Sunday, September 4, 2016

Folding Phish's Tweezer

At Dick’s Sporting Goods Park in Commerce City, CO, the rock group Phish is playing their last show of their 2016 summer tour as I post this. As phish.net says, "If there is a single Phish song that can be said to evolve with and exemplify Phish’s sound and artistry over the decades, it’s 'Tweezer.'" Below is a simplification of the famous opening lick, in "Tweezer Reprise"'s key of D. Here is a recording.

I've spent some time on this blog analogizing pitch and time. Here's one way that they are related:

The gamut of pitch register lays out linearly, from low to high, but it also circles back on itself via the concept of the octave: "do-re-mi-fa-so-la-ti...do." Even though the second "do" is higher than the first, we still call them both "do." In fact, of all of these notes within the octave, this "do" is special: it is the tonic, the note toward which all of the others are oriented.

The expanse of chronological time lays out linearly, from earlier to later, but it also circles back on itself via the concept of the measure: "1-2-3-4...1." Even though the second "1" is later than the first, we still call them both "1." In fact, of all of these beats within the measure, this "1" is special: it is the downbeat, the beat toward which all of the others are oriented.

Now these five notes from "Tweezer"' have a rare property, and they have this property not only because of how far apart they are from one another in pitch and in time, but also because of what the tonic and the downbeat are.

This video describes this property.


This was the "one from rock music" I was referring to a year ago here. The first four notes of Gershwin's "I Got Rhythm" have the same property, but I hope I explained it better this time around. I can think of another song released in 2015 with a similar lick and the same tonal and metric orientation.

Sunday, August 14, 2016

In Die Walküre, Space Becomes Time

Wagner's complete tetralogy Der Ring des Nibelungen was first presented 140 years ago yesterday. The one part of this music that has probably embedded itself into Western cultural consciousness the most is the beginning of the third act of the second opera Die Walküre, the so-called "Ride of the Valkyries." The image below provides a notation of the first presentation of the melody; I recommend clicking on it to get a closer look. While the severe distortion of the notation makes it harder to read, it makes the distance that a measure of time and a semitone of pitch takes up on the image more equal to one another. All of the melody's local maxima (B3-D4-F4#-A4-C#5) divide this 14-semitone span into a 3+4+3+4 organization. The dominant-(5)-to-tonic(1) moments of the melody divide this 14-measure span into a 3+4+3+4 organization.


P.S. My post title cheekily refers to the fact that, in Wagner's last opera Parsifal, Gurnemanz tells Parsifal that that in this realm time becomes space ("Zum Raum wird hier die Zeit").

Sunday, July 31, 2016

Film Music Style as Guide for Late Beethoven

There is something entertaining, if not a little anachronistically naughty, about hearing a kind of more recent music in the music of someone like Beethoven (like hearing this in Beethoven's 7th symphony, or hearing this in Beethoven's 8th piano sonata). But here's an instance where it can be quite helpful. (There's a relevant cognition experiment of mine here that I invite you to try, but, if you decide to take part, do so before reading on.)

A year ago on this blog, I shared this trend about recent popular film music: "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." This happens particularly when the two chords have been isolated from other tonal obligations and, without such obligations to gain tonal meaning, instead look toward one another.

In measure 13 of the first movement of Beethoven's op. 109 piano sonata, the notes of a D-sharp-major triad fill up most of the register of the piano and our attention with its sonorous and solid proportions, blocking from view much of what had occurred before. Then its firm surface suddenly shimmers and transforms into a B-major triad. To some film-music ears, this is a departure from stability and security: since B is four semitones lower than D-sharp, B is demoted in this two-character drama within a drama. I indicate this visually with a red arrow that fades into purple.
In sonata form -- a drama which this drama is within -- the recapitulation is both a return to and a reworking of the material from the exposition. In the first movement of Beethoven's op. 109, measure 62 in the recapitulation corresponds to measure 13 in the exposition. This time, it is a C-major triad that floods our senses much the same way that the D-sharp-minor triad did before. But its corresponding transformation is to an E-major triad, whose root is four semitones higher. The same harmonic conversion is now a homecoming, which I show with an arrow that changes from purple to red.
To be sure, this harmonic scheme is rather standard: the sonata is in E, and B is accustomed to its subservient role when E is the tonal sovereign. But local harmonies, together with a future music's chromatic tendencies, enrich this relationship and even offer a guide, however anachronistic, to the work's overall tonal plot.

Monday, June 27, 2016

Muse's Voice Leading at the Olympics

The 2016 Olympics are around the corner. The official song of the 2012 Olympics— “Survival,” by the British alternative rock band Muse—premiered on the radio on this day four years ago.

The lyrics of the opening are “Race, life’s a – race, That I’m gonna – win, yes I’m gonna – win, And I’ll light the – fuse, and I’ll never – lose…” The music accompanying these lyrics uses the triadic progression of BbM – Bb+ – Ebm – CbM – GbM, where M is major, m is minor, and + is augmented. The progression can also be considered in terms of smooth voice leading. First, one by one, each of the three voices in the BbM triad—on Bb, D, or F—moves up by a semitone, achieving the CbM triad. After this, two of the three voices in the CbM triad slip back down to make the GbM triad. Only the voice that started on the F and moved up to Gb never retreats, at least not until the progression starts over. Therefore, one can say that only this voice “wins,” as animated below, with the gold-colored figure as the winning voice.



What makes this more fitting is that, at least at the beginning of the song, Muse’s lead singer, Matt Bellamy, is intoning his first-person account of victory using precisely the notes of the “winning” voice. Now this, literally, is…VOICE…LEADING.

Saturday, May 28, 2016

Researching Some Music in Ligeti's Musica ricercata

The composer György Ligeti was born 93 years ago today. Here is a relationship between pitches and meters in some music from Ligeti's Musica ricerata that requires a little investment upfront.

1. A measure like 2/4 can be represented by the ordered series of positive integers 2-1, where 2 is the strong beat and 1 is the weak beat. Likewise, a measure of 3/4 can be represented as 2-1-1. A measure of 4/4, instead of being represented by 2-1-1-1, is often thought to have a metrical accent halfway through that is not as strong as the downbeat, but stronger than the weak beats: this could be represented as 3-1-2-1.

An uneven measure like 7/4 often breaks down as 4/4 + 3/4, but instead of simply concatenating 4/4's 3-1-2-1 with 3/4's 2-1-1 to make 3-1-2-1-2-1-1, a representation of 4-1-2-1-3-1-1 reflects the measure's initial division into 4/4 + 3/4 by making the beginning of the 3/4 measure into a "second-rank" downbeat.

When the quarter-note beat of one of these meters is subdivided into eighth notes, this inclusion can be represented by adding one to each of the numbers and then placing a 1 after each of them. For example, 3/4 (2-1-1) with eighth notes would be 3-1-2-1-2-1. 7/4 (4-1-2-1-3-1-1) would be 5-1-2-1-3-1-2-1-4-1-2-1-2-1.

2. Now, given a certain series of numbers, its cumulation is a series of numbers that sum the original series starting from the left and up to that point. For example, the cumulation of 2-1-1 is 2-(2+1)-(2+1+1) or 2-3-4. The cumulation of 3-1-2-1 is 3-4-6-7.

3. Lastly, a cumulation mod 2 of a series is a series's cumulation where every even number is replaced by 0 and every odd number is replaced by 1. For example, whereas the cumulation of 2-1-1 is 2-3-4, the cumulation mod 2 of this series is 0-1-0. The cumulation mod 2 of 3-1-2-1 is 1-0-0-1.

Below is an 7/4 example from the Overture to Leonard Bernstein's Candide -- placed into time signatures commensurate with the discussion above -- that shows 1) the 7/4 meter as a series of positive integers, 2) the culmination of the series, and 3) the culmination mod 2 of the series.



Below is a video with a score and recording of the second movement of Ligeti's Musica ricercata; attend to the first four measures.

I've re-notated these four measures to make a steady stream of eighth notes, and analyzed the music using the methods outlined above.



Notice how the alternation between 0s and 1s in the culmination mod 2 of the first measure, the second measure, and the third and fourth measures as a single measure of 7/4, corresponds exactly to these measure's alternations between the two pitches that open the movement. Each fermata both resets the metrical hierarchy and toggles the pitch-meter mapping.

Thanks to Nick Shaheed for encouraging me to think about this music.

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.