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.

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.

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.

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.

In Die Walküre, Space Becomes Time

Wagner's complete tetralogy Der Ring des Nibelungen was first presented 140 years ago yesterday. The one part of this music that has probably embedded itself into Western cultural consciousness the most is the beginning of the third act of the second opera Die Walküre, the so-called "Ride of the Valkyries." The image below provides a notation of the first presentation of the melody. While the severe distortion of the notation makes it harder to read, it makes the distance that a measure of time and a semitone of pitch takes up on the image more equal to one another. All of the melody's local maxima (B3-D4-F4#-A4-C#5) divide this 14-semitone span into a 3+4+3+4 organization. The dominant-(5)-to-tonic(1) moments of the melody divide this 14-measure span into a 3+4+3+4 organization.

P.S. My post title cheekily refers to the fact that, in Wagner's last opera Parsifal, Gurnemanz tells Parsifal that that in this realm time becomes space ("Zum Raum wird hier die Zeit").

Film Music Style as Guide for Late Beethoven

There is something entertaining, if not a little anachronistically naughty, about hearing a kind of more recent music in the music of someone like Beethoven (like hearing this in Beethoven's 7th symphony, or hearing this in Beethoven's 8th piano sonata). But here's an instance where it can be quite helpful. (There's a relevant cognition experiment of mine here that I invite you to try, but, if you decide to take part, do so before reading on.)

A year ago on this blog, I shared this trend about recent popular film music: "When two major triads whose roots are four semitones apart are adjacent, the triad with the root four semitones above is significantly more likely to be the tonal superior." This happens particularly when the two chords have been isolated from other tonal obligations and, without such obligations to gain tonal meaning, instead look toward one another.

In measure 13 of the first movement of Beethoven's op. 109 piano sonata, the notes of a D-sharp-major triad fill up most of the register of the piano and our attention with its sonorous and solid proportions, blocking from view much of what had occurred before. Then its firm surface suddenly shimmers and transforms into a B-major triad. To some film-music ears, this is a departure from stability and security: since B is four semitones lower than D-sharp, B is demoted in this two-character drama within a drama. I indicate this visually with a red arrow that fades into purple.

In sonata form -- a drama which this drama is within -- the recapitulation is both a return to and a reworking of the material from the exposition. In the first movement of Beethoven's op. 109, measure 62 in the recapitulation corresponds to measure 13 in the exposition. This time, it is a C-major triad that floods our senses much the same way that the D-sharp-minor triad did before. But its corresponding transformation is to an E-major triad, whose root is four semitones higher. The same harmonic conversion is now a homecoming, which I show with an arrow that changes from purple to red.

To be sure, this harmonic scheme is rather standard: the sonata is in E, and B is accustomed to its subservient role when E is the tonal sovereign. But local harmonies, together with a future music's chromatic tendencies, enrich this relationship and even offer a guide, however anachronistic, to the work's overall tonal plot.

Muse's Voice Leading at the Olympics

The 2016 Olympics are around the corner. The official song of the 2012 Olympics— “Survival,” by the British alternative rock band Muse—premiered on the radio on this day four years ago.

The lyrics of the opening are “Race, life’s a – race, That I’m gonna – win, yes I’m gonna – win, And I’ll light the – fuse, and I’ll never – lose…” The music accompanying these lyrics uses the triadic progression of BbM – Bb+ – Ebm – CbM – GbM, where M is major, m is minor, and + is augmented. The progression can also be considered in terms of smooth voice leading. First, one by one, each of the three voices in the BbM triad—on Bb, D, or F—moves up by a semitone, achieving the CbM triad. After this, two of the three voices in the CbM triad slip back down to make the GbM triad. Only the voice that started on the F and moved up to Gb never retreats, at least not until the progression starts over. Therefore, one can say that only this voice “wins,” as animated below, with the gold-colored figure as the winning voice.

What makes this more fitting is that, at least at the beginning of the song, Muse’s lead singer, Matt Bellamy, is intoning his first-person account of victory using precisely the notes of the “winning” voice. Now this, literally, is…VOICE…LEADING.

Researching Some Music in Ligeti's Musica ricercata

The composer György Ligeti was born 93 years ago today. Here is a relationship between pitches and meters in some music from Ligeti's Musica ricerata that requires a little investment upfront.

1. A measure like 2/4 can be represented by the ordered series of positive integers 2-1, where 2 is the strong beat and 1 is the weak beat. Likewise, a measure of 3/4 can be represented as 2-1-1. A measure of 4/4, instead of being represented by 2-1-1-1, is often thought to have a metrical accent halfway through that is not as strong as the downbeat, but stronger than the weak beats: this could be represented as 3-1-2-1.

An uneven measure like 7/4 often breaks down as 4/4 + 3/4, but instead of simply concatenating 4/4's 3-1-2-1 with 3/4's 2-1-1 to make 3-1-2-1-2-1-1, a representation of 4-1-2-1-3-1-1 reflects the measure's initial division into 4/4 + 3/4 by making the beginning of the 3/4 measure into a "second-rank" downbeat.

When the quarter-note beat of one of these meters is subdivided into eighth notes, this inclusion can be represented by adding one to each of the numbers and then placing a 1 after each of them. For example, 3/4 (2-1-1) with eighth notes would be 3-1-2-1-2-1. 7/4 (4-1-2-1-3-1-1) would be 5-1-2-1-3-1-2-1-4-1-2-1-2-1.

2. Now, given a certain series of numbers, its cumulation is a series of numbers that sum the original series starting from the left and up to that point. For example, the cumulation of 2-1-1 is 2-(2+1)-(2+1+1) or 2-3-4. The cumulation of 3-1-2-1 is 3-4-6-7.

3. Lastly, a cumulation mod 2 of a series is a series's cumulation where every even number is replaced by 0 and every odd number is replaced by 1. For example, whereas the cumulation of 2-1-1 is 2-3-4, the cumulation mod 2 of this series is 0-1-0. The cumulation mod 2 of 3-1-2-1 is 1-0-0-1.

Below is an 7/4 example from the Overture to Leonard Bernstein's Candide -- placed into time signatures commensurate with the discussion above -- that shows 1) the 7/4 meter as a series of positive integers, 2) the culmination of the series, and 3) the culmination mod 2 of the series.

Below is a video with a score and recording of the second movement of Ligeti's Musica ricercata; attend to the first four measures.

I've re-notated these four measures to make a steady stream of eighth notes, and analyzed the music using the methods outlined above.

Notice how the alternation between 0s and 1s in the culmination mod 2 of the first measure, the second measure, and the third and fourth measures as a single measure of 7/4, corresponds exactly to these measure's alternations between the two pitches that open the movement. Each fermata both resets the metrical hierarchy and toggles the pitch-meter mapping.

Thanks to Nick Shaheed for encouraging me to think about this music.

Approximating e Musical-e

The mathematician Leonhard Euler was born in this month 309 years ago. As he contributed in important ways to our understanding of some aspects of music, I thought I would use music to help understand some aspects of mathematics. The constant e, named after Euler, is one of the most important numbers in mathematics. One way of approximating e can be demonstrated using musical intervals:

At the least, this approach offers one answer to the question of when an approximation is good enough: it's good enough when you can't hear the difference between it and a better approximation.

Sonata-Principled Praises

Lent concludes during this month, and that means that many Easter performances of Beethoven's "Hallelujah," the popular chorus that concludes his unpopular oratorio Christ on the Mount of Olives, are just around the corner.

This is a rather unusual movement to bring up as an example of the sonata principle, because, as befitting a coda-chorus, it hardly strays from its main key of C major. In fact, it never provides an authentic cadence in any other key. The closest it comes to such a cadence, complete with a chromatic predominant and a 6/4 embellishment of the cadential dominant, is this:

If the sopranos had descended from E (A:^5) to A (A:^1) instead of ascended to G (enclosed in red above), then this would be a clear full close in A major. The G is both a surprise -- foiling a well-prepared escape from C major -- and not a surprise -- the A7 chord sends the music back, via a circle of fifths, to the movement's main key and tonic triad.

Toward the end of the movement, this happens:

If the basses had descended from G (C:^5) to C (C:^1) instead of ascended to B flat (enclosed in red above), then this would be a clear full close in C major. The B flat is both a (welcome) surprise -- foiling yet another authentic cadence in C major -- and not a surprise -- the lean toward the subdominant key is a common perorative technique in classical tonal music, and the basses's ascending-minor-third digression from the movement's main key recalls the soprano's earlier similar digression from a different key.

Sonata-Principled Kangaroos

This month and next, I’ll continue last month’s focus on sonata, particularly the “sonata principle,” Edward T. Cone’s name for common-practice-music’s propensity to present distinctive melodic material first outside of the movement’s main key and then later in the main key.

When you tonally attend to this leap of a minor seventh,

I suspect that you hear it in G, even though there’s no G there. The presence of the C natural keeps it from being in D (D as tonic would rather have C#), and the greater weight afforded to the D by its lower register, longer duration, and downbeat position keeps the C from materializing as tonic. Instead, the key of G matches these different weights well: the more weighted pitch of D is dominant, tonic’s second-in-command, and the less weighted pitch of C is a more subordinate subdominant.

So, with this in place, the “Kangaroos” movement of Carnival of the Animals, which Camille Saint-Saëns began composing 140 years ago this month, contains a rather subtle instance of the sonata principle. The movement clearly begins in C minor, but then the bass’s leap of a D-C seventh in mm. 4-5, especially when accompanied with notes taken from the G harmonic minor scale, leans toward G.

At the end, the minor seventh appears again in the bass, but transposed to G-F, accompanied by notes taken from the C melodic minor scale. Thus this minor-seventh leap is restored to the Kangaroo’s main key of C minor.

What appears even more clever is that the last three measures embed this minor-seventh recapitulation within a clear pattern in the bass: G--F#-G--F-G--E. This could be heard as a musical cross-dissolve: the repeated low G-major chords draw out the dominant of C, while the functional progression of higher chords—V/V, Fr. 6, V—prepare the key of A, which is the tonal center of “Aquarium,” the next movement where the same Fr. 6 returns as a less-functional coloristic harmony.

Schenker, Sonata Form, Mozart, and Meter

The Austrian music theorist Heinrich Schenker died 71 years ago today. The theory that Schenker founded 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. Often, similar forms have similar linear-harmonic essences. Take sonata form. When a major-mode late-eighteenth-century sonata-form movement has a prominent scale-degree 3 in its main theme, its linear-harmonic essence is often interpreted as something like this:

P = primary theme, TR = transition, S = secondary theme, C = closing section

This graph reflects how the primary themes in both exposition and recapitulation are basically, if not exactly, the same. It also reflects how the secondary theme in the exposition, whose treble line is distilled to D-C-B-A-G, is transposed back into the main key, with a distilled treble line of G-F-E-D-C. Schenkerian theory encourages the analyst to connect, as if part of a smooth melody, the prominent pitch that begins the primary theme and the prominent pitch that begins the secondary theme. This works quite well in the exposition: E is just a step higher than D. However, the stitching of the corresponding design in the recapitulation is a bit loose, particularly at the moment marked with an asterisk (*). On the one hand, the prominent pitch that begins the secondary theme in the recapitulation should be the same one that begins the secondary theme in the recapitulation, but down a perfect fifth or up a perfect fourth: in this case, D becomes G. After all, the secondary themes are typically copies of one another, differing only by their transpositional level. On the other hand, the prominent pitch that begins the primary theme in the recapitulation (E) and the prominent pitch that begins the secondary theme in the recapitulation (G) are not smoothly connected as they were in the exposition. Rather, if the prominent pitches of the recapitulation’s primary and secondary themes were to be closer together, the graph might look something like this:

But such a graph would signify a considerable change for the beginning of the secondary theme in the recapitulation from how it sounded in the exposition.

In the first movement of Mozart’s Piano Sonata in C Major, K. 279, there is such a change, although I will let the reader decide how considerable it is. This common-time movement is remarkable in how often it shifts the perceived downbeat from the notated downbeat to the spot halfway between notated downbeats. By my count, this shift happens 12 times: 6 times away from the notated downbeat, and 6 times back to the notated downbeat. Mozart does not notate these shifts, but they can be readily heard. I have notated them below for the entire movement.

I: = main key, V: dominant key, PAC = perfect authentic cadence, HC = cadence, --> = "becomes"

These shifts also affect how the beginning of the secondary theme is heard in both exposition and recapitulation. As shown below, in both exposition and recapitulation, the entrance of the secondary theme is preceded by a perception of downbeats halfway between notated downbeats. However, Mozart starts the secondary theme halfway through the measure in the exposition, while he starts it on the downbeat in the recapitulation. This changes which note in the secondary theme first receives metrical emphasis. In the exposition, that note is D. In the recapitulation, that note is not G, but E, as shown below.