Abstract
XronoMorph is a musical loop generator that opens up two huge spaces of unusual and interesting polyphonic rhythms: perfectly balanced rhythms and well-formed rhythms. These are rhythms that would often be hard to create in alternative software applications or with traditional musical notation. In this chapter, I explain the algorithmic principles used to generate the loops and how these principles have been parameterized and visualized to facilitate the exploration of paths within these two rhythmic spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The name XronoMorph is derived from the Ancient Greek nouns χρόνος (khrónos, “time”) and μορφή (morphḗ, “form, shape”) whilst also noting the recent English (Greek-derived) verb morph, which means “smooth change”. We leave its pronunciation to the reader!
- 2.
A hocket is a melody whose successive pitches are sounded by different musicians—a practice used in medieval vocal music and in contemporary African and gamelan music.
- 3.
Additional organizational principles, parametrizations and visualizations will likely be added in future versions.
- 4.
Changing the “swing ratio” usually delays every second column’s beats so they no longer sound precisely halfway between their previous and following beats.
- 5.
Indeed, XronoMorph is coded in Max and compiled as standalone applications for macOS and Windows.
- 6.
Well-formedness was originally developed to explain musical scales (Carey and Clampitt 1989).
- 7.
Alternative ways of independently visualizing each different rhythmic level include using only small disks (no lines) on the circumference of the circle but colouring them according to the level, or using a set of concentric circles—one for each level. Every method has advantages and disadvantages: we chose inscribed polygons because they make the geometrical relationships clear (concentric rings distorts the relationships, colours are not obvious enough).
- 8.
For a video demonstration of perfect balance using a physical bicycle wheel with attached weights, see https://youtu.be/ipiogoqBibw.
- 9.
Weights could also represent loudness, or probability of occurring, in which case real numbers are interpretable so long as they are nonnegative.
- 10.
The term “primitive” refers to their lack of rotational symmetry.
- 11.
The computational methods we have used are detailed in (Milne et al. 2018).
- 12.
The LCM value is calculated from the user-selected polygons; it shows the smallest possible regular grid size that can contain all polygons’ vertices when they are snapped to that grid.
- 13.
There are still other parametrizations that naturally derive from alternative ways of generating well-formed patterns through the use of the repeated iteration of an interval, where the size of the commonest interval (the generating interval) in the scale or rhythm is adjusted. For pitch-based scales this may be important because we may want this commonest interval to be itself consonant or a simple fraction of a consonant interval. This is the parameterization utilized in much microtonal music theory and in the Dynamic Tonality software synthesizers (http://www.dynamictonality.com).
- 14.
This regular metrical grid divides the period into \( mL + ns \) equal parts.
- 15.
Note that this is the same pattern as the pentatonic, diatonic, and chromatic scales. Other well-formed patterns are quite different to this hierarchy—they have differing numbers and arrangements of large and small steps.
- 16.
These higher-level r-values are calculated as a function of the r-slider’s value and m and n; the equations for this task are beyond the scope of this chapter but are detailed in (Milne and Dean 2016).
- 17.
The Stern-Brocot tree is a systematic enumeration of the rational numbers independently discovered in the 19th century by the mathematician Moritz Stern and the watchmaker Achille Brocot. Stern’s focus was mathematical whereas Brocot’s focus was the specification of gear ratios for clock design. The tree provides a method to iteratively generate all rational numbers, in reduced form, exactly once.
References
Amiot E (2007) David Lewin and maximally even sets. J Math Music 1(3):157–172
Arom S (1991) African polyphony and polyrhythm: musical structure and methodology. Cambridge University Press, Cambridge
Berstel J, Lauve A, Reutenauer C, Saliola FV (2008) Combinatorics on words: Christoffel words and repetitions in words. American Mathematical Society, Providence(RI)
Carey N, Clampitt D (1989) Aspects of well-formed scales. Music Theory Spectr 11(2):187–206
Clough J, Douthett J (1991) Maximally even sets. J Music Theory 35(1/2):93–173
Hamilton TJ, Doai J, Milne AJ, Saisanas V, Calilhanna A, Hilton C, Goldwater M, Cohn R (2018) Teaching mathematics with music: A pilot study. In Proceedings of IEEE International Conference on Teaching, Assessment, and Learning for Engineering (TALE 2018), University of Wollongong, NSW, Australia
Hofstadter DR (1985) Metamagical themas: questing for the essence of mind and pattern. Basic Books, New York(NY)
Kilchenmann L, Senn O (2015) Microtiming in swing and funk affects the body movement behavior of music expert listeners. Front Psychol 6(1232)
London J (2004) Hearing in time: psychological aspects of musical meter. Oxford University Press, Oxford
Milne AJ (2018) Linking sonic aesthetics with mathematical theories. In: McLean A, Dean RT (eds) The Oxford handbook of algorithmic music. Oxford University Press, New York, NY, USA, pp 155–180
Milne AJ, Bulger D, Herff SA (2018) Exploring the space of perfectly balanced rhythms and scales. J Math Music 11(2)
Milne AJ, Bulger D, Herff SA, Sethares WA (2015) Perfect balance: A novel principle for the construction of musical scales and meters. In: Collins T, Meredith D, Volk A (eds) Mathematics and computation in music. Springer, Berlin, pp 97–108
Milne AJ, Dean RT (2016) Computational creation and morphing of multilevel rhythms by control of evenness. Comput Music J 40(1):35–53
Milne AJ et al (2011) Scratching the scale labyrinth. In: Amiot E et al (eds) Mathematics and computation in music. Springer, Berlin, pp 180–195
Milne AJ et al (2016) XronoMorph: algorithmic generation of perfectly balanced and well-formed rhythms. In: Proceedings of the 2016 international conference on new interfaces for musical expression (NIME 2016), Brisbane, Australia
Toussaint GT (2013) The geometry of musical rhythm: what makes a “good” rhythm good?. CRC Press, Boca Raton
Acknowledgements
Dr. Andrew Milne is the recipient of an Australian Research Council Discovery Early Career Award (project number DE170100353) funded by the Australian Government.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Milne, A.J. (2019). XronoMorph: Investigating Paths Through Rhythmic Space. In: Holland, S., Mudd, T., Wilkie-McKenna, K., McPherson, A., Wanderley, M. (eds) New Directions in Music and Human-Computer Interaction. Springer Series on Cultural Computing. Springer, Cham. https://doi.org/10.1007/978-3-319-92069-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-92069-6_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-92068-9
Online ISBN: 978-3-319-92069-6
eBook Packages: Computer ScienceComputer Science (R0)