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.

When Canons in Inversion Don't Know When to Say When

Two years ago, I demonstrated a bit of serendipitous counterpoint with the Christmas carol "Deck the Halls." Luck this good is bound to run out.

Here are a couple of canons that did not make the cut, although the first few measures of each held promise.







Happy Holidays!

A Successful Prediction

In two posts from July and August of last year, I recomposed some music from The Hunger Games movie franchise to show various ways in which one could overthrow the "Panem National Anthem" and its I-bVI-I cliché. Furthermore, in the August 2014 post, I predicted that, during the last two Hunger Games movies, which had yet to be released, composer James Newton Howard would use something like one of my recompositions to accompany overthrowing. In short, I was predicting both the kind of music a composer would create in the future, and with what in the narrative the music would be associated.

In the penultimate Hunger Games movie (Mockingjay - Part 1), nothing like my recompositions can be heard. However, in the final Hunger Game movie (Mockingjay - Part 2), which was released three days ago, not only is something like one of my recompositions in Howard's score, but this bit of music appears to be associated with the unseating of the regime of the despotic President Snow, the aristocratic Capitol, and the I-bVI-I-featuring anthem that symbolizes their power.

Below I have transcribed the relevant excerpt, which you can hear here:

This music first occurs toward the end of the movie when (spoiler alert) the rebellion's President Coin has turned the tables on President Snow. The example below stepwise transforms the recomposition from my August 2014 post into much of the music above.

The first change is mostly cosmetic, notating the middle chord as #III instead of bIV. The second change transposes the music by a tritone, which is arguably the farthest away one can get from the original anthem's tonality. The third change replaces duple subdivisions with triple subdivisions, and alters the melody a little, although the important "up minor third, up augmented unison, down augmented unison" remains, as shown by the lines.

From here, things get interesting. Howard puts this FM-AM-FM music in an A-centered context. This would seem to upturn the upturned, promoting the A-major chord back to tonic and subordinating the F-major triad, creating the same clichéd harmonic subjugation that was present before. But it is not quite as it was before. A non-harmonic bass-clef D subverts the A-major harmony, and flips the tables once more: indeed, the I-bVI-I of Panem's former regime has been inverted. But the new authority that subsumes this inversion is essentially the same as the old -- the pitch A -- and the F-major harmonies still ultimately capitulate to its rule. Now in charge, President Coin also plans to assume totalitarian command, withhold elections, and conduct a "Symbolic Hunger Games," where children of the Capitol would suffer the same fate as children of the oppressed Districts had for 75 years.

Whether one thing is like another thing can be a matter of interpretation, and others may contend that my prediction did not come true. I believe it did, but I also believe that my original recomposition and Howard's music are unlike in a crucial way: his is truer to the story. The music shown above both upends the mediant relation of the old regime and nonetheless maintains the old tonal system, but my recompositions never saw Coin's coup coming. But, if you listen to the end of the cue, you will probably notice that the final harmony is rooted on F. Perhaps an end to tyranny is ultimately here, both once and for all.

Oh, and, for the record, my name is not in the credits.

Another Match in Reich's The Desert Music

This is the beginning of the third year of this blog. To celebrate, I am combining the idea from the first post of two years ago – relating pitch and time – with music by a composer featured in the post from one year ago – Steve Reich. A repeating five-chord progression occurs in the first and last movements of the five-movement The Desert Music. These chords are indicated below. In the first movement, the progression starts with the chord in the upper right and loops clockwise several times. The chord with the D in the bass sounds like a jazzy tonic harmony, whereas the next chord with the A in the bass sounds like a jazzy dominant harmony. In the first movement, adjacent chords after a while begin to overlap with and blur into one another. (This overlap is more conspicuous in the Michael Tilson Thomas recording than in, for example, the Kristjan Järvi recording.)

The ten circles outside the five chords use a circular piano keyboard to indicate notes irrespective of octave and diatonic spelling: C is at the top of each circular piano keyboard. The five circles next to the five chords simply translate the chords into this circular keyboard notation. The other five circles – the ones with arrows pointing toward them – show the union of two adjacent chords. The color coding – with reference to my blog post from a year ago – shows that the union of the dominant and tonic harmonies perfectly matches Reich’s characteristic rhythm, with the note C corresponding with the downbeat.

Furthermore, my Pythagorean slant of a year ago works on the pitch side: the union of Reich’s dominant and tonic chords is the smallest kind of note group that contains an augmented triad, a fully-diminished seventh chord, and a Guidonean hexachord (a major scale without ^4 or ^7). Moreover, the tonic chord is the constituent Guidonean hexachord (the blue star: in fifths, F-C-G-D-A-E), while the dominant chord contains both the constituent augmented triad (the red triangle: A-C#-F) and fully-diminished seventh chord (the purple square: C#-E-G-Bb), and both chords are the usual dominant-functioning harmonies of their respective chord qualities (V+ and vii°7) of the D tonic. In other words, if you trace the purple, red, or blue line in the figure above, you will put the colored dots in the same order that they are on the side of the right triangle from the October 2014 figure of length 3, 4, or 5, respectively: that is, green-purple-orange-purple-..., green-red-red-...,  and purple-red-blue-orange-blue-red, respectively.

I Got Rhythm and Pitch to Match

Today George Gershwin would have been 117 years old. (Imagine what other music he might have written if he had lived to even half that age. He didn’t even make it to a third of that number.)

I suspect the average person familiar with Western music, conceived rather broadly, knows just as well the four notes that follow these

as the four notes that follow these

What makes "I Got Rhythm" so catchy? There’s the pentatonic aspect of the opening pitch materials, which makes the tune quite user-friendly. And there’s the consistent dotted-quarter pulse overlaid on a common 4/4 meter, which makes the rhythm distinctive but uniform nonetheless. These features of pitch and duration have surely already been pointed out. However, what perhaps has not been pointed out is how the pitch and rhythm of the first four notes correspond in a significant way.

Rather than report this correspondence with a lot of words, I will use a couple of pictures, and let the reader figure it out.

This kind of match is rather rare among four-note motives. (Well, in full disclosure, it does happen with I Got Rhythm’s next four notes, but that’s just because they are the same thing backwards.) I will leave it as an exercise to the reader to discover how elements of these images (such as the stack of fourths, and the a-b-a rhythm) infiltrate the composer’s Variations on “I Got Rhythm.” I will leave it as an additional exercise for the reader to come up with a five-note motive with the same property. I can think of one from rock music.

More Mirrors: Mussorgsky Matches Mozart

This month last year, pianist Andrej Hoteev and singer Elena Pankratova released a recording of Mussorgsky’s Pictures at an Exhibition and Songs and Dances of Death, informed by original manuscripts. This recording brought Pictures and its various interpretations back into my forethoughts. The late professor of piano Nancy Bricard also consulted original manuscripts in her 2002 edition of Pictures. In her foreword to her edition, she claims that the composer “did not use the sonata form.” While this may be true, the opening of the last movement titled “The Great Gate of Kiev” resembles a sonata without development:

Exposition
mm. 1-29 Loud heraldic first theme in main key
mm. 30-46 Soft hymnic second theme in a subordinate key

Recapitulation
mm. 47-63 First theme again in main key
mm. 64-78 Second theme again, but transposed into the main key

(Hoteev even recommends, in his liner notes, inserting a brief silence before the start of the second theme, which is common at this juncture in traditional sonata forms.) However, the subordinate key is not the key a perfect fifth above the movement’s main key of E-flat, but the key of a perfect fifth below. This inversion of usual tonal practice extends beyond key choice, as the example below suggests. This example lines up a reduction of the exposition and recapitulation of a typical sonata-form movement—I chose the first movement of Mozart’s Eine kleine Nachtmusik—with a reduction of the beginning of “The Great Gate of Kiev.” Whole notes represent roots of chords. T, S, and D refer to tonic, subdominant, and dominant. I chose clefs that made the mirroring between the two movements visually apparent. The subscripted p refers to the “parallel” transformation in German-language music theory, which allows for a change of chordal root without change of function, as I have further shown with the baritone clef, the F clef in the middle of the staff.

[click image to enlarge]

Notice how authentic motions (D-T) in Mozart match plagal motions (S-T) in Mussorgsky, and how each second theme in its respective exposition reaches an apogee—in the thirty-ninth measure of each!—four perfect fifths away from the main key’s tonic, then gradually eases back toward this tonic in the measures that follow.