PERCEPTUAL ATOMIC NOISE

Kristoffer Jensen University of Aalborg Esbjerg Niels Bohrsvej 6, DK-6700 Esbjerg [email protected]

ABSTRACT signal point-of-view that creates with a uniform distribution and spectrum, or the physical point-of-view A noise synthesis method with no external connotation that creates a noise similar to existing . Instead, is proposed. By creating atoms with random width, an attempt is made at rendering the atoms perceptually onset-time and , most external connotations white, by distributing them according to the perceptual are avoided. The further addition of a frequency frequency axis using a probability function obtained distribution corresponding to the perceptual Bark from the bark scale, and with a magnitude frequency, and a spectrum corresponding to the equal- corresponding to the equal-loudness contour, by loudness contour for a given phon level further removes filtering the atoms using a warped filter corresponding the synthesis from the common signal point-of-view. to the equal-loudness contour for a given phon level. The perceptual frequency distribution is obtained by This gives a perceptual white resulting spectrum with a creating a probability density function from the Bark perceptually uniform frequency distribution. scale, and the equal-loudness contour (ELC) spectrum is created by filtering the atoms with a filter obtained in 2. NOISE the warped frequency domain by fitting the filter to a simple ELC model. An additional voiced quality Noise has been an important component since the parameter allows to vary the harmonicity. The resulting beginning of music, and recently noise music has sound is susceptible to be used in everything from loud evolved as an independent music style, sometimes noise music, contemporary compositions, to meditation avoiding the toned components altogether. The music. distinction between noise and tone has been clear for a long time. Helmholtz [5] and Schaeffer [6] opposed 1. INTRODUCTION sounds to unvoiced sounds. The mpeg 7 audio description [7] distinguishes between harmonic and The unvoiced sounds are used in many musical percussive, coherent and non-coherent, and sustained situations, from the abundance of cymbals and hihats in and non-sustained sounds. rhythmic music to the musique concrète of Schaeffer, Zwicker [8] found that band-pass filtered noise with the stochastic (random) processes of Xenakis, the large bandwidth has low relative pleasantness, as granular music of modern computer music or noise compared to sinusoids and band-pass filtered noise with music. low bandwidth. Three noise prototypes are identified here; random Noise as a component of music is found in its purest values (dice noise), random events (Geiger noise), and form in some percussion instruments, and in particular random (cymbal noise). The atomic noise in the unvoiced consonants of the human voice. The has previously been shown [1] to produce almost all futurist proposed, seemingly without success [9], a pure noise types, while permitting to vary the degree of series of instruments; the intonarumori that produced Geiger-, or cymbal-ness, thus easily creating a large rumbles, whispers, creaks, and other noises. Schaeffer variety of sounds with little natural connotation. This is went on to collaborate with Pierre Henry on musique done by adding atoms with random amplitude, width, concrète, in which recorded sounds, many of them frequency and onset time. unvoiced, were used in the compositions. Stockhausen Other noise synthesis methods include the - and others used electronic generators, sinusoids and wave-function (FOF) with random onset time [2], used white noise in their early works. Another composer, to resynthesize naturally occurring musical noises. The Xenakis, used stochastic processes in both compositions shaken instruments have a random event distribution but also in the creation of new sounds. The distinction that can be modelled through the stochastic event between composition and sound events have been modelling [3], while most other musical instruments blurred even more in the granular synthesis [10], a need random irregularities on the frequencies and method in which long music pieces can be obtained by amplitudes to produce an interesting sound [4]. random summation of time or frequency shifted short The atomic noise was created to produce sounds (10-50 msec) grains, often extracted from a short which demanded no control over the spectral or the sampled waveform using a random selection and temporal envelope. Contrary to the granular synthesis or manipulation process. the musique concrète, or the different noise synthesis The use of noise has been further enlarged by two models, there is no ‘original’ timbre that is retained in recent trends. In the noise music, adapts such as the resulting music. Merzbow or Caspar Brötzmann, use noise played In this work a further attempt is made to remove the extremely loud in a generally rather static way. ‘Noise ‘external’ influence of the music, by abandoning the

2 music becomes ambience not as you learn how to listen, ⎛ t−t0 ⎞ t − t −⎜ ⎟ or when you accept its refusal to settle, but when you atom(t) = a cos(2πf 0 )e ⎝ σ ⎠ . (1) are no longer in a position to accept or deny’ [11]. This sr stands in full opposition functionally with the use of noise to create ‘relaxation and calm, promoting sleep, The amplitude a is a random variable with a Gaussian and blocking annoying noises’ [12]. By mixing and distribution and the frequency f is random values with editing the perceptual atomic noises together, looping uniform distribution, t0 is the onset time and σ is the them, and ending up with an entire piece of music, it standard deviation (width) of the Gaussian. sr is the becomes obvious that the perceptual atomic noises can sample rate. The atomic noise is created by inserting also make up complex contemporary music. one new atom at time t0, if the random value drawn is greater than the probability threshold p. 3. ATOMIC NOISE As σ is approaching zero, the corresponding signal gets small duration and large bandwidth, thus The atomic noise [1] is a method used to easily create approaching the Geiger noise. As σ increases, the any kind of sounds between three kinds of prototypic sinusoids get more duration and less bandwidth, and the noises, without any attempt to model spectral or signal approaches the cymbal noise. Examples of the temporal envelope behaviour. These three prototypic atom noise as a function of σ and p are shown in figure noises are the dice, cymbal and Geiger noises. The 1. atomic noise is created by adding atoms with random Increasing width of atoms 15000

amplitude, frequency, duration and onset time. The ) z H

( 10000 different prototypic noises are created by setting the y enc distribution parameters appropriately. In addition, the 5000 equ r atomic noise harmonicity is varied by different f 0 procedures; either the time or frequency distribution is 0 1 2 3 4 5 time (sec) made periodic, or a short created noise (frozen noise) is Increasing probability of atoms repeated. 15000 ) z H

( 10000 3.1. Noise prototypes y enc 5000 equ r

The random values (dice) method of creating unvoiced f sounds is the most common method today. By using a 0 0 1 2 3 4 5 new, uncorrelated, value at each time sample, an time (sec) unvoiced, uncoloured sound is obtained. The distribution of the random values has not been found to Figure 1. Spectrum of atomic noise with increasing σ be very important, perceptually. (top) and p (bottom). The summation of a large number of sinusoids with random frequencies evenly distributed on the 3.3. Tone from noise frequencies also renders an unvoiced sound, if the Since the Geiger noise becomes harmonic when the number of sinusoids is high enough. As the sound probability of pulses increases at periodic times, and resembles that of a cymbal for a relative low number of similarly, the cymbal noise becomes harmonic when the sinusoids, this noise generation method is called cymbal probability of sinusoids increases at periodic noise. frequencies, a periodic distribution for the stochastic Pulses randomly distributed in time are heard as a signals is proposed [1]. The periodic distribution is ticking noise, reminiscent of a Geiger counter, when the based on a triangular window, raised to the wth power, number of pulses is low. Pierce [13] differentiates w between slow random pulses, which are heard as x p(x) = Λ x − 0 0 ≤ x ≤ x (2) separate pulses, whereas at a few hundred pulses per 2 0 second, not all pulses are detected individually. Above a few hundred pulses per second, a smooth noise is heard, Λ is a normalization necessary to obtain a power with no individual pulses perceivable. density function and w is the harmonicity coefficient. A 3.2. Atom noise synthesis harmonicity value of zero produces unvoiced sounds, while a higher value produces a more toned sound. The If the notion of random value, time and frequency is eq. (2) is repeated for the duration of the noise, or up to retained, much of the noise types can be obtained [1] by the Nyquist frequency, in case of time and frequency adding together a large number of sinusoids with periodic distribution, respectively. Another method for random amplitude and frequency, multiplied by a creating tone from noise is by using frozen noise (short Gaussian shape with random standard deviation at noise segment), and repeat it. Warren [14] used frozen random starting times, noise to show the perception of infrapitch (very low frequency pitch) and found whooshing, motor boating, and noisy pitch sound, with additional rattles, clangs

and other metallic types of sounds dependent on the the perception and possibly the attention to each repetition rate. component is weighted equally.

4. PERCEPTUAL ATOMS 4.2. Atoms Amplitude The relative amplitude of the atoms is also made so as to While the atomic noise was created with the intent of invert the sensitivity of the ear. As this sensitivity, making music with the least spectral and temporal commonly measured in phon as the equal-loudness envelope modification necessary, doubt arises as to the contour (ELC), is dependent on the loudness of the well-foundedness of the selected frequency distribution atom, the atom amplitude should also be dependent on and spectral envelope model. Indeed, the uniform the final loudness, as well as on the frequency. frequency distribution (in Hz), and the white spectral It is easy to multiply the individual atom amplitude envelope originates mainly from a signal point-of-view. with the equal-loudness contour directly, but this does Another solution is found if the inspiration came from not give a perceptual white spectrum for the short atoms either the physical world around us or from the that would still sound as clicks. Instead, it is necessary sensitivity of the auditory system. The solution chosen to filter the atoms after they have been created. In order here consists of creating an atomic noise with uniform to do this, a method to obtain the ELC from the absolute perceptual frequency distribution, and perceptually threshold in quiet is used. This threshold is white spectral envelope, thus prioritizing the perceptual given in dB as [16], point of view and creating a sound where all −0.8 −0.6()f −3.3 2 −3 4 components have the same relative perceptual space. Tq = 3.64 f − 6.5e +10 f , (4)

4.1. Atoms distribution where f is the frequency in kHz. In order to create the equal loudness contour (ELC) for a given phon, the As the frequency distribution should be consistent with hearing threshold is scaled linearly, the perceptual frequency scale, this scale has to be ELC ≈ T (1− phon /125) + phon + 3 determined. It can be found from several methods [8]; q . (5) for instance from the pitch ratio, from the number of hair cells in the cochlea allocated to each frequency, or The resulting equal-loudness-contour for phon values by measuring the just noticeable difference across the from 0 to 100 is shown in figure 3. frequency axis. Most of these scales give approximately the same frequency curves. Here, the frequency 120 distribution is found by creating a probability density 100 phon function (pdf) from the Bark scale [15]. The cumulative 100 density function (cdf) becomes, 80 phon 80 f sr

−1 −1 )

cdf = sinh ( ) sinh ( ) (3) B 60 phon

d 60 600 1200 ( el

Lev 40 phon where f is the frequency and sr is the sample rate. This 40 creates a frequency distribution where the low 20 phon frequencies are more likely to occur than the high 20 frequencies. Thus, the distribution of frequencies on the Threshold cochlea is approximately uniform. The resulting pdf can 0

2 3 4 be seen in figure 2. 10 10 10 -4 x 10 Bark probability density function frequency (Hz) Figure 3. Hearing Threshold and equal-loudness 3 contour obtained by a linear model. y t i l bi 2 The ELC is applied directly to the atomic noise by

proba multiplying with it in the frequency domain, or by 1 creating filter coefficients and filtering it. Traditional methods of creating filter coefficients weigh the high 0 2 3 4 10 10 10 frequencies too much; therefore a warped filter design frequency (Hz) method is used. The warped filter can be optimized so Figure 2. Probability Density Function of frequencies. as to weigh the filter magnitude according to the Bark scale [16]. Thus it is possible to create a filter with The distribution of frequencies corresponding to the relatively few coefficients, which adheres closely to the Bark frequency scale ensures that the subsequent desired filter magnitude, by prioritizing the curve tonotopically organized nerves all process according to the auditory frequency intervals. The approximately the same amount of information, and that resulting perceptual atomic noise created with a second-

order IIR filter with varying atom width and probability [2] Richard G. d'Allesandro C., Grau S. “Musical is shown in figure 4. noises synthesis using random formant waveforms”. Stockholm Music Acoustic Increasing width of atoms 15000 Conference, Sweden. pp. 580-583, 1993. ) z H

( 10000

y [3] Cook, P. R. “Physically Informed Sonic nc

ue 5000 Modeling (PhISM): Synthesis of Percussive eq r f Sounds”, Computer Music Journal, 21(3), pp. 0 0 1 2 3 4 5 38 – 49, 1997. time (sec) Increasing probability of atoms [4] Jensen, K. “Irregularities, Noise and Random 15000 )

z Fluctuations in Musical Sounds”, Journal of H

( 10000 y Music and Meaning, nc

ue 5000 www.musicandmeaning.net, 2004. eq r f

0 0 1 2 3 4 5 [5] Helmholtz, H. On the sensation of tone, (first time (sec) published in 1877) Dover publications, 1954 Figure 4. Perceptual atomic noise for a low phon value [6] Schaeffer P. 1966. Traité des objets musicaux. with increasing atom width σ (top) and probability p Editions de Seuil. (bottom). [7] ISO/MPEG N4224. Standard 15938-4 As the phon level increases, the equal-loudness Information Technology - Multimedia Content contour becomes more uniform, thus strengthening the Description Interface - Part 4, Audio, MPEG mid-range and treble relative to the bass. Audio Group, Sydney, 2001. 5. CONCLUSIONS [8] Zwicker, E., and H. Fastl, : Facts and Models, Springer-Verlag, Berlin, A simple, efficient noise synthesis model, the atomic Heidelberg, 1990. noise, produces morphing of dice (with no structure), [9] Chadabe, J. Electronic sound, the past and Geiger (clicks) and cymbal (inharmonic) noises by promise of electronic music, Prentice Hall, adding atoms with random onset time, width and 1997. frequencies. Furthermore, the noise harmonicity is varied using a periodic frequency or time probability [10] Roads, C. “Introduction to Granular distribution function, or by repeating short segments of Synthesis”. Computer Music Journal. 12(2) pp. frozen noise. The link to common signal theory, by the 11-13, 1988. assumption of linear frequency scale and the uniform spectrum, is removed in this work by changing the [11] Hegarty, P. “Noise threshold: Merzbow and the frequency distribution and spectrum magnitude. The end of natural sound”. Organised Sound 7(1) perceptually uniform frequency distribution is found pp. 193-200, 2002. from a probability density function obtained from the [12] Pure white noise. 2004. http://www.purewhite Bark frequency scale, and the perceptually uniform noise.com/, Visited 5/3-2005. spectrum is obtained by filtering the atoms with a warped filter, the coefficients of which have been found [13] Pierce, J. “Hearing in time and space”, in by fitting it to an equal-loudness contour model. The Cook, P. R. (editor), Music, Cognition, and equal-loudness contour is found by scaling linearly (in Computerized Sound: An Introduction to dB) the threshold in quiet. Psychoacoustics, Mit Press, pp 89-103, 2001. While composing with the perceptual atomic noise, [14] Warren, R. M. 1999. Auditory Perception, care must be taken that all simultaneous components are Cambridge University press. heard, since the random aspect sometimes blurs the sounds identity. Many variations of Geiger, cymbal and [15] Sekey A. & B. A. Hanson. “Improved 1-bark intermediate perceptual atomic noises can, however, co- bandwidth auditory filter”. J. Acoust. Soc. Am. exist, in particular if stereo, or surround mixes are used. 75(6), pp. 151-168, 1984. The low-frequency dominant spectrum of the perceptual [16] Terhard, E. "Calculating virtual pitch", atomic noise is more difficult to mix satisfactory, Hearing Research, pp. 155-182, 1979. however, without compression. [17] Härmä A., Karjalainen M., Savioja L.,. 6. REFERENCES Välimäki V, Laine U. K., and Huopaniemi J. ``Frequency-warped signal processing for [1] Jensen, K. “Atomic Noise”, Organised Sound, audio applications,'' J. Aud. Eng. Soc., 48(11) 10(1) pp 75-81, 2005. pp. 1011-1031. November 2000.