4.1 Introduction: tonal and twelve-tone music

In the previous chapter, a variety of methods, implicitly or explicitly related to twelve-tone techniques, as alternatives to traditional chord-scale improvisation, were discussed. They were taken from publications by saxophonists who combine their practices as composing improvisers with their careers as leading jazz educators. After summarizing their systems and comparing their theories and practical applications, I discussed a number of applications of some of them to my own musical practices.

   In this chapter I will summarize Peter Schat’s Tone Clock in theory and in practice. Examples of Tone Clock applications in jazz practices will be discussed using compositions by bassist Theo Hoogstins, pianist Frank Carlberg, and alto saxophonist John O’Gallagher. After the analyses of relevant fragments from their compositions and improvisations, I will discuss three movements of my saxophone quartet “Carillon” each of which is based on a distinct hour of the Tone Clock. The penultimate section of this chapter will display possible ways of constructing improvisational patterns with Tone Clock trichords, as an alternative and addition to existing improvisational idioms.

   The application of the twelve-tone system in this study is largely determined by my own search for a musical space that could serve as an alternative to my existing diatonic practices as a composing improviser. In section 1.4.4 I expressed certain dissatisfaction with my arrangements and improvisations, and a need for innovative harmonic issues as alternatives to my continuously filling in and ornamenting of existing forms within the tonal system. Inspired by Dutch bassist and composer Theo Hoogstin’s article “Peter Schat’s Tone Clock in Jazz and Improvised Music” I ended up experimenting with Schat’s compositional model that pretended to bridge the tonal and twelve-tone systems.

    Before discussing Schat’s Tone Clock I look back to Simple Composition (1994) by composer and theorist Charles Wuorinen introduced in chapter 3.2.3. In the context of my experiences with the Tone Clock, an important element in Wuorinen’s publication is his summary of both the differences and the similarities between tonal and twelve-tone music. According to Wuorinen, and put in its most simple way, tonality can be identified in terms of “content” and twelve-tone in terms of “order”. That is, in tonal music the content of pitches and intervals is determined by their functions in the diatonic scale and its implied triads. We are able to identify such content as “the tonic”, or “the root of the triad”, regardless of where the note or the interval appears in the melodic line, what register it is played in, or in whatever inversion of the chord. Contrarily, in twelve-tone music the fundamental structure of the tone row is identified by the order of the pitches and their connecting intervals. The pitches do not have any functional content, and do not dominate each other. Only the order of the pitches and their connecting intervals identify the fundamental structure of a tone row.

   Wuorinen explains how, both in the decades before the introduction of the twelve-tone system by the Austrian composer Joseph Matthias Hauer in 1920, and during its development until 1994, the two systems showed a number of similarities. Wuorinen uses the solemn term “reconciliation” to express these shared elements. Twelve-tone music can be seen as resulting from the evolution of the tonal system during the nineteenth century. New tones were added to the basic triads, resulting in extended chords that were hard to identify in terms of functional content. And by means of increasing melodic counterpoint, for instance in the music of Wagner, Stravinsky and the “freely” chromatic music by Schoenberg before he launched his 12-tone system, the amount of unresolved chromatic notes increased. So one could say that in the history of the development of Western classical music the two systems were intermingled before Arnold Schoenberg developed the idea of twelve-tone sets as a source for both the melodic and the harmonic material in a composition.

   As to the present-day practice, Wuorinen observes a blend of basic operations of twelve-tone composing, such as the derived tone row, the basic operations of transposition, inversion and retrogression of pitch successions, and the expansion of pitch successions to twelve-tone sets by adding other notes, with tonal elements. The “reconciliation” results from the re-introduction of principles of pitch organization derived from interval content in small segments of twelve-tone sets, for instance trichords or tetrachords. By their small sizes these segments can be perceived as content groups rather than as rigidly ordered pitch successions (Wuorinen 1994: 21).

   Wuorinen gives the following example as an illustration. The first line is a twelve-tone row in a strictly ordered succession. In bar 2 this line is divided into four groups of three notes. They are randomly arranged in vertical order. In the third line the notes of each group are internally re-ordered at pleasure, which in this case means in alternating ascending and descending directions.

ex 4.1.1

According to Wuorinen, the critical response to twelve-tone music is mainly due to its numerical notation allowing composers to translate the basic twelve-tone operations discussed above into arithmetic operations that could be applied to (segments of) the twelve-tone sets. In section 4.3.1, this will be illustrated by Schat’s ferocious criticism on the numeric aspect of twelve-tone music.


ex 4.2.2

The only exception is the tenth hour, representing the diminished trichord 3+3. This is displayed as three tetrachords, because it is impossible to include all twelve notes with four 3+3 trichords.

ex 4.2.3

ex 4.2.7


Schat could limit the number of possible trichords to a total of twelve by applying two rules. The first is: trichords containing similar intervals but in different order belong to the same Tone Clock hour. For instance the trichords 1+5 and 5+1 in the example below both belong to the fifth hour. As a consequence of this rule, a succession of trichords from the same hour, but in different orders will be named by the trichord with the smallest interval first.

   By alternating the order of the intervals in the trichords, Schat managed to derive twelve-tone rows with four trichords from the same Tone Clock hour. The next example shows his derivation of the row of the fifth hour. Of the four trichords in the twelve-tone row below, the first and the third are put in the order 1+5. The second and the fourth trichord are put in the order 5+1. Together they form a twelve-tone row.

ex 4.2.4

Another principle of the Tone Clock is that every hour is “steered” by at least one trichord that belongs to another hour. For instance, the following example shows a twelve-tone row in which the 3+4 trichord of the eleventh hour is steered by the 2+2 trichord from the sixth hour. The trichords of the row are written above the staffs, and the steering trichord is written under the staffs.

ex 4.2.5


The next example shows two variations of a twelve-tone row constructed with 1+3 trichords of the third hour. In the first variation the 1+3 trichords are steered by the 1+5 trichord of the fifth, and in the second variation by the 2+3 trichord of the seventh hour. As a result of these distinct steering hours, these rows of the third hour sound notably different.

4.2 Peter Schat’s Tone Clock

Schat introduced the Tone Clock as “a tool for scanning the relations between the different notes of our tone-system” (Schat 1993: 57). With this definition he intended to include both parts: the tonal system in which every pitch has its proper content at one side, and the twelve-tone system at in which the order and the intervallic structure are the main elements of determination at the other.

    At the core of what he calls his “system of pitch organization” is his inventory of twelve possible “triads” that can be combined in order to let all twelve pitches of the octave sound. He called each of these “triads” a “tonality” and marked them with the Roman numerals of the hours of the clock. To avoid confusion with traditional functional harmony, his term “triad” in this study is changed into the term “trichord”, in accordance with conventional twelve-tone terminology. For the same reason, the term “tonal color” is used instead of his term “tonality” to characterize the distinct sounds of the twelve trichords.

    Below are the twelve hours of his Tone Clock, marked by Roman numerals above the staff. The numbers connected with a plus sign under the staff refer to the number of semitones between the pitches.

For his second rule to limit the number of possible trichords, Schat defined trichords containing an interval larger than 5 semitones (a perfect fourth) as an inversion of a smaller trichord. For example, the 1+6 trichord c–c#–g is regarded as the first inversion of the 5+1 trichord g–c–c# and therefore belongs to the fifth hour. Likewise, the 2+6 trichord c–d–g# belongs to the eighth hour as the first inversion of the 4+2 trichord g#–c–d. As another example, the 8+3 trichord c–g#–b should be regarded as the second inversion of the 3+1 trichord g#–b–c and therefore belongs to the third hour. Trichords with two intervals larger than a perfect fourth should also be regarded as inversions of trichords of these intervals in their smallest form. For instance the trichord in wide voicing g–f#–c should be reduced to the first inversion of the 1+5 trichord f#–g–c, with the note c in octave displacement.

    The Tone Clock hours I, VI, X and XII are symmetrical, which means that the intervallic structure of the inversions of their trichord types is the same as that of their prime forms. The hours II, III, IV, V, VII, VIII, IX and XI are non-symmetrical, which means that their types of trichords can be put in three different positions: prime form, first and second inversion. In accordance with the conventional twelve-tone terminology the term “inversion” in this study is replaced by “rotation”. The next example shows the prime form (P), the first (R1) and the second rotation (R2) of trichords 1+2 and 2+1 from the second hour.

ex 4.2.6


The last example is about the row of the twelfth hour, containing four 4+4 trichords (augmented triads) that can be steered by the 1+1 trichord of the first hour. However, steering these trichords by the 1+2 trichord of the second, the 1+5 of the fifth, or the 2+3 of the seventh hour, results in notably distinct sounds.

ex 4.2.1

4.3 Theoretical context of Peter Schat’s Tone Clock

4.3.1 Allen Forte’s pitch set class theory

How does Schat’s Tone Clock relate to music theorist and musicologist Allen Forte’s pitch class set theory? The principle difference lies in the aims of the two systems. Forte designed his for the harmonic analysis of post-tonal music, meant to discuss twentieth-century music in an objective way by avoiding references to individual compositional techniques or traditional harmony (Forte 1973). Schat proclaims a similar analytical practice, but the emphasis of his model is more on its compositional quality.

    Concerning the layout of the models, the principal difference is that in Forte’s system each pitch of the chromatic scale has its own “pitch class” number independent of the register in which it sounds: c has number 0, c has number 1, d number 2, etc. The term “set” is used for any combination of two or more, equal or different pitch classes, independent of octave displacement and enharmonic spelling. Thus Forte’s three-pitch class sets correspond to Schat’s trichords although they are presented in quite a different way. In contrast to Schat’s relatively simple ordering of trichords alternating between prime and inverted forms in twelve-tone rows, Forte lists all prime forms of “cardinal number 3” according to their number of possible transpositions and inversions, to their interval vector (the total number of all intervals between the pitches in the set), and to a number of other categories enabling numerical analyses of post-tonal compositions.

    Another difference concerns the number of intervals identified within the set and the trichord. Schat considers a trichord as a collection of two intervals a–b, and b–c; Forte considers a three-pitch class set as a collection of three intervals: a–b, b–c, and a–c. As a consequence in Forte’s system the intervals between 1 (minor second) and 6 (tritone) are taken into account; in Schat’s system those between 1 and 5 (perfect fourth).

    In a reaction to an article by composer Maarten van Norden (1997) about possible compositional applications of Forte’s set theory, Schat admits that there is a connection between his Tone Clock and Forte’s set theory. The fact that Forte’s theory indeed gives a complete overview of all harmonic possibilities of the twelve tones, he calls “attractive at first sight“ (Schat 1998: 41). In the rest of this polemical essay he explains why he finds this theory unattractive from a composer’s standpoint. Apart from his rejection of Forte’s changing of the note names by numerical pitch notations, it was mainly the arithmetic concept underlying Forte’s set theory, in order to facilitate a composer to calculate various relations between different sets, that was emotionally disapproved of by Schat, because “[he] studied to be a composer, not a bookkeeper” (Schat 1998: 42).

    Considering the actual summary comparison between Forte’s extensive pitch–set class collection and Schat’s limitation to compositional applications I endorse the latter’s recommendation of the Tone Clock as a technique that is “self-explaining and therefore meets the most important requirement for a “common language”” (Schat 1998: 44).

4.3.2 The problem of the three-ness

Composer and organist Gerrit de Marez Oyens (1997) came up with the idea to divide the Tone Clock’s twelve-tone rows into three tetrachord segments. He constructed a list of tetrachords corresponding with Forte’s twenty-nine pitch class sets with cardinal number 4 and found seven tetrachords that could be manipulated to fit three times in the row. Six of these tetrachords also appear as the “all-combinatorial four-notes source sets” of composer Milton Babbitt that I found in music theorist George Perle’s publication Serial Composition and Atonality (Perle 1991: 98, 130).

    With his emphasis on tetrachords De Marez Oyens addresses a problem with the trichordal approach of the Tone Clock that was already mentioned in the evaluation of O’Gallagher’s method of twelve-tone improvisation in chapter 3.7.4. As to the rhythmical aspect, in their operations with this overwhelming “three-ness”, performers are forced to either phrase their lines in three-note patterns and triple signatures (¾, 6/8, 9/8, 12/8, bars), or to inventively re-phrase the trichordal patterns into four-note groupings. But by dividing twelve-tone rows in three tetrachords, the constructions by De Marez Oyens and Babbitt show a way to combine the creation of twelve-tone patterns with the quadrature that is such an important characteristic in most jazz meters. Sections 4.7.8 – 4.7.12 contain examples of patterns created with combinations of the actual tetrachords.

4.3.3 Controlled revolution and natural evolution

Music critic and author of Schat’s biography Bas van Putten’s (2015) argues that Schat in an early stage of his development had sacrificed the “natural” way of an artistic evolution to a “controlled” revolution. This assumption sheds an interesting light on the applications of his Tone Clock in this study. Schat’s choice for a kind of serialism that he embraced in the traces of the international avant-garde in those days, forced him to break with the music tradition he had grown up with until then. This break was a decision by which he deliberately ignored his musical intuition and the comfortable, open-minded leaning on his beloved predecessors.

    In the context of this study into the practices of jazz artists, such a break-up with existing tonal practices and historical examples is hard to imagine. As was mentioned at the end of chapter 3.7.1 in the discussion of O’Gallagher’s twelve-tone method, the application by (composing) improvisers of a post-serial model such as the Tone Clock is principally not meant to break with their existing harmonic practices. It is rather considered a potential skill to enrich their techniques to improvise outside the pre-given chords. It can either replace or it can be used alongside the musicians’ existing practices.  

    Yet, in order to at least learn to apply such a new tool, the students have to learn to sacrifice at least some of their “natural” habits to allow and embed new and “controlled” elements into their backpack of musical idioms. But is this not exactly what is expected from serious (composing) improvisers: importing “building blocks” taken from foreign musical idioms, in order to permanently refresh and enrich their expressive palettes? The obvious difference with van Putten’s opinion about Schat’s radical change as a young composer, is that for a saturated improviser, this process of innovation does not mean that existing elements should be radically discarded, but rather intelligently re-ordered.

Thus, this analysis of section C reveals how O’Gallagher managed to combine a twelve-tone ordering principle with the interval relations between the two lines. Thus, the dominance of the 1+5 trichord in section C is continued in the deep structure instead of at the surface of the composition.

After the solo section D, to be discussed hereafter, section E displays trichord 2+2 six times, versus 2+5 once. Bars 43–46 display an eleven-tone row, with the note c repeated.  Only the note d is absent in order to complete the twelve-tone row. 


“Green Room” – fragment tenor saxophone solo


Compared to sections A and B, where the alto saxophone melody and the bass part are played as independent lines in contrapuntal movement, section C displays rhythmic accents on two-voice vertical harmonies. The notes of the alto saxophone’s part together form the Ab major scale. The bass line displays two symmetric four-note groupings g–g–a–b and d–d–e–f, with the note b in between. The following nine-tone row containing trichords from the second could be derived.


It seems as if O’Gallagher, after the tonality of A7 in section B, with this choice to continue in the tonality of Ab takes distance from his twelve-tone approach and the Tone Clock. Is he intentionally alternating between a twelve-tone (read: a trichordal) and a purely tonal approach? Is this the interpretation of his idea in chapter 3.7.1, stating that his operations with twelve-tone techniques are not exclusively reserved to twelve-tone use, but also can be combined with tonal settings? That would be a disappointing finding, because he recommended this tune as an intended application of the Tone Clock. Or should it be worthy to take a closer look into the relationship between the lines of the bass and the alto saxophone to understand what operation is involved here?


The next example shows how from the original melody by the alto saxophone in bar 47 of the example above, a twelve–tone row with four 2+2 trichords can be derived. Interestingly, the 1+5 trichord appears again, this time as the steering trichord of this derived row.

The addition of this fourth trichord results from my interpretation of the twelve-tone technique called “derivation” which is defined by Wuorinen as “the generation of new sets from segments, which may themselves be segments of other sets previously employed in a composition” (Wuorinen 1994: 111). In this case the added trichord D is simply a transposition by a tritone interval (T6) of trichord B. 

ex – “7th Hour Blues” – bass solo

ex “Green Room” – trumpet melody


The example below shows the chord changes for the tenor saxophone soloist in Carlberg’s arrangement for the Clazz Ensemble. Instead of writing a fixed melody, he gave the instruction to start off with sparse filling notes and to gradually increase the density of the improvised lines along the way.


“Petulant Snoot” - section C melody

The example above shows how between the alto saxophone in the upper staff and the bass notes in the lower staff a succession of major and minor intervals in alternating directions can be identified. For ease of reading, they are notated as triads, but in fact the fifth is missing. This succession can be divided into two segments. The first segment sounds in descending direction: Am – Abm – Gm – Ebm – Dm. The second one sounds in ascending direction: Bbm – B – Dm – Ebm – Em. Now, if we consider the root notes of these intervals as pitches of a tone row, it is possible to derive the following ten-tone row, that hosts four 1+5 trichords. The re-appearance of these trichords obviously relates to the dominance of trichord 1+5 that was found in section A. 

4.4.3 “Petulant Snoot” (John O’Gallagher)

O’Gallagher’s composition “Petulant Snoot” on his CD The Honeycomb (2015) was released after publication of his method for improvisation that was discussed in subchapter 3.7.  In this subchapter I will discuss possible relations between the Tone Clock and the composition and the alto saxophone solo.

“Petulant Snoot” starts with a repeating bass ostinato that is constructed with trichords 1+5 only. Trichord 3+3 at the end of the second bar results from the note d that is added there just as a leading tone to the next bar.



Twelve-tone elements and operations can be identified throughout all composed and improvised examples discussed. But rather than following “rigid” serial rules dictating strictly defined operations with twelve-tone sets, derived rows, and segments from the row, the actual examples display combinations of selective operations with a pre-given or implied tonal context. Although it is impossible to “prove”, I consider these practices as examples of how Schat imagined his Tone Clock as a way to bridge the twelve-tone techniques of atonal music with the more intuitive and context-driven practice of tonal music. Likewise the examples seem to illustrate how the trichords from the Tone Clock hours seem to support Wuorinen’s ideas of small segments serving as quickly perceptible content groups rather than as rigidly ordered interval successions (see subchapter 4.1).

    In all examples both composers and improvisers apply the twelve-tone techniques transposition, inversion and retro-gradation of pitches, trichords, and combinations of trichords, with the emphasis on those from the Tone Clock hour at stake. Trichord analysis identified the dominating presence of trichords 2+3 (Hoogstins), 1+2 (Carlberg), and 1+5 (O’Gallagher). Of these trichords those containing a minor second interval worked better to create non-tonal sounds than trichords 2+2, 2+3, 2+4, 3+3, and 3+4 marking passages with a more diatonic character.

   In the context of the above-mentioned blending of twelve-tone with tonal techniques, all soloists were obviously challenged to connect the assigned trichord operations to their informed intuitive habits of embellishing pentatonic scales (Hoogstins), applying non-tonal superimpositions (de Graaf, O’Gallagher), or creating high-density chromatic textures (O’Gallagher). No matter what inconvenience this challenge may have created for the performing musicians, it seems to have helped them all to create meaningful solos.

    Expectedly, most applications of twelve-tone techniques were found in the composed parts. In the solos, the fragments where twelve-tone techniques are intentionally played are relatively short. As to O’Gallagher’s soloing I find this disappointing and in contrast to his well-structured method on twelve-tone improvisation discussed in subchapter 3.7. In his method he favors a comprehensive approach of trichords and trichord combinations to both non-tonal and tonal musical, but in his otherwise impressive solos, this approach is not as obviously present as I had expected. On the other hand, the trichord successions in his improvisations are often related to implied tonalities. Therewith he demonstrates their qualities as non-tonal superimpositions on tonal harmonies.

    To find out if separate applications of all distinct hours of the Tone Clock would lead to a better understanding of their potential musical space, I undertook the composing and playing of Carillon, which will be discussed in the next subchapter. My aims were to concentrate on operations with all basic tone rows by creating both horizontal and vertical successions of their trichords and trichord combinations. I also wrote suggestions for improvisations, to help performers making meaningful connections with the composed material. In the next sub chapter I will discuss three movements of Carillon. Chapter 6 contains the complete recording of this composition for saxophone quartet.

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After the theme is played twice, Hoogstins plays the first solo chorus on the bass. In this improvisation he shows a preference for the 2+3 trichord as well. Out of the twenty-one trichords that are identified in the example below, ten are members of the 2+3 trichord family. In contrast to the theme these trichords now do refer to the bluesy minor pentatonic sound. Then, the 3+2 trichords in bars 2 and 4, and the sequences in bars 5–6 and 7–8, also confirm the (pull to the) key of Cm. The same goes for the bars in which successions of different trichords could be identified.  For instance in bars 11–12, the 2+3 trichord is surrounded by 3+4 trichords, resulting in C minor and Eb major triads that explicitly emphasize the key of the tune.


In Hoogstins’ theme, twelve-tone operations are intentionally limited to trichord 2+3 from the row of the seventh hour of the Tone Clock. His choice for this trichord seems inspired by its tonal reference to the minor pentatonic scale although he manages to avoid this typical “bluesy” sound by his sparse permutations of trichords and rows. On the other hand, his improvisation is overwhelmed by his intuitive response to the compelling force of the Cm key, despite the numerical majority of the same 2+3 trichord. The minor pentatonic reference of this trichord together with the various trichords surrounding them confirm the soloist’s emphasis on the tonal content of the blues. This is slightly disappointing in comparison to his intelligent avoidance of this tonal reference in the construction of the theme. I would rather have taken this as a point of departure for a less root-oriented improvisation.


4.4.2 “Green Room” (Frank Carlberg)

Carlberg’s composition “Green Room” on the CD Federico On Broadway (2011) is composed with the second hour of the Tone Clock. It has an eleven bar form that is repeated and followed by another three bars. The next example shows a mini-score of this tune.


“Green Room” – fragment tenor saxophone solo


To get acquainted to the sound of the second hour and its trichords, I related it to the harmonic color of the diminished scale and to familiar practices of transposing motives by distances of minor seconds. These operations popped up in this solo as my intuitive response to the intended harmonic vagueness of the chord changes. The next example shows these two operations combined in bars 101–104.

    The two first beats of bar 100 show the Eb diminished scale, marked as a succession of four 2+1 trichords. From the third beat of the bar, an ascending sequence of conventional diatonic triads can be identified. The brackets under the staffs mark the triads of A, B, C and Db. Because the first note of every triad is approached by an accentuated leading note, and because the line is phrased as a sequence of four tetrachords, these four 2+1 trichords can be identified. Thus, my assumption that these familiar operations of tonal chromatism would connect well with the Tone Clock hour is confirmed by the trichord analysis of this line.


Just like “7th Hour Blues”, “Green Room” is based on operations with a distinct hour of the Tone Clock. “Green Room” has a more complex structure and shows a more comprehensive approach to the Tone Clock, because the row of the second hour is applied both in horizontal and vertical orderings.  Carlberg operates in the border area of twelve-tone and tonal music by combining the ordering principle of a twelve tone row with the tonal content of his trichords and the derived and extended chords.

    As to the improvisations, in “7th Hour Blues” Hoogstins managed to manipulate his prevailed trichord better than I did in “Green Room”, but he was unable to transfer the non-tonal structure of his theme into trichord manipulations in his solo. In my solo I deliberately operated in the musical space between a real chord-scale approach (almost impossible due to the rapid succession of the chords) and the yet poorly embodied knowledge of twelve-tone operations. As a result it contains a mixture of sparse twelve-tone operations, tonal references and familiar chromatic techniques.



To summarize my analyses of the composed parts of “Petulant Snoot”, O’Gallagher’s examples of the application of the Tone Clock are his operations with predominant trichords 1+5 from the fifth hour and 2+2 from the sixth hour. Trichord 1+5 can be identified both at the surface and in the deeper structures. Throughout the composition these operations are applied both in tonal and non-tonal contexts. Furthermore, multiple derivations of Tone Clock rows can be identified in section E.

In the following section three fragments of O’Gallagher’s alto saxophone solo will be analyzed to find out how his applications of the Tone Clock in the composed part are reflected in his improvisations. It should be noted that this tune was played in a free rhythm. For ease of reading and to give priority to the melodic and harmonic qualities of his melodic lines, they are transcribed into a 12/8-meter, and in some cases notation of the rhythms is simplified.

In the first eight bars of the following fragment, O’Gallagher is mainly concerned with the embellishment of a C pedal tonality by means of stressing certain notes. In bars 1–6 the notes e–e–g–g–a–b, together with the pedal point C played by the bass, evoke the harmonic color of Cmaj#5.

In bars 9–16 O’Gallagher displays a deliberate trichordal approach. In bars 9–11 the emphasis is on the trichord 1+5, and in bars 12–15 a number of 2+5 trichords. These trichordal operations sound as short excursions outside the C pedal tonality that, apart from two sideslips to Cm in bars 7 and 13, is omnipresent in the whole fragment.

Hereafter, in bars 28–37, fast sixteenth-note runs are alternated with short patterns that sound as rest areas between the dense textures of these fast passages. Apart from the perfect fourth interval in bar 29 that points forward to the trichord in bar 31, these short patterns are constructed with 1+5 trichords exclusively. The transcription also suggests a connection between the two 1+5 trichords in bar 28 because they are both played at the start of six-bar patterns. I assume that O’Gallagher played them deliberately, but due to the fast tempo these trichords have minimum effect.

    Finally, bars 38–46 show a large emphasis on the 1+5 trichord. This fragment displays a chromatically descending sequence of four trichords. The top note of each group, an f, remains unchanged, while the lowest notes are descending: g, g, and f. 

4.4.1 “7th Hour Blues” (Theo Hoogstins)

In 7th Hour Blues” on the CD Triosonic (1997) Hoogstins superimposed the 2+3 trichords of the seventh hour of the Tone Clock on a twelve-bar minor blues form. Its blend of twelve-tone and tonal elements allows analysis from both sides.


The following example shows how all four trichords can be identified in the rubato trumpet melody, put in their prime forms (P), first (R1) and second (R2) rotations.


The transcription of my tenor saxophone solo on “Green Room” displays a combination of conventional tonal techniques and dodecaphony. In the following fragment, bars 91–96, I intentionally refer to the row of the second hour of the Tone Clock. I start and end my phrase with trichord 2+1.  In bars 92–94 the trichords captured within the rectangles belong to the same twelve-tone row. The trichord in between, starting at the second half of bar 93, is only one note away from being the missing link to make this twelve-tone row complete. To achieve this, I should have played the 1+2 trichord a–b–c. The notes under the bracket in the second line are an embellishment of a descending chromatic scale leading to the 4+3 trichord that evokes the temporary tonality of Gbmaj.


“Petulant Snoot” – section A melody


In contrast to section A, bars 19–25 of section B do evoke a harmonic color: that of A7. Trichord analysis illustrates this by the types of trichords found. The alto sax plays seven 2+2, two 2+3, two 2+4, and two 3+4 trichords. The bass line contains five 2+2 and four 2+3 trichords. Except the two 3+4 trichords, all trichords played in section B, both by the alto saxophonist and by the bassist, are different from the ones they played before in section A.

4.4 Applications of the Tone Clock in jazz

In this subchapter, three compositions will be analyzed that are constructed with the Tone Clock. Hoogstins’ relation with the Tone Clock is confirmed by the title of the first tune to be analyzed in this section: “7th Hour Blues”. The second, “Green Room” was introduced by Carlberg to the Clazz Ensemble as being based on an hour of the Tone Clock, although he intentionally didn’t give away on which one. The third example is both written and performed by O’Gallagher. By email O’Gallagher advised “Petulant Snoot” as a convincing example of his operations with the Tone Clock. In order to disclose the compositions of and improvisations on the tunes in this section, I applied O’Gallaghers method of trichord analysis introduced in subchapter 3.7.1.


The next example shows the result at the surface of Hoogstins’ manipulation of trichord 2+3. The order of the pitches in the trichords is changed, and to add variety to the transpositions of the first line, the order of the pitches in trichords D in bars 6 and 9, and of trichord B in bar 10 is permuted (perm).

It should be noted that also in the trichord analyses in this study, trichords are represented in their root positions, i.e. in their smallest form in ascending direction. For instance in the first bar of this example the smallest ascending form of the first trichord is g–b–c. Therefore it is marked 3+2. And because the root position of the second trichord is a–b–d, it is marked 2+3. For ease of reading, markings of the positions (prime form, first and second rotation) and retrogrades of trichords are left out, except in cases where these are a relevant aspect of the discussion.


The next example displays the relationship between the basic row, written in the lower staff, the vertical orderings of the trichords (as in example, written in the upper staff, and the tenor saxophonist’s chord symbols, written above the upper staff. It shows how the trichords are considered as content groups and how the pitches of the twelve-tone row are attributed tonal content. In bars 5 and 6 the trichords A+B are put in a different vertical order than in bars 1 and 2, which allowed Carlberg to construct different chords.


    The lowest notes of the trichords have become the root notes of the chords. The remaining two pitches do not determine the tonality of the chords. They express the seventh (bars 1–3, and 5–6), the ninth (bars 1–4), and the sixth (bars 5–6) of the chords that Carlberg has created. Only in the fourth bar, the note d determines the Bbm tonality.


In the following example trichord analysis of the alto saxophone melody in section A reveals a dominating number of trichords 1+5: fourteen 1+5 trichords, versus one 1+4 trichord, one 2+5 trichord, and one 3+4 trichord. Meanwhile the contrapuntal melody played by the acoustic bass shows a wider variation of trichords: seven 1+5, seven 1+4, four 2+4, two 3+4, one 1+2, one 2+5, and one 3+4 trichords can be identified.


“Petulant Snoot” – section B melody

Although alto saxophone and bass play the same melodic line in octave intervals, bars 43–46 evoke the harmonic colors of the Eb – D – A – Ab triads. Between their root notes again two 1+5 trichords can be identified, as shown in the following example. The structural presence of trichord 1+5 that was confirmed in sections A and C is also evident in section E.


“Petulant Snoot” – alto saxophone solo fragment


“Petulant Snoot” – alto saxophone solo fragment


“Petulant Snoot” – alto saxophone solo fragment


To summarize his operations in the improvised part of “Petulant Snoot”, O’Gallagher shows coherence with the emphasis of the predominant trichord 1+5 in the composed part.  This appears both in tonal and in non-tonal contexts. In his improvisation he also demonstrates a tonal application of this trichord 1+5, by means of designing the order of the root notes of the chords implied in his solo lines.



The second and third rows are transpositions of the first: the second is a transposition of a perfect fourth (T5), and the third is a transposition of a minor seventh (T10). In the third row the trichords are put in displaced order: A–D–C–B.

In the following fragment of his solo O’Gallagher connects his improvisation to the composition in a different way. Instead of emphasizing the predominant 1+5 trichord in his melodic lines, it is now implied in the harmonic colors of Ab – A – D – Eb that I identify in section E of the composition. In the next example the chord symbols mark the tonal content of the melodic lines. All four lines contain four triads or chords built on the root notes a–a–d–e. They can be considered as non-tonal superimpositions on the C pedal point, creating tension before “landing” either on a C tonality in bars 97, 98, and 105, or on its chromatic sideslip C# (bar 101).

ex“7th Hour Blues” – theme

The theme consists of three twelve-tone rows with the same rhythmic structure. The following example shows the derived row, the basic row with the trichords put in prime


In the next example the basic chords of the minor blues form are added to the theme. Considered from the tonal point of view, the first three notes of each row each contain basic or altered chord notes. All three lines move outside the chords from the second half of their first bar, and land on a chord note again at the last beat of the second bar. The root note of C minor, the key of this tune, only appears as the first and the last note of the theme.


The last two trichords are repetitions of the first two. Leaving these out, the following twelve-tone row of 1+2, steered by 2+4, appears.


“Petulant Snoot” – section E melody


“Green Room” – theme


Bars 6–11 contain all trichords of the tune. The next example shows them in their prime forms.