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."

A Seasonal Musical Offering

In his contribution to C.P.E. Bach Studies (Cambridge UP, 2006), David Yearsley reprints a sixteen-measure composition by the composer that tackles a particularly challenging musical problem: the closed augmentation canon. In an two-voice augmentation canon (also called mensuration or prolation canon), one voice presents a melody and the other voice presents the same melody in counterpoint with the first, perhaps transposed higher or lower, perhaps starting earlier or later, but with the durations increased or decreased by a certain factor, usually a factor of two. If both voices begin together, and the augmentation factor is 2, then the faster voice will finish its melody in half the time that the longer voice needs to finish—typically, to continue the two-part counterpoint, the faster voice then uses unrelated material. However, in a closed augmentation canon, the faster voice repeats its first half verbatim; to construct not only well-formed counterpoint but an attractive composition under such fiendish restrictions requires great skill.

But to do so with preexisting materials that were ostensibly not designed for such canonic treatment is pure serendipity, as in the following, which achieves near-flawless* counterpoint.

Happy Holidays and Merry Christmas to you!

* The three instances of a harmonic fourth in mm. 2, 5, and 6 can be well justified as involving tones passing through measure-long prolonged harmonies. Perhaps the weakest point is an instance of "middleground" parallel octaves from beat 2 to beat 3 in m. 7. The similar motion to the octave at the beginning of m. 5 is not ideal, either.

A Chromatic Striving in Mahler's Second Symphony

In a solemn brass chorale from the last movement of Mahler's Second Symphony "Resurrection," starting at "Etwas energischer in tempo," the first six chords are Ab+, Eb-, C+, Bb-, Ab-, and F+ (+ for major, - for minor). During these chords, the bass line moves continuously downward, putting all the chords in root position, and the soprano line moves continuously upward: Ab, Bb, C, Db, Eb, F. The bass and soprano lines are entirely diatonic, either within the notated key signature of D-flat major or the secondary key of A-flat major. Apart from the fourth chord, which has a minor third between soprano and bass, the soprano and bass notes in the other chords span perfect intervals, which means the quality of the triad could be either major or minor. Therefore, in fixing only the outer voices, there are 2^5 or 32 possible progressions of consonant triads.

There are multiple ways to measure the distance between two consonant triads. One way—call it Tonnetz distance—is to count the minimum number of L, P, and R transformations needed to change one triad into the other. Among the possible 32 progressions, the only completely diatonic progression (Ab+, Eb+, C-, Bb-, Ab+, and F-) minimizes Tonnetz distance for every triadic adjacency, and is the only one whose five triadic-adjacency distances sum to the smallest possible value of 11. Among the possible 32 progressions, Mahler's progression is the only one that maximizes Tonnetz distance for every triadic adjacency, and the only one whose five triadic-adjacency distances sum to the largest possible value of 18.

Later in the movement, one hears these words: "Mit Flügeln, die ich mir errungen, In heißem Liebesstreben, Werd'ich entschweben Zum Licht, zu dem kein Aug' gedrungen!" ("With wings which I have won for myself, In love’s fierce striving, I shall soar upwards To the light which no eye has penetrated!")

A Pitch-Time Analogy in Carl Vine's First Piano Sonata

Australian composer Carl Vine's Piano Sonata from 1990 begins with the pianist silently depressing A0, C1, E1, G1, and B1 and keeping them undamped with the sostenuto pedal for the first 12 measures. In a very short preface, the composer tells the performer that the tempo markings are "not suggestions but indications of absolute speed." The first 103 measures of the work, before a "poco allargando" in m. 104 and "Poco meno" tempo change involving a factor of 11 in m. 105, establish various isochronous pulses with wavelengths longer than 250 ms. These pulses group into exactly five "tempo classes" by equivalence of 2:1 ratios. If the first "tempo class" in the work is represented as 1, then the other four tempo classes can be represented as 1.2, 1.5, 1.8, and 2.25.  If the frequency of the pitch class A is represented as 1, then the other four pitch-class frequencies in the opening "silent" chord can be represented in just intonation as 1.2, 1.5, 1.8, and 2.25. (Such an analogy could also be demonstrated using the method David Lewin used in his 1987 book Generalized Musical Intervals and Transformations to analyze a passage from Carter's First String Quintet.)