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.

Mirrored Rhyming in "The Star-Spangled Banner"

I have used this blog to make little musical observations and even little musical predictions. This month I will use it to make a little suggestion. This is the month when those in the United States celebrate their independence from the United Kingdom; hence, the “Star-Spangled Banner,” the national anthem of the United States, gets a little more airtime in July than in other months.

There are countless arrangements of this anthem. The music to the left is not in four-part harmony, but is rather the superimposition of melody-bass arrangements of two snippets of this anthem: the music in blue is near the beginning, and the music in red is near the end. Both melodic snippets are from the original tune, and the blue bass line is by far the most common choice for arrangers, although not all, like here. The red bass line, while less common in arrangements, can still be found, like here. This red bass line, when matched with the other notes of my example, produces a neat symmetry if you like that sort of thing: the counterpoint of the two blue lines perfectly reflect around a mirror to produce the counterpoint of the two red lines. When the anthem is in B-flat major—a frequently selected option—this mirror can be placed at middle C, and the treble and bass clefs of the grand staff can well display this symmetry. This symmetry engages not only meter (weak-weak-strong) and relative duration (short-short-long), but also the only internal rhyme (“see,” “free”) in the anthem’s first and, for many, only stanza.

Twisted Tristan

Speaking of anniversaries in multiples of fifteen, Richard Wagner’s Tristan und Isolde premiered 150 years ago this month. Wagner in general, and the introduction to this opera in particular, is well known for withholding the tonic note or tonic triad of a key whose presence is made clear. A very efficient way of doing the former—with two voices, three pitch classes, and three total semitones of voice-leading work—is the progression F3D4 to E3E4 (call it what you want), which is in A although no A sounds. (The C clef, while not as user-friendly for some as other clefs are, will be useful later.) Now take this progression, and skew it by displacing the top voice forward in time. This puts the D above the E, and excludes a D above the F.

Now fill out this twisted progression in four parts, while maintaining some implication of an A tonality. An E7 is the best way to harmonize the second moment. As for the first chord, exactly two of the twelve half-diminished seventh chords have an F and also do not have a D; they are shown below. (My unusual spelling of each half-diminished (hd) chord matches Wagner’s.)

These are exactly the two solutions, allowing for transposition (shown using the roving clef below), used in the famous opening eleven measures of the opera's instrumental introduction: Wagner’s first two progressions use the first solution, and his third progression uses the second solution.

Moreover, both chords contain a note—either C# or D#—that is only a half step away from the D: a clear “not-D” note through its half-step displacement. Lastly, this derivation from a two-voice model also jibes well with the voice leading: the voices with the F or the “not-D” note in the first chord (the solid noteheads) each move by a single half step into the second chord, while the two added voices (the hollow noteheads) do not.

Rare Sibling Harmony at the End of Holst’s Solar System

Let us say that a three-note chord’s sibling is a transposition or inversion of the chord; that is, siblings have the same three intervals between their three pairs of notes, allowing for change of octave. For example, F-A-C and C-Eb-G are in the same family—each contains a minor third, major third, and perfect fourth—but C-E-F is in a different family. There are twelve three-note-chord families. Let us further say that the relationship between two (non-identical) siblings in the same family is harmonious if there is no half step, allowing for change of octave, that exists between a note in one chord and a note in the other.

The family of major and minor triads has been shown to be special for many reasons. Here is one more: of the twelve three-note-chord families, the percentage of harmonious types of sibling relationship among the family of major and minor triads is, perhaps surprisingly, the smallest. (In truth, it is tied with the family to which C-E-F belongs.) Shown below are the twenty-three possible types of relationship a major triad can have with its siblings, up to transposition and inversion. The top system shows all eleven non-zero transpositions, the second system shows six inversions around C, and the third system shows six inversions around C/C#. A notehead is filled in if its pitch forms a half step, allowing for change of octave, with a note in the other chord in the same measure: the two clashing notes have the same notehead shape.

Only two out of these twenty-three relationship-types are harmonious: they are the measures without any filled-in noteheads. If you combine together the two triads in each pair into a richer harmony—an F9 chord, and a C#m7 (or EMadd6)—you have the last two chords of Gustav Holst’s The Planets. These are the chords, sung by an offstage female chorus, that alternate with one other until they fade out to silence, that is, unless Colin Matthews’s Pluto, the Renewer follows on their heels, a piece that premiered fifteen years ago today. Pluto also ends with the same choir singing essentially the same C#m7 chord.

For more on this, see Táhirih Motazedian and Scott Murphy, "Holst’s Planets, The Final Frontier:

Interplanetary Voyage as Intrapersonal Escape" Intégral 36: 1–17.

https://www.esm.rochester.edu/integral/36-2023/36-motazedian-murphy/

Verdi Under the Radar

The innovative-chromatic-harmony radar that musicians bring to their hearing of nineteenth-century opera tends to ping more often with the music of Wagner than of that of his contemporary Verdi. However, I like to think of some of Verdi’s innovations as more stealthy than showy.

Take this cadence-ending treble line.

It’s not too hard to imagine this bass line, with an implied Neapolitan chord, underneath it.

Now take this cadence-ending bass line.

It’s not too hard to imagine this treble line, with an implied secondary dominant, above it.

By themselves, these lines imply staples of chromatic harmony. But toward the end of a chorus from La forza del destino ("Nella guerra, è la follia"), Verdi puts them together.

On the one hand, Verdi puts a root-position G-major chord in C-sharp minor music: ping! On the other hand, the stylistically normal soprano and bass lines -- enharmonics aside -- fly right by, sotto il radar. Viva Verdi indeed.

Zero-Sum Games in Dvořák

A year ago, I had something to say about Dvořák's Ninth Symphony in E Minor. With the return of March, I thought I would do so again. The end of the symphony is a little unusual. Not the final nine bars, but the three bars right before it, shown in reduction on the left side of the example below. The musical rhetoric—the entire orchestra loudly alternating between the major tonic chord and some other chord—would seem to suggest that tonic is being affirmed, as we would expect in a tonal symphony’s concluding measures. However, the “other chord” is enharmonically a half-diminished seventh chord built upon D, far from a chord rooted on B (or maybe A) one would expect. If anything, this chord, coupled with an E-major triad, destabilizes E as tonic: the notes in these chords fully constitute an A harmonic-minor scale, except the A, as shown on the right side of the example below. Play an A-minor chord after all of this and it might justify this oddity, but the symphony’s imminent end will probably feel farther away, not closer.

But, from another point of view, this harmonic alternation is consistent with other concluding aspects of the symphony. In the traditions of the per aspera ad astra symphony and the cyclic symphony, Dvořák lifts many of the work’s minor-mode themes into the major mode during the work’s final minutes. However, one theme goes the opposite way: the major-mode chorale at the beginning of the second movement, of which the first four chords are provided on the left side of the example below, is brazenly put into the minor mode toward the end of the fourth movement. Never after does the final movement restore this chorale to major, but one could hear the D half-diminished seventh chord as offering such a restoration in the abstract.

If one considers the E-major triad with a doubled root, the idealized voice leading among adjacent chords at the beginning of the chorale is always balanced; that is, the amount of total semitones voices each that go down and go up are the same, as shown in the example below with the left-hand-side columns of numbers that each sums to zero. (This is also true for the six-voice realization discussed a year ago.) This balance also occurs in the music with the D half-diminished seventh chord, as shown in the example below with the right-hand-side columns of numbers that each sums to zero. This would also be true for any of the four half-diminished seventh chords rooted on D, F, Ab, or B. But only the D half-diminished seventh chord brings back two of the three notes in each of the two balanced non-E-major chorale chords (Bb major and Db major), as shown with the four slurs below. In fact, it is the only chord—regardless of quality—that 1) brings back two of three pitch classes in these two triads, 2) can achieve balanced idealized voice leading with the root-doubled E-major triad, and 3) does not contain a half step between any of its notes.

In the Metric Wastelands of Led Zeppelin's "Kashmir"

Led Zeppelin’s “Kashmir” was released 40 years ago this month. The meter of the opening of this song is famously not straightforward, although it appears that 3/4 (quarter = 82), with the first sound placed on the downbeat, is a possible choice, as can be seen with transcriptions here and here and here. And yet this choice is unusual, because there is nothing in the music—like a boom-chuck-chuck—that exactly projects this meter.

I have a theory about why the music can nonetheless be heard this way. Below are twelve possible metrical interpretations of the opening of the song.  (You will want to click on the image to see detail.) The ordering from left to right corresponds to the relation between the first sound of the song and the first notated downbeat: in the middle, they match; to the left, the first sound is an eighth note earlier than the first downbeat; to the right, the first sound is an eighth note later than the first downbeat. The ordering from top to bottom corresponds to the length of duration that can be grouped into threes: eighth note for the highest, quarter note for the second highest, half note for the second lowest, and whole note for the lowest.

Music that is not faded represents a line in the texture that works with its notated meter.  The fact that there is no single interpretation that is completely not faded signifies that the meter is not straightforward. However, note that, although the brown-bordered metric interpretation is entirely faded, this interpretation is the only one adjacent (like a chess king is adjacent) to five of the six interpretations (thinly bordered) that have a line that is not faded. This brown-bordered metric interpretation is their center and their best approximation (in two dimensions!), even though the music has no content that would directly support this metric interpretation. This is like the fact that there is usually no town located at the mean population center of a country.

This metaphor seems appropriate given the song’s extra-musical content and context. The Kashmir region in South Asia is not a separate state, but rather defined politically only through its division between India and Pakistan. However, according to Wikipedia, Led Zeppelin’s singer Robert Plant was initially inspired not by this part of the world but by a place similarly liminal: the barren “waste lands” of Southern Morocco between the cities of Guelmim and Tan-Tan.

An Anti-Aging Harmonic Formula from Brahms

This month I was kindly invited to produce a guest post for the important blog of Dr. Timothy Chenette, a fellow music thinker. But I'll offer a follow-up here.

In my post for Dr. Chenette, I suggested how pop music and classical music approach the diatonic-diminished-triad-made-major-triad in ways that both mirror one another, and still sound considerably different from one another. For example, the I - bVII - IV - I (or, in another interpretation, I - IV/IV - IV - I) is a pop-rock trademark, but this kind of progression is quite rare in classical music, as is the closely related blues-based V - IV - I, whose kinship to I - IV/IV - IV - I can be shown by renotating (and hearing) it as V/IV - IV/IV - I/IV - I.

But this is not completely unheard in classical music. Brahms's choral work Nänie appears to make frequent and purposive use of the rare V - IV progression, more so than any other work of Brahms to my knowledge. Moreover, the use of this "retrogression" befits the text, which juxtaposes perfection, beauty, and their apparent immortality with their eventual and inevitable demise. When IV goes "as it should" to V, the arrow of time and aging points "as it should," driving toward tonic's end with normative predominant-dominant syntax. But when V is followed by IV, one could hear a reversal of this process. With this in mind, consider this excerpt and my analysis. The back-to-back V - IV and IV - V may not perfectly coincide with the back-to-back "die Schöne [the beautiful]" and "vergeht [perishes]," but it's close enough for me.