The transcription of the first twenty-four bars of Nils van Haften’s baritone saxophone solo doesn’t show any concrete links to my suggestions for improvisation above. However, thanks to his well-informed intuition and his substantial experience as a jazz improviser he manages to play a solo that displays evident connections with the structure of the fifth hour’s row and its application in “Pontiac”. In bars 1, 9, 18, and 24 he relates explicitly to the 1+5 row of the fifth hour, but also to trichords 1+2 from the steering trichord that is important in the melody of this tune. This is evident in bar 14 that sounds like an echo of the alto sax melody in section A. It is also illustrated in bar 8, were the 1+2 trichord is played in the melody, while the basic 1+5 trichord is the steering interval.
An important ingredient of the 1+5 trichord, the perfect fourth interval is emphasized several times in Van Haftens’ solo. Beside in the 1+5 trichords from the row, this interval is also implied in the 2+5 trichords in bars 7, 9, and 21. Likewise, in the interval successions that are marked by trichords 1+4, 4+1 (bar 2, 13) and 2+3 (bar 5, and 11) the perfect fourths might lead him to use them as a references to the fifth hour of the Tone Clock. Although this striking element in his solo does not comply strictly with the appointed 1+5 trichords of the fifth hour, it is tenable in musical terms. In the next section I will come back to the relations between trichords 1+5, 2+5, and 2+3.
As a first suggestion for improvisation in bar 2, I stay close to the basic row of the ninth hour, by putting the second (B) and fourth trichord (D) of the row in retrograde. In bar 3 I suggest a sequence of a trichord combination A+B, transposed by major second intervals. Then, in order to escape to the “three-ness”, the continuous presence of 12/8 triplets in the twelve-tone rows, I play the 5+2 and 2+5 trichords in bars 4 and 5 in extended forms. The sequence in bar 4 is created by repeating the root notes of all 5+2 trichords and by transposing the resulting tetrachords by minor third intervals. In bar 5 I extend the trichords by repeating their first intervals: trichord 2+5 becomes tetrachord 2+5+2, and trichord 5+2 becomes tetrachord 5+2+5. The resulting tetrachords are steered by minor thirds and by minor second intervals. Here I show again my preference to create differences between the intervals in the trichords and the steering intervals.
The example below shows my suggestions for improvisation with the first hour. The first three bars display complete twelve-tone rows. To prevent from an abundant use of the chromatic scale, I suggest the (combination of) trichords 1+1 and 1+3 in bar 2 as a possible point of departure. To help getting acquainted with the sound of this row I point at the tonal reference of the combination of trichords 1+1 and 3+1 to what is called the harmonic minor of the fifth degree, the second tetrachord of a minor (or major) harmonic scale. To highlight this characteristic sound of the first hour my advice is to create sequences by transposing this trichord combination up and down by intervals of (combinations of) minor seconds, minor thirds and major seconds. Extended forms of these trichords in bar 3 result in three tetrachords completing a twelve bar row. Bars 4 and 5 contain simple manipulations of the form and the order of similar trichords. Bar 4 shows a succession of the extended 1+3 trichords steered by trichord 3+3. The presence of this steering trichord refers to the tonal color of the diminished chord. In bar 5 trichord 2+2 is used as steering trichord to order the extended trichords of the basic row in bar 1. Here the steering trichord refers to the tonal color of the whole-tone scale.
In my suggestions for improvisations in the example below, I intend to present ideas of meaningful melodic lines. I mention for instance in bar 2 that a transposition of trichord 5+1 by minor third intervals will create a stronger melodic line than applying the same operations to trichord 1+5. The latter will result in a pattern evoking a “cliché embellishment” of a diminished seventh chord, a pattern that in my opinion should be avoided in order to keep distance from obvious tonal references. In bar 3, I suggest the retrograde of trichord 5+1 played in descending direction, transposed by major second intervals. I find this line attractive because in this ordering the major second interval is added to the minor second and perfect fourth of the trichord, to result in a higher harmonic vagueness and at the same time in a larger emphasis on the basic trichord. The suggestions in bars 4 and 5 are related to elements in the theme of “Pontiac”. In bar 4, I suggest to pick the tetrachord played by the baritone saxophone in bar 5–6 of section A, a characteristic pattern in the composed part, and to transpose this by minor second intervals. In bar 5 the extended 2+1 steering trichord of the row, that plays an important role in the vertical structures of the composition, is transposed by descending minor second intervals.
Section E repeats the twelve-tone row from section C. After being played in a number of variations, it ends in the form of a canon. The first line in the next example shows the melody in its simplest form. The second line displays its basic twelve-tone row.
4.5.2 “Pontiac”
In the basic row of “Pontiac” the 1+5 and 5+1 trichords of the fifth hour are steered by trichord 2+1 from the second hour.
The solo form in section C is a traditional A1 (bars 1–4) - A2 (bars 1–4 repeated) - B (bars 5–8) – A3 (bars 9-12) form that is – although usually arranged in lengths of eight bars per section – quite common in jazz. The accompanying part is made of stacked 1+5 trichords in second rotations, resulting in intervals of augmented and perfect fourths.
Section C is open for a tenor saxophone solo. The accompaniment is a succession of vertically ordered steering trichords 1+2 and 2+1. This results in a harmonic reference of a C pedal point and a G pedal point. The tutti part starting at bar 5 is finished abruptly by a long “fall” played by all four saxophones.
Trichord analysis of my tenor saxophone solo reveals a large number of deliberate applications and variations of the 2+5 trichord; there are no clear references to the suggestions for improvisation above. The encircled fragment of bars 17–29 summarizes a number of characteristic applications. First, in bars 17–20 the first rotations (R1) of trichord 2+5 draw the attention by the symmetric combinations of two perfect fourth intervals. These are followed by several 2+5 trichords in prime forms in bars 23–27.
Bars 21–28 also show combinations of trichord 2+5 and trichords 2+3, and 2+5. This again recalls Schat’s statement that the ninth hour easily makes contact with the seventh and the fifth. The trichord analysis indeed shows a connection between trichord 2+5 and the 2+3 and 1+5 trichords of these hours. I admit that at the moment I played this solo I was not aware of this connection. Just as in Van Haften’s baritone saxophone solo in "Pontiac", the connection seems to result of my emphasis on the interval of a perfect fourth which is also a component of trichord 1+5, and the “sum” of the trichord 2+3.
Evaluation
Two different twelve-tone rows, two different ten-tone rows, and one eleven-tone row can be identified in “Dicke Pitter”. Both twelve-tone rows are constructed with trichords from various Tone Clock hours. Only the eleventh-tone row is constructed with ninth hour trichords exclusively. Despite the sparse references to the ninth hour in the melodic lines, the vertical ordered fragments contain separate stacking of both its rather consonant basic 2+5 trichord and the more dissonant 1+2 steering trichord. Schat’s assumption of an “easy contact” between trichord 2+5, 2+3, and 1+5 is confirmed both in the composed part and in the tenor saxophone solo. Although not mentioned in the suggestions for improvisation, the second rotation of trichord 2+5 resulting in a stacking of two perfect fourth intervals proved to be an interesting tool to create combinations of 2+5 trichords.
The combination of trichord 2+5 (rather consonant) with its steering trichord 1+2 (rather dissonant) on the one hand, with friendly trichords 2+3 (rather consonant) and 1+5 (rather dissonant) on the other, make this ninth hour of the Tone Clock an attractive tool in the border area between twelve-tone technique and tonality.
ex 4.5.1.1
In section A, the alto saxophone plays an e♭ octatonic scale. Trichord analysis shows it construction with 1+2 and 2+1 trichords. The vertical orderings in bar 1 display 3+2 trichords c♯–e– f♯, and d–f–g; and tetrachord e♭–f♯–c♯–g marked as 2+3+1 (prime form). Then, after his two pick-up notes at the end of bar 1, the baritone saxophone plays a twelve-tone row marked by the rectangle in bar 2 combining the trichords 1+1, 3+1, and 1+3. The remaining saxophones play stacked 3+1 trichords.
In the next fragment of my solo the symmetry of the first hour’s row is illustrated by the 3+3 steering trichords in bars 16–17 and 26–27. These bars point back to the idea of ordering segments at minor third distances as expressed in bar 4 of my suggestions for improvisation. In bars 18–21, the basic 1+1 trichord of the first Tone Clock hour itself serves as a steering hour of a sequence of 1+2 and 1+3 trichords.
From bar 5 in section A, the alto saxophone continues this basic line, while the soprano and tenor saxophones play 5+1 and 1+5 trichords. So does the baritone saxophone, but he adds a chromatic pick-up note. This combination of trichords 1+1 and 1+5 results in fact into a 1+1+5 tetrachord.
In the continuation of section B the stacked 1+2 and 1+5 trichords, with the soprano part in octave displacement, keep emphasizing the dissonant sound of the fifth hour.
At the beginning of section B the baritone saxophone repeats the twelve-tone row, transposed from C to G# (T8, in the rectangle). Then, another transposition of this line to D (T6) seems to appear, marked by dotted line, but, with the note b left out, this is not a complete twelve-tone row. Trichord analysis of the clusters played by the remaining saxophones displays a collection of 1+3, 2+3 and 1+2 trichords. Bars 5–6 are dominated by incomplete, complete and stacked 3+3 trichords.
From bar 5 the following melody is distributed over the four voices. The first four bars display a twelve-tone row, the second half a ten-tone row with one note repeated.
ex 4.5.1.5
The following example shows section C. The encircled dyads and triads in bars 2–7 all contain minor second intervals that emphasize the dissonance of the first hour of the Tone Clock. The unison parts in the last four bars are constructed with trichords 3+3, the steering trichord of the first hour.
This melody is harmonized with vertical orderings of 2+5 and 5+2 trichords. Trichords 5+2 are stacked on 2+5 trichords and 2+5 trichords stacked on 5+2 trichords, resulting in tetrachords of 2+5+2 and 5+2+5.
From bar 9 until the end of section A, the baritone saxophone plays a melodic line that is mainly constructed with 2+5 trichords. The fragments in bars 10–11 and in bars 30–34 display combinations of trichord 2+5 with trichords 2+3 and 1+5. These bars can serve as demonstrations of Schat’s statement that trichord 2+5 “easily makes contact with other tonalities [… such as that of the Seventh Hour [… or more distantly, that of the Fifth Hour” (Schat 1993: 67).
In the first four bars, and in the first note of the fifth bar of section C, the third twelve-tone melody is introduced, a descending C chromatic scale that is obscured by octave displacements and by its distribution over the four voices, in such a way that the last note of a melodic fragment overlaps the first note of the next.
Section D, the solo section of this tune, serves as a continuously repeated background line behind the solos of the baritone, alto and soprano saxophonists. Only after the last soloist has finished his part, it is played by all four horns. The soprano and alto saxophones play crossing lines that together display a ten-note row, while the tenor and the baritone saxophones combine the root note c with the notes a, f♯, and e♭. Together they create a C-diminished chord by stacking two rotations of the 3+3 steering chord: c– e♭–f♯, and e♭–f♯–c.
The alto saxophone opens section A with the following fragment, created with the steering trichord of the second hour, that will serve as a basic line all through this movement.
Evaluation
Compared to “Onsa”, the focus in “Pontiac” has been changed from manipulations of complete twelve-tone rows to operations with the trichord segments of the row. As a result the horizontal relationship in the melodic lines between the 1+5 basic trichord of the row and the 2+1 steering trichord, ordered in alternating positions, is more intense. In their vertical orderings both trichords effectively support the characteristic dissonant sound of the fifth hour.
In his solo, Van Haften’s reference to the Tone Clock hour is more distant than mine in my solo in “Onsa”. Instead of deliberate operations with a single predominant trichord he intuitively refers to the appointed intervals of a minor second and a perfect fourth. This is the direct result of my additional advice to the soloists, to refer intuitively to the composed parts while following my written suggestions for improvisation.
4.5 My applications of the Tone Clock
I have called my saxophone quartet Carillon in response to Schat’s idea to construct a carillon that every hour plays the appropriate Tone Clock row (Schat 1993:73). During initial finger exercises in order to become acquainted with the distinct sounds of a number of Tone Clock hours, I became fascinated by the surprising sounds that occurred from applying these trichords. Therefore I decided to broaden my individual experience with the Tone Clock by completing all twelve movements.
The choice for the line-up of a saxophone quartet was made for three reasons. First, it follows from my personal experiences as a saxophonist and composer, being familiar with the instrument and with this particular combination of instruments. Second, after I had rehearsed and recorded the first exercises with my students at Codarts University of the Arts, the professional saxophone quartet Koh-i-Noor showed an interest in rehearsing and performing the first five movements I had written (using the first, third, ninth, eleventh and twelfth hours) with myself as the soloist. Their comments stimulated me to finish the rest of the twelve movements. In 2013 I recorded Carillon with my fellow saxophonists of the Clazz Ensemble at that time. Third, I considered that working with this small line-up and without a rhythm section, would help us to leave our comfort zones and to fully concentrate on the “new stuff”, without the opportunity to “lean back” on the rhythm section, and just rely on their common jazz idioms.
My aim was to compose twelve compact pieces, one to every hour of the Tone Clock, which could be performed within a time period of forty-five minutes, so that listeners could experience the distinct sound qualities of the hours of the Tone Clock in a coherent way. To highlight these sonic qualities, I wrote the twelve movements in the form of “riffs”, short and repetitive fragments. From the early days of jazz music, riffs have been played as compact melodic phrases to create continuous movement and, played as backgrounds in solo sections, to support the efforts of the soloists. A number of these riffs have found their way into the codex of jazz originals, such as for instance “C-jam Blues” (Duke Ellington) and “Bemsha Swing” (Thelonious Monk). Thus, I agree with composer Gordon Delamont’s ideas expressed in a publication about twelve-tone techniques: “In formal jazz composition, using serial techniques, it is desirable that the characteristic rhythms of jazz be retained. Also, repetition of melodic elements is likely to occur more frequently in jazz composition” (Delamont 1973: 24).
I realized that, besides the challenge to get acquainted with the Tone Clock hours, this carillon would contain rhythmic challenges for the performers. But thanks to their long time professional experiences in various musical styles, they responded quite naturally to the rhythmic requirements such as distinguishing between swing timing and straight timing in all tempos, and mastering Latin American rhythms, odd meters, and flexible (accelerating and retarding) tempo’s.
Schat’s metaphors of his Tone Clock hours, such as “Pale sun through grey cloud-cover. Or also: a sharp, icecold moon” (Schat 1993: 59), associated with the seventh hour, and “Much dust in the atmosphere. Rush hour” (ibidem: 63) with the fifth, are not taken into consideration. The same goes for his suggestions to combine different Tone Clock hours to complete twelve-tone rows. As I discussed before, my aim with the construction of “Carillon” was to experiment with each individual hour and I was afraid that intentionally blending distinct hours would obscure this idea.
As to the arrangement of the Tone Clock’s trichords, I mainly took their prime forms as points of departure. This seemed the best way to optimize the expression of their distinct colors. In my opinion, deliberately obscuring the rows by complex permutations and (retrograde) inversions would reduce the listeners’ ability to distinguish the differences between the hours.
To help find patterns for solos I devised a sort of guided improvisation by attaching a number of subjective and rather random suggestions to the distinct hours. These suggestions contain a selection of melodic patterns that are constructed with the trichords from the rows. Initially they were meant for my saxophone students to show them possible applications of the Tone Clock hours and trichords, as introductions to the distinct sounds of the rows and trichords, and as examples of how to connect these twelve-tone techniques to their existing improvisational practices.
In the following sections my individual interpretations of the Tone Clock are discussed in three movements of Carillon. First is “Onsa”, based on the first hour, the symmetric row with the highest dissonance and the most obvious twelve-tone structure: the chromatic scale. Next is “Pontiac”, based on the fifth hour, the 1+5 trichord. The attractive dissonance of this trichord was already obvious in the previous examples of O’Gallagher’s music. And finally “Dicke Pitter”, based on the 2+5 trichord from the ninth hour. By its intervals of a major second and a perfect fourth it sounds less dissonant than “Onsa” and “Pontiac”.
As to each movement I will discuss the applications of the Tone Clock in the composed sections, the suggestions for improvisation, and how the improvisations relate to the compositions. As to the analysis of both the composed parts and the solo transcriptions, trichord analysis will be applied. For the sake of reading ease I have removed all indications concerning tempos, dynamics, phrases and directions from the examples.
Trichords are displayed both in horizontal and in vertical orderings. Similar to the sections above, trichords in horizontal orderings are marked by legato lines above or below the staff, and notated with spaces between the numerals and the plus signs. Trichords in vertical orderings are marked above the notes that sound as similarities, and notated without these spaces between the numerals and the plus signs. The figures mark the trichords in their prime forms, the smallest possible form of the trichord. For the ease of reading the octave disposition of the pitches is not indicated. Trichords in which one pitch is repeated are considered incomplete trichords or “dyads”.
In section B the centrifugal sound of the fifth hour is emphasized by repeating the complete basic twelve-tone row as a canon over all four voices. As a result of this arrangement of the rows, vertical orderings of steering trichord 2+1 and trichord 3+3 could be created.
4.5.3 “Dicke Pitter”
In the basic row of “Dicke Pitter” the 2+5 and 5+2 trichords of the ninth hour are steered by trichord 1+2 from the second hour.
ex 4.5.3.1
In section A of “Dicke Pitter” the bells of the cathedral in Cologne are sounding. This fragment mocks the rhythmic effects of bells in a clock tower ringing at the same time. In the first eight bars the soprano saxophone plays a series of accentuated notes constructed with 1+1 and 1+2 trichords.
ex 4.5.2.3 – “Pontiac” – section A fragment
Then, the baritone saxophone continues playing perfect fourth intervals, largely at distances of minor seconds, again resulting in 1+5 and 5+1 trichords. In bars 16 and 20 the 1+1+2 tetrachord is played again, with the note c in octave displacement.
Tenor, alto, and soprano saxophones play vertical orderings of trichords 1+5 and 5+1.
In section B the tenor saxophone sets up a melodic line that in bars 1–2 displays a twelve-tone row ordered with a succession of trichords 1+5, 1+2, 2+4 and 1+1. The other saxophones successively answer this line with rows of different lengths and different trichords. The soprano and baritone saxophones together in bars 5–6 play a combination of 5+2, 5+1, 3+2, and 5+2 that results in a ten-tone row. The alto sax in bars 4–5 plays a twelve-tone row that combines trichords 2+5 and 5+2 with trichords 2+3 and 3+2, and is repeated by the tenor saxophone in bar 8. These lines again provide evidence for Schat’s statement about the easy contact between these trichords. The baritone saxophone plays a ten-tone row in bars 2–3 and repeats this in bar 7. The line in bar 9, played in unison and octaves, almost displays the complete twelve-tone row of the ninth hour. The last note b is changed into the note d, to end this short theme with a trichord of two perfect fourth intervals, the first rotation of 2+5 trichord d–e–a.
4.5.1 “Onsa”
In the basic row of “Onsa” the 1+1 trichords of the first hour of the Tone Clock are steered by the 3+3 trichord of the tenth hour.
The following fragment of my own tenor saxophone solo shows examples of trichord operations as suggested above. In bars 8–11, I play an ascending sequence of 3+1 trichords. The steering hours of these trichord combinations are quite random: 2+1, 3+1, and 3+2. In the following bars, I play a descending succession of 3+1 trichords in retrograde. This sequence is steered by the symmetry of the extended trichord 2+2+2. This fragment refers both to my ideas of both repeating trichords by freely transposing them up an down, and to create orderings by applying symmetric steering trichords, as demonstrated in bar 5 of my suggestions for improvisations. Both operations are meant to illustrate and emphasize the sound possibilities of the trichords and trichord combinations at stake rather than to manipulate the complete Tone Clock hours.
Evaluation
The composed parts in “Onsa” display conscious applications of various twelve-tone operations. Operations with three distinct twelve-tone rows are applied to construct the melodic lines. They are manipulated by octave displacements, by re-ordering trichords and trichord combinations from the row, and by transpositions of these elements. Vertical orderings are created by the stacking the trichords and trichord combinations from the first Tone Clock hour and from other hours. The trichord terminology appears to be an insightful tool to identify these vertical orderings that are difficult to express in conventional chord symbols.
The transcription of my tenor saxophone solo demonstrates a deliberate use of successions of trichords from the basic row and from related rows. It provides clear illustrations of ordering these trichord successions by the steering chord of the row or by other symmetric steering trichords. These operations illustrate the possible applications of my suggestions for improvisation.
ex 4.6.4
From the chord in the last bar of the fragment of section D (see example 4.6.1) an eleven-tone scale can be extracted as is shown in bar 2. To complete a twelve-tone row the note a has been added in bar 3. This chromatic scale allows improvisers to superimpose the trichords 1+3 again, adding continuity to the previous rows. But they could also derive various rows at pleasure, as shown in bars 3–5. Bar 3 shows the derivation of a twelve-tone row constructed with 1+1 trichords from the first hour, steered by the 3+3 trichord from the tenth hour. Bar 4 displays an alternative derived tone row with trichords 2+2 from the sixth hour, steered by trichord 1+5 from the fifth hour. The derived row in bar 5 is constructed with the same trichord, steered by trichord 2+1 from the second hour.
Regarding the improvisations Liebman suggests to connect his “triple polychords” to a number of “composite scales” (Liebman 1993: 174), I have concerns with the complexity of these suggestions for improvisation. In my opinion Liebman’s scales as such sound like twelve-tone rows, without a clear link to the structure of the accompanying chords. This, combined with the fact that the separate “tonalities” of the last four bars each, tends to evoke an “anything goes” approach to improvisation that I dislike. As an alternative to his approach I will demonstrate how twelve-tone techniques with the Tone Clock’s trichords can be applied as a useful alternative to Liebman’s random mixture of twelve-tone and tonal approaches.
The example below displays the rows, ranging between ten-tone and twelve-tone rows, as well as the implied fragments of diatonic scales in each row. The polychords (slightly different from those in the fragment above) are notated as root note with, stacked upon them, a combination of either two triads or a triad with a seventh chord.
ex 4.6.5
Evaluation
For sure, Liebman’s complex scales justify the hybrid tonal colors that can be distinguished in his chords. But, as an addition to this, the trichord applications demonstrated above could help the improviser to create consistent melodic lines in complex harmonic contexts. Experienced improvisers will manage to superimpose these twelve-tone techniques to express their individual harmonic colors as an alternative to, or alongside the advanced chord-scale approach that Liebman is pointing at. They can create continuity between complex chords even when they are not intentionally composed with these twelve-tone techniques. The continuation of the 1+3 trichord between the first and second chord is a clear example of this. The actual application of manipulating the Tone Clock’s trichords proves the value of this system as a self-employing tool for improvisation in a complex harmonic context.
4.6 Trichord operations instead of “composite scales”
In chapter 3.2.3, I discussed the first three sections of Liebman’s composition “Invocation” (Liebman 2013: 170). The example below displays a fragment of the slow ballad in section D of this composition, I will take this as an example to discuss trichord operations as a simple alternative to Liebman’s complex improvisational approach.
ex 4.6.2
The following examples show my trichord operations to the first, the second and the last accompanying chords in the fragment of section D shown above. In order to maximize the connection between composition and improvisation, I chose the complex chords from that section, instead of Liebman’s slightly modified solo chords.
The first line contains all pitches from the melody and the chord in the first bar, in displayed order. By moving the note d♭ to the end of the row, this line starts with a combination of trichords 1+3 and 3+1. By transposing this combination a minor seventh up (T10), so that it starts with the note d♭, the twelve-tone row in bar 3 is created, containing a succession of 1+3 and 3+1 trichords from the third hour of the Tone Clock. This row is steered by tetrachord 5+5+5, a combination of two 5+5 trichords. The notes that are added to complete the row are encircled. As seen in “Dicke Pitter” in section 4.5.3, trichord 5+5 is the first inversion of trichord 2+5. With this in mind, the correct row that is displayed in the fourth line can be derived. It is steered by tetrachord 2+5+2, a combination of trichords 2+5 and 5+2. Thus, improvisers who are familiar with the Tone Clock can use this row as the basis for their improvisations on the stated complex chord.
ex 4.6.3
In the next example all seven pitches present in the second bar of “Invocation” are ordered in a succession of trichords 1+4 and 4+1 plus the note c, the root note of the chord, which is displaced to the last position of the row. By changing the positions of the notes f♯ and f, a combination of trichords 1+3 and 3+1 is created. By transposing this trichord combination a major second up, a twelve-tone row is created with the note c at the last position. The added notes are encircled. The fourth bar shows how I re-ordered these trichords with steering hour 2+5. This row sounds the same as the fourth one in the previous example. Thus, improvisers could create continuity between the chords in bar 1 and 2 by taking this row as their reference.
ex 4.7.7.2
By combining these two trichord collections based on trichord combination A+B, and by putting the B trichords in retrograde the following sequence results.
ex 4.7.2.5
Trichord combination B+C in the example below shows an embellishment of an incomplete octatonic scale. In the second bar this octatonic scale is completed and enharmonized. By adding the notes c and b, it evokes the sound of B7 (♭9, ♭10, ♯11, 13).
ex 4.7.6.11
4.7.7 Trichord 2+5
The last trichord family to be discussed here is that of 2+5. In the next example the basic row of trichord 2+5 is steered by trichord 1+2.
ex 4.7.1.2
In the next example an F minor pentatonic scale (shown by the accentuated notes) is embellished with this first rotation of trichord 1+2. Just as the example above, the interval between the first and the last note of every trichord is a major seventh, and the F minor pentatonic scale sounds as an upper structure of the F♯ minor pentatonic scale.
ex 4.7.5.3
By ordering trichord pairs such as in the following line, the obvious pentatonic sound of this trichord can be slightly obscured. The hexachord with the trichord combination A+C is arranged in ascending order, at distances of two ascending and one descending minor second intervals. In bars 3 and 4 they are put in the retrograde form.
ex 4.7.7.1
The example below shows two patterns in which the trichord is transposed at distances of major seconds. In the first line this is done with trichord 2+5, and in the second line this is done with trichord 5+2. In the first line, the first two notes of all trichords together belong to the c whole tone scale, and all third notes of the trichords together belong to the g whole tone scale. In the second line, all first notes together form the c♯ whole tone scale, while all second and third notes together form the f♯ whole tone scale.
ex 4.7.7.10
4.7.8 Tetrachord 1+5+1
The following example displays tetrachord 1+5+1, the second of Babbitt’s all combinatorial four-notes sets (see section 4.3.2). It is steered by trichord 4+4.
ex 4.7.6.6
The next row shows trichord 2+4 steered by 4+1. Trichord combination A+B shows an embellishment of the C whole tone scale and trichord combination C+D does the same with F whole tone scale.
ex 4.7.6.9
The same trichord combination in second rotation triggers the interval of an augmented fourth. Both lines in the example below show operations with trichord combination B+A, with B put in retrograde. In both lines the interval combinations are transposed at distances of major seconds. In the second line the intervals between all trichords are minor seconds. In the first line the intervals between the trichords inside the combination are minor seconds, but those between the last note of each combination and the first note of the next one are ascending minor thirds.
ex 4.7.9.2
The next example shows the minor third intervals implied in these tetrachords, played in ascending and in descending directions. In bar 3, I have again changed the order of the tetrachords to obscure the compelling augmented color of steering hour 4+4.
ex 4.7.10.2
4.7.11 Tetrachord 5+1+1
The next example displays the sixth basic row of Babbitt’s all comprehensive four-notes source sets, a permutation of row 1+5+1 discussed in section 4.7.8. The steering trichord is again 4+4.
ex 4.7.12.1
The next two lines are examples of permutations of the tetrachords. These permutations again create a more smooth transition from one tetrachord to another by replacing the major second interval in the basic row by a minor second interval. As a result of these permutations in the tetrachords the 4+4 steering chord is again obscured in bar 1. Due to the permutations of tetrachord A at the end, bar 2 has a steering tetrachord 5+4+4.
4.7 Generative compendium of melodic patterns
The improvisations by the saxophonists in “Carillon” display a combination of guided and informed intuitive improvisation. The guided part results from my written suggestions, my verbal explanations, and my sonic demonstrations of how to operate with the basic trichords, trichord combinations and row variations. The intuitive part results from how the saxophonists launch their individual musical experiences to connect this new information to their existing improvisational practices.
Saxophone quartet “Carillon” was composed and recorded before the publication of O’Gallagher’s method on twelve-tone improvisation, discussed earlier in subchapter 3.7. There I expressed my concerns with his rather encyclopaedic approach resulting in a large quantity of barely meaningful patterns by which his method shows rather a theoretical than a practical quality. As another drawback I mentioned that O’Gallagher didn’t offer clear solutions for a major rhythmic problem with the trichordal approach: the predominant presence of three-note groupings.
I consider my suggestions in the following sections as an addition to O’Gallagher’s method. I share the same goal as O’Gallagher, “to acclimate and train the ear to a new way of hearing harmonic and intervallic space” (O’Gallagher 3013: 8), but considered from a different perspective. I intend to show how I derived this selection of patterns from particular trichords and trichord combinations.
In this section I present a selection of melodic lines that are created with the trichords in the twelve hours of the Tone Clock. I constructed these lines as row patterns, to express the characteristic sound of each trichord family. The trichord patterns are followed by patterns of tetrachords brought up by Babbitt and De Marez Oyens as discussed in section 4.3.2.
I realize that my presentation of these lines as patterns consisting of symmetrical sequences of intervals can raise problems. In chapter 2, I mentioned Liebman’s difficulty with the excessive use of patterns or “licks” in improvisations. Although, by their specific contours, “these shapes can act as a kind of filler material as a connecting phrase between main musical ideas [… there is always the danger of overuse leading to a mechanized and predictable musical statement” (Liebman 2013: 69). I emphasize that, just as the basic bebop patterns a beginning improviser repeatedly plays to internalize the harmonic, rhythmic and melodic aspects of the traditional jazz idioms, my suggested trichordal patterns are not meant as “ready-mades” to be quoted literally on stage, but rather serve as basic structures and a point of departure for personal embellishments and variational techniques. Jazz students are used to working like this in order to permanently understand, organize, and embody new musical knowledge. This manipulation of sequential patterns is omnipresent during this learning process, from the two quadrants in a conventional major scale, to the multiple divisions of the chromatic scale in composer Nicholas Slonimsky’s Thesaurus of Scales and Melodic Patterns (1975). And therewith practically: repeating and transposing scales and scale patterns is a sequential effort that helps to memorize new musical information. In section 3.3.2, I discussed this in the context of memorizing interval combinations in Bergonzi’s Thesaurus of Intervallic Melodies (2000).
In the first rotation (R1) of trichord 1+2, the interval between the two outer notes is a major seventh. Because of the “contemporary” sound of this wide and dissonant interval, I use this to embellish single notes or scales, such as the C major scale (shown by the accentuated notes) in the following example. As a result of this operation the resulting line sounds as an upper structure of the scale of D♭ major, with two notes adapted. To comply with the demand of merely playing 1+2 trichords in first rotation, I play G instead of G♭ in the third grouping, and D instead of D♭ in the penultimate grouping.
ex 4.7.2.6
The relationship of trichord family 1+3 with the octatonic scale is even more obvious when we take the next row, steered by trichord 1+5, as a point of departure.
ex 4.7.3.3
By playing trichords D – C in reverse order and transposing C by major second intervals down, a cycle of fourths results: F# – B – E – A – D – G – A – F.
ex 4.7.7.3
The lines in the following example show variations of the sequence above, using extended trichord combinations. The first line is based on trichord A+B, with B played in retrograde. After the last note of trichord B, the C♯, I have added two chromatic notes (encircled) to lead to the second trichord combination that starts at a distance of a major third above the first. The same operation is repeated and results in the third pattern.
The second line also starts with trichord combination A+B, but now, after the last note of trichord B, enharmonized into D♭, two descending chromatic notes are added, to lead to the next trichord combination, starting a major second below the first.
With these operations I combine my preference to connect patterns by using chromatic leading tones with the creation of four-note groupings to solve the dominance of three-note groupings in the operations with the Tone Clock so far.
ex 4.7.7.4
The following examples show trichord combinations in prime forms, displaying obvious tonal colors. In the first example trichord combination D+A evokes the harmonic color of Am7. It is followed by its transposition a major second up, and put in retrograde, sounding as Bm7.
ex 4.7.8.1
The next example shows the implied augmented fourths intervals. In bar 1 they are played in ascending directing and in bar two in alternating directions. In bar 3 the second trichord of the row (B) is played in retrograde and in bar 4 in the second rotation of the retrograde. By this rotation, the 4+4 steering that is obvious in bars 1–3, is obscured. As a result of this the lines sound less predictable than those in the bars before.
ex 4.7.10.1
In the following examples three different dispositions of the tetrachords are combined with three different retrogrades.
4.7.2 Trichord 1+3
In the following example the basic row of trichord 1+3 is steered by trichord 2+3.
ex 4.7.3.1
Both in their prime forms in the row above and in their first and second rotations in the examples below, these trichords display the harmonic color of maj7/omit3 chords, transposed in ascending direction by minor third intervals.
ex 4.7.3.5
The next three sequences of 1+4 trichords evoke the harmonic color of maj7/omit5 chords. The first line shows 1+4 trichords in prime forms in a descending line transposed by a major second, with the trichords in alternating directions. The second line shows the trichords in first rotation, transposed in distances of a major second, in ascending direction. The same transposition is again used in the third line, in descending direction, with the trichords in second rotation.
ex 4.7.4.1
The centrifugal sound of the basic 1+5 trichord is illustrated in the following trichord combinations. The first example shows orderings of trichord combinations A (prime form) and B (retrograde). In the first line four hexachords are ordered at minor third distances. Because the last note connects back to the beginning of this sequence, the line evokes the idea of a “turnaround” in a C tonality. The same could be said about the second line, but I consider the abundance of major second intervals making this line more predictable and therefore less interesting.
ex 4.7.4.2
Next is a pattern of hexachords of B (prime form) and C (retrograde). It is transposed a perfect fourth up, and then descends by major second intervals. I imagine that this pattern evokes a turnaround on, for instance, a Dsus chord.
ex 4.7.5.4
4.7.6 Trichord 2+4
In the following example the basic row of trichord 2+4 is steered by trichord 1+3.
ex 4.7.6.1
Combinations of trichords from this row tend to evoke tonal colors of whole tone scales, and intervals of augmented fourths and fifths. The following hexachord with trichords B+C emphasizes this tonal reference. It is transposed twice, by tritone intervals, in ascending direction.
ex 4.7.6.2
In the next example a hexachord with trichords A+B is transposed a major third up twice. The steering tetrachord 4+4+4 emphasizes the augmented tonal color of this eighth hour of the Tone Clock. The combination of this augmented tonal color of the steering trichord, the minor second intervals between the trichords inside each hexachord, and the minor third intervals between the last and first notes of the adjacent hexachords, creates an interestingly vague harmonic color.
ex 4.7.6.7
The following paragraphs show patterns resulting from the inversions of the 2+4 trichord family. The example below shows the 2+4 row steered by 1+2. The second line shows the first rotations, and the third line shows the second rotations of the trichords in the row.
ex 4.7.7.5
The second example shows trichord combination B+C, with C put in retrograde. This evokes the sound of F♯maj7. The next trichord combination starts a minor third interval above the first, as a result of my preference to play a leading tone to connect both parts. This second part of the sequence sounds as Amaj7.
ex 4.7.7.7
As a variation to the sequence above, it is played with the trichords in alternating directions, by which the steering interval of a minor third sounds at all connections of the trichords.
ex 4.7.8.2
In the following example tetrachord B has been extrapolated to avoid the compelling sound of the 4+4 steering trichord. It is true that it is still present in the steering trichord C–G♯–E, but the minor second intervals between the connecting notes between the intervals obscures the sound of the two augmented fifths (C–G♯–E) in the permutated steering trichord.
ex 4.7.1.3
The following example shows a pattern with the second rotation (R2) of trichord 1+2 put on the (accentuated) chord notes of C7. This operation results in a melodic line that also implies the chord tones of A7 and Bb7, while each separate trichord evokes the harmonic color of a major7/sus2/omit5 chord.
ex 4.7.2.4
Combining trichords C+D, with C played in retrograde, creates a chromatic sound with an interesting harmonic vagueness. In the next example this trichord combination is transposed with minor second intervals. The contrast between the ascending direction of the trichord combination and the descending minor second interval between its last note and the first note of the next combination creates an interesting effect in this line.
ex 4.7.4.3
The final example shows a hexachord of trichords D and A. These are steered by 3+1+1 to evoke a turnaround of a Fsus chord.
ex 4.7.5.1
A hexachord with the combination A+B evokes the harmonic color of Cmaj6, or (Gm7–) C7. In the following line this hexachord is transposed down by a major second interval, evoking the iii – vi – ii – V progression G7sus – C – F7sus – Bb.
ex 4.7.6.5
The next example shows two lines evoking an intriguing sound. The first shows a hexachord of 2+4 trichords A+B, the second a combination of 4+2 trichords C+D. The steering trichord in both lines is 4+4, while the intervals between the trichords inside the hexachords are minor thirds and the intervals between the last note of each hexachord and the following is a minor second. As a result, both lines combine the diminished sound of their hexachords with the augmented sound of the steering trichords.
ex 4.7.8.3
4.7.9 Tetrachord 2+1+2
The following example displays trichord 2+1+2,the fourth of Babbitt’s all combinatorial four-notes sets. It is steered by trichord 4+4.
ex 4.7.9.1
By playing the second tetrachord in retrograde, it sounds a bit less predictable, and the obvious minor tonal color gets slightly obscured.
ex 4.7.12.2
Evaluation
The examples in this section represent my subjective selection of numerous possibilities of combining trichords, clarifying the thinking behind their construction. They are meant to serve improvising musicians as an inspiration to add variations or to create different combinations according to their individual musical preferences.
In general, the examples given show a preference for connections at distances of minor seconds, because they create the effect of leading tones and therewith suggest a type of conventional voice leading. Second, the examples show a preference for steering intervals that contrast with the prevailing harmonic color of the trichord and the row at stake. Sequences containing these contrasting steering intervals, such as for example major seconds and major thirds contrasting with trichord 1+3, minor third intervals contrasting with trichord 2+4, and minor seconds contrasting with trichord 2+5, evoke an interesting harmonic vagueness. Thereby they have the potential to create new ways to move outside the expected sounds of certain tonal or non-tonal contexts.
ex 4.7.9.3
4.7.10 Tetrachord 2+3+2
The next tetrachord is 2+3+2, the fifth of Babitt’s all combinatorial source set. Bar 1 shows the basic row, steered triad 4+4, while bar 2 the basic row is put in retrograde. Bar 3 displays the basic row with tetrachords A and C in retrograde.
From here on the trichords are named by their interval combinations, no longer by the numbers of the Tone Clock hours. The series of examples starts with the non-symmetric trichords 1+2, 1+3, 1+4, 1+5, 2+3, 2+4, 3+4, and 2+5. It is followed by the tetrachords 1+5+1, 2+1+2, 2+3+2, 5+1+1, and 1+2+3. To facilitate the reader’s practice, every section starts with the display of the basic twelve-tone row in one or more steerings. Next, its trichords, trichord combinations, and tetrachords are presented in different positions and transpositions. In a number of cases their potential tonal colors are discussed. Patterns constructed with symmetric trichords and tetrachords have been left out because they refer predominantly to diminished and augmented tonal colors. By applying them as steering trichords these harmonic references are less obvious because then the emphasis is on the trichords they connect. Moreover the symmetric steering hours serve well to order the trichords and tetrachords along the five possible equal divisions of one octave. Trichord 1+1 refers to the chromatic scale itself and points at transpositions at distances of a minor second interval. Trichord 2+2 refers to the division of the octave in six equal parts, trichord 3+3 to the division in four equal parts, and trichord 4+4 to three equal parts. The division into equal parts results from tetrachord 2+2+2.
My operations to avoid obvious tonal references go hand in hand with my intentions to evoke a certain amount of harmonic vagueness. Thus, it appeared to work best by creating contrasts between intervals in the basic trichords and tetrachords of the row on the one hand, and those in the symmetric steering hours on the other. As expected even this maximum emphasis on interval variation would not rule out any tonal references, but it appeared to be an effective strategy to avoid too obvious chord-scale applications. In his identification of the content of his Tone Clock hours and their fragments, Schat loosely alternates between notions of (meteorological) atmospheres and unfavorable associations to examples in classical music. For the time being, I prefer to remain connected to Liebman’s notions of implied tonalities, temporary tonal centers, tonal anchor, linear tonality and diatonic lyricism discussed in chapter 3.1.
ex 4.7.2.1
In the example below, trichord combination A+B from the row above, with trichord B put in retrograde, results in hexachords transposed in ascending steps of major thirds.
ex 4.7.5.2
As I already discussed in section 4.4.1, the trichords 2+3 and 3+2 are basic elements of the minor pentatonic scale.
ex 4.7.6.8
The combination of trichords A+B in their first rotations evokes the tonal color of D7(omit5) – Eb7(omit5). In the first line the example below, the trichord combinations are transposed by a major third up. In the second line the retrograde of the trichords are transposed down by minor seconds. The result sounds as a hexachord that is transposed by major seconds in descending direction.
ex 4.7.7.6
Both the first inversion of trichord 2+5 and the second inversion of trichord 5+2 display the trichord 5+5, a stacking of two intervals of a perfect fourth. In the example below, the first line I applied these inversions to the basic row steered by trichord 1+2. In the second line the same inversions are applied, but the order of occurrence in the basic row has been changed into A – C – B – D. As a result, the intervals sound at a distance of a minor third interval.
ex 4.7.7.8
Two variations of this sequence are played in the example below. Once again I have added two chromatic leading tones at the end of both trichord variations A+C and B+D above, to create four-note groupings and to suggest conventional voice leading at the connection of both parts of the sequence.
ex 4.7.11.1
4.7.12 Tetrachord 1+2+3
The next example shows the basic structure of a twelve-tone row constructed with tetrachord 1+2+3 and steered by trichord 4+4. This row does not belong to Babbitt’s all combinatorial four-notes sets, but was suggested by De Marez Oyens (1997) (see section 4.3.2).
ex 4.7.6.10
Both inversions of the trichords C and D contain the augmented fourth interval as well. Both the first and the second line display transpositions of trichord combination C+D, with C put in retrograde. In the first line they are transposed a minor third up, in the second line a major second down. The intervals between the trichords inside the combinations are again minor seconds.
The third line shows variations with trichord combination C+D in their second rotations. The first bar displays C+D ascending and, transposed a major third up, descending chromatically. The second bar continues the descending line by two combinations of trichords in alternating directions, a major second apart.
ex 4.7.7.9
Another way to create four-note groupings is to extend the trichord by repeating its first interval. For instance trichord 2+5 is extended into trichord 2+5+2 and trichord 5+2 is extended into trichord 5+2+5. In the next example this is applied to the trichord combinations A+B and C+D and their transpositions of a major second up.
ex 4.7.1.4
Next, both rotations of trichord 2+1 are shown. Both can be used to embellish the first three notes of a minor triad. In the first sequence minor triads are transposed in distances of minor thirds. In the second sequence they are arranged at distances of a major second.
ex 4.7.2.2
The descending variation of this sequence shows hexachords of trichord combination B+A, now with A in retrograde. It is interesting to hear how in this arrangement, the genuine minor/major7 harmonic color of the 1+3 trichord is overlaid by the whole-tone/augmented color that results from the steering trichord 4+4 and the major second intervals between the last and first notes of the hexachords.
ex 4.7.2.7
Trichord combination A+C shows a fragment from the C octatonic scale, while trichord combination B+D shows a fragment from de C# octatonic scale.
ex 4.7.3.4
The next example shows two chord embellishments that result from two different trichord combinations from the row above. Combining trichords B+D in their prime forms (with E♭ enharmonized to D♯) results in an embellishment of Bmaj7#5. The combination of the second rotation of B, and the prime form of C evokes a harmonic color of Amaj7/omit3.
ex 4.7.2.3
Other interesting trichord combinations with the trichord 1+3 family are A+C that sounds as a composite triad of C♯ augmented over C augmented, and D (retrograde)+A (retrograde) that evokes the sound of a C7b9 chord. Both trichord combinations sound particularly well as an ascending-descending riff that is phrased in five-note groupings, as shown in the second and third line of the example below.
ex 4.7.2.10
4.7.3 Trichord 1+4
The basic row of trichord 1+4 is steered by trichord 2+2.
Due to the sound of the perfect fourth in the outer voices of these trichords in prime form, in particular the 4+1 trichord refers to conventional V–i cadences. This is for instance the case in the row with steering 3+3 in the example below, which contains 4+1 trichords exclusively.
ex 4.7.6.3
The hexachord of trichords D (in retrograde) + C in the following example shows a variation in descending direction, transposed by distances of a major second.
ex 4.7.2.8
Both rotations of trichord A and C, each transposed in steps of minor thirds (T3), result in quite common embellishments of the octatonic scale, as we see in the next example.
ex 4.7.2.9
The same goes for both rotations of trichord combination B+D as is shown in the next example.
4.8 Conclusion
The applications of the Tone Clock in this chapter confirm the utility of its twelve-tone operations and elements for the practice of (composing) improvisers in jazz. They show compositional and improvisational techniques based on characteristic operations both with complete twelve-tone rows and with segments of these rows. In improvisations however the emphasis is more on manipulating these segments, namely the trichords and tetrachords of the Tone Clock hours. From the perspective of my research aim to develop improvisational techniques beyond tonal limitations, these operations with trichords and tetrachords appear to be attractive.
O’Gallagher (2013) provides examples of trichords and trichord combinations with explicit references to tonal colors of scales and chords, by which they are supposed to “blend” with the content characteristics of diatonic phrases. Just like Bergonzi (2000) and Garzone (2009), both discussed in chapter 3, I do not intend to refer to explicit tonal colors although I am aware that I cannot prevent them from emerging spontaneously. Few examples of these references can be found in my suggestions for improvisation in subchapter 4.5 and in the patterns presented in subchapter 4.7. The identification of implied tonal colors depends on the individual performers’ informed intuition (defined in section 3.3.1) and on their intention to deliberately avoid or create tonal connections. In the latter case the twelve-tone operations can be merged with fragments of conventional chord-scale approaches. In this respect I agree with Liebman who likewise expressed the purpose of his chromatic approach as “… to integrate this material alongside already established and familiar tonal ideas … into a coherent musical statement to satisfy both the intellectual and emotional needs of artistic creation" (Liebman 2013: 9).
The process of learning twelve-tone improvisational idioms is at large comparable with the traditional language acquisition of a jazz student. It includes the creation and manipulation of patterns as deep structures for "spontaneous" variations. Most jazz musicians are familiar with this learning strategy from the day they started to develop their improvisational skills. Similarly, this application of variation techniques is meant to enrich one's “backpack” with a choice of individual patterns that, after being securely memorized, will pop up “spontaneously” during improvisations. However, due to the absence of a substantial body of existing examples, developing twelve-tone skills will appeal more to the student’s individual curiosity than to his imitation and emulation of existing examples.
Beside the fruitful application of the Tone Clock as a model to generate interesting melodic patterns, its quality as an analytic tool also helps the (composing) improviser to enrich his artistic palette by applying interventions to existing repertoire. In subchapter 1.4.2, I explained how I re-composed two jazz standards into contrafacts as a step to develop my personal sound. How trichord analysis likewise can help to re-compose a diatonic melody I demonstrated in subchapter 3.7.2, with my transformation of Parkers “Quasimodo” into the contrafact “Quasi Mad Though”. There, I also showed how existing chord changes can be re-composed by twelve-tone ordered successions of alternative chord changes. The reharmonization of Davis’ “Tune Up” becoming the harmonic structure of “Count Your Blessings” can be considered as a convincing example. Next, in my discussion of Liebman’s “Invocation” I have shown how trichord analysis can be successfully applied to identify complex simultaneities that result from multiple contrapuntal lines and how such analysis can serve as a thoughtful point of departure for improvisations.
One could argue that trichordal analysis leads to a simplification of the complex web of intervallic structures in a composition or improvisation. It is obvious that selecting trichords, and reducing them to their prime form, neglects the surface of the music with its complexity of distinguishing features as registers, texture, tempo, rhythm, and dynamics. But it is also obvious from the transcriptions of the improvised fragments in this chapter that experienced (composing) improvisers are not at all tied down by this simple format in the deep structures of their musical grammar.
The theory of the Tone Clock strives for simplicity instead of complexity, allowing the performer to freely employ its “self-explaining quality” that Schat considered as “the most important requirement for a ‘common language’” (Schat 1998: 44). Thus it appears not only a useful, but also an easily accessible tool to embed twelve-tone techniques in the improvisational languages of contemporary jazz artists. By its limited number of intervals and orderings it looks like a relatively simple musical grammar that can be quickly memorized, without the compulsions and restrictions of a complex arithmetical system, but with the basic elements and operations of the twelve-tone system. Tonal references can be avoided or easily obscured by the twelve-tone techniques in its DNA, giving priority to orderings of notes and intervals above their tonal meanings. The next chapter highlights a compositional technique combining serial elements with tonal elements that are more obvious: Messiaen’s modes of limited transpositions.
