Tone Clock

Introduction

The Tone Clock, and its related Tone-Clock Theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Because it emphasises the role of three-note chords (trichords or 'triads') in creating harmonic fields, it results in a high economy of musical intervals, despite the chromatic nature of its musical language.

Relationship to pitch-class set theory and serialism

While Tone-Clock Theory has many similarities to Allan Forte's pitch-class set theory, it places greater emphasis on the creation of pitch 'fields' from multiple transpositions and inversions of a single set-class, while also aiming to complete all twelve pitch-classes (the 'chromatic aggregate') with minimal, if any, repetition of pitch-classes. While the emphasis of Tone-Clock Theory is on creating the chromatic aggregate, it is not a serial technique, as the ordering of pitch-classes is not important, although it bears a strong similarity to the technique of 'serial derivation', which was used by Anton Webern and Milton Babbitt amongst others, in which a row is constructed from only one or two set-classes.

Peter Schat

The term 'tone clock' (toonklok in Dutch) was originally coined by Dutch composer Peter Schat, in reference to a technique he had developed of creating pitch 'fields' by transposing and inverting a trichord so that all twelve pitch-classes would be created once and once only.^[1] Schat discovered that it was possible to achieve a trichordally partitioned aggregate from all twelve trichords, with the exception of the diminished triad (036 or 3-10 in Forte's pitch-class set theory). Schat called the 12 trichords the 'hours', and they became central to the harmonic organization in a number of his works.

Jenny McLeod and Tone-Clock Theory

In her as-yet-unpublished monograph 'Chromatic Maps', New Zealand composer Jenny McLeod extended and expanded Schat's focus on trichords to encompass all 223 set-classes, thus becoming a true 'Tone-Clock Theory'.^[2] She also introduced new terminology in order to 'simplify' the labelling and categorization of the set-classes, and to draw attention to the specific transpositional properties within a field.

McLeod's terminology

The following terms are explained in McLeod's Chromatic Maps I:

• Intervallic Prime Form (IPF): the prime form of a pitch-class set, expressed as a series of interval classes (e.g. set-class (037) is called 3-4 in Tone-Clock Theory, as these are the interval classes between successive pitches in the prime form). Where possible, IPFs should be labelled using hour-group notation (see below). Furthermore, if an IPF can be rewritten so that the number of different interval classes in the title is one or two, then this is the preferred notation: e.g. IPF 143 (0158 in pc-set theory) can be rewritten as 414 or 434, which is to be preferred, as it makes the relationship to the trichords clearer — see below.
• Hours: the 12 trichordal set-classes, called 'triads' in Tone-Clock Theory. The 'first hour' is therefore IPF 1-1 (in pc-set theory, this would be set-class 3-1 or (012)), while the 'twelfth hour' is IPF 4-4 (in pc-set theory, this would be set-class 3-12 or (048)). In Tone-Clock Theory, the hours are often referred to using Roman numerals — so IV is IPF 1-4, while IX is IPF 2-5.
• Major/minor form: For 'asymmetrical' hours (hours that are formed from two different interval classes), the 'minor' form is the inversion of the triad with the smallest ic on the bottom, while the 'major' form is the inversion with the largest ic on the bottom. So, XIm is equivalent to a standard minor triad (3-4), while XIM is equivalent to a major triad (4-3).
• Hour Groups: IPFs with only one or two interval classes can often be related to a single hour, and relabelled using the Roman numeral hour notation to make this relationship clear. For instance, the tetrachord IPF 242 clearly relates to the 'eighth hour', IPF 2-4 (set-class 3-8 in pc-set theory). It can therefore be labelled as VIII⁴ — the 4 relating to its cardinality, a tetrachord. Note that some IPFs cannot be labelled as hour-groups if the distribution of intervals is ambiguous: e.g. for IPF 2232, it is unclear as to whether the generating trichord is 2-2 (VI) or 2-3 (VII). However, 2232 can be rewritten as 3223, 5225 or 5555 or 2323, all of which are valid hour groups (see 'Multiple-Hour Groups' below).
• Oedipus Groups: The commonest kind of hour-group, in which two interval classes alternate (e.g. the octatonic scale, in which the interval classes proceed 1212121, relating to the second hour (II, or IPF 1-2). These are simply written in the form: II⁸.
• Multiple-Hour Groups: Some IPFs can be rearranged so that while they are no longer in prime form, they do display a different hour relationship — for instance, 414 (IVM⁴) can also be rewritten as 434 (XIM⁴). In Tone-Clock Theory, this is considered to show that an IPF has multiple relationships to different hours, which can be brought out by the composer depending on how they are voiced and utilized.
• Symmetrical Pentads: A pentachord/pentad that has a clear relationship to an asymmetrical hour, but in which the two interval classes are arrayed symmetrically rather than alternately (e.g. 2442) is called a 'symmetrical pentad', and is written thus: SP VIII.
• Steering: one IPF transposes by another (i.e. IPF a 'steers' IPF b). If IPF a and b are the same, then this is 'self steering'. Note that the IPF does not necessarily remain in its prime form, but can also appear inverted. In Tone-Clock Theory, the 'steering group' (the IPF that is underlying the transpositional levels) has a kind of 'deep structure' status — the listener does not necessarily hear its immediate effect, but it governs elements such as voice-leadings.
• Reverse Steering: the 'steering group' becomes the 'steered group' and vice versa — i.e. IPF b 'steers' IPF A. In Tone-Clock Theory, this is considered to have a kind of 'symmetry', and often appears to provide contrast or 'closure' to a passage.
• Twelve-Tone Steering or Tone-Clock Steering: a specific steering of an IPF so that the chromatic aggregrate is create with no repetition of pc. All of the triads except the tenth hour (the diminished triad) can be steered in this way. Some tetrachords, and all hexachords that are self-complementary (i.e. not Z-related) can also be steered in this way.
• Anchor Form: the creation of the twelve-tone aggregate with no pc repetition, typically from a tetrachord, but using a second IPF to complete the aggregate.

Mathematical generalizations of 'tessellating' set-classes

New Zealand composer and music theorist Michael Norris has generalized the concept of 'tone-clock steering' into a theory of 'pitch-class tessellation', and has developed an algorithm that can provide tone-clock steerings in 24TET. He has also written about and analyzed Jenny McLeod's 'Tone Clock Pieces'.^[3][4]

References

1. ↑ Schat, Peter (1993). Tone Clock (Contemporary Music Studies, vol. 7). Routledge. 
2. ↑ McLeod, Jenny. "Chromatic Maps I & II". SOUNZ. 
3. ↑ Norris, Michael (2006). "Tessellations and Enumerations: generalizing chromatic theories". CANZONA: The yearbook of the Composers Association of New Zealand: 92–100. 
4. ↑ Norris, Michael (2006). "Crystalline Aphorisms: commentary and analysis of Jenny McLeod's Tone Clock Pieces I–VII". Canzona: the yearbook of the Composers Association of New Zealand: 74–86. 

Atonality and post-tonality
Scales and tuning	Equal temperament Modes of limited transposition Mystic chord Octatonic scale Whole tone scale
Genres and schools	Twelve-tone technique Row List Second Viennese School Serialism Spectral music
Techniques and concepts	Chromaticism Cyclic set Dissonant counterpoint Dynamic tonality Emancipation of the dissonance Klangfarbenmelodie Polymodal chromaticism Distance model Quartal and quintal harmony Tone Clock Tone cluster Unified field
Compositions	List of atonal compositions