Friday, June 9, 2017

Broken Symmetries 3.1

A Tale of Two Tunes

1. Rhythmic structure
in the themes of Messiaen's Île de feu I & II

The following rhythmic analysis comparing the two themes in Île de feu I (herein Idf1) and Île de feu II (Idf2) is a rather unexceptional description based on the notion of fuzzy rhythmic contours. But while it doesn't present any surprises taken by itself, this rhythmic analysis lays the groundwork for the deeper connections that will be presented in the following two entries in this thread. The two themes are shown in Examples 1 and 2.

Example 1
Theme, Idf1 (mm.1-2)

Example 2
Theme, Idf2 (mm.1-7)

The basic rhythmic cell in both themes is a pair of notes which we'll designate [>~], where '>' indicates a short stressed note and '~' indicates a long unstressed note.[1] And usually the accented note is higher in pitch than the unaccented one. These pairs can be heard in the comparatively brief theme of Idf1, but they are especially obvious in Idf2's lengthier theme. They take three forms (Example 3). Both themes are built rhythmically from these pairs, the added-note features in both themes notwithstanding.[2]
Example 3.

Added-note rhythmic segments as quasi-palindromes – especially obvious in the Idf2 theme – are probably more in keeping within Messiaen's compositional conception. This would make the basic building block [~>~], which is also the characteristic incipit in both themes. But trying to fit every foot into the shoe of Messiaen's theory of non-retrogradable rhythms here will leave out other significant structural features. E.g., the Idf2 theme can be parsed further as [~>~~], a basic four-note rhythmic contour with the hammer stroke falling on the second note as marked in Example 4. There are two forms: those circled in blue all have similar basic pitch contours, [^vv], while the pitch contours circled in red are inversionally related [^v^] / [v^v]. This also approximates Messiaen's barring.

Example 4.
(Idf2 theme)

A comparable rhythmic parsing of the Idf1 theme (Example 5) produces less straightforward and ultimately less convincing results.

Example 5.
(Idf1 theme)

The pairs circled in orange are [>~], as in Idf2's theme, and that almost works. However, going beyond those rhythmic pairs in Idf1 strains the basic analytical idea we're testing. Circled in purple at either end are [~>~] structures. Beyond that, things get implausible. We might try to force the theme to fit into this analytical mold by identifying [~>~]'s inverse, [>~>], enclosed by the purple dashed circle around E-F-C#. But then (assuming our ear hasn't stopped us by now) we have to account for the two consecutive sixteenths, F-F#. This rhythmic anomaly could be 'fixed' analytically by eliding the sixteenths into a single 'background' eighth note, reasserting the [~>~] analysis (gray dashed segment with the question mark). This would clearly be overanalyzing only for the sake of asserting a theory, and we can hear Messiaen laughing.

But there is no need to find in Idf1's theme all the rhythmic symmetries found in Idf2's theme in order to establish a strong rhythmic similarity.  It is enough to assert conservatively that the two themes are built from [>~] pairs, and beyond that each takes on its own personality.


[1] I was tempted to use terminology from scansion in poetry, but I quickly abandoned this idea. What I was after would be, in poetry, a confusion between the iamb and the trochee due to the accent falling on the short duration and lack of accent on the longer duration. As far as I know, adequate terminology is for the most part missing from analogous musical rhythmic analysis, at least since the ars antiqua. So I opted for using a binary n-tuple notation consisting of the symbol '>' for a variably short, accented hammer note and '~' for an unaccented (or relatively less accented) note of duration longer than or equal to its associated '>'. Beyond this fuzzy 'notation rule', specific measures of accent and duration (e.g., 'accented staccatissimo sixteenth note') are left undefined. This was a pro tem solution, but one might consider it a provisional basis for an analytical theory for rhythmic contour analogous to that for pitch contour.

[2] One can also hear these characteristic pairs emerging in the rhythms generated by the pairing of interversions in Idf2's B material, but it is difficult to say whether the theme statement, after several hearings, has trained the ear to select for these pairs during the interversion sections, or whether they were pre-compositionally built into, or discovered in, the interversions by Messiaen.