The psychological effect that polytonality can have on the listener is often governed by the relationship between the keys that are presented. Vincent Persichetti elaborates:
“The fundamental quality of polytonal texture is determined by the key relationship set up by the tonics. In major-key combinations, a polytonal order of tension from consonant to dissonant is secured by combining two keys that lie a perfect fifth, major ninth, major sixth, major third, major seventh apart – and so on up the cycle of fifths . . . Those keys that are not closely related according to the circle of fifths will more easily set apart the tonal key spheres.” 1
The following is an elaboration of Persichetti’s visual representation of his theory with regard to the consonance or dissonance of key combinations:
Although the case can be made that this is a subjective classification system, and although Persichetti does not provide a definition of or reasoning with regard to how he defines resonance within the confines of polytonality, the examples provided in the chapter of his book display a thorough understanding of his experience with polytonality. If I can offer a criticism, it is that I feel as if Persichetti could have provided a much more in depth analysis of why certain key relationships are more consonant, dissonant, or resonant than others, rather than mere stating that this is so.
1 Persichetti, Vincent, “Polytonality.” Twentieth-Century Harmony: Creative Aspects and Practice. New York: Norton, 1961, p. 255-261.
In the following examples, repeated melodic fragments (motives) factor importantly in how new keys are introduced into an established tonal environment. Further, one of the tonal environments (either the established or the newly introduced) can be purposefully ambiguous, suggesting perhaps two or more tonal possibilities.
In what is one of Lennie Tristano’s most widely acclaimed improvised melodies, the chord progression to All of Me is visited with a torrent of running eight-note lines at a fast-paced tempo. By the beginning of the third chorus, the listener has had ample opportunity to have become acclimated to the chordal framework of the composition, complete with solo material that has been largely diatonic to the key of B-flat major. It is at this moment, however, that Tristano chooses to briefly introduce a repeating motive, beginning in the key of B major and again in the parent key of the composition.
While the tempo and usage of continuous eighth notes (common throughout the entire solo) provides the melody with a definite sense of momentum, it is the repetition of a six-beat motive, once in the foreign key and again in the original key that provides the listener with a fragment to identify with. Additionally, the fact that this motive crosses the bar line adds to the sense of momentum.
Leading up to the excerpt above from Kneebody’s Clime Pt. 2, the melody establishes itself as being firmly within an E minor or G mixolydian tonality. This is reinforced by the eight-measure ostinato in the bass, repeated throughout the five-minute composition. Once established, however, the melody briefly visits the keys of C major, A-flat lydian, and D-flat major, using the repeated ascending scale, descending arpeggio motive throughout. Once these new keys are introduced, the door is opened for additional tonal environments to be brought in. And while the E and F-sharp major melodic segments do not follow a specific motive, they do maintain the eighth-note triplet momentum of the previous melodies.
While Darius Milhaud was the first to use the term “polytonality” as a means of describing his own compositional styling, it would perhaps be more accurately regarded as instruction toward composers, performers, and listeners alike with regard to this technique’s clear differentiations from strict atonality. Milhaud makes the distinction by pointing out the similarities of his techniques with those of more traditional harmonic environments:
“It is easy to find the sources of polytonality. From the harmonic point of view, they are found in passing tones, unresolved appoggiaturas, and foreign notes of chords that one can consider as members of another key.”1
A clear example of this can be found in the Botafogo from Milhaud’s Saudades do Brasil:
Here we can see a clearly articulated F minor Latin dance ostinato in the left hand, while the right hand begins with the ascending form of the F# melodic minor scale, beginning on C#, and continuing through B, after which alternating F# minor and E# diminished triads explore their respective inversions. It isn’t until the right hand’s C# falls to C natural in measure 13 that a point of resolution is reached between these two conflicting keys.
While Milhaud sought to explain polytonality as a technique he was using at the time, the concept is explored retrospectively in Ludmila Ulehla’s Contemporary Harmony: Romanticism through the Twelve-Tone Row, citing examples in works from Prokofiev, Hindemith, Barber, and others. Her exploration of polytonality focuses primarily on the vertical structures assembled by juxtaposing disparate elements, as seen in Copland’s Piano Sonata2:
While Ulehla points to a simultaneous sounding of B major and D major tonalities with interspersed events in C major throughout these first several measures of the movement, the block chord movement and cluster style voicing of these tonalities make it difficult (if not impossible) to clearly decipher them as anything beyond polychords, the presence of which does not necessarily constitute polytonality. Ulehla calls the cluster in measures three and four an F#13. But without the presence of an A# in this voicing, a crucial tone of this chord is missing, and there is no leading tone to suggest that it belongs to B major. Additionally, none of the tonal environments suggested by Ulehla account for the E# in measure 14.
Composer Adrian Allen similarly states in the Polytonality volume of Harmonic and Melodic Music Theory and Method for the 21st Century3 in that, by stacking block chords with notes from different keys, clear, discernable polytonality is evident. Throughout the volume, Allen operates solely within the vacuum of his own compositions to reinforce his theorem, as seen below:
Allen’s String Quartet No. 2 is meant to illustrate how voices derived from different keys can be used for a desired polytonal effect. However, with the ensemble key signature changing at every measure for the first five measures of the piece, there is no opportunity for the establishment of a stable environment with which to contrast any others. Further, as established in the earlier Copland example, block chord movement between voices makes it more likely that the combined sonorities of the ensemble will form little more than unrelated polychords or, as is the case in measure three above, simple triads that, given a slightly more stable backdrop, would be diatonic to a single given key.
In his book Polytonal Triad Etudes4, Ed Byrne offers a practical approach to his method, in that he would hope that playing through the etudes themselves would provide the wisdom needed to distinguish polytonality, as there is no explanation in his volume with regard to how the studies in his book are polytonal. An excerpt follows:
This eight-measure excerpt, being nothing more than a string of major triad arpeggios with a passing tone thrown in as the second-to-last note, is part of a study in which several nearly identical iterations of the same triads in the same places of each measure are repeated ad nausea. Composed and performed without accompaniment, a stable tonal environment is never established (which begs the question, “Why the B-flat key signature?”). Further, this method is notably similar to George Garzone’s Triadic Chromatic Approach, a method that has gained a reputation within jazz pedagogical circles as being a revolutionary method for ear training and improvising.5 Garzone’s method, while making ample use of linking strings of non-related triads as a rule, makes no claims with regard to polytonality.6
For the sake of comparison, an excerpt of Garzone’s contrafact written over the chord progression to Have You Met Miss Jones is below:
By virtue of the fact that Garzone’s triadic treatment of a pre-existing tonal landscape is meant to be performed within a tonally stable chordal accompaniment, this melody is actually more polytonal than Byrne’s, which has no such harmonic association.
In his book Twentieth-Century Harmony, Vincent Persichetti defines polytonality as, “a procedure in which two or more keys are combined simultaneously.” He goes on to specify, “Each melodic line should retain its own individuality.” And, while I don’t want to be so bold as to put words into Persichetti’s mouth, I would choose to expound on his definition/instruction by stating that polytonality is the horizontal manifestation of multiple ideas, or multiple iterations of a single idea, clearly articulated in multiple distinguishable keys, modes, or tonalities.
Continuing with Persichetti’s advice regarding effective execution of polytonal situations, he states, “For maximum clarity in the projection of different tonalities, one key is introduced and as the next key is added.” In summary, polytonal events are most effectively implemented after a single tonal environment (key) has been well established, so as to provide adequate contrast between existing and new material. An example from Bob Brookmeyer follows:
The tonal environment, clearly defined with a ii – V7 – I – VI7 – ii – V7 – I progression in E-flat major, only appears to be clearly honored in the melody as a point of resolution during the last two measures. This is possibly due not only to the commonality of the diatonic progression, but also because this particular improvised melody occurs after several iterations (choruses) of the same progression, providing the listener with an ingrained, expected harmonic environment.
Analysis of the excerpt above will show a half-step relationship between different environments, somewhat reminiscent in the Milhaud example above. And while there are substitution-based occurrences of multiple key events, I would like to also bring the reader’s attention to the asterisks, demarcating a downward, Schenkerian-esque “züge-like” pull towards the tonic, which may account for Brookmeyer’s selection of notes more so than a deliberate “key-against-key” technique.
1 Milhaud, Darius, “La Mêlodie,” Melos 3 (1922): p. 195.
2 Ulehla, Ludmila, “Bichordal Writing and Polytonality.” Contemporary Harmony: Romanticism through the Twelve-Tone Row. Advance (1994). p. 284-286.
3 Allen, Adrien, “Polytonality.” Harmonic and Melodic Music Theory and Method for the 21st Century. Vol. 10. Maestro Press, 2014. p. 59.
4 Byrne, Ed, Polytonal Triad Etudes, ByrneJazz, 2008. Linear Jazz Improvisation.
5 Garzone’s method, while not copyrighted or trademarked, is documented by himself in the DVD, The Music of George Garzone & the Triadic Chromatic Approach (Jody Jazz, Inc., 2008). The year of publication of this DVD is the same as that of Ed Byrne’s Polytonal Triadic Etudes.
6 I feel as if this can be said with some authority, having studied the Triadic Chromatic Approach method with Garzone in a private-lesson and ensemble format over the course of two years, and having received positive feedback on multiple occasions from Garozne with regard to my grasp of his concept.
As an event, polytonality appears to be much easier to recognize than it is to define. Within jazz improvisation pedagogy circles, recent attempts have been made to instruct musicians who are curious about improvising “outside” of a prescribed harmonic framework, while at the same time not playing purely intervalically. And while each author appears eager to put his stamp on an arguably self-developed method, he would appear to be operating under the assumption that the reader has without discussion chosen to adopt the same definition of what polytonality is and where it occurs.
To arrive at a standardized (or perhaps just more widely accepted) definition of polytonality, it would be prudent to trace the term back to its origin. Within the framework of the music of composers such as Darius Milhaud or Maurice Ravel, polytonality is a compositional technique introduced in the early 20th century*, and is sometimes recognized as a trend easily overshadowed in an era of more radical and perhaps more noticeable techniques and soundscapes. Indeed, much in the way that cubism can shift the attention of the art world at large to focus from impressionlsm, conventional musical material articulated in multiple keys can scarcely be considered modern in the wake of radical departures such as atonal serial composition, aleatoric performance-derived music, or a handful of other concurrent techniques and genres.
Even within accepted and respected codices specializing in modern composition, polytonality, while usually included, does not appear in my view to have been sufficiently explored, or in some cases even explained. This is perhaps because, in many instances of polytonality, it is often taken for granted that the standardized rules of harmony and voice leading usually still apply, only with the modification that events happening in multiple keys simultaneously. As such, it is feasible that these events can be perceived, analyzed and understood in manners akin to those applied to a single tonal environment. Bearing in mind that a differentiation between simultaneous tonal events is not always so clear, it is my aim to examine a variety of types of occurrences, seeking applications of common methodologies.
As a means of achieving maximum function, this series will focus on composed concert music from the early to mid-20th century, and improvisations from the late 20th and early 21st centuries, providing reasoning behind why certain choices are made as opposed to others, and how those choices may affect the listener’s psychology of perception.
*While earlier occurrences of the articulation of multiple keys can be found in Bach and even in pre-tonal music, “polytonality” as a term and formal concept was not formally articulated until 1922.
I’ve been spending a lot of time with triad pairs recently. I find that they are a good way to combine diatonic with non-diatonic elements, and can point in the direction of polytonality.
While previous entries have used the terms “trichord” and “tetrachord” to refer to cells derived from supra-tonal scales/modes, I’m hesitant here to use the term “hexachord” for a variety of reasons. First and foremost, in order to maximize combinations, it is useful to be willing to consider two triads that may share one or more notes. Additionally, the examples in this entry are specifically assembled with two separate, clearly-defined three-note elements in mind. Likewise, I’m hesitant to use “trichord” because the three-note elements are among the most fundamental building blocks of traditional diatonic music (major, minor, augmented, and diminished triads). I may change my mind about this later.
I’ve spent some time practicing major traids in different relationships, beginning with the least amount of distance, and venturing out. In the case where two major triads share a note with each other, I’ve been practicing it both with re-articulating the shared note, and also not re-articulating the shared note. The latter approach can upset the rhythmic symmetry of a pattern, which is almost never a bad thing.
While I won’t take up space here with diagrams of triad-pair exercises (Bergonzi’s Hexatonics and Gary Campbell’s Triad Pairs for Jazz are excellent sources), I would like to illustrate a couple of practical applications that I am finding useful.
Beginning with half-step relationships, F major and E major triads can be useful over ii and V in the key of C, while also generating interest with non-diatonic tones. The same can be said for C major and B major triads over the I chord:
As is the case with most things I practice, I find that practicing inversions is a good way to become as thorough as possible with an idea:
Likewise, two major triads separated by a whole step can reinforce diatonic stability while at the same time adding a small amount of non-diatonic color:
Experimenting with different types of triads and with relationships beyond half-step and whole-step will provide more melodic possibilities, some leaning more towards polytonality than others. In my experience, peppering a diatonic melody with a rogue triad from another key can help guide my ear towards melodic possibilities that would not occur in a purely diatonic environment.
I thought I had posted this months ago. Regardless, here are a couple of exercises (“licks”) that I came up with that help to demonstrate some melodic possibilities of Messiaen’s Mode 3 (see previous entries for an in-depth explanation):
The following ii V I exercise (Exercise 1) begins by utilizing the D Dorian scale as a means of firmly establishing our overall C major tonal framework. Generally speaking, I often believe that it’s usually a good idea to let listeners (and at times, myself) know that I’m comfortable with adhering to a key before venturing off into supra-tonal territory. In the second measure, the root of the chord is played first, again as a means of continuing to establish a C major foundation, and also as a means of transitioning into Messiaen Mode 3. From the second note forward, triads are played in a “down-up” directional motif. These triads are of the Type 2 variety (see previous entry, Messiaen Mode 3 – Matrix & Trichords).
The following exercises are inversions of Exercise 1. The same Dorian and Mode 3 material is utilized in the same places, but the exercises begin on the third and fifth of the D Dorian scale, respectably:
Along the same lines as my previous post on Messiaen Mode 3, Mode 4 possesses some tonal colors that are interesting and perhaps even a little more practical.
Whereas Mode 3 is a symmetrical arrangement of three identical trichords, Mode 4 is a symmetrical arrangement of two identical tetrachords, each made up of three half steps. The tetrachords are separated by the distance of a tritone:
Parsed out into groups of four notes, the Mode 4 matrix looks like this:
By establishing a pattern of steps and skips between notes, we can build the following trichords (among others):
Note the similarities between Type 3 and Type 4 (think of them as inversions of each other). These two types also contain two major and two minor triads.
Like Mode 2 (octatonic scale), Mode 4 can have applications over dominant chords. The following lick can be heard by a number of modern jazz masters, including Michael Brecker, Donny McCaslin, and others:
The following is an extension of my previous post. Whereas the last post looked at a series of trichords derived from Messiaen’s Mode 3, the grids below depict tetrachords derived from the same. The first grid is all possible combinations of any four pitches (pitch 1, pitch 2, pitch 3, and pitch 4). The second grid is a collection of tetrachords that occur when applying certain step/skip patterns to the Mode. In comparing these tetrachords to traditional harmony, you’ll see Major 7 chords in Type 5, Dominant 7 in Type 4, Minor 7 in Type 3, Major 6ths, Minor 6ths, and more.
Continuing on from my last post about the subject, the following matrix depicts one transposition of Messiaen’s Mode 3:
In keeping with the last post’s concept of looking at the mode as a symmetrical collection of trichords, the matrix below charts all possible arrangements of three pitches (listed here as pitch 1, pitch 2, and pitch 3):
Much in the same way that triads are made from skipping notes within major scales, the following is a series of trichords that are made up of different combinations of skipping and stepping through the scale:
These trichords are selected and named arbitrarily, with the intention of displaying some of the vertical possibilities of Mode 3. For example, Type 4 contains a series of trichords that can be looked upon as major triads. Likewise, Type 3 contains trichords that are identical to minor triads, Type 2 contains diminished triads, and Type 5 contains augmented triads. There are other trichords within each of these grids that can suggest some other tonalities, or better yet, can be used as a means of combining tonalities.
Extended tonality, polytonality . . . however you choose to define it, Mode 3 provides a systematic way of combining three triads (such as Type 4’s C major, Eb major, E major, G major, Ab major, and B major) that would not be possible within the traditional major scale-based system.
In my last post, I mentioned a fondness for Messiaen’s knack for using symmetrical pitch sets as a means of generating melodic and harmonic material. This post will focus on introducing Messiaen’s Mode 3:
There are texts that refer to this collection of notes as the “nine-note augmented scale.” And while they’re not incorrect, I hesitate to use the term “augmented,” as it implies that the scale is only useful over augmented chords. If looked at vertically, we’ll find that there are many major, minor, and diminished triads, seventh chords, and plenty of other functional harmonic sets available within this collection of notes.
As a means of highlighting both the symmetry and melodic possibilities of the scale, I like to look at this mode as a series of three identical trichords. A trichord, plain and simple, is a collection of three pitches, arranged in any order. The pitches in question for this discussion are:
Messiaen’s Mode 3 consists of nine notes. These nine notes can be broken down into three trichords.
By starting on a different note within the mode, or by skipping notes, several trichords are possible. My choice of this one in particular is arbitrary, as is my use of numbers and shapes to differentiate one pitch from another, and one trichord from another.
The trichord used above consists of a whole step (C to D) followed by a half step (D to Eb). Since C is a major third from E, and E is a major third from G#, and G# is a major third from C, using each of these notes as a starting point will provide symmetrical relationships from trichord to trichord.
See next post for more.