 12 slurs above the line). 


But what if the voice leading used the same intervals as those of the harmonies? This would avoid any double standard of diatonic reckoning. This is exactly what we get with the ident music for The Good Place, as shown below. Not only are the outer voices involved in the diatonic irreconcilability, but the top line—played by the oboe—leaps directly from C#5 to the white note four semitones higher, while the bass leaps from A down to F (best spelled as such since the A remains a common tone). Either this new note is an E#5 and the melody uses a diatonic major third but the harmony contains an augmented seventh...


...or the new note is a F5 and the harmony is diatonic but the melody spans a diminished fourth.


I remember hearing this music for the first time and disbelieving that it was an F-major triad, so strong was my sense that the oboe was playing an E# in a consonant major triad. Here was a rare realization of this progression indeed, one that uses voice leading to cover as much “surface area” on a line of perfect fifths as the harmonic progression allows: in the bass, the flattest note in the A major harmony (A) moves down a major third—which is the one directed consonance that transports a note the farthest in a flatward direction—and, in the treble, the sharpest note in the A-major harmony (C#) moves up a major third—which is the one directed consonance that transports a note the farthest in a sharpward direction. The result is music that fits even less well under diatonic carpets than standard realizations of the progression. It’s no wonder certain deceits were exposed at the end of Season One.

What is even more remarkable is how this music compares to, and contrasts with, the music that plays right as the biggest deceit of all is revealed toward the end of the final episode: although it uses a transposition (by major third!) of the same down-by-major-third progression, both its harmony (consonant triads) and voice leading (all moving voices move down by major third) are entirely diatonic, as shown below. No deceits here (apparently).


Wednesday, January 25, 2017

Haydn, Sonata Form, and Schenker

A year ago, I proposed a perspective that correlated two oddities. The first oddity is a voice-leading and formal irreconcilability that arises when applying Schenker's theory to sonata-form recapitulations, particularly secondary themes. The second oddity is a set of interesting alterations that Mozart made in the lead-up to the recapitulation of his secondary theme in the first movement of his Piano Sonata in C Major, K. 279. The second oddity appears to mitigate the first oddity, because scale degree 3, which is the primary treble tone for the entire movement, receives more emphasis at the beginning of the subordinate theme in the recapitulation than it did in the beginning of the subordinate theme in the exposition. This makes the recap's subordinate theme a clearer vehicle for the movement's ultimate tonal closure: transporting scale degree 3 down by step to the final scale degree 1.

Something similar happens in the last movement of Joseph Haydn's last (and, according to some, his greatest) piano sonata in E-flat major. Here is the beginning of the movement, which presents the primary theme. In the third issue of his self-authored periodical Der Tonwille, Schenker designates scale degree 5 (B flat) as the primary treble tone for the movement. I choose scale degree 3 (G) instead.


In general, Haydn is well known for making witty and sometimes surreptitious alterations to the recapitulations of his secondary themes, rather than merely producing the secondary theme of the recapitulation by "copying, pasting, and transposing" the exposition's subordinate theme into the tonic key. The examples below compare two moments in the subordinate theme ("S-theme") of the exposition to their corresponding moments in the recapitulation. In both of these moments, an F (scale degree 2 in the main key, and scale degree 5 in the subordinate key) is replaced by G (scale degree 3 in the main key) instead of a B flat, which is the note that a simple "copy-paste" transposition would have delivered. Both of these moments are initiating moments, and the first in particular, with its five repeated eighth notes, harkens back to the primary theme's initiation. They encourage a reinterpretation of the recap's secondary theme's treble line as tracing an overall 3-2-1 path instead of an overall 5-4-3-2-1 path as it more likely did in the exposition. Like my interpretation of K. 279/i, this interpretation of Haydn XVI:52/iii better connects the primary treble tone of the primary theme in the exposition and recapitulation to the secondary theme's initiating tone in the recapitulation.




Thursday, December 1, 2016

Octaves Above Milstein's Prokofiev is...Sort of the Same Prokofiev

It is not too far-fetched, or at least not unprecedented, to identify an even division of time too slow to be heard as a pitch with a label that customarily is assigned to a pitch. For example, the sound waves emanating from a black hole in the Perseus Cluster have been identified as a B flat, although they are too low to be heard: indeed, the crests of the waves are millions of years apart. But to label this frequency with a pitch is simple enough. Multiplying or dividing a frequency by 2n moves it up or down, respectively, by n octaves. This transfer by one or more octaves both preserves its letter name and potentially puts it within the range of human hearing where we typically categorize frequencies with letters. For example, 440 Hz is an A, and so is 880 Hz, 1760 Hz, 220 Hz, 110 Hz, and the rate at which the moon goes around the earth (the sidereal month of 27.321661 days). Well, actually the sidereal month is a rather high A (30 octaves lower than ~459 Hz); in fact, it is closer to the black hole's B flat (of which one is ~466 Hz).

With this in mind, consider the opening of Prokofiev's Second Violin Concerto in G minor, which was premiered in Madrid on this day 81 years ago. It begins with a G, B-flat, D, E-flat, C sharp, and another D played by the soloist, and then this rising motive repeats in its entirety. The frequency at which the rising motive appears can also be labeled as a pitch. Prokofiev indicated a tempo of quarter = 108, which makes the motive's frequency a slightly low F sharp. While some commercially recorded performers roughly take that tempo, many others tend to go slower: around an F or even around an E for the rising motive. But Nathan Milstein, in a live recording from 51 years ago, not only plays it faster than what Prokofiev requests (and with relatively little rubato), but also plays the rising motive "at a G," several octaves below the open-string G he plays to begin the concerto, a G that matches the opening tonal center of the concerto.

Prokofiev follows this rising motive and its repetition with a descending motive (D-C-Bb) and its embellished repetitions that extend the motive lower and longer by one suffixed note with each repetition. However, thanks to durational reductions of interior notes, each descending motive still takes up the same amount of time; therefore, the frequency of the descending motive can still be labeled by a single pitch letter. Milstein's tempo for this portion puts the descending motive's frequency at around a B. Although the descending motives neither contain a B nor are in B, the immediately following orchestral statement of the ascending and descending motives is transposed to start on, and position the tonal center on, B.

The movie below demonstrates these pitch-tempo relationships.


Thanks to Debbie Rifkin for encouraging me to think about this concerto.

Friday, November 4, 2016

Some Unused Counterpoint in Brahms's First Symphony

Brahms's First Symphony in C Minor premiered 140 years ago today. Its first movement finds ingenious ways to combine its motives and themes together in counterpoint. (Julian Horton's video for the Society for Music Analysis provides an insightful introduction to these motives, themes, and combinations.)

In the exposition, Brahms shows how an inverted form of the main theme meshes well with the closing theme. In the instance below, the inverted main theme is in the bass, while the closing theme is in the treble. Eight measures later, he swaps their registral positions.




In the development, Brahms shows how the closing theme can be well combined with itself in canon at the octave a half-measure later. In the instance below, the dux (leading voice) is in the treble, while the comes (the imitating voice) is in the bass. Eight measures later, he swaps their registral positions.



What Brahms never does in the symphony is combine these two instances of two-part counterpoint into an instance of three-part counterpoint, which works quite well in forming seven complete triads while maintaining independence among all three voices. One possible version, transposed to the key of the symphony, is below.

Tuesday, October 11, 2016

Carter Burwell's Musical Accuracy in The Chamber

Perhaps the most appreciated part of The Chamber, a film adaptation of the John Grisham novel by the same name that was widely released in theaters exactly twenty years ago today, was the musical score by Carter Burwell. One of the parts of Burwell's score that I especially appreciate is the music that accompanies the end of an impassioned closing argument (around 1:09:00) that young lawyer Adam Hall (Chris O'Donnell) is making in court on behalf of his client and grandfather Sam Cayhall (Gene Hackman), who is scheduled to be executed for a racially motivated murder.

The musical content is quite straightforward: it is in E-flat major, the harmonies alternate back and forth between an E-flat major triad and a G-minor triad, the melody rises do-re-mi three times, and the texture thickens and dynamics rise gradually over the course of the cue. The tonal-harmonic aspects involve what I have called a "loss gesture" and have written about here and here and demonstrated here.

But that's not what I especially appreciate. The "loss gesture" works well when a listener well perceives the transition from one triad to another. The soundtrack is dominated by Chris O'Donnell's dialogue. What I admire is how Burwell's harmonic changes find holes in the dialogue. He finds not only big holes in between sentences, of course, but also two smaller holes within sentences: the 0.8-second hole between "It's a tragedy that" and "has destroyed three lives already," and the even smaller 0.4-second hole between "He was" and "raised by his family and this state to become the man that he became."

Below is a transcription of the music and the dialogue that notates time exactly proportional to space. (Sts. = strings, Brs. = brass, +W.W. = woodwinds are added.) The holes are indicated with enclosures whose color matches the description above.


Sunday, September 4, 2016

Folding Phish's Tweezer

At Dick’s Sporting Goods Park in Commerce City, CO, the rock group Phish is playing their last show of their 2016 summer tour as I post this. As phish.net says, "If there is a single Phish song that can be said to evolve with and exemplify Phish’s sound and artistry over the decades, it’s 'Tweezer.'" Below is a simplification of the famous opening lick, in "Tweezer Reprise"'s key of D. Here is a recording.

I've spent some time on this blog analogizing pitch and time. Here's one way that they are related:

The gamut of pitch register lays out linearly, from low to high, but it also circles back on itself via the concept of the octave: "do-re-mi-fa-so-la-ti...do." Even though the second "do" is higher than the first, we still call them both "do." In fact, of all of these notes within the octave, this "do" is special: it is the tonic, the note toward which all of the others are oriented.

The expanse of chronological time lays out linearly, from earlier to later, but it also circles back on itself via the concept of the measure: "1-2-3-4...1." Even though the second "1" is later than the first, we still call them both "1." In fact, of all of these beats within the measure, this "1" is special: it is the downbeat, the beat toward which all of the others are oriented.

Now these five notes from "Tweezer"' have a rare property, and they have this property not only because of how far apart they are from one another in pitch and in time, but also because of what the tonic and the downbeat are.

This video describes this property.


This was the "one from rock music" I was referring to a year ago here. The first four notes of Gershwin's "I Got Rhythm" have the same property, but I hope I explained it better this time around. I can think of another song released in 2015 with a similar lick and the same tonal and metric orientation.