Alma Mater Studiorum University of Bologna, August 22-26 2006
SINGLE TIME- CONTROLLED FRACTIONAL NOISE ALGORITHM Costantino Rizzuti
Riccardo Santoboni
[email protected]
Conservatorio di Musica “N. Piccinni” Bari
[email protected] examples of music of different styles. This research shows how, in many compositions, loudness and pitch are distributed according with a trend which is statistically similar to a noise characterized by a power spectral density function:
ABSTRACT Musical composition has been deeply influenced by mathematic processes during the centuries. Since 80s, chaos and fractal geometry have strongly affected the development of a new fields of musical research, with the use of fractional noises. In this work we will present a single algorithm producing a family of fractional noises with changing power spectral density on time. The algorithm was implemented by VST Plug-In technology, in order to have wide range audio software compatibility. The results of our plug-in were compared with several integer index (white, pink, and brown) fractional noise algorithms available from most common audio software, with an excellent data matching.
P( f ) =
(1).
This kind of noise, also called pink noise, is characterized by the same time correlation trend as the music analyzed by Voss and Clark, for this reason they hypothesized it might be possible to use this noise to generate musical compositions exhibiting pleasing musical characteristics. Pink noise, together with white and brown noises, are a members of the family of random processes called fractional noises. Fractional noises are a special class of random processes characterized by a power spectral density:
INTRODUCTION New methods in computer music are constantly affected by experiences and knowledge, in particular, from science and mathematics. Researches carried out by Voss and Clarke at California University on temporal correlation between events in musical compositions induced the growth of new fields of musical research by using fractional noises [1, 2]. Voss and Clarke evaluated long-term variations of pitch and loudness through the analysis of the audio signal in many
P( f ) =
1 fβ
0≤β ≤2
(2)
this is the way the power density decreases with increasing frequency over a spectral range. Fractional noises exhibit a wide range of autocorrelation trends associated with the variation in the index β, these being produced by random processes characterized by different memory degrees. Fractional noises with integer index (white: β=0, pink: β=1, and brown: β=2) have been widely used either for sound synthesis or/and to control musical parameters such as pitch and duration or complex patterns time distributions. In literature, we found specific and different approaches for each integer index β fractional noise generation. Even though a unique mathematical formula already exists to describe this process, at the moment we could find no mention of any method to generate all kinds of fractional noises with real index β, using a single algorithm.
In: M. Baroni, A. R. Addessi, R. Caterina, M. Costa (2006) Proceedings of the 9th International Conference on Music Perception & Cognition (ICMPC9), Bologna/Italy, August 22-26 2006.©2006 The Society for Music Perception & Cognition (SMPC) and European Society for the Cognitive Sciences of Music (ESCOM). Copyright of the content of an individual paper is held by the primary (first-named) author of that paper. All rights reserved. No paper from this proceedings may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information retrieval systems, without permission in writing from the paper's primary author. No other part of this proceedings may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information retrieval system, without permission in writing from SMPC and ESCOM.
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ICMPC9 Proceedings The purpose of this work is to present a single algorithm that generates fractional noises characterized by every value of the exponential real index β defined in the interval [0, 2]. The β coefficient can be controlled during the process in order to produce a family of fractional noises with changing power spectral density on time. The algorithm was implemented by VST Plug-In technology, in order to have wide range audio software compatibility.
FRACTIONAL NOISES GENERATION A spectral synthesis process has been implemented in a single algorithm for fractional noises generation. Therefore, all properties of the signal are entirely defined in the frequencies domain. Freed proposes an interesting review, chronologically ordered, of many spectral synthesis applications; moreover supplies a wide bibliography useful to understand contributions and developments of this technique in the time [3]. Power spectrum density (PSD) is the starting point for the fractional noises generation process; the algorithm can be listed in five steps: • • • • •
Figure 1. White noise amplitude
Generation of a power spectrum density table Evaluation of the amplitude spectrum Generation of the phase spectrum Computation of the spectrum in complex form Computation of IFFT to obtain the signal in the time domain.
Figure 2. Pink noise amplitude
In the first step we create a table of PSD values over frequency, for any fixed value of β, through the relation:
PSD(n ) =
1 Sr n N
β
(3)
where Sr is the sampling rate of the audio signal, N is the number of points used to sampling the spectrum, then equal to the number of point utilized to perform the IFFT. Index n = {1,2,......, N } is used to scan the frequency axis from 0 to the sampling rate value. Second step can be realized by performing a square roots operation on the values of the PSD(n) table:
A(n ) = PSD(n )
Figure 3. Brown noise amplitude An uniform distribution of random numbers in the range [0, 2π] define the phase spectrum. L’Ecuyer [4] proposes a wide and interesting review of methods and implementations of random number generators. Third step, involving phase spectrum generation, has been performed by using the random number generator Mersenne Twister proposed by M. Matsumoto [5], because was the best matched for our task. Fourth and fifth steps allow obtaining the audio signal in time domain; by performing a transformation between polar to Cartesian coordinate it is possible, combining
(4).
This step allows to evaluate the amplitude spectrum of the noise with a particular β index value; figure 1-3 show the log (amplitude) vs. log (frequency) graphics for integer index fractional noises, the slope match the value of β.
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ICMPC9 Proceedings amplitude and phase spectrums, computing the noise spectrum in complex form. Finally performing an IFFT of the signal spectrum in complex form allows obtaining the audio signal in the time domain. Figure 4 shows the noise generation process flow chart.
paragraph. In the Plug-In VST class were defined two methods to perform the noise generation: •
•
GenerateNoiseSpectrum() IFFT(),
First method generate the noise spectrum in complex components for the current value of β index; the second one implement the Inverse Fourier transform to obtain the audio signal’s sample values in the time domain. The IFFT is a time discrete process; in order to have a time continuous stream of audio samples the Plug-In VST process() method was defined reading the audio samples from two buffers alternatively. Figure 5 shows the flow chart of the noise generation process implemented in the Plug-In.
Figure 4. Noise generation process flow
ALGORITHM IMPLEMENTATION The algorithm was implemented as Plug-In VST in order to realize a real time control on the parameter β during the generation process [6]. This real time control allows producing a family of fractional noises with changing power spectral density on time. The Plug-In presents two audio output and none input; no MIDI interface was implemented because not useful in controlling noise generation process. Two parameters control the Plug-In: first one is used for output volume of audio signal, (also working as switch to turn on the audio generation process); second one is used to control the β value in the noise generation process. Chancing second parameter values it is possible to generate fractional noises characterized by every value of the exponential real index β defined in the interval [0, 2]. The audio signal generation process is realized in accordance with the algorithm shown in the previous ISBN 88-7395-155-4 © 2006 ICMPC
Figure 5. Plug-In flow chart
VALIDATION AND CALIBRATION GENERATION PROCESS
OF
Validation and calibration of the noise generation process was realized comparing noises produced with different audio software. Comparing was made taking using the integer index fractional noise: white, pink and brown 346
ICMPC9 Proceedings noise; as comparison term were used three audio software: AdobeAudition ver. 1.5, Max/MSP Cycling ‘74/IRCAM, and Litter Power Starter Pack, an “Extras” package for Max/MSP realized by Peter Castine [7]. Figure 6 shows the comparison between white noise spectra generated with different software; spectrum trends are very similar, the graphic show a remarkable difference in the mean amplitude of the signals. Figure 7 shows comparison among pink noise spectra: pink noise spectrum from Litter Power differ from the
others because this kind of generator use the Voss-Gardner method that approximate the pink noise by summing many white noise sources. Plug-In noise spectrum is more linear than which generated with Audition and MSP, as is possible to notice in figure, differences are remarkable in high frequencies range in the right side of the graphic. Moreover noise spectra produced with Audition and MSP exhibit many ripples in the high frequencies range, probably due to the generation of pink noise by filtered white noise.
Figure 6. Comparison between white noise spectrums
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ICMPC9 Proceedings Figure 7. Comparison between pink noise spectrums
Figure 8 shows comparison between brown noise spectra. Spectra of noises produced by Audition and Litter Power present a deviation from a linear trend, more than the one produced with our Plug-In. In fact, as shown in figure,
differences between spectral trends are more evident in high frequencies range.
Figure 8. Comparison between brown noise spectrums [2] W. Kirby, Compositional Chaos And Music Pleasure, Proc. of 20th Int. Technological Directions in Music Learning Conference, 2003. [3] A. Freed, Real-Time Inverse Transform Additive Synthesis for Additive and Pitch Synchronous Noise and Sound Spatialization, AES 104th Convention, San Francisco, CA, AES, 1998. [4] P. L’Ecuyer, Uniform Random Number Generators: A Review, Proc. Of Winter Simulation Conference, IEEE Press, Dec 1997. [5] M. Matsumoto, http://www.math.sci.hiroshimau.ac.jp/~m-mat/eindex.html, 2002 [6] http://www.steinberg.de/Steinberg/ygrabit/ [7] http://www.bek.no/~pcastine/Litter/LPStarterPackW.zi p
CONCLUSIONS In this work we have described a new techniques for fractional noises generation through only one algorithm. The algorithm has been implemented as Plug-In VST in order to have a dynamically controllable fractional noises generator. The results of our plug-in were compared with several integer index (white, pink, and brown) fractional noise algorithms available from most common audio software. Our comparison of fractional noises sources shows an excellent data matching by performing a qualitative analysis of amplitude spectrum, we need moreover performing a quantitative comparison which could give us a more consistent validation of the results. We are trying using our Plug-In in musical application controlling either event time parameters involved in the compositional process or audio time scale parameters for complex sound synthesis.
REFERENCES [1] C. Dodge, T.A. Jerse, Computer Music II ed., Schirmer.
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