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Unmögliche Räume: Anmerkungen zu Steven Daversons jüngsten Werken


Impossible Spaces: Notes on Steven Daverson's Recent Works

Markus Böggemann - University of Kassel

Markus Böggemann's article in the German journal MusikTheorie introduces the preoccupations and techniques of my recent work, focused on Arkanar: A Report from the Interior for tuba and live electronics and the ongoing projects La Nitroglycerine des Lumières for saxophone, percussion, electric guitar, piano, devices and live electronics, and the piano works Studies for Figures Outside a Dacha, and Figures Outside a Dacha, with Sunrise. He alludes to a software program I am compiling in Max to assist in the pre-compositional aspects of writing in multiple tempi, and links to this site for more information about the ideas behind the software and its functions.

Purchase from Laaber-Verlag 

Tempus in Machina (or TiMa, pronounced “timer”) is a method currently in development for planning and executing complex tempo relationships within a musical work that have hitherto been fiendishly difficult to achieve, including music in multiple independent tempi that accelerate or decelerate and then resynchronise accurately. Previously, this has been easy for composers to imagine, but the number of calculations required to compose with independent tempi in this way has frequently been too great to be practical. TiMa is intended to be released in the next couple of years as a series of Max objects that will be made available for free, should research support become available.


The process I have devised for planning the temporal relationships in music is deceptively simple, using simple Pythagorean equations to calculate the amount of music that needs to be written over a period of time of a given duration and given starting and ending tempi. With straight tempo trajectories this is quite straightforward (Fig.1), and can tell us with some accuracy how many beats are needed and where they should be located.

Pythagoras Triangle.png

Figure 1. A simple example of a Pythagorean equation applied to musical parameters.

c = sqrt (a^2 + b^2)           therefore           number of beats = sqrt ([change in tempo]^2 + [time]^2)

However, this linear trajectory often results in a non-integer number of beats, which can make the music difficult to perform or interrupt a metrical flow. TiMa can handle so-called “irrational” time signatures of the sort found in works by Thomas Adés, but they are not necessary to ensure that the music synchronises properly. By changing the amount of curvature in the acceleration we can change the number of beats in a musical passage to ensure that there is always an integer number of beats, possible with exponential and S-shaped curves (Fig.2). 

Pythagoras Curve.jpg

Figure 2. Multiple Pythagorean equations performed over a curved trajectory, with the total of each small hypotenuse (c1, c2, c3... above) will produce the number of beats. Changing the amount of curve will allow greater control and flexibility over the number of beats over a given period of time. TiMa will perform an equation for every millisecond over the course of a section, far more than in the example above.


Previously, composers were often tied to particular tempi that were mathematically related. Fascinating and effective as it is, Stockhausen’s “chromatic tempo scale” as described in his seminal article How Time Passes... and demonstrated in GRUPPEN, has some tempi that can appear unusual (such as quarter-note = 113.5) to musicians who prefer the more familiar numbers found on traditional mechanical metronomes. Again, TiMa can handle decimal tempo markings like Stockhausen’s, but they are no longer a necessity; it is now possible to choose any metronome marking as the starting and end points of a musical passage and calculate the trajectory from there.


This allows composers to write multitemporal music that is less problematic for performers, while still maintaining the rigorous approach required to make it work logically. Of course, this still requires some additional work on the part of the composer. 


At the time of writing, no mainstream software program is capable of producing a score like this without some considerable compromises. Score, the DOS-based software used to produce the score for Harrison Birtwistle’s Theseus Game and Panic (Fig.3), is able to handle these kinds of scores and produces beautiful results, but is hard to use and is not available on Mac. LilyPond is also capable of realising complex metrical relationships, including irrational time signatures, but it is also hard to use and rarely produces results of a publishable standard. Sibelius and Finale require one to notate each stave as an individual system, which produces issues when one reaches the end of a page part-way through a bar, which happens frequently. Dorico comes closest as it supports independent time signatures that produce bars of different lengths within a single system, which can then be subdivided with irrationals hidden, but full support for completely independent strands of music in different tempi is still not available. Even if these programs were easy to use, they would not be intuitive enough to compose with. This is leaves us with no choice but the old-fashioned method of pencil and paper, but there one must plot every beat in the piece individually to write notes and gestures proportionally in the manuscript.

Harrison Birtwistle Panic Score

Figure 3. Harrison Birtwistle, Panic. bb.295-305. Published by Boosey & Hawkes. Permission sought.

Typeset in Score, note the ways that the barlines do not synchronise between the orchestra and solo saxophone (top), and the solo drum kit and trombone section (bottom), a typographical feat that is very difficult to achieve in mainstream music software.


Once again, TiMa has a method to help produce accurate manuscript paper. The composer can specify the length of the space starting after the time signature and ending at the end of the system in millimetres, and how many pages they want their score to be. TiMa will then produce a chart that says where each beat should be placed from the edge of each page, for each part, for the whole score (Fig.4). There is also space left at the beginning and end of each system for points where whole beats would not fit. Currently, this data needs to be transferred to the paper or the software manually (Fig.5), but I would be excited to hear from anybody who knows an effective way to automate this! It would be especially useful to be able to import this into Sibelius, Finale, or Dorico via an XML file, or into a graphics programme like Illustrator or GIMP, though that would require additional efforts to build the staves.


[Figure 4. Fragment of chart of numbers]

Nikel Manuscript.jpg

Figure 5. Excerpt from the manuscript paper for La Nitroglycerine des Lumiéres, with beat durations marked.

The application of these processes in my own music is based on the combination of structures that are derived from subdivisions of time periods measured in “clock time” (minutes, seconds, milliseconds etc.), and how they map onto the more complex tempo-based structures produced by TiMa measured in a more flexible “musical time” (beats). Thus, it is essential that one can translate a particular instant in “clock time” within a series of beats which themselves might be of changing durations, which allows for synchronisation of individual attacks between different parts in independent tempi. TiMa is capable of performing these calculations with great precision, and will tell the composer where within a series of beats a desired attack should occur as a decimal (Fig.6). 


It is important that TiMa leaves the translation of these attacks into notation to the composer, so they can decide the degree of rhythmic complexity or flexibility for themselves.

Lumieres SCORE fragment from beginning.j

Figure 6. Moments of alignment in La Nitroglycerine des Lumières (electric guitar and piano only)


All music that features (or simulates) multiple tempi inevitably has some form of compromise. Stockhausen has some peculiar tempo markings in GRUPPEN as already described, also requiring multiple conductors and precluding any kind of acceleration of deceleration of tempo unless all three conductors are in unison; Carter is limited by his decision to simulate other tempi using basic subdivisions of the beat in a single global tempo (cf. his First String Quartet); Adés’ use of irrational time signatures, and little guidance given to performers to resynchronise at important moments make his Piano Quintet fearsomely difficult to perform, belying its classicist surface; Birtwistle also calls for two conductors, and uses a system of pauses and cues that allow for greater metrical flexibility, but does not always allow for a consistent musical flow. TiMa also requires a compromise: to perform with anything approaching the accuracy that can be achieved, one must use a click track. However, TiMa can use the data produced during the pre-composition process to create a click track quickly and easily. Furthermore, by manually inputting time signatures (or extracting them from an XML file) a composer can stipulate where a different sound is heard on the first beat of every bar, and remove unnecessary clicks to permit players greater freedom in performance if desired. This part of TiMa’s functionality is still in its early stages and will be tested for the first time in a new work for Ensemble Nikel, La Nitroglycerine des Lumières

Steven Daverson, December 2019

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