Saturday, April 30, 2016

Approximating e Musical-e

The mathematician Leonhard Euler was born in this month 309 years ago. As he contributed in important ways to our understanding of some aspects of music, I thought I would use music to help understand some aspects of mathematics. The constant e, named after Euler, is one of the most important numbers in mathematics. One way of approximating e can be demonstrated using musical intervals:


At the least, this approach offers one answer to the question of when an approximation is good enough: it's good enough when you can't hear the difference between it and a better approximation.

Tuesday, March 8, 2016

Sonata-Principled Praises

Lent concludes during this month, and that means that many Easter performances of Beethoven's "Hallelujah," the popular chorus that concludes his unpopular oratorio Christ on the Mount of Olives, are just around the corner.

This is a rather unusual movement to bring up as an example of the sonata principle, because, as befitting a coda-chorus, it hardly strays from its main key of C major. In fact, it never provides an authentic cadence in any other key. The closest it comes to such a cadence, complete with a chromatic predominant and a 6/4 embellishment of the cadential dominant, is this:



If the sopranos had descended from E (A:^5) to A (A:^1) instead of ascended to G (enclosed in red above), then this would be a clear full close in A major. The G is both a surprise -- foiling a well-prepared escape from C major -- and not a surprise -- the A7 chord sends the music back, via a circle of fifths, to the movement's main key and tonic triad.

Toward the end of the movement, this happens:


If the basses had descended from G (C:^5) to C (C:^1) instead of ascended to B flat (enclosed in red above), then this would be a clear full close in C major. The B flat is both a (welcome) surprise -- foiling yet another authentic cadence in C major -- and not a surprise -- the lean toward the subdominant key is a common perorative technique in classical tonal music, and the basses's ascending-minor-third digression from the movement's main key recalls the soprano's earlier similar digression from a different key.

Monday, February 15, 2016

Sonata-Principled Kangaroos



This month and next, I’ll continue last month’s focus on sonata, particularly the “sonata principle,” Edward T. Cone’s name for common-practice-music’s propensity to present distinctive melodic material first outside of the movement’s main key and then later in the main key.

When you tonally attend to this leap of a minor seventh,
I suspect that you hear it in G, even though there’s no G there. The presence of the C natural keeps it from being in D (D as tonic would rather have C#), and the greater weight afforded to the D by its lower register, longer duration, and downbeat position keeps the C from materializing as tonic. Instead, the key of G matches these different weights well: the more weighted pitch of D is dominant, tonic’s second-in-command, and the less weighted pitch of C is a more subordinate subdominant.

So, with this in place, the “Kangaroos” movement of Carnival of the Animals, which Camille Saint-Saëns began composing 140 years ago this month, contains a rather subtle instance of the sonata principle. The movement clearly begins in C minor, but then the bass’s leap of a D-C seventh in mm. 4-5, especially when accompanied with notes taken from the G harmonic minor scale, leans toward G.



At the end, the minor seventh appears again in the bass, but transposed to G-F, accompanied by notes taken from the C melodic minor scale. Thus this minor-seventh leap is restored to the Kangaroo’s main key of C minor.



What appears even more clever is that the last three measures embed this minor-seventh recapitulation within a clear pattern in the bass: G--F#-G--F-G--E. This could be heard as a musical cross-dissolve: the repeated low G-major chords draw out the dominant of C, while the functional progression of higher chords—V/V, Fr. 6, V—prepare the key of A, which is the tonal center of “Aquarium,” the next movement where the same Fr. 6 returns as a less-functional coloristic harmony.


Thursday, January 14, 2016

Schenker, Sonata Form, Mozart, and Meter

The Austrian music theorist Heinrich Schenker died 71 years ago today. The theory that Schenker founded distills tonal structures to their linear-harmonic essence, where prominent starting notes and normative cadential notes are stitched together into a well-formed contrapuntal design. Often, similar forms have similar linear-harmonic essences. Take sonata form. When a major-mode late-eighteenth-century sonata-form movement has a prominent scale-degree 3 in its main theme, its linear-harmonic essence is often interpreted as something like this:


P = primary theme, TR = transition, S = secondary theme, C = closing section

This graph reflects how the primary themes in both exposition and recapitulation are basically, if not exactly, the same. It also reflects how the secondary theme in the exposition, whose treble line is distilled to D-C-B-A-G, is transposed back into the main key, with a distilled treble line of G-F-E-D-C. Schenkerian theory encourages the analyst to connect, as if part of a smooth melody, the prominent pitch that begins the primary theme and the prominent pitch that begins the secondary theme. This works quite well in the exposition: E is just a step higher than D. However, the stitching of the corresponding design in the recapitulation is a bit loose, particularly at the moment marked with an asterisk (*). On the one hand, the prominent pitch that begins the secondary theme in the recapitulation should be the same one that begins the secondary theme in the recapitulation, but down a perfect fifth or up a perfect fourth: in this case, D becomes G. After all, the secondary themes are typically copies of one another, differing only by their transpositional level. On the other hand, the prominent pitch that begins the primary theme in the recapitulation (E) and the prominent pitch that begins the secondary theme in the recapitulation (G) are not smoothly connected as they were in the exposition. Rather, if the prominent pitches of the recapitulation’s primary and secondary themes were to be closer together, the graph might look something like this:



But such a graph would signify a considerable change for the beginning of the secondary theme in the recapitulation from how it sounded in the exposition.

In the first movement of Mozart’s Piano Sonata in C Major, K. 279, there is such a change, although I will let the reader decide how considerable it is. This common-time movement is remarkable in how often it shifts the perceived downbeat from the notated downbeat to the spot halfway between notated downbeats. By my count, this shift happens 12 times: 6 times away from the notated downbeat, and 6 times back to the notated downbeat. Mozart does not notate these shifts, but they can be readily heard. I have notated them below for the entire movement.


I: = main key, V: dominant key, PAC = perfect authentic cadence, HC = cadence, --> = "becomes"

These shifts also affect how the beginning of the secondary theme is heard in both exposition and recapitulation. As shown below, in both exposition and recapitulation, the entrance of the secondary theme is preceded by a perception of downbeats halfway between notated downbeats. However, Mozart starts the secondary theme halfway through the measure in the exposition, while he starts it on the downbeat in the recapitulation. This changes which note in the secondary theme first receives metrical emphasis. In the exposition, that note is D. In the recapitulation, that note is not G, but E, as shown below.


Thursday, December 3, 2015

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!

Monday, November 23, 2015

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.

Thursday, October 1, 2015

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.