Démasqué” – my tenor saxophone lines in the collective solo on coda

3.4.3 Twelve-tone techniques


Although Garzone states that he is borrowing from the twelve-tone scale, he does not intend to create any twelve-tone collections. He designed his own limited applications of basic twelve-tone operations such as transposition, inversion and permutation instead. His most obvious ordering principle is that all triads should be ordered at distances of minor seconds, but he does not apply this strictly either.

Other elements are rooted into diatonic harmony, such as his limitation to the use of the four diatonic triads, and his avoidance of obvious twelve-tone terminology such as retrograde and rotation. Instead of the latter, he uses traditional tonal terminology such as first and second inversions of the triad. Likewise the applications of Garzone’s approach as an arranging tool to the jazz standard in the example above look more like random superimpositions than conscious twelve-tone operations.


3.4.4 Evaluation


In fact, Garzone is another author to discuss an individual approach aimed at the creation of tone rows by experienced performers. Just like Bergonzi’s model discussed in the previous section, Garzone’s approach addresses the advanced (composing) improviser who has both the curiosity and the technical ability to blend this distinctive model with his existing knowledge. Just like Bergonzi, a meaningful application of Garzone’s approach largely depends on the composing improviser’s informed intuition. Less experienced students might yet be confused by the quite strict playing rules as to the transposition, inversion and permutations of the four diatonic triads on the one hand, and the absence of any theory of its harmonic implications on the other.


At first sight, by its playing rules that are so easily explained, Garzone’s triadic approach offers attractive techniques for creating melodic lines beyond functional harmony. But the simplicity of this approach is also deceptive. In order to generate meaningful lines one needs to develop an agility to make multiple decisions. After the last note of one’s initial triad, in root position, first or second inversion, one has to take four decisions. The first is about the direction of the half-note coupling: either ascending or descending. The second is about the position of the next triad. Here, one again has  two choices because copying the position of the previous triad is not allowed. The third decision to make is about the direction of the triad: again two choices, ascending or descending. The fourth choice is to either play the triad “straight”, i.e. with the three notes in natural order, or to apply a permutation. Actually, with every step to the next triad this procedure has to be repeated.


Interestingly, the author himself presents his application of the model in a composed solo instead of, for instance, in a number of examples transcribed from solos.


As I experienced myself, after taking some time to embody Garzone’s way of manipulating triads chromatically, it enabled me to create surprising melodic lines particularly in a context without a stated harmony or with only a limited number of stationary chords.  For example as an intro on a pedal point or an ostinato in the bass part, as a solo over a stationary chord, or as an harmonic superimposition over the chord changes in a blues or in the A sections of “I Got Rhythm”, or other familiar forms that can be reduced to a single overlapping tonal color.

My personal experiences with Garzone’s method were surprising from the first time I started practicing it. At every attempt, I felt challenged to find a meaningful balance between its formally correct application and my spontaneous responses to the surprising tonal colors it inspired. The intense concentration this process requires sometimes results in a feeling of meditation. 

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3.4 George Garzone (2009). Basics of the Triadic Chromatic Approach.

Downbeat 76/5: 58.

3.4.1 Theory
Garzone calls his chromatic triadic approach “a conceptual theory […] put together on the blackboard to improvise freely in the way I would like myself and the students to improvise”. His method is an example of a simple model to create chromatic melodic lines. The author uses the four traditional types of diatonic triads: major, minor, augmented and diminished, to create melodic lines by connecting these triads in random inversions at a distance of a minor second. Apart from his “wish to be able to improvise freely” he does not share his thoughts on the tonal or post-tonal aspects of his method except that with his conceptual theory “you borrow from the 12-tone scale”.

3.4.2 Practical applications
In Garzone’s approach triads can be played with the notes in their traditional order of appearance, either in ascending or in descending direction, or in “displaced permutation”: with the notes sounding in alternating order. In the following example the C triad in root position is followed by the same triad in displaced permutation.

ex – displaced permutation (Garzone 2009:58)

The playing rules of Garzone’s Triadic Chromatic Approach are as follows: start with any triad in any position. From its last note either go one minor second up or one minor second down and play this as the first note of the next triad. This can again be played in any position, but one should avoid repeating the same position of the previous triad.

    The following example illustrates a correct application of Garzone’s approach. The C triad in root position is descending to the B triad in second inversion. The latter ascends back to the C triad in its first inversion that ascends to the Db triad in a displayed permutation of its root position.

ex – correct application

Because Garzone considers any displaced permutation of a triad as an inversion, he allows the same inversion to be used in adjacent triads as long as one of them is played in displaced permutation. In the following example the adjacent triads C and B are both put in second inversion, and the D and B triad are both in root position. Garzone allows them to be coupled this way because the B and D triads are put in displaced permutation.

Although Garzone presents his approach as a generative model for improvisers, the example of his composed solo above illustrates that it also can be used as an interesting arranger’s tool.


    The next example shows my correct application of the triadic chromatic approach in my improvisation on the coda of Démasqué, on pianist Andrea Pozza’s CD Gull’s Flight (2011). The rhythm section repeats eight times the notes as written in bars 1–4, while I freely improvise together with alto saxophonist Christian Brewer over a Bb pedal, ending on the Bmaj7 chord in the two final bars.  


     In bars 7–9, I play a chromatically descending line with the triads B (second inversion ascending, the fifth is repeated), Bb (second inversion descending, the third and fifth repeated) and A (root position descending). For the half-note coupling between the B and Bb in bar 8, the note e is played as an extra leading note. The half-note coupling between Bb and A is achieved after repeating the notes d and f from the Bb chord.


     The line in bars 11–15 contains the triads of F augmented (ascending, first inversion), Db (ascending, root position, displaced permutation), C (second inversion, displaced permutation), E (root position, displaced permutation, root note e is repeated), Bm (root position descending, the fifth is repeated), and A (root position, note of F added, third repeated, fifth repeated). Not all couplings in this line follow Garzone’s instructions literally. Half-note couplings are played between F+ and D, and between D and C. However the coupling between C and E is made by repeating the common note e. Next, the coupling between E and Bm is done by adding the note g, as a chromatic passing note to the note f in Bm. The coupling between Bm and A is done by adding the note c as the ninth of Bm and connecting this to the common note as the third in A6. The adding of extra notes to Bm and A in fact turns the triads into tetrachords. These operations are not provided in Garzone’s model, but turned out to be effective in generating a melodic line like this.


    In bars 19–20 the triad of C (first inversion, the third is repeated) connects to Abm (second inversion, displaced permutation) by a half-note coupling. Bar 18 can be considered a Gmaj7#5 chord with the third omitted, connected to C by a half-note coupling, but this is again an operation beyond the scope of Garzone’s triadic approach.

Another exemplary line in bars 21–25 shows two additions I made to Garzone’s triadic approach. The first is the coupling of the triads Bm in bar 23 and A in bar 24, by the interval of a major second. This whole-note coupling is also applied in bar 27 to connect B and A. The second addition is the use of incomplete triads, shown as C(omit3) in bar 21, B(omit3) at the transition of bars 24-25 and F7(omit5).

ex – displaced permutation

ex –  appropriate approach of the method (Garzone 2009:59)


In a composed solo over the chord changes of the jazz standard “Have You Met Miss Jones” Garzone puts his conceptual theory into practice. Garzone plays this composed solo as a countermelody in octaves with pianist Joey Calderazzo, while tenor saxophonist Joe Lovano plays the original melody. In the improvisation that follows Garzone loosely alternates between echoes of this composed triadic countermelody and lines that are firmly rooted in traditional linear improvisation.


    The next example shows this composed solo. While Garzone's article uses major triads exclusively to demonstrate his triadic chromatic approach, in this solo forty-five triads of different sorts can be found: twenty-eight major, four minor, four augmented and ten diminished triads. The connecting intervals between the triads are not only half steps, such as in bars 4 and 9, but also whole steps, such as in bars 15–16. The connections between the triads are created by repeating a note from the triad, such as in bars 22–23, or by inserting passing notes that are not in the triad. Examples of this can be found in bars 24 and 31. The triads are alternated with short diatonic and chromatic scale patterns.

    Apparently, the shape of the melodic lines prevails over application of the correct scales to the chords, which results in interesting examples of outside notes. For instance, in bar 8 where the note b is played instead of the correct b in the chord of C7, and in bar 14 where a B triad is superimposed on Dmin7. By choosing these notes, Garzone explicitly distances himself from a rigid chord-scale approach.

In the following melodic line Garzone demonstrates a proper approach of his method.


Have You Met Miss Jones” – composed solo (George Garzone)

ex – melodic pattern constructed with triad pair F+G (Weiskopf 2009:4)


Weiskopf discusses examples of a large number of triad pairs that are derived from the major and minor scales, the diminished, whole tone and augmented scales and the application of his method on polychords. The example above shows the most inside (i.e. diatonic) application of his approach on a Dm7 chord: a pair of major triads placed, a whole step apart, on the third and the fourth degree of the D Dorian mode. The next example deals with the dominant seventh sus4 chord. Here the most inside sounding triad pair is placed on the root and on the flat seventh, resulting in triad pair Bb+C on the chord C7sus4

ex – intervals resulting from triad pair operations (Weiskopf 2009:4)


As a practical application of his method Weiskopf expects the player to learn the different permutations of the triad pair and to use these to create melodic patterns such as the following.

ex – triad pairs on C#7alt (Weiskopf 2009: 10,11)


Another illustrative example of creating such an outside effect results from the adaptation of tritone substitutions to dominant chords. The next example shows the superimposition of a triad pair that belongs to the tritone substitution of the original G7 chord by Db7 in the ii-V-I progression Dmin7 – G7 – Cmaj7. 

3.5 Walt Weiskopf (2009). Intervallic Improvisation. The Modern Sound: A Step Beyond Linear Improvisation. New Albany: Jamey Aebershold Jazz.


3.5.1 Theory


Weiskopf’s book codifies a method of intervallic improvisation for advanced improvisers who want to modernize their vocabulary. It is motivated by his observation that jazz players, although they have a certain competence in the chord-scale approach that is the basis of linear improvisation, “often feel a large piece of the puzzle is still missing” (Weiskopf 2009:3). They hear modern jazz musicians improvise more intervallically and lack a method for developing this vocabulary.


    Weiskopf argues that intervallic improvisation has the same principle as traditional chord-scale improvisation in that the artist chooses material that sounds good over an underlying harmony. The difference is that in intervallic improvisation, selected intervals from the scales are played instead of traditional scale embellishments. Thus, in contrast to Liebman’s non-tonal superimpositions, Bergonzi’s intervallic melodies and Garzone’s chromatic triadic approach, Weiskopf’s concept of intervallic improvisation is explicitly based on existing harmonic structures.


3.5.2 Practical applications


The intervals discussed in Weiskopf’s book result from arranging pairs of triads that are derived from scales. These triad pairs – defined as “two triads that do not share any notes” (Weiskopf 2009:5) – can be derived from the modes of the major scale: the melodic and harmonic minor scales, the diminished, augmented, and whole tone scales. Every mode hosts one or more triad pairs that capture the tonal color of that particular mode. By playing these triad pairs, alternating their root positions and inversions, a series of intervals is represented.


    For instance the F major and G major triads capture the tonality of Dm7 because together these six notes contain all notes of the scale except the note e. 

ex – triad pairs on C7sus4 (Weiskopf 2009:6)


On major seventh chords the inside triads should be put on the fourth and the fifth degree of the scale,

ex – triad pairs sounding from inside to outside F#mb5 (Weiskopf 2009: 13,14)


In the last example Weiskopf’s strategy offers an interesting solution to make a Bb7#5 chord sound more interesting than by simply playing a whole tone scale over it. By superimposing the augmented scale (b, d, d, f, f, a) or the inverted augmented scale (b, b, d, e, f, g) he creates alternatives that sound more outside the chord.

ex – triad pair F+G with inversions



With the two triads played in root position, first and second inversion, the following succession of intervals of major thirds (M3), minor thirds (m3), perfect fourths (P4) and perfect fifth (P5) occurs. 


"Bohemia After Dark" –fragment tenor saxophone solo in first B-section

Then, the line in the B-section of the next chorus, as shown in bars 2–5, shows a variation on Weiskopf’s model. Here the triad pair C (first inversion) + B (second inversion, the third is repeated) is followed by a series of four pairs of dyads that can be considered incomplete C and B triads. The effect is the same as with the triad pairs in the previous example: the original C7 sound is obscured, and in this case delayed to bar 5, where a diatonic melodic pattern confirms its original tonal color.

ex – triad pairs on Cmaj7 (Weiskopf 2009:7)


or, in the case of an augmented fourth (#11) sounding in the chord, on the root and the second degree of the scale.


ex – triad pairs on Cmaj7#11 (Weiskopf 2009:7)





ex – triad pairs on ii-V-I (Weiskopf 2009:19)


The musical examples in the introduction are followed by a table of triad pairs, listing eight categories of scales and their derived triad pairs: major, minor and augmented triads ordered at a whole step, a half step, a tritone and a minor third apart, in twelve keys. In the left column of his table the possible triad pairs are listed, in the right one the appropriate keys. The four columns in between mark the possible chords symbols that the triad pairs can be applied to.

    The next example from his table of triad pairs shows the author’s options of triad pairs in harmonic major scales.

ex – intervals resulting from triad pair operations


In contrast to Liebman’s non-tonal superimpositions, Bergonzi’s intervallic approach, and Garzone’s triadic chromatic approach, Weiskopf’s method is intended to create tonal superimpositions on chord changes. That is why it should be considered as an advanced chord-scale approach rather than as a technique to deliberately play outside the chord changes. Thus, at first sight, there are no intended elements of twelve-tone theory or operations in his method. However, Weiskopf's way of manipulating diatonic triad pairs helps the improviser to create lines with characteristic shapes, with high densities of texture and with intentionally vague harmonic colors that according to Liebman could be considered as twelve-tone characteristics. Furthermore, although Weiskopf’s triad pairs match the scales from which they are derived correctly, he also manages to create an amount of harmonic vagueness by gradually obscuring the underlying harmonic content as the triad patterns become “more advanced”.


For instance, of the six examples of triad pairs derived from half diminished chords the first and fourth sound the most inside because the triad pairs include the root, the minor third, the flatted fifth and the minor seventh of the Fm7b5 chord. The second has the major ninth instead of the more common flatted ninth. The third triad pair misses the flatted fifth of the chord and the fifth triad pair misses the minor third. As a result of the absence of these characteristic notes, these triad pairs sound outside the expected Fm7b5 tonality. The sixth triad pair is derived from the E augmented scale that does not imply the root or the minor third of the tonality, but it implies both the flatted ninth and the major ninth instead. As a result this triad pair sounds pretty much outside but still has a discernable relation to the underlying chord.

ex – triad pairs derived from augmented scales (Weiskopf 2009:16)



3.5.4 Evaluation


Compared to the authors discussed earlier in this chapter, Weiskopf’s publication is exclusively addressed to performing improvisers. His approach is useful as a step-by-step method to extend one’s improvisations beyond conventional linear improvisation, because it guides the reader from the most inside to the more outside sounding triad pairs.


    Between Garzone’s triadic chromatic approach and Weiskopf’s pairs of triads there are more differences than similarities. Garzone’s approach largely depends on the performer’s informed intuition and stands far away from Weiskopf’s didactical approach of arranging triad pairs. For the coupling of his triads, Garzone principally uses minor seconds. Weiskopf uses four intervals, coupling his triad pairs by intervals of a minor second, a major second, a minor third and a tritone apart. Due to this difference, there are only two overlaps between the two: Weiskopf’s harmonic major derivations (one major and one augmented triad, a minor second apart) and his inverted augmented scale derivations (two augmented triads, a minor second apart) could also result from applying Garzone’s method.


    In the context of my research into developing improvisational techniques beyond the conventional chord-scale approach I am principally interested in the possibilities of Weiskopf’s method to create, in Liebman’s terms, a certain harmonic vagueness. Despite its obvious connections to the functional harmonic fabric it can serve as an effective tool to start making "educated steps" outside the stated chords. I experience this both in my own practice as with the students in my classes of saxophone improvisation at Codarts University of the Arts. However, the fact that this technique facilitates the improviser to create approved melodic lines just by playing variations of the same six notes implies the danger of a mechanical approach. “Remembering” the right triad pair might prevail over “actually hearing” its musical meaning in relation to the underlying harmonies. But again, considering the aim of my research to help improvisers sound outside the chords, this drawback seems less momentous. From experience I have noticed the advantage of this tool to evoke a tonal color that suggests an outside effect, while at the same time I “trust” that there is a plausible connection with the underlying harmonic structure. It is hard to cut my roots in chord-scale improvisation. 

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ex – triad pairs on Dmmaj7 (Weiskopf 2009:10)


Different triad pairs derived from the same melodic minor scale will result in different tonal colors. This is shown in the next examples of two different triad pairs, derived from the D minor melodic scale that is imposed on the C7altered chord. Both lines contain the triad pairs G and A, but in the first, the note e in the A triad is changed to an f. This f, together with the notes d, a, and c, confirms the tonal color of d minor melodic more convincingly than in the second example. where the triad pair G and A is used without this altered note. This triad pair creates an ambiguous tonality. It can sound either as G#11, or as an A7sus4 chord on a C# pedal point.



Master Slow Feet” – fragment tenor saxophone solo


The following example shows my application of triad pairs in the B-sections of three solo choruses on Oscar Pettiford’s Bohemia After Dark.The transcription shows a development in my operations with the triads. The line in bars 3–7 is an exemplary illustration of turning the tonal color of the C7 chord into C7sus4, by playing the triad pair C+Bb in alternating positions and directions. 


“Bohemia After Dark” – fragment tenor saxophone solo in third B-section



3.5.3 Twelve-tone techniques


In Weiskopf’s approach, a succession of intervals is defined by the internal structures of the four diatonic triads in prime form and inversions. His model does not specify the intervals between the trichords that result from his triad operations. For instance, looking once again at example above, the intervals inside the F and G major triads are major and minor thirds and perfect fourths, as a result of their positions in prime form, first and second inversion. However, the intervals between the first notes of the triads, four major seconds and one minor second – and those between last and first notes of the triads– two perfect fourths, a tritone and two perfect fifths – are not taken into account. However characteristic the sounds of these additional intervals are, they just occur by accident, as an interesting “bycatch” after the arrangement of the triads. 

Weiskopf extensively discusses examples of triad pairs that are derived from the harmonic and melodic minor scales, for instance in the following example. Because in the ii-V-I progression Emin7b5 – A7b9 – Dm, all chords are in the key of D harmonic minor, the triad pair Gm, Amaj#5 will sound well and inside all three chords - recalling a common embellishment of the minor chord in the bebop style.

ex – harmonic major derivations: a major and an augmented triad a half step (Weiskopf 2009:21)


The next fragment of eight bars from my tenor saxophone solo in a recording of my composition “Master Slow Feet” (2015) shows my application of triad pairs.

    Instead of grouping two triads on top of the D7 chord in bars 1–2, I group three triads. The triad pair E+D, superimposed on the D7 results in the tonal color of D7#11, while the triad pair C+D evokes the sound of D7sus4. Apparently the slow straight rhythmic groove of this tune allowed me the opportunity to combine both ideas.

    In the line in bars 3–6 the triad pair E+D is played over both the Bm7 and again over the D7 chord. As superimposed on Bm7, these triads define the tonal color of the B Dorian mode. In bars 5–6, the triad pair C and D again follows the superimposition of the triad pair D+E on D7, just as in the first line.

ex – triad pairs referring to the C7 octatonic scale (Bergonzi 2009: 117, 281,247)


Concerning the phrasing of his triad combinations Bergonzi also manages to create more diversity than Weiskopf (2009). In order to avoid an abundance of three-note groupings or four-note groupings in which one of the pitches of the triad is repeated, he creates four-note groupings with fragments of two notes from the triads as illustrated in the first line of the example below. The second line shows how he creates five-note groupings by skipping a note in one of the triads.

ex – variations of triad combination Gm+Am (Bergonzi 2006: 276)


Both examples illustrate how Bergonzi’s method serves to create diversity in superimpositions with triad combinations.



3.6.3 Twelve-tone techniques


Although Bergonzi does not discuss any relations to twelve-tone techniques, some of his patterns can be considered as one half of a twelve-tone row. The next example shows his category “augmented over augmented a half step apart”. Due to its symmetry it refers to a number of chords arranged at distances of major third intervals. 

3.6 Jerry Bergonzi (2006). Hexatonics. Advance Music.


3.6.1 Theory


In Hexatonics (2006) Bergonzi displays his method of creating melodic devices by combining two triads that do not have any common tones. His system looks similar to Weiskopf (2009), but there is one major difference. In contrast to Weiskopf who derives his triad pairs from existing diatonic scales and church modes, Bergonzi defines his collection of sixteen triad pairs and hexatonic scales simply as “the ones that are practical for the improviser and composer” (Bergonzi 2006:6). His collection is randomly ordered along the tonalities of his combined triads (major, minor, and augmented) and the distant intervals between them (major second, minor second, minor third, and tritone). Again, just as with his intervallic system elaborated upon in subchapter 3.3, Bergonzi exposes a relatively simple device generating unlimited possibilities. And again, the suggested operations are related to the musician’s informed intuition rather than to a comprehensive theoretical concept. As a consequence it is left to the discretion of the individual artist to decide which tonal references can be associated to which triad combinations.


3.6.2 Practical applications


Bergonzi categorizes his collection of sixteen hexatonics along the tonal qualities of their triad pairs and the intervals between them.  To each category he adds a résumé of the various chords that they are supposed to match.

For instance in the next example, in the category “major over major a whole step apart” he puts a D major triad over a C major triad (D/C). In the first bar these triads are put in succession, and in the second bar they are intertwined to form a hexatonic scale. The chords that Bergonzi considers this hexatonic refers to are written above the bar.

ex –  “major over major a whole step apart” (Bergonzi 2006:9)


The next example illustrates how diversity results from Bergonzi’s approach taking a variety of triad pairs as a point of departure instead of just picking two triads from an existing scale as in Weiskopf (2009). The triad pairs in bars 1, 3, and 5 sound really different but, as the chords written above bars 2,4, and 6 indicate, they all refer to the C octatonic scale. 


Cmaj chords resulting from stacked triad pairs


The two highlighted chords above are illustrated in the following examples.  The first illustrates two variations of triad pair Bm+C(5), superimposed on C major. Both lines contain five-note groupings constructed with three notes of the C♯5 triad and two notes of the Bm triad in alternating ascending and descending directions.

ex – transposition of hexatonic scale completes a twelve-tone row


This, and other types of twelve-tone techniques will be discussed in detail in chapter 4.  


3.6.4 Evaluation


After what has been said about Garzone (2009) in subchapter 3.4 and about Weiskopf (2009) in subchapter 3.5, I cannot disagree with Bergonzi considering triads as “incredibly strong melodic devices. They are easy to think of and combining them is an accessible task for the soloist” (Bergonzi 2006: 6). In my experience it is the diversity of Bergonzi’s suggested triads, rather than the related hexatonic scales that is most valuable here. Treating the hexatonic scales as separate objects risks a reduction of not only the accessibility but also the effectiveness of this method. Because they can easily be identified as fragments of existing diatonic, octatonic and whole-tone scales they contain the danger of emphasizing obvious tonal elements instead of moving beyond them.


    In contrast to Weiskopf’s presenting his method as a strategy to modernize the techniques of the improvising performer, Bergonzi advocates his method as a tool not only suitable for improvisation, but also for arranging, and composing. Experience with application by my students and myself has proven it to be a valuable improvisational tool, offering improvisers freedom of choice as to how and when it should be applied to the underlying harmonies. Moreover it also incorporates an advanced chord-scale approach that can be very effective in creating outside sounding patterns on pre-given chord symbols.


    Thus, by these various options to move beyond the limits of tonality, Bergonzi’s method relates very well to the subject of my research. Weiskopf’s Intervallic Improvisation, by its focus on finding triad pairs in familiar diatonic scales and church modes, could serve as a pre-stage for Bergonzi’s Hexatonix’s creating a more varied sense of harmonic vagueness. But just like Weiskopf, also Bergonzi's model also contains the danger of a rather mechanical approach, namely by playing the combinations one has studied by muscle memory instead of actually hearing them in the actual moment. Therefore these steps outside the pre-given chords should always go hand in hand with patient and conscious ear training. The same might go even more so for another method enriching the (composing) improviser’s palette to be reviewed in the following chapter: O’Gallagher’s application of twelve-tone techniques in improvisation.

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ex – variations of triad pair Bm+C♯5 (Bergonzi 2006:276)


In the second example the superimposition of triad combination Gm+Am manages to create an even more outside sound. 

ex – “augmented over augmented, a half step apart” (Bergonzi 2006:91)


By transposing its hexatonic scale up a tritone and adding the resulting scale to the original one, the twelve-tone row in bar 2 is derived. It consists of three triads with the same interval structure of a minor second and a minor third.

ex – alternatives to three-note groupings (Bergonzi 2006: 124, 282)



In the context of my research goal to create improvisational patterns that sound outside the chords, I consider a reverse application of Bergonzi’s model to be more fruitful. Instead of Bergonzi’s associating a single combination of triads to a variety of possible chords, I suggest to map a variety of triad combinations that could be associated to a single chord. I would for instance take a single major chord and map the vertically stacked hexatonics in which (according to Bergonzi) this chord is identified, ordering them from more inside to more outside the sound of the basic chord. The next example shows this in a succession of C major chords arranged in increasing complexity. Below the staff are the triad combinations that according to Bergonzi refer to these chords. 

ex – trichord 1+4 and 4+1 (O’Gallagher 2013: 97)


The remaining five trichords are symmetric and contain the same distances between the pitches: 1+1; 2+2; 3+3; 4+4 and 2+5; the first four are presented in only one version because rotation doesn’t change their intervallic structure. 

ex – combinations of trichords 1+2 and 2+1  (O’Gallagher 2013: 37-39)


After the exposition of all trichords, O’Gallagher thoroughly discusses their diatonic applications in three steps. First, he superimposes single trichords as chord voicings over all 12 tones as bass notes and attributes the appropriate harmonic qualities as chord symbols to each of them. Next, he superimposes single tri-chord and combined tri-chord applications that fit the chord tones and tensions from the associated modes on the “common chord types” Gm, C7 and Fmaj7. The next example shows trichord 1+3 in prime form and in three transpositions, fitting respectively the modes of C mixolydian b2; C blues; C Lydian b7; and C half-whole tone diminished.

ex – twelve-tone row 1+5 and 5+1 (O’Gallagher 2013:24)


O’Gallager’s method discusses all trichords in five steps. First he lists the trichords in their prime forms, first and second rotations, followed by a survey of all possible combinations of two trichords. Seven trichords are non-symmetric, with different intervals between the three pitches: 1+2; 1+3; 1+4; 1+5; 2+3; 2+4 and 3+4. They are presented in two versions. The first starts with the smaller interval followed by the larger interval. The second goes the other way around. 

ex – trichord 4+4 (O’Gallagher 2013:244)


Trichord 2+5 is an exception. O'Gallagher considers this a symmetric trichord because its first rotation turns the non-symmetric 2+5 into a symmetric 4+4.


Next O’Gallagher constructs trichord combinations from the row. In the following example the trichords in row 1+2 are ordered as A B C D. 

ex  – row 1+2 (O’Gallagher 2013:50)


From this row, the following trichord combinations can be derived: A+B; C+D; A+C; B+D; B+C; and D+A. In the resulting six-note patterns displayed in the following example each pattern is transposed to start from the root note c. Trichord combinations C+D and D+A are left out because, as a result of this transposition, they would sound exactly the same as combinations A+B and B+C.

ex – single trichord 1+3 over C7 (O’Gallagher 2013:84)


As his third step, O’Gallagher superimposes all possible sets of two trichords on each of the three common chord types. Many of the diatonic applications of these sets depend on the ear of the performer. They can contain a mixture of inside and outside trichords, for instance the next example, superimposed on Gm.  To my ears it sounds as a D7b9b10 chord, with a pull to Gm.

ex – diatonic applications of trichord 2+3 on Fmaj7 (O’Gallagher 2013: 151)


Only after his extended discussions of the trichords and trichord combinations from each row, does O’Gallagher display its complete form, its steering trichords and the number of unique transpositions, i.e. the number of transpositions that is possible before the row repeats itself in another key. To some of the rows he also connects their characteristic tonal colors, such as diminished or augmented, or a color that suggests the sound of any of Messiaen’s modes of limited transposition.


    Finally, he discusses the related or embedded rows. These rows are constructed with the so-called steering trichords that are found by grouping the first notes of the trichords in the basic row. Five of the non-symmetric rows and two of the symmetric rows can appear in different orderings that are related to different steering trichords. The reader is advised to use these relationships first in compositions and then later in improvisations. I will come back to the issue of the steering hours in chapter 4.

     The following table summarizes the trichords, their number of possible constructions, the number of unique transpositions, their steering trichords and their tonal colors. 

ex – combined 1+3 trichords superimposed on Gm (O’Gallagher 2013:85)


As to the diatonic application of his trichords and combinations of trichords on common chord types, O’Gallagher’s method is able to define levels of dissonancy by manipulating the trichords. The next example illustrates diatonic applications of the trichord 2+3 on Fmaj7. The first line shows row 2+3 and 3+2 in prime form. The second shows a succession of combinations of prime form and transposed trichord combinations A+B (2+3 and 3+2). The ones marked with an asterisk are the most important in defining the sound of the chord. In other words they sound the most inside. The third line shows combinations of set A+C. The interval of an augmented fourth between the initial notes of the trichords cause these combinations to sound more outside.

ex – re-composition on the basis of tri-chord analysis of “Quasimodo”


The next example shows my application of trichords to reharmonize the chord changes of the jazz standard “Tune Up”(Davis) by superimposing combinations of trichord 1+5. The following line shows how the prime form (P) of trichord combination A+B from the 1+5 row is transposed by T7 (a perfect fifth), T5 (a perfect fourth) and T3 (a minor third). In my re-composition the pitches resulting from these operations become the root notes of the new chord changes.

ex – transpositions of A+B from row 1+5


In the next example the upper staff shows the original melody and chords, the lower my superimposed harmonies. The notes in the original melody that conflict with the new accompanying chords are crossed.


ex – table of trichord characteristics



3.7.2 Practical applications


The analysis of trichord types in a melodic and harmonic context as discussed by O'Gallagher in his introduction inspired me to apply a trichord reduction on Charlie Parker’s “Quasimodo”and to redistribute the pitches in random permutations and intuitive rhythmic phrasings, while I kept to the original chords. Next I re-harmonized Miles Davis’ “Tune Up” by superimposing a new chord structure based on a succession of 1+5 trichords on the original chord changes.


The following three examples illustrate my trichord operations as applied to “Quasimodo”. The first example is the lead sheet for tenor saxophone in Bb. The second example is my trichord analysis of the theme. The third example shows the lead sheet of “Quasi Mad Though”, my re-composition of “Quasimodo” on the basis of this trichord analysis.

ex  – trichord analysis of “Quasimodo”

ex - “Tune Up” (Miles Davis) – reharmonization with combinations of trichord 1+5)



The following example displays my composition “Count Your Blessings” in an arrangement for tenor saxophone, trumpet, trombone, bass guitar and drums. The upper staff shows the trumpet and the tenor saxophone melodies. The lower staff shows the trombone part. In between the staffs are the basic accompanying chords that also serve as chord changes for the solos. I imagine that, just as with the concept of “Coltrane changes”, tenor saxophonist John Coltrane’s superimposed chord structures on ii-V-i cadences, my chord changes derived from the fifth hour can be used alike to add a fresh sound to these conventional chord changes. This application of the Tone Clock serving as a bridge between dodecaphony and tonality can be considered a useful technique to help improvisers enriching their artistic palette.

ex – trichords from the row in various shapes (O’Gallagher 2013:54)


In addition to the five publications discussed earlier in this chapter, O’Gallagher’s book offers a new perspective on playing outside the chords that highly meets my motivation to undertake this research project. With his pragmatic approach of an analytical model related to twelve-tone music, he introduced an alternative approach to intervallic improvisation, in the form of a detailed and systematic device with clear playing rules. His method is applicable both in twelve-tone as in tonal contexts. Thereby, in the improvisers’ “backpacks” it can be both connected to, and stored alongside their existing knowledge and practices. Because O’Gallagher’s publication combines the characteristics of a thesaurus with those of a didactical method, it offers the student multiple directions towards the selection of his personal applications.


    My analysis of O’Gallagher’s method showed three major problems. First, as I mentioned before, in the practical applications of his twelve-tone approach, composing and arranging don’t receive as much attention as improvisation. Second, the excessive collections of tonal references of all trichords and trichord combinations illustrate the drawback of presenting entire numbers of options in the form of a thesaurus.   Because these patterns are not related to musical examples, for less experienced readers they will look like barely meaningful patterns that display a more theoretical than practical quality. Third, and more importantly, O’Gallagher did not offer clear solutions for a major rhythmic problem with his trichordal approach: the overwhelming three-ness, or more precisely: the predominant presence of three-note groupings. These problems will be addressed in chapter 4.7.

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ex – row with 1+2 R1, embellishing the scale of F major


The second example shows my application of the trichord 2+5 in combination with two whole-tone scales. In bar 1 the trichord 2+5 is placed on every degree of the C whole-tone scale and in bar 2 the trichord 5+2 is placed on every degree of the Db whole-tone scale. By merging these two scales and by alternately playing the trichords in ascending and descending directions the row in bars 3–4 results. This row contains two twelve-tone rows, each with six notes repeated. These repeated notes, marked in grey colors, together form the C whole-tone scale. With its high density this scale pattern lacks an obvious tonal reference, making it an attractive pattern to be superimposed on using a multitude of pre-given or imaginary chord changes.

ex - rotations from common tones (O’Gallagher 2013: 32,33)



3.7.4 Evaluation

As a composing improviser, O’Gallagher created a method for twelve-tone improvisation based on the analytic model of the Tone Clock that was designed by Schat, a twentieth century composer of modern classical music with no connections to jazz and improvised music. The relationship between composition and improvisation based on this actual model is however missing in O’Gallagher’s book; he mainly addresses the improvising performer. The act of composing is merely mentioned as a pedagogical tool to learn to apply the trichords, their combinations and their variations. His advice to examine the relationship between the rows, by applying them first in compositions, before they are transferred into improvisations is quite open-ended. Furthermore, although the players of chordal instruments are advised to “play these trichord exercises as simultaneities as well” (O’Gallagher 2013:70,96), the polyphonic application of the method is also given few attention.


    As to the aesthetic aspect of his approach, I consider it an inspiring tool that enables a performer to create infinite quantities of interesting sounds by manipulating a relatively small number of basic intervals, rows, and operations. It helped me to deepen my hearing and understanding of interval relationship, as a useful addition to my existing skills.  Because of the large number of detailed examples, O’Gallagher’s book can be considered a thesaurus of operations with trichords and related rows. “Whether it works” depends on the artistic motivation of the individual artist who wants to find out “how it works” for him.  


    One could argue that the melodic lines that result from applying O’Gallagher’s complex operations don’t always allow the informed listener to clearly identify the distinct trichords and trichord combinations. I assume that O’Gallagher’s preferred audience has a level of experience with listening to complex music, by which it is able to appreciate his twelve-tone approach as an interesting way of playing outside the beaten tracks of jazz improvisation. However, particularly in the initial stage of mastering these techniques of intervallic improvisation, performers should be able to hear and play the distinct trichords clearly before they start blending them with members of the other trichord families.


    Unfortunately, the implications of the predominant three-ness as a result of the trichord manipulations are not given attention in his book. The same goes for the directions of the lines. Although in the summary of trichord exercises the student is advised to practice the trichord and its inversions in various directions, it is striking that the exercises are basically written in ascending directions. Only at the end of the exposition of row 1+2 does O'Gallagher display examples of trichord combinations in both ascending and descending directions. For instance in the next example he creates a melodic line using the trichords from row 1+2 in alternating directions. The connections between the trichord sets are made by using the next closest neighboring tone and continuing the direction of the line. There are a variety of rotations in this line that follow the set pattern ABCD repeatedly (O’Gallagher 2013:67). 

ex – twelve-tone pattern of trichord 2+5 combined with two whole tone scales



3.7.3 Twelve-tone techniques

In contrast to Liebman’s treatise on chromatism, Bergonzi’s systematic intervallic method, Garzone’s chromatic triadic approach, and Weiskopf’s and Bergonzi’s applications of triad pairs and, O’Gallagher’s method is explicitly based on twelve-tone techniques. Both in his examples and in his practical instructions, he thoroughly addresses the twelve-tone operations of transposition, rotation and extension. For example, he advises to apply the following seven exercises to every type of trichord as follows. (O’Gallagher 2013:36) I have marked the appropriate techniques between parentheses.


1. Practice root position trichords chromatically in various directions [transposition].

2. The same with first rotation trichords [rotation, transposition].

3. The same with second rotation trichords [rotation, transposition].

4. Practice longer patterns of each trichord using all rotations [rotation and extension].

5. Practice trichord rotations from common tones [rotation, transposition].

6. Practice freely combining the various transpositions and rotations [rotation, transposition].

7. Practice all two trichord combinations using all transpositions [transposition, extension]


    Furthermore, in his exposition of row 1+2 he demonstrates retrograde, inversion and retrograde inversion as “basic operations on the row” (O’Gallagher 2013: 52) listing all possible combinations of two, three and four trichord combinations from the row and outlining comprehensive row exercises.


    Compared to Bergonzi, Garzone and Weiskopf, the performer who wants to become familiar with O’Gallagher’s operations will have to spend substantially more time and effort mainly on the typical composer’s operations that are suggested. For instance the following rotation from a common pitch – referred to as “one technique by Stravinsky and other composers to develop a row” (O’Gallagher 2013:6) - is an interesting addition to the technique of mere transposition to get familiar with the trichords, but it takes time before a composer’s operation like this will pop up spontaneously in an improvised line. 

3.7.  John O’Gallagher (2013). Twelve-Tone Improvisation. A Method for Using Tone Rows in Jazz. Mainz: Advance Music.


3.7.1 Theory


O’Gallagher’s publication connects twelve-tone applications to jazz improvisation. His method is based on the system of Schat’s Tone Clock. This analytic model consists of 12 twelve-tone rows, each of them constructed by sets of similar tri-chords types. The twelve-tone rows are interrelated because they share common interval connections. In chapter 4 applications of the theory of Schat’s Tone Clock will be discussed in detail.


    O’Gallagher transfers Schat’s compositional system into a practical tool for the improviser. As the theoretical background for his trichord and row operations he briefly discusses Anton Webern’s use of derived rows based on trichords and his trichord transformations: inversion, retrograde, retrograde inversion and transposition. He recommends transposition as the key technique to get familiar with the trichords and their inversions, with the triad combinations and with the rows as a whole. The other three he considers “operations composers typically applied to twelve-tone rows.” Of these, retrograde is more familiar to improvising performers – as the descending way of playing traditional scales and modes – than inversion and retrograde inversion, which are more current in composer’s practices. O'Gallagher's connection with twelve-tone techniques is also obvious by his terms and definitions such as pitch (note); set (a collection of pitches either ordered or unordered); tri-chord (any set or segment of 3 pitches); tetrachord (any set or segment of 4 pitches, in traditional jazz harmony known as a seventh chord); prime form (the fundamental form of a trichord, a set of trichords or a row from which all variations are derived, in traditional harmony called root position); rotation (a cyclic permutation of either pitches or sets in which the first pitch of a set is moved to the last positions or the first set is moved to the last position of the set order, in traditional harmony called inversion).


    Despite these connections to the domain of twelve-tone composition O’Gallagher explicitly discards the basic rule that rows should be careful not to imply any kind of traditional harmonic movement. Moreover, the student is advised to use his method creatively and to not limit himself exclusively to the twelve-tone system, because, according to O'Gallagher, any combination of tri-chords can generate a musically valid statement without necessarily using a twelve-tone row. With this he confirms the observation of Wuorinen that, in the current state of twelve-tone practice, the main determinants of musical coherence are “the ordered successions of intervals” and that  “even twelve-ness […] cease[s] to have the fundamental significance [it was] once thought to possess” (Wuorinen 1994:8).


    O'Gallagher presents his devices as practice tools “to help the ear create interesting sounds.”His exercises are intended to train the ear to a new way of hearing harmonic and intervallic space, as an addition to the linear improvisation skills that are taught in traditional jazz education. The goal of this method is to expand the reader’s melodic content and understanding of interval relationships. Working with his exercises would enable a musician to recognize the distinct sound of each tri-chord and its corresponding row in the same way he recognizes harmonies from major and minor scales. O’Gallagher claims that although his method can be approached as a tool that generates musical lines and chords that are not related to traditional music theory, it can also be applied to functional harmony. The way his exercises are arranged represents this duality. 

ex - “Quasimodo” (Charlie Parker) – original theme (Bb-instruments)

    The three main elements in O’Gallagher’s method are intervals, trichords and twelve-tone rows. These rows consist of four tri-chords and the tri-chords consist of two intervals.

The sizes of the intervals in the trichords range between a minor second and a perfect fourth. They are indicated by figures: 1 means a minor second; 2 a major second; 3 a minor third; 4 a major third; and 5 a perfect fourth. Each trichord is marked by the two figures of its intervals – the smallest interval first –, connected with a plus sign. For example, interval 1+4 means a minor second, followed by a major third; for instance c–c–f. The twelve-tone rows each consist of four members of the same trichord family, arranged in such a way that a complete twelve-tone row results. For instance, of the four trichords of the 1+5 family in the twelve-tone row below, the first and the third display the order 1+5. The second and the fourth trichord display the order 5+1. 


Count Your Blessings



The following applications of trichords and rows in a tonal context are examples of operations that quickly became part of my personal improvisational language. The first is based on row 1+2. Because of the combination of the sound of a major second between the first and the second note and the sound of a major seventh between the first and the third, I became attracted to the first rotation of row 1+2, as an operation to embellish pitches, groups of pitches and complete scales. The next example shows the embellishment of the scale of F major. All trichords are constructed as a first inversion of 1+2. As a result of this intervention, the scale is represented in the top notes while the lowest notes form the scale of G major. 

3.8 Comparing the models


The authors of all six publications reviewed in this chapter express a variety of motivations to help (composing) improvisers learn new musical techniques. The effectiveness of their methods depends on the attitude of the students. Liebman expresses that an exploratory attitude in his students is crucial. He compares their artistic motivation to extend their playing beyond the limitations of functional harmony to their response to contemporary music, favoring those who prefer the dissonant sounds of the music of the twentieth century above the ordered diatonicism of earlier periods. Bergonzi’s remarks on the importance and value of the musician’s intuition allowed me to propose the concept “informed intuition” in order to express that this “precious gift” can probably never be isolated from the knowledge the students have achieved through study, experience and taste. Garzone addresses the motivational issue in plain language, saying that his method should simply enable him and his students to improvise freely.  Weiskopf presents his book as a response to the actual need of his students to learn how to improvise with intervals because they hear that many modern jazz artists do so. O’Gallagher aims at those students who are motivated to learn a new way of hearing harmonic and intervallic space.


As to the applications of twelve-tone techniques, all authors in one way or another present their personal strategies to create tone rows. The term tone rows is not used in the traditional sense of a set of twelve pitches with strict serial principles, but to express that these lines are not based on diatonic or modal scales, but on successions of pitches that are ordered by intervals. Liebman, Garzone and O’Gallagher mention their relations to twelve-tone techniques explicitly, while Bergonzi and Weiskopf only implicitly refer to these techniques by taking intervallic constructions of melodic lines as their focal points. All five authors leave it to the discretion of the individual musician to make his choices from their models.


Remarkably, all authors consider their models as an addition to existing chord-scale improvisation. Liebman and O’Gallagher emphasize the importance of substantial expertise in linear improvisation before making steps into their new areas. Just as Weiskopf, they are able to precisely define the relationships between their operations and the underlying chord structures. O’Gallagher’s, Bergonzi’s and Garzone’s intervallic approaches basically intend to not imply tonal references, but Bergonzi argues and Garzone demonstrates that their systems can not be seen as totally distracted from conventional functional harmony.


O’Gallagher even emphasizes that, although his system provides his readership with the tools to hear and think in the twelve-tone system, it should not be limited exclusively to twelve-tone usages. Thus, his model is meant, in Wuorinen’s terms, to create interactions between content and order of pitches and intervals. Weiskopf assumes that learning intervallic improvisation will improve his students’ linear improvisation. Bergonzi and Garzone are more in favor of an intuitive approach. Bergonzi loosely formulates that “everyone internalizes and applies concepts in a unique way and [his] system is wide open for interpretation” (Bergonzi 2000:7). Likewise the performers applying his operations discussed in Hexatonics (2006) are expected to rely on their informed intuition in order to find their way through the wealth of possible melodic and harmonic possibilities with this actual method.


Liebman, Bergonzi, and O’Gallagher explicitly advocate didactical applications of their methods such as ear training, rhythmic development, and composition of individual exercises. Although in the context of the actual study I am mainly interested in the creative applications of these and of related approaches, I now take a moment to summarize my experiences with applying their models in my own practice as an educator.


Liebman’s treatise supported my educational practice just like it had supported my playing skills. It served as a roadmap for the journey into the more advanced improvisational practice beyond the chord-scale approach. His analysis of this developmental process, his chromatic concepts of tonal and non-tonal superimpositions and his sketch of the parallels with the development of classical music through the ages served as an illustrative theoretical background.  Bergonzi’s intervallic approach appeared to work well in a small ensemble class when the students were asked to write short compositions using a collection of three intervals. All students took advantage of having this clear context as a point of departure. Because I left it open how strictly they should apply the operations with the orders and directions of their selected intervals, the results showed a large variety of compositions. The approach served particularly well to accelerate their process of getting started with composing and arranging their notes. Bergonzi’s model appeared easily accessible as a compositional device, but any spontaneous application in the improvised part caused serious problems to most of the students. Small exercises constructed with random selections helped them to develop their awareness and hearing of the interval combinations.


The introduction of O’Gallagher’s approach in my improvisation classes was met with interest, but also with suspicion. To some students the thorough and comprehensive structure of this publication caused certain pessimism, because they felt that they should master his tone rows and trichords completely before they would be able to apply them to their everyday improvisational language. On the other hand, some of them easily  got used to manipulating certain trichords and started using them in little compositions. Examples from my compendium of generative patterns from the Tone Clock, discussed in subchapter 4.7, helped them to blend some of the trichord techniques with their existing improvisational languages.


Regarding Garzone’s chromatic triadic approach I can say nothing  then the students in my improvisation classes had a lot of fun with it. With its simplicity it is a quick and suitable tool to make the students feel comfortable playing outside the chords. Moreover we experienced that, by repeatedly playing certain lines, tonal colors would emerge. Consequently it also served as a demonstration of Liebman’s concepts of linear tonality, tonal anchors and harmonic lyricism.


In my improvisation classes, Weiskopf’s operations combining triad pairs from underlying scales and modes appeared to be the most accessible of all models in this chapter. Bergonzi’s hexatonics served well as a next step after Weiskopf resulting in more advanced trichord combinations evoking more ambiguous tonal colors.

Thus, apart from my own practical experiences with the actual methods I have noticed how they have inspired my students to add them to their existing skills. I assume that this will apply even more so to the techniques I will be discussing in the following chapters.

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