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MULTIPHONICS ON VIBRATING STRINGS

Published online by Cambridge University Press:  19 December 2019

Caspar Johannes Walter
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Abstract

Multiphonics on vibrating strings have been an important element in my compositions since the early 1990s. In order to calculate the frequency components of so-called pure multiphonics (multiphonics consisting of harmonic partials of the fundamental) on vibrating strings, I developed my fraction windowing algorithm. The first section of this article details the use of multiphonics in my compositions and the second section discusses how the fraction windowing algorithm works and its relationship to the closely related mathematical concept of a continued fraction. The article also discusses the online apps I have developed as tools for composers and performers who are interested in using these methods in their own work on string multiphonics.

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Type
RESEARCH ARTICLE
Information
Tempo , Volume 74 , Issue 291 , January 2020 , pp. 7 - 23
Copyright
Copyright © Cambridge University Press 2019

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References

1 The term ‘pure multiphonic’ was coined by Ellen Fallowfield on her website Cello Map: www.cellomap.com/de.html.

2 See Walter, Caspar Johannes, ‘Mehrklänge auf dem Klavier: vom Phänomen zur mikrotonalen Theorie und Praxis’, in Mikrotonalität – Praxis und Utopie, ed. Pätzold, Cordula and Walter, Caspar Johannes (Mainz: Schott, 2014)Google Scholar.

3 If a particular point on the string is touched while the string is being played, two, three, four or even more clearly perceptible pitch can be heard simultaneously.

4 The apps are freely accessible and work on most modern web browsers installed on computers and mobile devices today without the need to install additional software. Since 2015 these apps have been developed in JavaScript in collaboration with Florian Bauckholt. See www.casparjohanneswalter.de/research.

5 www.casparjohanneswalter.de/research/piano_multiphonic_calculator.

6 In the example given in Figure 23, the numbers 7 and 12 have been typed into the two fields Partial 1 and Partial 2 on the left side of the upper section of the interface labelled red Multiphonic One (not shown).

7 The numerator and denominator of the mediant are the sums of numerators and denominators of the given fractions respectively, i.e. where a/c < b/d and a, b, c, d > 0, the mediant is defined as (a+b)/(c+d).

8 The procedure can also be extended to include negative whole numbers.

9 For a detailed explanation, see Walter, ‘Mehrklänge auf dem Klavier’, n. 2.

10 Fibonacci fractions are those with Fibonacci numbers in the numerator as well as the denominator. A sequence of Fibonacci fractions in which the Fibonacci number in the denominator is one term further than the numerator is a sequence without any mutation.

11 The most comparable source I found in the material available to me was the denominator algorithm developed by Bodo Werner in his ‘Fibonacci-Zahlen, Goldener Schnitt, Kettenbrüche und Anwendungen, für Lehramtsstudierende, SoSe 06’, www.math.uni-hamburg.de/home/werner/GruMiFiboSoSe06.pdf (accessed 19 August 2019).