"... I even tried a bit of twelve-tone composition and struck up a correspondence with the pianist Lennie Tristano, the most 'advanced' jazz musician of that time [the early 1950s], in which I helpfully pointed out that the twelve-tone method obviously also held the key to the future of jazz. (Tristano gently replied that I seemed not to have too much acquaintance with jazz improvisation, and was kind enough to invite me to come and talk with him. We shared a turkey sandwich one Thanksgiving evening.)"William H. Youngren, "Schoenberg, Rosen, and the Common Listener," 1978.
Monday, October 28, 2013
Lennie Tristano on Twelve-Tone Improvisation
Thursday, September 5, 2013
Coltrane Cycles and the Blues
Jazz is a syncretistic music, and I therefore believe that it requires a heterogeneous approach to analysis.
In a previous post, I discussed Coltrane's chromatic major third cycles in terms of harmonic implications derived from the principles of tonality. Scholars like William Rothstein, Matthew Bribitzer-Stull, and David Kopp have traced the history of the chromatic major third full cycle (I-bVI-III#-I) back to examples in Schubert and Rossini, and they demonstrate how such chromatic mediant relationships abounded throughout nineteenth-century music. It is apparent that Coltrane's application of the major third cycle belongs in this lineage, perhaps as mediated by Slonimsky.
It would be an obvious mistake, however, to neglect to have an ear to the blues when analyzing jazz music, especially the music of an artist like John Coltrane.
In his book Origins of the Popular Style, Peter van der Merwe persuasively describes the melodic practice of the blues as consisting partly of a "process of piling up thirds" (124):
Listening to any of his improvisations on "Impressions" reveals that what is typically misconstrued as "Dorian modality" is instead Coltrane's version of just such a blues-derived, "ladder-of-thirds" melodicism. The so-called "mode" is most often a six-note scale abstracted from the stacks of thirds on top of and below the tonic: i.e., D-F-A-C and D-B-G.
There is, then, a sense in which it might be appropriate to think of Coltrane's emphasis on chromatic (major and minor) third relations and cycles as a translation into harmonic terms of the "ladder of thirds" melodic concept that Van der Merwe finds central to blues practice. When Coltrane prolongs a harmony with digressions through third-related keys, I suggest that he is "composing-out" the pendular or stacked thirds of a higher-level blues melody.
Another element of blues melody comes to the fore in conjunction with Coltrane's mediant-based harmony: the variable third. According to Van der Merwe,
When Coltrane moves from a minor tonic to the major mediant, for example, or from a major tonic to the major submediant, the major third and minor third of the tonic key will be intermixed. As just one example from Coltrane's playing, this excerpt from the eleventh and twelfth bars of "Countdown" illustrates:
Here Coltrane is in the act of traversing from D major to B-flat major via F7, but in these five beats one hears only the collection (D E F F# G A). These are simply scale steps 1-2-3-4-5 in D, where scale step 3 includes both the major and minor variants.
When taken all together, the pitch classes of the three triads contained in a chromatic major third cycle (I-bVI-III#-I) yield the hexatonic collection, set class (014589). This collection consists of two augmented triads a semitone apart. Coltrane used two hexatonic scales as a basis for his composition "One Down, One Up," for instance.
Notably, the hexatonic collection contains major and minor triads built on shared roots. Thus a possible non-discrete tetrachordal subset of the hexatonic collection is set class (0347), shown in the example above. Bearing the intervals of a minor third, major third, and fifth, an example of this set class is (D F F# A). This is a "triad" with the variable third that Van der Merwe describes as a hallmark of blues melody.
It appears therefore that the hexatonic collection and the blues have a certain affinity, which John Coltrane succeeded in making audible.
In a previous post, I discussed Coltrane's chromatic major third cycles in terms of harmonic implications derived from the principles of tonality. Scholars like William Rothstein, Matthew Bribitzer-Stull, and David Kopp have traced the history of the chromatic major third full cycle (I-bVI-III#-I) back to examples in Schubert and Rossini, and they demonstrate how such chromatic mediant relationships abounded throughout nineteenth-century music. It is apparent that Coltrane's application of the major third cycle belongs in this lineage, perhaps as mediated by Slonimsky.
It would be an obvious mistake, however, to neglect to have an ear to the blues when analyzing jazz music, especially the music of an artist like John Coltrane.
In his book Origins of the Popular Style, Peter van der Merwe persuasively describes the melodic practice of the blues as consisting partly of a "process of piling up thirds" (124):
"In short, the blues mode takes the form of a ladder of thirds, but it is a flexible ladder that can be extended up or down at will." (125)
Listening to any of his improvisations on "Impressions" reveals that what is typically misconstrued as "Dorian modality" is instead Coltrane's version of just such a blues-derived, "ladder-of-thirds" melodicism. The so-called "mode" is most often a six-note scale abstracted from the stacks of thirds on top of and below the tonic: i.e., D-F-A-C and D-B-G.
There is, then, a sense in which it might be appropriate to think of Coltrane's emphasis on chromatic (major and minor) third relations and cycles as a translation into harmonic terms of the "ladder of thirds" melodic concept that Van der Merwe finds central to blues practice. When Coltrane prolongs a harmony with digressions through third-related keys, I suggest that he is "composing-out" the pendular or stacked thirds of a higher-level blues melody.
Another element of blues melody comes to the fore in conjunction with Coltrane's mediant-based harmony: the variable third. According to Van der Merwe,
"...[in] genuine blues tunes there will be no question of a clear-cut contrast such as we find between the classical major and minor... major and minor or neutral thirds may be juxtaposed throughout." (120).
When Coltrane moves from a minor tonic to the major mediant, for example, or from a major tonic to the major submediant, the major third and minor third of the tonic key will be intermixed. As just one example from Coltrane's playing, this excerpt from the eleventh and twelfth bars of "Countdown" illustrates:
Here Coltrane is in the act of traversing from D major to B-flat major via F7, but in these five beats one hears only the collection (D E F F# G A). These are simply scale steps 1-2-3-4-5 in D, where scale step 3 includes both the major and minor variants.
Notably, the hexatonic collection contains major and minor triads built on shared roots. Thus a possible non-discrete tetrachordal subset of the hexatonic collection is set class (0347), shown in the example above. Bearing the intervals of a minor third, major third, and fifth, an example of this set class is (D F F# A). This is a "triad" with the variable third that Van der Merwe describes as a hallmark of blues melody.
It appears therefore that the hexatonic collection and the blues have a certain affinity, which John Coltrane succeeded in making audible.
Thursday, August 29, 2013
Harmonic Tension in Coltrane's Chromatic Major Third Cycles
It is interesting to note that Coltrane’s chromatic major
third cycle was applied compositionally and improvisationally (too sharp a
distinction should not be drawn) in both the ascending and descending directions.
Ascent and descent have accordingly different musical effects. “Giant Steps”
and an excerpt from “Venus” may serve as examples.
The first seven bars of “Giant Steps” feature two incomplete descending chromatic major third cycles which are paired with descending melodies. The remainder of the composition reverses that direction for an ascending major third cycle with rising melody.
This excerpt from “Venus” (1:33-1:57 in the video below) illustrates an instance of the
ascending major third cycle complemented by an ascending melody, followed by a major third harmonic descent with descending melody.
In both cases, the ascending third cycle is accompanied by
the feeling of an increase in tension, while the descending third cycle brings
about the feeling of a dissipation of tension. This is partly due to the rising
and falling melodies, but there is also a harmonic component to the effect of
accumulating or lessening musical tension.
In Coltrane’s music, the chromatic third cycle has harmonic
implications inherited from the principles of tonality. The ascending major third
cycle (I-III#-bVI-I) moves each harmony four ticks in the sharp-key direction
along the circle of fifths (C-G-D-A-E...). In tonal music, harmonic
movement in the sharp-key direction is associated with an increase in tension. Conversely, the descending major third cycle (I-bVI-III#-I) moves each
harmony four ticks in the flat-key direction along the circle of fifths (C-F-B-flat-E-flat-A-flat…). Shifts to the flat-key area are generally associated with
the relaxation of musical tension. In addition, the III triad contains the
leading tone and (raised) fifth, and is therefore related to the dominant chord;
whereas the bVI triad contains the tonic and (flatted) sixth, thus relating it
to the subdominant chord.
The sense of building and releasing tension in these
examples, then, is overdetermined: both melodic contour and the harmonic implications
of the third cycle play important roles in creating the effect.
Wednesday, July 10, 2013
Prolongation in "Take the Coltrane"
A friend who is a brilliant trombonist as well as a medical doctor draws my attention to two outstanding excerpts from John Coltrane's solo on "Take the Coltrane." These turnaround phrases start in measures 9 and 8 of the blues form and can be heard in the video below at 1:49 and 2:10, respectively.
We hear Coltrane going "outside the changes" in a fashion not at all untypical of his playing in this period. It is clear, however, that there is no merely mechanical superimposition of "Giant Steps" or "Countdown" harmonies, as is so often and so glibly asserted. Rather, what we hear in these two examples is a fluid and spontaneous chromatic prolongation of a particular governing harmony.
The term "prolongation" is borrowed from Schenkerian theory and is meant to suggest that the chromatic harmonies heard in these excerpts represent not simple substitutions but an elaboration of a single higher-order harmony that obtains over the course of four and five measures. To illustrate this I have verticalized Coltrane's melody, adjusted note register to idealize the voice-leading, and used Schenkerian symbols to highlight higher-level relationships. (In the graphs below, the bass voice is derived from the implied root movement of the melody. Accidentals apply only to the pitch immediately following.)
In Example A, the foreground cycle of chords (C-B-E7-A-C7) prolongs a C chord, the dominant to the tonic harmony of this blues in F. Prolongation of the dominant is appropriate since this phrase begins in the ninth measure of the form. The sense of orderly departure from and return to the governing harmony gives the phrase a kind of centripetal logic. Coltrane uses a chromatic upper-neighbor motion to prolong the conceptual top voice C while the harmony moves through the chromatic lower mediant, itself preceded by secondary dominants. The conceptual bass voice returns to C via the subdominant minor, creating a III-IV-V ascent that effectively prepares the subsequent arrival of the tonic.
The phrase illustrated in Example B begins in the eighth measure and prolongs an F-sharp seventh chord, which is the "tritone sub" of the dominant or the Phrygian II, spelled enharmonically.
In this phrase, Coltrane pivots from the F-sharp seventh to a local resolution on B, followed by an incomplete cycle of chords whose roots descend by major thirds (B-G-E-flat). The cycle is related to "Giant Steps," although in this case the major chords are not mediated by intervening dominant chords. Through this prolongation the upper voice C-sharp descends by step to B-flat/A-sharp. Here the F-sharp harmony returns and is additionally expanded, this time by a lower neighbor tone in the conceptual upper voice and a passing tone in the conceptual inner voice. The melodic tones of the last measure can be heard as 6-5-2-3 in F-sharp or as 4-3-7-1 in B-flat minor, the subdominant to the tonic F which returns at the top of the form in the following measure. The latter hearing suggests a particularly grave sort of plagal cadence, and the leading-tone transformation of F-sharp major to B-flat minor reflects the type of major-third relation that Coltrane explored to great effect in this period.
This rising third also inverts the falling third from earlier in the phrase. It is worth noting that in both examples, the goal chord reached before the return of the chord of the governing harmony is the chromatic lower mediant (A in Example A, where the prolonged harmony is C, and E-flat in the F-sharp prolongation of Example B.).
Also notable in Example B is the stepwise descending third motive, heard twice at the foreground level in the inner voice (F-sharp-E-D-sharp, then E-D-sharp-C-sharp) and once at the middleground level in the upper voice (C-sharp-B-A-sharp). The descending third motive is also suggested by the bass movement from F-sharp to E-flat on the middle ground level and more loosely (in augmentation) by the foreground cycle, B-G-E-flat.
In my view, these excerpts offer evidence of hierarchical structure in Coltrane's improvisations, inviting analyses that make use of Schenkerian concepts like prolongation.
We hear Coltrane going "outside the changes" in a fashion not at all untypical of his playing in this period. It is clear, however, that there is no merely mechanical superimposition of "Giant Steps" or "Countdown" harmonies, as is so often and so glibly asserted. Rather, what we hear in these two examples is a fluid and spontaneous chromatic prolongation of a particular governing harmony.
The term "prolongation" is borrowed from Schenkerian theory and is meant to suggest that the chromatic harmonies heard in these excerpts represent not simple substitutions but an elaboration of a single higher-order harmony that obtains over the course of four and five measures. To illustrate this I have verticalized Coltrane's melody, adjusted note register to idealize the voice-leading, and used Schenkerian symbols to highlight higher-level relationships. (In the graphs below, the bass voice is derived from the implied root movement of the melody. Accidentals apply only to the pitch immediately following.)
In Example A, the foreground cycle of chords (C-B-E7-A-C7) prolongs a C chord, the dominant to the tonic harmony of this blues in F. Prolongation of the dominant is appropriate since this phrase begins in the ninth measure of the form. The sense of orderly departure from and return to the governing harmony gives the phrase a kind of centripetal logic. Coltrane uses a chromatic upper-neighbor motion to prolong the conceptual top voice C while the harmony moves through the chromatic lower mediant, itself preceded by secondary dominants. The conceptual bass voice returns to C via the subdominant minor, creating a III-IV-V ascent that effectively prepares the subsequent arrival of the tonic.
The phrase illustrated in Example B begins in the eighth measure and prolongs an F-sharp seventh chord, which is the "tritone sub" of the dominant or the Phrygian II, spelled enharmonically.
In this phrase, Coltrane pivots from the F-sharp seventh to a local resolution on B, followed by an incomplete cycle of chords whose roots descend by major thirds (B-G-E-flat). The cycle is related to "Giant Steps," although in this case the major chords are not mediated by intervening dominant chords. Through this prolongation the upper voice C-sharp descends by step to B-flat/A-sharp. Here the F-sharp harmony returns and is additionally expanded, this time by a lower neighbor tone in the conceptual upper voice and a passing tone in the conceptual inner voice. The melodic tones of the last measure can be heard as 6-5-2-3 in F-sharp or as 4-3-7-1 in B-flat minor, the subdominant to the tonic F which returns at the top of the form in the following measure. The latter hearing suggests a particularly grave sort of plagal cadence, and the leading-tone transformation of F-sharp major to B-flat minor reflects the type of major-third relation that Coltrane explored to great effect in this period.
This rising third also inverts the falling third from earlier in the phrase. It is worth noting that in both examples, the goal chord reached before the return of the chord of the governing harmony is the chromatic lower mediant (A in Example A, where the prolonged harmony is C, and E-flat in the F-sharp prolongation of Example B.).
Also notable in Example B is the stepwise descending third motive, heard twice at the foreground level in the inner voice (F-sharp-E-D-sharp, then E-D-sharp-C-sharp) and once at the middleground level in the upper voice (C-sharp-B-A-sharp). The descending third motive is also suggested by the bass movement from F-sharp to E-flat on the middle ground level and more loosely (in augmentation) by the foreground cycle, B-G-E-flat.
In my view, these excerpts offer evidence of hierarchical structure in Coltrane's improvisations, inviting analyses that make use of Schenkerian concepts like prolongation.
Thursday, June 20, 2013
"Brasilia": Octatonicism, Self-Similarity, and Blues
In a previous post I looked at twelve-tone elements in the composition “Brasilia.” Another remarkable feature of that piece is the octatonic collection OCT1,2 embedded within the full chromatic set that makes up its first phrase. After the initial E-flat minor triad, all eight members of OCT1,2 are stated in (almost uninterrupted) succession, and the prepenultimate B-flat links OCT1,2 with the returning statement of E-flat minor.
Of the four pitch-classes excluded from OCT1,2, E-flat and G-flat comprise the tonality-defining beginning and ending gestures of the first phrase. The middle portion of the phrase can be considered OCT1,2 if non-members C and A are heard as subordinate embellishments to melodically predominant members of OCT1,2. From one possible voice-leading perspective, C is a neighbor-passing tone to B, and A is a neighbor tone to B-flat.
The use of the octatonic collection here is significant insofar as it can be conceived as a substitute for B-flat7, the dominant of the governing E-flat minor tonality of the piece. Jazz musicians sometimes use an octatonic scale over a dominant seventh chord when improvising, and theorists like Schoenberg, Berger, and others have shown the derivation of the octatonic collection from the double-semitonal transformation of a cycle of seventh chords whose roots are separated by the interval of a minor third. “Brasilia” manifests this melodically, stating the octatonic collection with pitches from B-flat7 (F, D), G7 (F, D, G, B), and E7 (B, G-sharp, E) as well as C-sharp minor sixth or B-flat half-diminished seventh (G-sharp, E, C-sharp, B-flat).
As I point out in the previous post, the B-phrase of this piece seems to prolong the dominant harmony of a tonic E-flat minor. Thus, like much tonal music, “Brasilia” is hierarchically self-similar: the tonic-dominant-tonic structure of the A-phrase is reflected on a higher level by the tonic-dominant-tonic (ABA) form of the whole piece.
For one final thought, notice how the register of melody notes in the A-phrase can be rearranged to depict the entire phrase as a long and almost unbroken string of falling thirds. This brings to mind the "ladder of thirds" which Peter van der Merwe argues is the underlying form of what he calls the "blues mode." It is crucial to recognize that the application of twelve-tone technique in "Brasilia" does not thereby jettison the norms of tonal music and the blues.
Thursday, May 9, 2013
The Golden Ratio in Improvised Music
Analysts such as Ernő Lendvai and Roy Howat have claimed to discover signs of the golden ratio in compositions by Bartók and Debussy, respectively. It has also been discussed with regard to Mozart's piano sonatas, and elsewhere. But is the golden section strictly an architectural principle, or could it be a natural, emergent property of music? If the latter is true, one might expect to find it in purely improvised music.
This improvisation by trombonist Ben Gerstein is approximately 115 seconds long. The ostensible climax occurs on G3, the note with the highest pitch, greatest intensity, and longest duration of the entire piece, at roughly 71 seconds in. The ratio of the duration of the whole improvisation to the duration preceding the climax, then, is 115/71, or 1.6197... This is very close to the golden ratio value of 1.6180... The ratio of the duration of the piece preceding the climax to the duration following the climax is 71/44, or 1.6136... Since the ratios of the larger and smaller durations are not exactly equal, they do not precisely represent the golden proportion. Nonetheless, the climax of Gerstein's improvisation is a remarkably close approximation of the sectio aurea.
Is this example a solitary coincidence, or is it often the case that a proper sense of balance in a piece of music will spontaneously approximate the golden ratio?
A more extensive analysis of Gerstein's improvised music can be found here.
Is this example a solitary coincidence, or is it often the case that a proper sense of balance in a piece of music will spontaneously approximate the golden ratio?
A more extensive analysis of Gerstein's improvised music can be found here.
Wednesday, May 1, 2013
Music and Brains
Evidence from recent brain studies suggests that the pleasure in listening to music is derived at least partly from "pattern recognition and prediction," which would seem to lend support to Leonard Meyer's theory of musical expectation.
Does this have critical implications for non-developmental or non-teleological music? For music that is effectively static, or without recognizable patterns? Or for ad hoc music of the type which bears no relation to an established tradition or corpus, for which it is impossible for the listener to make any kind of prediction?
In The Aesthetics of Music, philosopher Roger Scruton points out one of the problems with thinking about music in terms of pattern recognition and prediction, expectation and frustration: "The pattern of expectations and fulfilments would change from hearing to hearing, to the point where, knowing the piece by heart, we should assign probability 1 to every event in it, and therefore cease to distinguish it from other pieces in the repertoire." (332)
"As an unfamiliar piece unfolds in time, our brains predict how it will continue to unfold," says Valorie Salimpoor, the neuroscientist who authored the study linked to above. Ostensibly it is the satisfaction of these predictions that gives us pleasure. Unfortunately, this tells us nothing about our understanding and enjoyment of a piece of music that we already know note for note, of which for musical Kenner as well as Liebhaber there are many.
Furthermore, the conclusions of this study suffers from a certain circularity. The study tells us that listening to music releases dopamine in the brain. The brain releases dopamine when we like something. So this study confirms that we like listening to music. This might strike some as less than earth-shattering news.
While Pinker's notion of music as "auditory cheesecake" may well explain its biological origins, it remains to be seen whether neuroscience will succeed in explaining our experience of music and its subjective meanings. Scruton expounds on his skepticism in this regard here.
Does this have critical implications for non-developmental or non-teleological music? For music that is effectively static, or without recognizable patterns? Or for ad hoc music of the type which bears no relation to an established tradition or corpus, for which it is impossible for the listener to make any kind of prediction?
In The Aesthetics of Music, philosopher Roger Scruton points out one of the problems with thinking about music in terms of pattern recognition and prediction, expectation and frustration: "The pattern of expectations and fulfilments would change from hearing to hearing, to the point where, knowing the piece by heart, we should assign probability 1 to every event in it, and therefore cease to distinguish it from other pieces in the repertoire." (332)
"As an unfamiliar piece unfolds in time, our brains predict how it will continue to unfold," says Valorie Salimpoor, the neuroscientist who authored the study linked to above. Ostensibly it is the satisfaction of these predictions that gives us pleasure. Unfortunately, this tells us nothing about our understanding and enjoyment of a piece of music that we already know note for note, of which for musical Kenner as well as Liebhaber there are many.
Furthermore, the conclusions of this study suffers from a certain circularity. The study tells us that listening to music releases dopamine in the brain. The brain releases dopamine when we like something. So this study confirms that we like listening to music. This might strike some as less than earth-shattering news.
While Pinker's notion of music as "auditory cheesecake" may well explain its biological origins, it remains to be seen whether neuroscience will succeed in explaining our experience of music and its subjective meanings. Scruton expounds on his skepticism in this regard here.
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