Crystalline Aphorisms: commentary and analysis of Jenny McLeod’s Tone Clock Pieces I–VII

This article was originally published in Canzona 2006. It was revised in 2026; revisions were made to acknowledge the completion of the set of pieces in 2012, and the passing of Jenny McLeod in 2024.

Introduction

The 24 Tone Clock Pieces by New Zealand composer Jenny McLeod are a series of short preludial works for solo piano composed between 1988–2012.¹ They are important milestones in New Zealand composition, as McLeod was the first New Zealand composer to have developed and articulated, like her mentor Olivier Messiaen before her, a fully-fledged compositional theory resulting in a highly individual musical language. This theory, called tone-clock theory, is a post-tonal system first developed by Dutch composer Peter Schat.² After meeting Schat in the late 1980s, McLeod discovered its connections to her own compositional approaches, as well as to Allen Forte’s seminal pitch-class set theory, and spent several years extending both Schat’s and Forte’s ideas well beyond their initial conceptions.³ Given the poise and refinement of McLeod’s music, as well as the somewhat unusual fact that the theory itself is explicitly contained in the title of the collection, the 24 Tone Clock Pieces have become important exemplars for composers, musicologists and educators in Australasia, and are becoming more widely known around the globe.

Tone-clock theory is, at heart, an attempt to develop a helpful vocabulary for chromatic post-tonal composition. While Schat’s original formulation adheres to using all twelve chromatic pitches, McLeod generalised the underlying concepts to focus less on attaining the twelve-tone aggregate, and focused more on the underlying principle of generating pitch collections that are highly economic in the number and type of constituent intervals. This means that pieces written using tone-clock principles attain a very clear sense of harmonic ‘colour’ or ‘character’. In this regard, the 24 Tone Clock Pieces have a hermetic, self-contained musical unity to them, thanks primarily to this ‘intervallic frugality’: the same intervals are heard constantly, even though the pitches vary greatly, giving each piece a clear sense of harmonic unity. The theory also places special emphasis on certain symmetrical transpositions of pitch collections, which result in some very interesting and unusual harmonic effects.

The soundworlds that McLeod creates in the 24 Tone Clock Pieces might be best described as ‘crystalline’ or ‘jewel-like’. I mean this to be more than just a pretty description, for I think this aural interpretation can be traced to four specific characteristics of the music:

1. the nonfunctional, nonteleological harmonies that result from McLeod’s use of chromatically saturated pitch collections and symmetrical divisions of the octave;
2. their articulation through a relatively free, often ametric rhythmic scheme; (both 1 and 2 discouraging a linear narrative in favour of an interpretation governed by ’events’ or ‘sound objects’ articulated in resonant space)
3. the high degree of internal order and prevalence of symmetrical pitch and rhythmic forms mimic the internal patterning of a crystal’s symmetrical structure, a similar metaphor to that found in works such as Varèse’s Hyperprism or Messiaen’s ‘Liturgie du Cristal’ from Quatuor pour la fin du temps;⁴
4. the fixed attack-decay character of the piano’s timbre, the relative brightness and ‘clarity’ of its overtone spectrum (particularly given the stretched, slightly inharmonic relationship between partials), allied with its strictly mechanical nature, further suggest a non-organic metaphor.⁵

These hazy, undirected musical events are often distributed in a somewhat episodic manner, similar to Messiaen’s compositions. Unlike Messiaen’s harmonic language, however, the 24 Tone Clock Pieces tend to use perfect and major intervals, particularly seconds, thirds and fourths. This gives the music a slight jazz inflection—indeed, Schat’s initial system was picked up by a number of Dutch jazz improvisors and incorporated into their harmonic language.⁶

Although the 24 Tone Clock Pieces do not systematically explore harmony in a graduated fashion, as do other famous collections of keyboard music such as Bach’s Das Wohltempierte Klavier, Bartók’s Mikrokosmos or Ligeti’s Musica Ricercata, they do, from piece to piece, display different levels of sophistication and complexity in the application of tone-clock theory. Tone Clock Piece I features the most straightforward application of tone-clock principles, while Tone Clock Piece III is based on a highly detailed harmonic scheme. Despite the variety in tone-clock approaches, however, the pieces share a number of unifying commonalities, discussed in later sections.

Tone-clock Theory & the Intervallic Prime Form

McLeod’s development of tone-clock theory is based primarily on taking Schat’s original terminology and post-serialist techniques, and expanding them to all possible combinations of chromatic notes, by drawing on Allen Forte’s idea of the ‘set class’.⁷ A set class is a family of all possible unordered chromatic collections that through transposition and/or inversion map onto the same set of pitch classes (pc). Using this idea, we can compare chords or melodic cells and demonstrate relationships in their intervallic construction. Because constituent interval classes do not change under transposition or inversion, the set class became a key tool in analysing atonal and post-tonal repertoire, especially that of the Second Viennese School, for whom intervallic relationships were an important force in creating musical coherence.

Forte has two ways of cataloguing and labelling a set class: the first, prime form notation, uses the most compact form of both the set and its inversion, listed as a series of pc integers transposed to pc 0.⁸ The set of pitches {C, D, E♭, G} would, for example, be labelled [0237]. The second method involves looking up the prime form in Forte’s table, and using its catalogue number, or ‘Forte name’. In this case, the previous tetrachord [0237] has the (admittedly somewhat insipid) title ‘Set Class 4-22’.

In total there are 220 set classes in Forte’s theory, including 12 three-note classes (trichords). Peter Schat discovered that of these 12 trichords, 11 can be transposed and/or inverted in such a way that every note of the chromatic scale is generated once and once only.⁹ After exploring Schat’s ideas, McLeod discovered this property in other-sized set classes as well. In fact, she systematically examined every set class in Forte’s list, annotating whether or not it had this property, as well as noting many other transpositional and symmetrical properties that were not included in Forte’s lists of set classes. In addition, she also cross-referenced her findings with other chromatic techniques and theories, notably those of Messiaen, Boulez and Xenakis.¹⁰ This table—her ‘chromatic map’—is an extremely important, if little-known contribution to chromatic musical theory.

McLeod’s considerable maturation of Schat’s tone clock is, therefore, also a development and repackaging of pc set theory, with extensive additional observations and information. She also set out to improve on certain aspects of pc set theory—firstly, by abandoning Forte’s often confusing nomenclature and, secondly, by including tabulated information about transpositional attributes of chromatic collections. Instead of Forte’s rather cumbersome Forte name and prime form notation, tone-clock theory uses a single label derived from the list of interval classes (ic) between subsequent pc of the prime form. This has the immediate advantage that it does not require mental transposition (as does prime form notation) nor the use of lookup tables (as does the Forte name). In tone clock notation, then, tetrachord [0237] would be labelled 214—that is: major 2nd (2 semitones), minor 2nd (1 semitone), major 3rd (4 semitones). Triads are always labelled with an intervening hyphen, such as 2-3 (not to be confused with Forte name notation). Because of the emphasis on intervals, rather than pitches, McLeod calls this label the Intervallic Prime Form, or IPF for short.¹¹

McLeod’s rationale for IPF notation is that it allows both composer and musicologist to instantly see the intervals between neighbouring pc, making it both more easily comprehended and hence more useful for the composer, whilst also providing ‘clues’ as to the sort of ‘flavour’ of the prime form.¹² For instance, we can see straightaway that IPF 1-2 contains both a minor second (interval class 1) and a neighbouring major second (interval class 2). This fact leads us to certain conclusions about the relative dissonance of the chord (for example, the two seconds will give the chord a certain modal ‘spice’, though not, perhaps, the percussive bite of a true tone-cluster). In order to get the same information from prime form notation, Forte’s nomenclature requires the extra step of ‘mental unravelling’, or at least subtraction, to determine the intervallic content from the list of pc.

There is one property of IPFs and set classes that is somewhat controversial. Because groups of notes that relate to each other through inversion share the same prime form, major and minor triads are considered ’equivalent’. It might be argued that this is a fatal flaw of the whole concept of set classes. My feeling is, however, that this objection misses the point. If we consider common-practice tonality as merely one small subset of the total chromatic universe, then the fact that major and minor triads ‘belong together’ in this way demonstrates, to some degree, the success of the theory. In fact, McLeod extends the concept of ‘major’ and ‘minor’ to other bi-interval IPFs, such as 1-2. The ‘major form’ of the IPF is voiced with the larger interval on the bottom (e.g. 2-1), while the ‘minor form’ is has the smaller interval on the bottom (e.g. 1-2).

Given the fact that tone-clock theory has not really adopted the more advanced analytical tools from pc set theory, such as set complexes and pc set genera, and given the ubiquity of pc set theory in current musicological publications and tertiary analysis courses, it seems unlikely that it will ever be widely adopted by the musicological community. But what really sets it apart from pc set theory, is that it was conceived by a composer as a compositional strategy. It shows us ways we might ‘instantiate’ IPFs through special transposition levels that have aurally and conceptually attractive symmetrical properties. Pitch-class set theory, on the other hand, is analytical, and does not really suggest to us any particular approach to the transposition of prime forms. We might say, then, that tone-clock theory is a conceptual tool for composers working in an equal-tempered, post-tonal context, a tool that not only provides us with more ‘user-friendly’ naming conventions than pc set theory, but also gives specific suggestions for transpositions and inversions of IPFs.

Proximity and Segmentation

As Forte remarks, ‘the process of segmentation… determining which musical units of a composition are to be regarded as analytical objects… often entails difficulties in the case of an atonal work.’¹³ Forte seems to opt for a largely intuitive approach to deciding which pc ‘go together’, although he does draw on conventional musical concepts of ‘distinctiveness’ (for instance, rhythmic distinctiveness, rests, beam groups, and other clustering and individuation mechanisms). However, if one is composing with prime forms, as opposed to analysing, then a simple principle of proximity is sufficient to ensure the effect of intervallic relationships. In other words, we need to place the constituent pc of each IPF within a short enough span of time such that our short-term aural memory is able to perceive the relationships between them. If notes are strung too far apart in time or register, or interspersed too drastically with notes from other transpositions, then essential intervallic characteristics of the prime form can be lost.¹⁴ Of course, the question that arises here is: exactly how proximate do pc need to be? And, what is the perceptual continuum on which the effects of proximity operate? Unfortunately, these questions are too rooted in the field of psychoacoustics and perception research for me to try and tackle them in this article. It is interesting, nevertheless, to see that in the 24 Tone Clock Pieces, there is often deliberate and clear partitioning and clustering of constituent pc, making segmentation, even under Forte’s criteria, reasonably straightforward.¹⁵

Relationship to Serialism

Due to tone-clock theory’s emphasis on creating twelve-note aggregates, it might be conflated with serialism.¹⁶ While there is some superficial similarity, there are also significant differences. The ordering of a set of pc is not part of tone-clock theory; in fact, tone-clock theory has no concept of order, only using the terms ‘collections’, ‘groups’ and ‘fields’ rather than ‘rows’. An IPF may appear in any order, as long as its fundamental intervallic flavour is preserved (through the proximity principle mentioned above). In this sense, tone-clock theory is less constrictive than serialism, and, it could be argued, more ‘integrated’ due to the emphasis placed on the perceptibility of intervallic content as a primary unifying force.

While it is true that, in one sense, serialism also preserves the perceptibility of intervallic content, if we concede that serial operations of transposition, inversion and retrograde do not change the proximity of the intervals to each other, composers (particularly Schoenberg) would often construct their rows from a variety of different intervals. This would lead to an ‘overabundance’ of intervallic flavours.¹⁷

Tone-clock theory, on the other hand, operates on smaller collections, typically triads or tetrads, which ‘saturate’ the larger collections. Within each small collection, there is no limit placed on order, but there is specific emphasis placed on intervallic consistency. Proponents of tone-clock theory argue, therefore, that it creates a lower-level, more perceptible harmonic structure than serialist techniques, while allowing for greater freedom in the way the generative material is treated.

Terminology

Before embarking on the analysis and commentary of the first seven Tone Clock Pieces, I will need to introduce some important terms used throughout the remainder of the text. Space does not permit more than a rudimentary explanation of these terms, so for further information the reader is referred to [McLeod, unpub.]

Tone-clock steering

Tone-clock theory’s primary advance on pc set theory is a detailed examination of the fascinating process of transposing and inverting (or ‘steering’ in tone-clock parlance) a set of pc such that all twelve chromatic pc are generated without repetition. McLeod calls this technique ’tone-clock steering’, and it is a cornerstone of tone-clock theory. The example below, from Tone Clock Piece I, shows a set of three pc (C♮–D♮–G♮, IPF 2-3) transposed and inverted so that every note of the chromatic scale is generated once and once only:

This technique has the potential to generate a rich, chromatically-saturated harmonic language, while simultaneously economising intervallic content. Unfortunately, only 44 of the 222 IPFs possess this ability.¹⁸

The steering group

The term ‘steering’ is a nice metaphor for transposition, capturing the idea that there is a group of notes (the ‘steering group’) which is responsible for transposing the other. In the example above, this group of notes is indicated by downstems and hollow noteheads (IPF 121). Although beyond the scope of this article, the concept of the steering group raises interesting psychoacoustical questions: does the brain subconsciously register the presence of the steering group, and if so, in what way? Is it something we could articulate, even if we had not heard the steering group on its own? Or is the link between the two quite arbitrary in practice?¹⁹

In any case, McLeod uses the principle of steering in every aspect of the harmonic organisation of the 24 Tone Clock Pieces. The clearest example is in Tone Clock Piece I: the entire harmonic structure, except for the last two bars, is reducible to the tone-clock steering above: 2-5 steered by 121. In the last two bars, however, we hear instead the ‘reversed’ steering or the ‘steering partner’: that is to say, 121 steered by 2-5 (see Appendix I)

Note, however, that 121 is incapable of being tone-clock steered. Instead, McLeod has to use what we might call an ‘incomplete twelve-tone steering’: firstly, two complementary transpositions of 121, then the ‘remainder’, 131. This kind of steering, dubbed an ‘anchor form’ by McLeod, is often deployed throughout the 24 Tone Clock Pieces (this will be covered in more depth shortly).

The codetta-like expression of the steering group in Tone Clock Piece I can be thought of as the bringing to the surface an otherwise ineffable force—the sudden awareness of a previously sublimated sound. This reification of a ‘deep structure’ is not dissimilar to a final I-V-I cadence of a Classical tonal piece echoing Schenker’s notion of the Bassbrechung, the large-scale 1 ## ˆ ## –5 ## ˆ ## –1 ## ˆ bass motion that he claims controls overall harmonic motion. The ‘mirror effect’ of this functional reversal between macro- and micro-structure is also an attractive conceptual feature for McLeod, recalling Messiaen’s preoccupation with musical symmetry.

The twelve hours

Tone-clock theory pays particular attention to the twelve ’triads’. As there are exactly twelve triadic prime forms in total, Schat calls these the ‘hours’, and arrays them circularly, in a shape similar to a clock-face. This he calls the ‘zodiac of the hours’:

The hours represent a gradual intervallic ‘opening up’ from the tightly chromatic hour I (1-1) to the open augmented chord of the hour XII (4-4). For each triad, the transpositions and inversions required for tone-clock steering are represented as rotated and reflected versions of the same triangle.

Fields

The term ‘field’ is used by McLeod to indicate a group of prime forms that, like key areas in common-practice tonality, define the harmonic content of some segment of the work (typically a phrase, section, and sometimes an entire piece). A field is usually defined by one or two IPFs undergoing transposition and inversion, the resulting pitch collections used repeatedly to form a sense of ‘chromatic tonality’ or, more abstractly, a ‘compositional pitch space’. Ex. 3 shows an example of a triad that has been transposed and inverted repeatedly to form a set of pitches from which we might compose a section of a work (McLeod uses this particular field extensively in Tone Clock Piece I).

Many of the fields in the Tone Clock Pieces comprise transpositions and inversions of an hour or hour-group that together form the twelve-note chromatic collection, although there are instances of fields that contain, for example, thirteen or sixteen notes. Twelve-note chromatically saturated fields are normally formed either by symmetrical transpositions of a single IPF (this technique is discussed in the ’tone-clock steering’ section above), or by non-saturating transpositions of an IPF plus the chromatic complement of the aggregate (this technique is discussed in the ‘anchors and shells’ section below). It turns out that tone-clock steering does not, in fact, appear in every Tone Clock Piece. This circumstance is surely due to the overly limited intervallic construction of tone-clock steerings, not to mention the fact that only a few IPFs can be tone-clock steered. Because of that, McLeod uses so-called ‘anchor forms’ more frequently, as any IPF can be used to create them.

Anchor forms

Only a relatively small number of IPFs can form a complete tone-clock steering. Of the 29 tetrads, for instance, only 7 are able to be tone-clock steered. To reach chromatic saturation, most IPFs require another IPF to complete saturation without repeating pcs. The use of two IPFs to reach saturation, in which one IPF appears multiple times and the second only once, is called an ‘anchor form’.

Ex. 4 shows an anchor form appears in the opening Tone Clock Piece I. In the final bars tetrad 121 is steered by the interval class 5, but this does not form chromatic saturation on its own. The chromatic ‘remainder’ is IPF 313 (the ‘minor-major chord’). When presented in the form given in Example 1B of Appendix I, we can see that the remainder IPF (called the ‘anchor’) forms an axis of symmetry around which the transpositions of 121 (called the ‘shells’) are arrayed in a mirror-like fashion.²⁰

It can be shown that an anchor form can be derived from any IPF by following the following strategy:

If an IPF is symmetrical, then an anchor form can be created by steering the IPF with a symmetrical steering group of any size. If the IPF is asymmetrical, however, then you must steer it with a symmetrical steering group that contains an even number of pc, and you must use symmetrical inversions of the IPF when transposing ( for instance, mmMM, MMmm, MmMm or mMmM) As long as these criteria are followed, then the aggregate of the transpositions (the shell) will be symmetrical, as will be the chromatic complement (the anchor). Although a complete generalised mathematical proof of this is beyond the scope of this article, a sample proof for the case where the IPF is a triad and the steering group is a symmetrical tetrad—a case commonly seen in the Tone Clock Pieces—is given in Appendix II.

Developmental procedures

IPF Superposition/Aggregation

In the most straightforward application of tone-clock theory, found in Tone Clock Piece I, the triad 2-5 is used exclusively except, as previously noted, for the final two bars. The result is a high degree of intervallic coherence—the major second and perfect fourth plus their respective inversions, the minor seventh and perfect fifth, saturate the harmonies. This has the disadvantage, however, of potentially limiting musical variation, as the set of intervals is essentially static. McLeod therefore uses a fairly simple procedure to extract a greater variety of intervallic material from a single IPF: by superimposing and revoicing two or more transpositions of a single IPF, she forms a larger superset out of which new intervallic configurations can be emphasised. Ex. 5 shows how this strategy is used in b. 22 of Tone Clock Piece IV, where the main IPF we hear earlier in the piece, 232 (VII4) is transposed twice to form (G, A, C, D) and (C♯, D♯, F♯, G♯), which is then aggregated, split and revoiced to form two transpositions of V4 (151) (C♯, F♯, G, C) and (D, A, G♯, D♯).

The ability for two different IPFs (232 and 151) to share a superset (111-3-111) is a useful tool for the development of the harmonic structure, and is regularly used by McLeod.

Harmonic Stability/Instability

In order to develop a sense of tension and release through harmonic structure, composers have often sought to vary the degree of ‘harmonic stability’ in their music. There are two ways in which we control this: firstly, the length of time over which we hear the same set of pc (the ‘harmonic rhythm’), and secondly, the relative ‘distance’ between successive pc sets—that is to say, the number of new pc introduced by the new set. In the Tone Clock Pieces, the tendency is to use chromatic aggregates in a fairly short space of time, thus diminishing the sense of harmonic rhythm, creating a more nonteleological musical language. However, there are moments when subtle uses of ‘common tones’ (pc kept in common between subsequent transpositions of an IPF) can give the impression of a more goal-oriented unfolding of harmony.

Formal Considerations

Although pitch structures are the focus of this analysis, they do not, of course, provide a complete picture of a piece’s construction. Having said that, there is a very close relationship between the harmonic structure of the Tone Clock Pieces and large-scale form-defining properties such as texture, tempo, motif, and rhythmic materials.

This is borne out when we examine the boundaries between different tone-clock collections. In Tone Clock Piece VII for instance, there is a clear division of material between the sedate homophonic quaver/quaver triplets ‘chorale’ of Field A (1–12) and the sprightlier, more angular semiquaver material of Field B (13–23), that becomes increasingly contrapuntal in texture. This close alliance between tone-clock field and the other non-pitch parameters is found throughout the Tone Clock Pieces. By creating this alliance, McLeod engenders a clear and unambiguous sequence of structural landmarks.

The clarity of the fairly conventional formal designs is presumably intended as a counterpoint to the more complex chromatic harmonies. By deliberate non-pitch signalling of changes in IPF, we can safely assume that McLeod wants the listener to appreciate the different ‘flavours’ of the tone-clock collections. If we could respond synaesthetically to the intervallic construction of tone-clock collections—if one could ‘hear’ colours in response to aural stimuli, as Messiaen famously did—then we might imagine a visual analogy: rather than Messiaen’s stained-glass effect of multiple refracted colours at once, McLeod’s tone-clock collections—carefully partitioned areas of intervallic colour—would be heard as a sequence of ‘colour fields’.

The Analyses

McLeod’s analyses

In the composer’s private collection are a number of unpublished analyses of the Tone Clock Pieces, primarily intended to assist presentations and lectures on her working methodologies. These are not reprinted here, and I did not examine these analyses in advance of writing this article. This fact turned out to be quite useful in examining the relationship between her detailed compositional strategies and the resulting musical outcome and my own perceptual segmentation of that. By avoiding McLeod’s own analyses, except for reading her introduction to tone-clock theory, I was able to compare my own hunches and intuitions with her specific compositional choices. This resulted in a number of deviations between her intentions and my interpretations. In the analyses below, I note these deviations and attempt to give possible explanations for them.

Analytical notation

Due to the recurrence of certain principles and procedures throughout the composition of the Tone Clock Pieces, I will use a shorthand notation in my analyses of the pieces:²¹

1. A = [2-5⁴ / 121]

Tone-clock steering field — the square brackets here indicate that Field A is a complete twelve-note aggregate, formed such that each note appears once and once only (the so-called ’tone-clock steering’ discussed previously). The superscript denotes the number of times the IPF appears. The broken-bar symbol ‘/’ is shorthand for ‘steered by’, with the IPF following the symbol being the steering group. Thus, in this case, the IPF 2-5 is transposed four times to complete the chromatic scale, with 121 giving the transposition levels.

2. A = [121² + 131] / 2-5

‘Anchor’ form field — here the square brackets indicate the same as before—a complementary steering—except that in this case Field A is made up of two different IPFs in order to complete the chromatic scale (two transpositions of 121 and one of 131). This is the ‘anchor form’ that was discussed previously. A steering group is given as well (2-5), controlling the pitch levels of both IPFs.

3. A = [(2-5⁴ / 131) + 6]

Anchor form field variation — similar to above, except the steering group applies only to the IPF 2-5; the remainder (a tritone) is not included as part of the steering group.

4. A = {232⁴ / 131}

General twelve-tone steering field — the curly brackets indicate that although a complete twelve-tone aggregate is formed, pc may appear more than once (i.e. the field contains more than 12 notes in it).

5. B = A⁺⁵

Field transposition — identifies field B as being a transposition of field A, in this case by 5 semitones (a perfect fifth)

6. B = A!

Field reordering — identifies field B as being a clear reordering of field B, usually through aggregation and revoicing (see section on aggregation above).

7. B = «A»

Steering partner — identifies field B as being the ‘steering partner’ of field A (see previous discussion)

Note on analyses: All analyses have an accompanying harmonic reduction given in Appendix I, which allows the listener to follow the fields as they appear in the piece.

Tone Clock Piece I

Tone Clock Piece I is the most straightforward of the Tone Clock Pieces. A clear ’theme and accompaniment’ texture predominates, with the harmony based upon a single tone-clock steering of 2-5, and a final two-bar ‘coda’ based on the steering partner anchor-form, 121 steered by 2-5:

1–29: A = [2-5⁴] / 121

30–1: B = «A» = [121² +313] / 2-5

Due to the relative simplicity of the application of tone-clock theory, the piece is saturated with a small number of intervals: open intervals of perfect fourths and fifths and major seconds are particularly prevalent. This saturation remains fairly much in force throughout, although resulting voice-leadings often hint at other intervallic constructions: the triad 1–2 appears in the tenor voices of 8–9 and 13–14, and its related tetrad 121 in the same voice in 1–3 and 15–16 (a result, of course, from the underlying steering). 131 appears in the bass voice of 1–3, subliminally relating to the anchor group 313 of Field B, while the triad 2–3 appears in the tenor voice in b. 10 and 22–25.

The rhythmic material displays a great deal of the ‘written rubato’ approach, primarily through flexible subdivisions of minim beats into patterns of 2 (crotchets), 3 (crotchet triplet), 4 (quavers), 5 (quaver quintuplet), 6 (quaver sextuplet), and 7 (quaver septuplet), as well as similar subdivisions of the crotchet beat. These subdivisions often appear next to each other to give the impression of expressively accelerating or decelerating music. A good example appears in the right hand of b. 8: crotchet triplet–quaver quintuplet–quavers (3–5–4). Interestingly, the ‘written rubato’ is coupled with a high number of written tempi changes, primarily rits at phrase endings followed by a tempos.

Registral distribution is fairly conventional, with low bass resonances being used either as emphasis of structural arrival points (such as in b. 15) or cadential phrase-endings (such as in b. 29). The extreme high registers of the piano are not explored in this piece, with the exception of a single chord in the right hand at b. 12. The overall spectral breadth of the piece, therefore, is contained within a span of about 3½ octaves—an almost chorale-like distribution of voices, an unsurprising distribution given McLeod’s background in choral composition.

Tone Clock Piece II

Tone Clock Piece II is of particular interest as it demonstrates that a listener can arrive at quite a different interpretation from that intended by the composer. This is particularly the case here because of McLeod’s intuitive approach to voicing and aggregation, which can sometimes lead to unexpected and unintended groupings.

My analysis of Tone Clock Piece II was written before I had consulted the composer’s own sketches, and I discovered that it appeared to vary quite considerably from the actual steering plan used. This might seem like a flaw of tone-clock theory, or, more likely, of my analytical skills, but I think on closer inspection the variation is perhaps not so surprising.

McLeod’s own sketches and analyses illustrate that the piece derives from two fields: Field 1, which, similar to Tone Clock Piece I, features triad 2-5, this time being steered by - The steering is incomplete, however, and the remainder is the tritonal anchor A–E♭. Field 2 features exactly the same steering, but with the triadic inversions reordered (mmMM instead of mMmM). It too is incomplete, this time with only a single pc F♯ forming the anchor.

My own analysis interprets the same set of notes differently. I hear an aggregate being formed over the first four bars: IPF 212-212, otherwise known as the diatonic collection²². Indeed, if we take F♯ as being the tonal centre, which it appears to be given its prominence in the opening bars, then we could say the opening passage is in F♯ Dorian.

That I should arrive at such a different analysis than the composer herself is perhaps not so surprising when we see that the triad 2-5 can be voiced as two stacked perfect fifths. These, when transposed to form a seven-note cycle of fifths, give us a diatonic segment. To my ears, this larger aggregate is more perceptually ‘audible’ from a structural point-of-view than the lower-level 2-5 steering. In any case, the ambiguity between the two readings is typical of a more complex use of tone-clock theory.

At the change of harmony in b. 7, a C Dorian aggregate is formed. In fact, the Dorian collection seems to permeate the score. My analysis, then, is much simpler than McLeod’s. I hear the entire piece as reducible to a single field:

A = {212-212⁴} / 333

This also gives us some interesting points of note: between transpositions 1–2 and 3–4, there are two pc in common (a tritone). These common tones feature at prominent moments, often fulfilling a ‘pivoting’ role between the different transpositions, in much the same way Debussy often uses common pc to ‘pivot’ between different diatonic or modal collections.

The primary flaw in my analysis is that it fails to recognise the degree to which 2-5 does in fact control much of McLeod’s partitioning and voicing of 212-212. This discrepancy between segmentation of a superset and a subset points to an area of future research: is there a perceptual tendency to collate smaller surface forms into larger background patterns, and, if so, under what conditions does this occur?

Incidentally, McLeod’s analysis demonstrates the use of a permutational technique which she calls ‘intrinsic spin’; discussion of this technique, and the particular perceptual properties of it, are unfortunately beyond the scope of this article.

Tone Clock Piece III

Tone Clock Piece III is by far the most complex of the collection. All twelve hours, except I and IX, are deployed in a kaleidoscope of shifting chromatic colours. The form is tied together primarily by rhythmic and textural means: the additive rhythmic homophony of Section A (1–29), the open spacious arpeggios of Section B (30–35) the chorale-like ‘melody + accompaniment’ of Section C (36–41, which later incorporates aspects of Section A), and the ‘floating’ quintuplets of Section D (78–81).

Using this simple segmentation, the form can be summarised as:

ABC–ACB–DBCD–A

Section A opens with…

[222^2 + 212] / 34

…developing secundal harmony as a key force. The presence of many minor and major seconds imbues the piece with a constant clang that only in brief developmental passages ‘unfurls’ into wider, non-secundal hours (namely 3-3, 3-4 and 4-4).

For this Tone Clock Piece, however, I abandoned my own analysis once I had seen McLeod’s. This is because she employs a rather diabolical system—one that I would have been hard-pressed to have discovered on my own. The ‘system’ at work is that the steering-groups are derived from an abstraction of the harmonic structure of Bach’s famous Prelude no. 1 in C major from Das Wohltempierte Klavier.⁸

The analysis below—a ‘merger’ between mine and McLeod’s–demonstrates at a glance the complexity and richness of the piece as it traverses almost the full range of hours. As can be seen, McLeod not only uses tone-clock steerings and anchor forms to generate chromatic saturation, but also so-called ‘dual tonalities’. These are twelve-note sets generated from two transpositions of two different IPFs. (This technique only appears in Tone Clock Piece III.)

Column 1 identifies the range of bars in which the field is used; Column 2 gives the Field name and its tone-clock derivation; Column 3 gives the name of the steering group in conventional Western triadic terms; and Column 4 refers to the bar in the original Bach Prelude from which the chord is derived.

The derivation is not literal: repetitions of a chord within a section are ignored, and, as can be seen from the last column in the analysis, the chords are also occasionally out of sequence

To the listener, there might appear to be no discernible pattern to the progression of tone-clock collections. It is possible we might identify predominant interval classes as being ‘bunched’ together, binding otherwise disparate collections. For instance, many of the fields in Section A seem to contain the major second (ic2), while a second group of fields featuring the minor second (ic1) follows in Sections B and C (30–42); and the final section from 66 until the end features the major third (ic4) ## 29 . Using a prominent interval as structural demarcation is certainly not unusual in the Tone Clock Pieces; nevertheless, the degree to which the idea is developed to knit together otherwise unrelated collections is unprecedented. I would argue that although the Bach-derived ‘deep structure’ is interesting, it is, most likely, perceptually insignificant. The surface level intervallic constructions are more immediately ‘form-creating’ in terms of perceptual linkages formed by the listener between different materials. Nonetheless, I would argue that we do get a sense of harmonic narrative, as the music ranges from relative consonance through an extended passage of dissonance and chromatic complexity, resolving in a passage of relative ‘stasis’ and sense of resolution. It is true, however, that McLeod does in the latter bars attempt to bring out an extended IV–V–I cadence: this starts at b. 82 with an emphasis on F, A and C, followed by G–B–D–F outlined in b. 86, and the pedal point C throughout the last page.

Tone Clock Piece IV

Tone Clock Piece IV uses the symmetrical tetrad 232 in a tone-clock steering as its primary material. In field A it is steered by a major third then a semitone, forming a 16-note saturated field. Other fields are generated using different steerings and alternative aggregations of 232. The piece is strophic in form: the first two transpositions of 232 are aggregated to form a symmetrical octad (22111-22), which appears, arpeggiated, as a two-bar refrain (McLeod calls this the ‘sea chord’), around which choralelike verses are arrayed. 1–4: A¹ = (232² / 4) = 22-111-22 5–6: A² = (232² / 1) 7–8: A¹ ( A¹ + A² = A = {232⁴ / 144} ) 29 Even the major seconds of Field A ## 1 are reordered on the last page to feature major thirds instead. ## 82 | 9–11: B = {232⁴ / 121} 12–13: A ## 1 14–15: B ## 1 = subset of B = (232² / 3) 16–17: A ## 1 18–19: B ## 2 = subset of B = (232² / 1) 20–23: C = {232⁴ / 333} 24–25: A ## 1 26–31: B 32–37: A ## 1 McLeod’s own analysis is similar: an opening field of [232] ## 3 / 4-4, a second 16-note field of 232 ## 4 / 121, and a third 16-note field of 232 ## 4 / 333 (my only disagreement then being with the interpretation of field A, a fairly minor point). The technique of IPF superposition is demonstrated in 22–23, where two transpositions of 232 are superimposed and revoiced to create two transpositions of 151, developing ‘closer’ semitonal harmonic elements from the otherwise open intervals of 232. This passage acts as a subtle ‘mini-climax’, a point of somewhat higher tension in an otherwise refined and unruffled calm. The other example is in 27–30, where two transpositions of 232 are used to create IPF 414, another semitonal tetrad⁰ .

Tone Clock Piece V

Tone Clock Piece V, subtitled ‘Vive Messiaen!’, is a brilliant and colourful tribute to McLeod’s ‘maître’, with numerous small nods toward the thumbprints of Messiaen’s langage musicale—most notably the birdsong-like melodic patterns of the first subject. The form can be characterised as a quasi-rondo form (or, perhaps, a double ternary form): ABA-ABA. The A material is based on the opening IPF in the right hand, constructed from two tritones a major second apart. This forms tetrad 242, a subset of Messiaen’s Mode I (the whole-tone scale). Another transposition of 242 in the right-hand at b. 3 combines with the first to form a complete Mode II (the octatonic scale). Other intervallic constructs become important later on, particularly the IPF 232 and the augmented triad (4-4), both seen emerging in b. 19. The chromatic complement of the octatonic scale is the diminished seventh chord. McLeod uses the two 242 transpositions plus the diminished seventh in a typical anchor group form: 1–6 A¹ = [(242² / 3)+333] McLeod then proceeds to put the octatonic scale through the remaining two possible transpositions, varying Field A by transposing it down consecutive semitones: 7–10 A² = A¹ -1 11–14 A³ = A¹ -2 15–18 A ## 2 In the B section, a second field is used: a tone-clock steering of 232 and its steering partner the augmented triad: 19–23 (RH) B¹ = [232³ ] / 44 (incomplete) 19–23 (LH) B² = «B¹ » = [44⁴ ] / 232 24–29 B ## 1 Then follows a series of vacillations between the three A fields (with a short interruption from the B ## 2 field). Due to 30 This could also be thought of as (5³ / 22) space reasons, I am unable to go beyond the simple analysis below, which does not quite do justice to the various aggregations and revoicings that McLeod puts them through (such as the emergence of (4 / 333) in b. 38): b. 30 A ## 1 b. 31 B ## 2 b. 32 A ## 3 33–36 A ## 2 36–38 A ## 3 39–40 A ## 2 At b. 41 we return to the opening material and the opening field, although the pcs McLeod chooses are in a different order than its first appearance at b. 1. This is followed by a reordering of A ## 2 in which the tritonal relationships from the opening are brought out: 41–46 A ## 1 47–52 A⁴ = A² ! = [(6⁴ / 333) + (6² / 5)] 53–62 A⁵ = [(242² / 5)+151] 63–78 A ## 3 The B section returns with a fairly conventional re-establishment of the B fields: 79–83 (RH) B ## 2 79–83 (LH) B¹ (complete) 84–85 (RH) B¹ (incomplete) 84–85 (LH) B² (incomplete) 86–88 B ## 2 The final refrain is followed by a coda featuring various reorderings of the original A field to bring out new intervallic features, particularly major thirds and tritones: 89–99 A ## 1 100–104 A⁶ = A¹ ! = [(4⁴ / 333) + 333] b. 105 A ## 4 106–107 A⁷ = A¹ ! = [313² + 333] / 2-4

Tone Clock Piece VI

After the brilliance and vitality of Tone Clock Piece V, Tone Clock Piece VI returns to the more serene waters of earlier pieces. In this case, the anchor form is extremely clear: the left-hand chords in 1 and 2 derive from a single IPF ## (323) ## 31 , while the right hand plays the symmetrical anchor ## (313): 1–7: A¹ = [(323² / 1) +313] However, from 8–20, things become a little more complicated. According to McLeod’s analyses, the left hand uses an aggregate of the two 323 IPFs (transposed up a fifth), but reordered to create a new field: 8–20: B¹ = A¹ ! ## +7 ## = [{7⁴ / 121}+313] The open fifths, however, are voiced and grouped in such a way that we might be tempted to interpret this passage as simply a sequence of symmetrical, but otherwise unrelated IPFs, even though this does not conform to a typical tone-clock field construction. 8–20: B ## 1 = 414+232+313 (alternative reading) According to McLeod’s analysis, from 21–24 the original transposition of the 323 IPFs returns, albeit reordered. 21–24: B² = A¹ ! = [{7⁴ / 121}+313] 31 In McLeod’s analysis, she refers to this as 343; this is in fact the same as 323, although 323 is technically the more ‘compact’ form. ## | 83 On the other hand, we might simply choose to interpret the reordering as a ‘revoicing’ of A, or a deliberate ‘playing’ with the effects of proximity. This is a good example of a passage in which it is unclear whether or not a new field has truly been introduced. From 25–28, field B ## 1 returns, followed by a short passage featuring A ## 1 (29–30). At b. 31, we seem to be getting a repeat of B ## 1 , as the same 131 anchor group is heard (Af-## B–C–E f). However, the shell has in fact been reordered. An IPF analysis quickly shows us that this is, in fact, just a transposition of field A¹ : 31–33: A² = B¹ ! = A¹ +7 The last 4 bars of the piece are codetta-like, featuring the anchor group from B ## 1 and A ## 2 now becoming the shell of a new anchor form: 34–37: C = {(313² / 5)+1441} No matter how you interpret the specific tone-clock configurations, the macrostructure is fairly clear: ## ABB ́–BAA ́–C

Tone Clock Piece VII

Common-practice major and minor triads form the primary material in Tone Clock Piece VII. However, their treatment is very different from common-practice tonal procedures, controlled instead by tone-clock steering principles. The first subject comprises a tone-clock steering of four triads—two major and two minor—presented in a homophonic, parallel ‘planing’, reminiscent of writing found in the preludes of both Debussy and Messiaen: 1–10: A¹ = [3-4⁴ ] / 242 The major triad has the tessellation property that it can only be tone-clock steered through ‘TI-tessellation’: that is, the inversion, the minor triad, must be used. Therefore, in this case, we have the following series of triads: C maj, F s maj, D min, Af min McLeod combines pairs of triads a tritone apart to form a subset of the octatonic scale, IPF 12312 ## 32 . Both the use of IPF 242 and the overriding octatonic flavour relate back to material found in the opening of Tone Clock Piece V. In 11–12 the same triads appear, although transposed by a fourth: 11–12: A² = A¹ +5 Following this short passage, the subject area B appears, comprising two fields: Field B ## 1 , the steering partner of A¹ , and B ## 2 , a transposition of B¹ . B ## 1 shares the same construct ion as Field A ## 5 in Tone Clock Piece V. The rhythmic material is developed from multiples of semiquavers, drawn from the Fibonacci series (1, 2, 3, 5, 8, 13). 13–17: B¹ = «A¹ » = [242² +151] / 3-4 18–20: B² = B¹ +5 21–23: B¹ (incomplete) At b. 24, we return to Subject A material, although the A ## 1 and A ## 2 fields are revoiced to bring out an underlying structure based on perfect fifths. Note that the aggregate of A ## 3 is also a mode of limited transposition, the hexatonic scale. ## 33 32 Incidentally, the only asymmetrical mode of limited transposition. See McLeod, unpub., p. 48 33 One of the few so-called ‘phi clusters’. See following article. 24–26: A³ = A ## 1 ! (revoiced as 7³ / 4-4) b. 27 A ## 1 28–29 A ## 3 b. 30 A ## 1 31–32 A⁴ = A³ -1 b. 33 A⁵ = A³ -3 31–36: A⁶ = A² ! = [7⁶ / 22222] The final section, marked Tempo primo, recapitulates the two generative fields, A ## 1 and B ## 1 , in a delicate oscillation: 37–40 B ## 1 41–44 A ## 1 45–46 B ## 1 47–48 A¹ (incomplete) The ‘blending’ of A and B material at the end makes for an slight twist on a conventional ternary form: ABA’’-C, where C is formed from a quick succession of A and B.

Conclusion

These analyses only scratch the surface of this complex and beautiful music, and its complex and beautiful theory. Although I have chosen to focus primarily on the technical constructs, I acknowledge with some regret that there is not time or space enough to thoroughly delve into areas such as phraseology, rhythmic structure, texture, tension and release, voice-leading, and ‘deep structure’. These aspects will, I hope, be picked up by other authors. A contextual view of the impact of Jenny’s introduction of tone-clock theory to an Australasian audience would also be timely. Sadly, it seems unlikely the tone-clock theory will ever replace pc set theory in musicological journals as the analyst’s terminology of choice. Nevertheless, to those who choose to absorb even some of its basic lessons, tone-clock theory will leave permanent imprints on their harmonic organisation, most notably in its insistence on the economy of intervallic content and its exploration of the alluring effects of musical symmetry, while advocating a colourful exploration of multiple chromatic transpositional levels.

Bibliography

• Boulez, Pierre. Penser la musique aujourd’hui. Gonthier. Geneva, 1963.

• Croft, John. ‘Earth and Sky: The Music’, Music in New Zealand, Spring 1993. pp. 20–24

• Forte, Allen. The structure of atonal music. Yale University Press, 1973

• McLeod, Jenny. Tone-Clock Theory Expanded: Chromatic Maps I & II. Unpublished (available as a free download from http:// sounz.org.nz/catalog/BO1095DS.pdf)

• —. 24 Tone Clock Pieces. Score. Wai-te-ata Music Press.

• Messiaen, Olivier. Technique de mon langage musicale. Alphonse Leduc. Paris, 1956.

• Schat, Peter. (trans. McLeod). The tone clock (Contemporary Music Studies, Vol. 7). Harwood Academic Publishers, 1993.

• Xenakis, Iannis. Musique formelles. Revue Musicale 253–4, Éditions Richard-Masse, 1981.

Appendix I

TONE-CLOCK REDUCTIONS OF TONE-CLOCK PIECES I–VII

Appendix II

PROOF OF CRITERIA FOR FORMATION OF SYMMETRICAL ANCHOR GROUPS

In this proof, I demonstrate one particular case of anchor group formation: steering a triad by a tetrad. The generalisation of this proof to any arbitrary IPF is not too difficult, but left to the reader. Lemma: any triad can be steered by a symmetrical tetrad using symmetrical inversions, resulting in a symmetrical aggregate (the ‘shell’) and a symmetrical complement (the ‘anchor’). Proof: Let A = {a,b,c}, where A is a triad and a, b and c are its constituent pc ints. Its inversion A ́ will then be {a,c-(ba),c} or {a,a+c-b,c}. Let B = {0, x, y, z}, where B is a tetrad steering group, and x, y and z are its constituent pc ints. In order to steer A by B, we use the transposition operator of addition (i.e. p ## 2 = p ## 1 +k, where k is the interval of transposition) upon every member of an IPF. We also have to use symmetrical inversions: for this example proof we will use the original form (A) for the first two transpositions and the inversion (A ́) for the remaining two—{A, A, A ́, A ́}—but we could also use the other possible arrangement of inversions {A, A ́, A, A ́} and the proof would be very similar. Now, A transposed by 0 and x gives us A+0 = {a,b,c} and A+x = {a+x,b+x,c+x}. For the two inverted triads, we transpose A ́ by y and z which gives us A ́+y = {a+y,a+c-b+y,c+y} and A ́+z = {a+z,a+c-b+z,c+z}. However, if B is a symmetrical tetrad, then we know that the interval between 0 and x is the same as that between y and z. This means that x = z-y, or y = z-x. Substituting this into A ́+y, we get the following: A ́+y = {a+z-x,a+c-b+z-x,c+zx}. This gives us the following four transpositions, which together will form our aggregate collection: ## {a,b,c}1. ## {a+x,b+x,c+x}2. {a+z-x,a+c-b+ z-x,c+z-x}3. ## {a+z,a+c-b+z,c+z}4. To prove that this aggregate is symmetrical, we use the following definition: two IPFs are related symmetrically if for every pc in one IPF, there exists a pc in the other IPF that relates via the same inversion equation: p ## 2 = k-p ## 1 , where k is some constant, and p ## 1 and p ## 2 are the two pc. (Incidentally, the axis of symmetry on the chromatic circle would run between k/2 and (k/2)+6.) We can use this definition to show that the steering aggregate is indeed symmetrical, by rewriting the last two sets to bring out a common term: {(z+a+c)-(c+x),(z+a+c)(b+x),(z+a+c)-(a+x)} and {(z+a+c)-c,(z+a+c)-b,(z+a+c)-a}. If we now set k = z+a+c, then we can rewrite these to prove the inversion relation between the first two and second two sets: {a,b,c} & {a+x,b+x,c+x} AND k-{a,b,c} & k-{a+x,b+x,c+x} which are therefore related symmetrically by inversion (around the axis {k/2,(k/2)+6}). Therefore, we can say that both the aggregate (the ‘shell’) is symmetric, and, by corollary, its complement (the ‘anchor’) is also symmetric. (A proof that the complement of a symmetric set is also symmetric is not included here). Note that if the aggregate generated by the transpositions is the complete chromatic scale, then the complement will of course be empty. In this case we will have formed a tone-clock steering rather than an anchor form. This does not affect the truth of the lemma. If we wish to generalise to IPFs other than triads, from this proof we can intuitively generate some necessary and sufficient criteria for forming an ‘anchor group’ construct from any IPF (although I have not included the general proofs of these here): The steering group must be symmetrical;1. If the IPF is also symmetrical, then a steering 2. group of any size can be used; If the IPF is asymmetric, then the steering 3. group must contain an even number of pc, and you must use a symmetrical system of IPF inversions when transposing (e.g. if there are four levels of transposition, transpositions 1 and 4 must be inversions (m-M or M-m), and transpositions 2 and 3 must be inversions).