Saturday, November 1, 2014

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.

Wednesday, October 1, 2014

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 sets 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.

Saturday, September 20, 2014

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."

Friday, August 1, 2014

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.

Tuesday, July 1, 2014

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.


Tuesday, June 3, 2014

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.

Saturday, May 10, 2014

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.