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.

A Twist on an Old Holiday Classic

Below are eight signature measures from a traditional holiday tune that, in the manner of last month's post, have been turned slightly counterclockwise, adding a diatonic step up for every quarter note after the first downbeat. I've added a bass drone on E that produces good counterpoint with this rotated tune: every note that is dissonant with the bass is linked by step to a consonant tone. This means that, when rotated, the counterpoint can remain good, at least with regards to the relationship between the lines. I encourage you to try to figure out the tune before scrolling down to see the answer; a dashed plumb line has been added to assist you.

Happy Holidays!

More Score-Turning

Last month I dabbled in a bit of score-turning, mostly just for show, as the visuals were mostly beside my point. This month the score-turning is less just for show, although mostly for fun. Below is the first cantus firmus from Johann Joseph Fux’s Gradus ad Parnassum of 1725.

An interesting thing happens when you rotate everything but the staff a little bit counterclockwise to make a new melody:

Now there are only the two different notes of D and G — octave differences aside — rather than five different notes.

When combining lines in counterpoint, Fux recommends, all things being equal, that one use more imperfect intervals (thirds, sixths, and their octave compounds) than perfect intervals (unisons, perfect fifths, and their octave compounds) between simultaneous notes. This two-tone melody makes this quite easy: one could simply place a drone of B flat or B natural under or over this two-tone melody and only thirds and sixths will sound. When you rotate these two lines the same little bit back clockwise, the old melody returns and the drone becomes a line that completely descends by step. Here it is with the added line below; only thirds and sixths sound.

Fux wouldn’t like how it begins and ends, but, as I said, this is mostly for fun.

Next month I will share a winter holiday tune that basically has this same property as Fux’s cantus firmus.

Reich and Pythagoras

In many of his compositions, Steve Reich employs the following rhythm of eight notes, which repeats over a twelve-unit span. The colors are my own, intended to facilitate the reader’s orientation: the white dots indicate the absence of onsets.

Music researchers have proposed several features that set this rhythm apart from other rhythms; some are summarized in Gottfried Toussaint’s recent book The Geometry of Musical Rhythm (CRC Press, 2013), along with the author’s own ideas. I propose another such feature.

On the one hand, this figure is the well-known smallest-integer illustration of the Pythagorean Theorem. On the other hand, and admittedly rather incidental to Pythagoras, each of the three squares is a metrical interpretation of some segment of the repeating Reich rhythm. The side of a square shared with the triangle serves as the “left” side of the square’s “score”: the rhythm is then read as usual, first left to right, then top to bottom. Each square represents a “pure” meter: a triple of triples (like three measures of 3/8), a quadruple of quadruples (like four measures of 4/8), and a quintuple of quintuples (like five measures of 5/8). The first two are quite easy to experience; however, the last takes a little practice, but is possible.

The triangle in the middle is completely lined with dots. This means that, in each of the three metrical interpretations, there is an onset on every downbeat. No rhythm besides Reich's (up to rotation) with eight or fewer onsets in the same repeating span has this property. While there exist other eight-onset rhythms over longer repeating spans that have this property, only Reich's rhythm among these has this property plus the equivalent for sextuples; that is, all downbeats for a "pure" sextuple meter.

A Balancing Act (and its Dual) in Some Measures of Schubert

In much tonal music that is triadic and metered, stable events like tonic (I) triads and consonant (C) melodic tones tend to occur at relatively metrically strong (U) moments, and unstable events like dominant (V) triads and dissonant (D) melodic tones tend to occur at relatively metrically weak moments (/). However, in the “classical” variety of this kind of music, this synchrony is not overdone and is sometimes deliberately undercut; the suspension, which offsets these bedfellow arrangements, is a quick way to evoke a “classical” sound. The four-measure idea that begins each of the famous first two phrases in the first of Schubert’s Trois marches militaires balances synchrony (shown below with characters tracing parallel trajectories) with offset (shown below with characters tracing divergent trajectories); moreover, the means by which this balance is obtained is dual to that of the other phrase.

Something similar happens in an even more iconic classical melody, Beethoven's "Ode to Joy."