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."

A Music-Theoretic Prediction

Theory, in general, is as useful in predicting the future as it is in explaining the past. For example, Stephen Hawking has theorized about an upcoming big crunch as well as about a previous big bang. Music theories and theorists predict too, but these predictions usually involve notes already composed in the past that then become someone else’s possible future. Music theorists are less known for predicting the details of the music that a composer has yet to create.

But I will do so, right now. There are two Hunger Games movies left: the last book, like the last book from the Harry Potter and Twilight series, is being divided into two movies. IMDb tells me that Mockingjay Part 1 is currently in post-production, Mockingjay Part 2 is due out next year, and James Newton Howard is continuing as the series composer. There are ample references to overthrowing in the last book. Mr. Howard is a sophisticated film composer. I predict that, in the underscore for at least one of these two final movies, he will use something like one of the sketches in last month’s post, and/or something like below, which is the ultimate and most compact tonal reversal. Notice that only with a second-beat melodic note exclusively on scale degree flat-6 can the melody stay the same and the chromatic-mediant relationship invert. (The smaller added staff plays the same melodic game that the music from last month played.)

I suppose there is an infinitesimal chance that, by publicly declaring my conjecture, this post affects the outcome, either by 1) providing an idea that did not already exist for Mr. Howard (or whomever this may concern), or 2) discouraging the use of an idea already existing (perish the thought of a composer, especially a film composer, adopting the ideas of another). If this is the case, I ask, purely in the name of science, that this influence be recognized. A note would suffice. (If the influence follows the first scenario, a film credit would also work: two t's in Scott, and no e in Murphy.) But I predict that my little experiment will safely proceed untainted in this way.

I optimistically lay 10:1 odds.

Potential Tonal Overthrows in The Hunger Games

A theme by James Newton Howard, called the “Panem National Anthem,” recurs in the first two Hunger Games movies. It can be heard most overtly during the tribute parade in the first movie; the opening of this music is transcribed below. It opens with a standard Hollywood musical cliché: a tonic triad (in this case, B major) flanking and firmly subordinating a different major triad (in this case, G major) whose root is four semitones below. This tonal-triadic scheme is so conventionalized in film music that, 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.

But not always. A few seconds later in the theme, this situation is reversed: following the B-major tonic triad is a D-sharp-major triad, whose root is four semitones above. Yet this reversal of fortune is fleeting: B as tonic persists. These two moments – what I transcribed as the first and third measures – have something else in common, perhaps by design, perhaps by chance: the triad with its root four semitones above puts the fifth of the triad in the melody (F# in m. 1, A# in m. 3), and the triad with its root four semitones below puts the root of the triad in the melody (G in m. 1, B in m. 3).

This melodic-harmonic correspondence permits a kind of two-sided musical game in which each moment can be crossbred with the other. On the one side, the following recomposition depicts how the fleeting D sharp (respelled as E flat) could oust B as tonic, using B-major’s own tactics of clichéd subjugation and thematic banner-waving.

On the opposite side, the following recomposition depicts how G could turn the tables on B by permanently transforming B into the selfsame fugitive insurrectionist that D sharp briefly was.

Schoenberg and Another Parsimonious Tritone Progression

Last month I promised to bring this multi-post thread to Schoenberg: here goes. The six-note collection of notes that can voice lead smoothly to its tritone partner in the most ways—36, to be exact—is some transposition of {C, C#, D, F#, G, G#}. Tied for second place at 20 ways are the whole-tone scale—some transposition of {C, D, E, F#, G#, A#}—and the hexatonic scale—some transposition of {C, Db (or C#), E, F, G# (or Ab), A}. The 20 permutations that correspond to the 20 possible voice leadings of the hexatonic scale are given below, organized into rows by the kind of chord in each permutation partition (or "orbit"):

Three perfect fifths:                               (CF)(C#G#)(EA)
Two major thirds, one perfect fifth:         (CE)(C#G#)(FA) || (CF)(C#A)(EG#) || (CG#)(C#F)(EA)
Two augmented triads:                          (CEG#)(DbFA) || (CEG#)(DbAF) || (CG#E)(DbFA) || (CG#E)(DbAF)
One major-7th chord, one perfect fifth:    (CEAF)(DbAb) || (CF)(C#G#EA) || (CF)(C#AEG#) || (CFAE)(DbAb) || (CFDbAb)(EA) || (CAbDbF)(EA)
One hexatonic collection:                      (CEG#C#AF) || (CEAFDbAb) || (CFC#AEG#) || (CFAC#G#E) || (CG#EAC#F) || (CG#C#FAE)

However, of these three kinds of collections, only the hexatonic scale and its tritone transposition have different notes. In fact, they are complements, in that they share no notes, and, together, they contain all twelve tones. Moreover, of all the kinds of six-note collections, only the hexatonic scale can voice lead smoothly to its complement in the most possible ways of 20—the next highest number of ways is 10. This might make the hexatonic collection attractive to a twelve-tone composer who is interested in a variety of smooth voice leadings.

Schoenberg’s twelve-tone work Ode to Napoleon frequently juxtaposes complementary hexatonic-scale collections, and, at times in the work, each note in one hexatonic collection is no more than a pitch-class whole step away—in register, in a single instrument, or both—from a note in an adjacent complementary hexatonic collection. It remains to be shown whether or not Schoenberg uses, or at least implies, all twenty possible voice leadings, but he does so for at least one voice leading in each of the five rows above, in mm. 23 (strings), 21 (piano), 21 (strings), 219 (piano), and 187 (strings), respectively.

Permutations and Parsimonous Tritone Progressions

Last month’s method of partitioning into dyads falls short as a way to enumerate the different ways a chord can voice lead smoothly to its tritone partner. For one example, the major seventh chord divides into “large” dyads in two ways, but it can voice lead to its tritone partner in four different ways. A better, more generalized way to enumerate the number of possibilities is to use permutations. A permutation of a set can be understood as its partition into one or more cyclic orderings. For example, (spring, summer, fall, winter) is a permutation of the seasons, which is equivalent to (winter, spring, summer, fall) but different from (spring, fall, summer, winter) and from (spring, fall)(summer, winter). Y is “followed by” X in a permutation if (…XY…) or (Y…X).

Many permutations—21, to be exact—can be invoked on a chord like a CM7 {C, E, G, B}, but present purposes favor only those permutations in which some registral realization of every note is followed by another note that can be realized 4, 5, 6, 7, or 8 semitones higher. There are four such permutations of {C, E, G, B}: not only (CE)(GB) and (CG)(EB), which reproduce the dyadic partition from last month, but also (CEBG) and (CGBE). These four possible permutations match one-to-one with the four possible smooth voice leadings to CM7’s tritone partner.

Caught between a stated commitment to bring up Ode to Napoleon during this entry and an unstated commitment to keep these entries relatively short, I have decided to honor the latter and delay meeting the former until next month.

Other Parsimonious Tritone Progressions

My previous posts have recognized that the major tritone progression is the only progression between consonant triads that can allow both exclusively smooth voice leading and root doubling. One way to understand why the major tritone progression is capable of this is to observe that there are exactly three pitch-class dyads that can each connect with their tritone pitch-class transpositions with exclusively smooth voice leading: the major third or minor sixth (both voices move by whole step), the perfect fourth or fifth (both voices move by half step), and the tritone (both voices stay put). Any chord that is a disjoint union of these dyads can voice lead smoothly to its tritone partner. The root-doubled major triad is one such chord: it is the combination of a perfect fifth and a major third such that the "bottom note" of each doubles that of the other. (Substitute "top note" for "bottom note" and you have the fifth-doubled minor triad, which can also enjoy a parsimonious tritone progression.)

Doublings set aside, there are 18 different four-note chords that can do this, and 35 six-note chords that can do this, up to transposition and inversion. Of these chords, the ones I find most compositionally suggestive are those that can be disassembled into these "large" pitch-class intervals in multiple ways. For example, the major seventh chord, which can partition into two perfect fifths or two major thirds, can voice lead to its tritone partner in at least two different ways. Next month, I'll generalize this (and, in so doing, explain the "at least" in the previous sentence) and use this generalization to make an observation about Schoenberg's Ode to Napoleon.

Dvořák and the Parsimonious Major Tritone Progression

Last month, I revealed how the major tritone progression is the only progression that permits stepwise-equivalent voice leading between root-doubled complete triads. If more than four voices are involved, it is common (all other things being equal) in voicing major triads to weight the root the most, the fifth the next most, and the third the least. The second movement of Dvořák's Symphony No. 9 features three tritone progressions in measures 1, 22, and 120. The second instance is higher than the other two, and involves exclusively woodwinds -- six woodwind players, to be exact. Dvořák assigns two players each to the half-step lines (Ab-G-Ab and Db-D-Db), and one player each to the whole-step lines (Db-B-Db and F-G-F). This results in a doubling of both major triads such that there are three roots, two fifths, and one third -- exactly the distribution among the first six partials of the natural overtone series.

Next month, I will explore what it takes for a chord to voice lead smoothly to its tritone partner.

The Parsimonious Major Tritone Progression

Around ten years ago, I started circulating an idea regarding the association between two major triads a tritone apart and depictions of outer space in recent Hollywood movies. It eventually ended up here. In the article, I noted that the major tritone progression, if voiced with a texture of complete triads and three voices, is tied for last place as the least parsimonious of all triadic progressions. For example, in this texture, one of the voices must leap -- that is, move by more than two semitones.

However, the typical number of voices in common-practice music is four, not three. Moreover, in such four-voice textures, the general preference is to double the root of consonant triads. In this more customary environment, the freakish major tritone progression makes a Disney-style comeback worthy of the most epic of space operas: it is the only progression in four voices in which 1) the chords are complete, 2) the roots are doubled, 3) no voice duplicates any other, and 4) all voices can move by no more than two semitones. For any other four-voice triadic progression that meets the first three conditions, the fourth condition will necessarily not be met.

Next month, I will show how this property relates to some music of Dvořák.

An Instance of "Anti-Composing-Out" in "The Alcotts"

One type of "composing out" is when the intervals within a group of notes correspond to transpositions among groups of notes. A simple and well-known example is in "Nacht" from Arnold Schoenberg's Pierrot Lunaire: E4-G4-Eb4 | G4-Bb4-Gb4 | Eb4-Gb4-D4. (The 4's mean that these notes are all in the octave above middle C.) The melodic intervals of minor-third-up and major-third-down between a note and the one that follows it within each three-note cell correspond to the transpositions between a cell and the one that follows it.

The second system of "The Alcotts," the third movement of Charles Ives's Piano Sonata "Concord," contains a single phrase that begins and ends with three-note chords that are transpositions of one another. A third transposition occurs soon after the phrase begins, using a duration that is longest of any three-note chord in this phrase before the end. These three chords are Ab4-F5-Bb5 | F4-D5-G5 | Bb3-G4-C5. The harmonic intervals of major sixth and perfect fourth between registrally adjacent notes within each chord correspond, but in inversion, to the minor-third and perfect-fifth transpositions between adjacent chords. Because of the inversion, I call this "anti-composing-out."