Sunday, February 1, 2015

In the Metric Wastelands of Led Zeppelin's "Kashmir"

Led Zeppelin’s “Kashmir” was released 40 years ago this month. The meter of the opening of this song is famously not straightforward, although it appears that 3/4 (quarter = 82), with the first sound placed on the downbeat, is a possible choice, as can be seen with transcriptions here and here and here. And yet this choice is unusual, because there is nothing in the music—like a boom-chuck-chuck—that exactly projects this meter.

I have a theory about why the music can nonetheless be heard this way. Below are twelve possible metrical interpretations of the opening of the song.  (You will want to click on the image to see detail.) The ordering from left to right corresponds to the relation between the first sound of the song and the first notated downbeat: in the middle, they match; to the left, the first sound is an eighth note earlier than the first downbeat; to the right, the first sound is an eighth note later than the first downbeat. The ordering from top to bottom corresponds to the length of duration that can be grouped into threes: eighth note for the highest, quarter note for the second highest, half note for the second lowest, and whole note for the lowest.


Music that is not faded represents a line in the texture that works with its notated meter.  The fact that there is no single interpretation that is completely not faded signifies that the meter is not straightforward. However, note that, although the brown-bordered metric interpretation is entirely faded, this interpretation is the only one adjacent (like a chess king is adjacent) to five of the six interpretations (thinly bordered) that have a line that is not faded. This brown-bordered metric interpretation is their center and their best approximation (in two dimensions!), even though the music has no content that would directly support this metric interpretation. This is like the fact that there is usually no town located at the mean population center of a country.

This metaphor seems appropriate given the song’s extra-musical content and context. The Kashmir region in South Asia is not a separate state, but rather defined politically only through its division between India and Pakistan. However, according to Wikipedia, Led Zeppelin’s singer Robert Plant was initially inspired not by this part of the world but by a place similarly liminal: the barren “waste lands” of Southern Morocco between the cities of Guelmim and Tan-Tan.

Saturday, January 24, 2015

An Anti-Aging Harmonic Formula from Brahms

This month I was kindly invited to produce a guest post for the important blog of Dr. Timothy Chenette, a fellow music thinker. But I'll offer a follow-up here.

In my post for Dr. Chenette, I suggested how pop music and classical music approach the diatonic-diminished-triad-made-major-triad in ways that both mirror one another, and still sound considerably different from one another. For example, the I - bVII - IV - I (or, in another interpretation, I - IV/IV - IV - I) is a pop-rock trademark, but this kind of progression is quite rare in classical music, as is the closely related blues-based V - IV - I, whose kinship to I - IV/IV - IV - I can be shown by renotating (and hearing) it as V/IV - IV/IV - I/IV - I.

But this is not completely unheard in classical music. Brahms's choral work Nänie appears to make frequent and purposive use of the rare V - IV progression, more so than any other work of Brahms to my knowledge. Moreover, the use of this "retrogression" befits the text, which juxtaposes perfection, beauty, and their apparent immortality with their eventual and inevitable demise. When IV goes "as it should" to V, the arrow of time and aging points "as it should," driving toward tonic's end with normative predominant-dominant syntax. But when V is followed by IV, one could hear a reversal of this process. With this in mind, consider this excerpt and my analysis. The back-to-back V - IV and IV - V may not perfectly coincide with the back-to-back "die Schöne [the beautiful]" and "vergeht [perishes]," but it's close enough for me.

Wednesday, December 3, 2014

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!

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.