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Page 00000001 PERCEPTUAL ATOMIC NOISE Kristoffer Jensen University of Aalborg Esbjerg Niels Bohrsvej 6, DK-6700 Esbjerg krist@cs. aaue. dk ABSTRACT A noise synthesis method with no external connotation is proposed. By creating atoms with random width, onset-time and frequency, most external connotations are avoided. The further addition of a frequency distribution corresponding to the perceptual Bark frequency, and a spectrum corresponding to the equalloudness contour for a given phon level further removes the synthesis from the common signal point-of-view. The perceptual frequency distribution is obtained by creating a probability density function from the Bark scale, and the equal-loudness contour (ELC) spectrum is created by filtering the atoms with a filter obtained in the warped frequency domain by fitting the filter to a simple ELC model. An additional voiced quality parameter allows to vary the harmonicity. The resulting sound is susceptible to be used in everything from loud noise music, contemporary compositions, to meditation music. 1. INTRODUCTION The unvoiced sounds are used in many musical situations, from the abundance of cymbals and hihats in rhythmic music to the musique concrete of Schaeffer, the stochastic (random) processes of Xenakis, the granular music of modern computer music or noise music. Three noise prototypes are identified here; random values (dice noise), random events (Geiger noise), and random frequencies (cymbal noise). The atomic noise has previously been shown  to produce almost all pure noise types, while permitting to vary the degree of Geiger-, or cymbal-ness, thus easily creating a large variety of sounds with little natural connotation. This is done by adding atoms with random amplitude, width, frequency and onset time. Other noise synthesis methods include the formantwave-function (FOF) with random onset time , used to resynthesize naturally occurring musical noises. The shaken instruments have a random event distribution that can be modelled through the stochastic event modelling , while most other musical instruments need random irregularities on the frequencies and amplitudes to produce an interesting sound . The atomic noise was created to produce sounds which demanded no control over the spectral or the temporal envelope. Contrary to the granular synthesis or the musique concrete, or the different noise synthesis models, there is no 'original' timbre that is retained in the resulting music. In this work a further attempt is made to remove the 'external' influence of the music, by abandoning the signal point-of-view that creates sound with a uniform distribution and spectrum, or the physical point-of-view that creates a noise similar to existing sounds. Instead, an attempt is made at rendering the atoms perceptually white, by distributing them according to the perceptual frequency axis using a probability function obtained from the bark scale, and with a magnitude corresponding to the equal-loudness contour, by filtering the atoms using a warped filter corresponding to the equal-loudness contour for a given phon level. This gives a perceptual white resulting spectrum with a perceptually uniform frequency distribution. 2. NOISE Noise has been an important component since the beginning of music, and recently noise music has evolved as an independent music style, sometimes avoiding the toned components altogether. The distinction between noise and tone has been clear for a long time. Helmholtz  and Schaeffer  opposed harmonic sounds to unvoiced sounds. The mpeg 7 audio description  distinguishes between harmonic and percussive, coherent and non-coherent, and sustained and non-sustained sounds. Zwicker  found that band-pass filtered noise with large bandwidth has low relative pleasantness, as compared to sinusoids and band-pass filtered noise with low bandwidth. Noise as a component of music is found in its purest form in some percussion instruments, and in particular in the unvoiced consonants of the human voice. The futurist proposed, seemingly without success , a series of instruments; the intonarumori that produced rumbles, whispers, creaks, and other noises. Schaeffer went on to collaborate with Pierre Henry on musique concrete, in which recorded sounds, many of them unvoiced, were used in the compositions. Stockhausen and others used electronic generators, sinusoids and white noise in their early works. Another composer, Xenakis, used stochastic processes in both compositions but also in the creation of new sounds. The distinction between composition and sound events have been blurred even more in the granular synthesis , a method in which long music pieces can be obtained by random summation of time or frequency shifted short (10-50 msec) grains, often extracted from a short sampled waveform using a random selection and manipulation process. The use of noise has been further enlarged by two recent trends. In the noise music, adapts such as Merzbow or Caspar Br6tzmann, use noise played extremely loud in a generally rather static way. 'Noise
Page 00000002 music becomes ambience not as you learn how to listen, or when you accept its refusal to settle, but when you are no longer in a position to accept or deny' . This stands in full opposition functionally with the use of noise to create 'relaxation and calm, promoting sleep, and blocking annoying noises' . By mixing and editing the perceptual atomic noises together, looping them, and ending up with an entire piece of music, it becomes obvious that the perceptual atomic noises can also make up complex contemporary music. 3. ATOMIC NOISE The atomic noise  is a method used to easily create any kind of sounds between three kinds of prototypic noises, without any attempt to model spectral or temporal envelope behaviour. These three prototypic noises are the dice, cymbal and Geiger noises. The atomic noise is created by adding atoms with random amplitude, frequency, duration and onset time. The different prototypic noises are created by setting the distribution parameters appropriately. In addition, the atomic noise harmonicity is varied by different procedures; either the time or frequency distribution is made periodic, or a short created noise (frozen noise) is repeated. 3.1. Noise prototypes The random values (dice) method of creating unvoiced sounds is the most common method today. By using a new, uncorrelated, value at each time sample, an unvoiced, uncoloured sound is obtained. The distribution of the random values has not been found to be very important, perceptually. The summation of a large number of sinusoids with random frequencies evenly distributed on the frequencies also renders an unvoiced sound, if the number of sinusoids is high enough. As the sound resembles that of a cymbal for a relative low number of sinusoids, this noise generation method is called cymbal noise. Pulses randomly distributed in time are heard as a ticking noise, reminiscent of a Geiger counter, when the number of pulses is low. Pierce  differentiates between slow random pulses, which are heard as separate pulses, whereas at a few hundred pulses per second, not all pulses are detected individually. Above a few hundred pulses per second, a smooth noise is heard, with no individual pulses perceivable. 3.2. Atom noise synthesis If the notion of random value, time and frequency is retained, much of the noise types can be obtained  by adding together a large number of sinusoids with random amplitude and frequency, multiplied by a Gaussian shape with random standard deviation at random starting times, (t-to 2 t - to atom(t) = a cos(2 f )e sr (1) The amplitude a is a random variable with a Gaussian distribution and the frequency f is random values with uniform distribution, to is the onset time and a is the standard deviation (width) of the Gaussian. sr is the sample rate. The atomic noise is created by inserting one new atom at time to, if the random value drawn is greater than the probability threshold p. As o is approaching zero, the corresponding signal gets small duration and large bandwidth, thus approaching the Geiger noise. As a increases, the sinusoids get more duration and less bandwidth, and the signal approaches the cymbal noise. Examples of the atom noise as a function of a and p are shown in figure 1. Innrsinn wanidth nftnm 1500( 1000( sooo 500( 1500( S1000( S500 . 500( time (sec) Increasina Drobabilitv of atoms time (sec) Figure 1. Spectrum of atomic noise with increasing a (top) andp (bottom). 3.3. Tone from noise Since the Geiger noise becomes harmonic when the probability of pulses increases at periodic times, and similarly, the cymbal noise becomes harmonic when the probability of sinusoids increases at periodic frequencies, a periodic distribution for the stochastic signals is proposed . The periodic distribution is based on a triangular window, raised to the wth power, w p(x)=Ax x- X- O <x:!x0 2 (2) A is a normalization necessary to obtain a power density function and w is the harmonicity coefficient. A harmonicity value of zero produces unvoiced sounds, while a higher value produces a more toned sound. The eq. (2) is repeated for the duration of the noise, or up to the Nyquist frequency, in case of time and frequency periodic distribution, respectively. Another method for creating tone from noise is by using frozen noise (short noise segment), and repeat it. Warren  used frozen noise to show the perception of infrapitch (very low frequency pitch) and found whooshing, motor boating, and noisy pitch sound, with additional rattles, clangs
Page 00000003 and other metallic types of sounds dependent on the repetition rate. 4. PERCEPTUAL ATOMS While the atomic noise was created with the intent of making music with the least spectral and temporal envelope modification necessary, doubt arises as to the well-foundedness of the selected frequency distribution and spectral envelope model. Indeed, the uniform frequency distribution (in Hz), and the white spectral envelope originates mainly from a signal point-of-view. Another solution is found if the inspiration came from either the physical world around us or from the sensitivity of the auditory system. The solution chosen here consists of creating an atomic noise with uniform perceptual frequency distribution, and perceptually white spectral envelope, thus prioritizing the perceptual point of view and creating a sound where all components have the same relative perceptual space. 4.1. Atoms distribution As the frequency distribution should be consistent with the perceptual frequency scale, this scale has to be determined. It can be found from several methods ; for instance from the pitch ratio, from the number of hair cells in the cochlea allocated to each frequency, or by measuring the just noticeable difference across the frequency axis. Most of these scales give approximately the same frequency curves. Here, the frequency distribution is found by creating a probability density function (pdJ) from the Bark scale . The cumulative density function (cd) becomes, the perception and possibly the attention to each component is weighted equally. 4.2. Atoms Amplitude The relative amplitude of the atoms is also made so as to invert the sensitivity of the ear. As this sensitivity, commonly measured in phon as the equal-loudness contour (ELC), is dependent on the loudness of the atom, the atom amplitude should also be dependent on the final loudness, as well as on the frequency. It is easy to multiply the individual atom amplitude with the equal-loudness contour directly, but this does not give a perceptual white spectrum for the short atoms that would still sound as clicks. Instead, it is necessary to filter the atoms after they have been created. In order to do this, a method to obtain the ELC from the absolute hearing threshold in quiet is used. This threshold is given in dB as , Tq= 3.64f08 -o 6.5e-0.6(f-3.3)2 +10-3 f4, (4) where f is the frequency in kHz. In order to create equal loudness contour (ELC) for a given phon, hearing threshold is scaled linearly, the the ELC Tq(1 - phon /125) + phon + 3 (5) The resulting equal-loudness-contour forphon values from 0 to 100 is shown in figure 3. ign.1 cdf= sinh )sinh( s) 600 1200 (3) 100 80 60 40 20 0 - I I I I I T I 11100 phon I I -ri-------FT li T IF II _______ iii180 phon I III _ Thresh I II I 60phon1 1 1 1 1 140 phI I I I I I I II II 1 1 120phon 1 -1-1i T -1 T -F -Ii- V liii I F'~'H~IJ~.Ij~jThreshoI where f is the frequency and sr is the sample rate. This creates a frequency distribution where the low frequencies are more likely to occur than the high frequencies. Thus, the distribution of frequencies on the cochlea is approximately uniform. The resulting pdf can be seen in figure 2. X 10-4Bark probability density function 103 frequency (Hz) 104 3 - 7 --7,11 -1 7 --- - -1Ti l [II - IT - - - -- T F -1 r- - - T -1 1 -1 -L TIJL - -- -1T - - 7 T 71 - - T 102 103 104 frequency (Hz) Figure 2. Probability Density Function of frequencies. The distribution of frequencies corresponding to the Bark frequency scale ensures that the subsequent tonotopically organized nerves all process approximately the same amount of information, and that Figure 3. Hearing Threshold and equal-loudness contour obtained by a linear model. The ELC is applied directly to the atomic noise by multiplying with it in the frequency domain, or by creating filter coefficients and filtering it. Traditional methods of creating filter coefficients weigh the high frequencies too much; therefore a warped filter design method is used. The warped filter can be optimized so as to weigh the filter magnitude according to the Bark scale . Thus it is possible to create a filter with relatively few coefficients, which adheres closely to the desired filter magnitude, by prioritizing the curve according to the auditory frequency intervals. The resulting perceptual atomic noise created with a second
Page 00000004 order IIR filter with varying atom width and probability is shown in figure 4. 15000 10000. 5000 0 15000 10000 S5000 time (sec) Increasing probability of atoms time (sec) Figure 4. Perceptual atomic noise for a low phon value with increasing atom width cr (top) and probability p (bottom). As the phon level increases, the equal-loudness contour becomes more uniform, thus strengthening the mid-range and treble relative to the bass. 5. CONCLUSIONS A simple, efficient noise synthesis model, the atomic noise, produces morphing of dice (with no structure), Geiger (clicks) and cymbal (inharmonic) noises by adding atoms with random onset time, width and frequencies. Furthermore, the noise harmonicity is varied using a periodic frequency or time probability distribution function, or by repeating short segments of frozen noise. The link to common signal theory, by the assumption of linear frequency scale and the uniform spectrum, is removed in this work by changing the frequency distribution and spectrum magnitude. The perceptually uniform frequency distribution is found from a probability density function obtained from the Bark frequency scale, and the perceptually uniform spectrum is obtained by filtering the atoms with a warped filter, the coefficients of which have been found by fitting it to an equal-loudness contour model. The equal-loudness contour is found by scaling linearly (in dB) the threshold in quiet. While composing with the perceptual atomic noise, care must be taken that all simultaneous components are heard, since the random aspect sometimes blurs the sounds identity. Many variations of Geiger, cymbal and intermediate perceptual atomic noises can, however, coexist, in particular if stereo, or surround mixes are used. The low-frequency dominant spectrum of the perceptual atomic noise is more difficult to mix satisfactory, however, without compression. 6. REFERENCES  Jensen, K. "Atomic Noise", Organised Sound, 10(1) pp 75-81, 2005.  Richard G. d'Allesandro C., Grau S. "Musical noises synthesis using random formant waveforms". Stockholm Music Acoustic Conference, Sweden. pp. 580-583, 1993.  Cook, P. R. "Physically Informed Sonic Modeling (PhISM): Synthesis of Percussive Sounds", Computer Music Journal, 21(3), pp. 38 - 49, 1997.  Jensen, K. "Irregularities, Noise and Random Fluctuations in Musical Sounds", Journal of Music and Meaning, www.musicandmeaning. nt, 2004.  Helmholtz, H. On the sensation of tone, (first published in 1877) Dover publications, 1954  Schaeffer P. 1966. Traite des objets musicaux. Editions de Seuil.  ISO/MPEG N4224. Standard 15938-4 Information Technology - Multimedia Content Description Interface - Part 4, Audio, MPEG Audio Group, Sydney, 2001.  Zwicker, E., and H. Fastl, Psychoacoustics. Facts and Models, Springer-Verlag, Berlin, Heidelberg, 1990.  Chadabe, J. Electronic sound, the past and promise of electronic music, Prentice Hall, 1997.  Roads, C. "Introduction to Granular Synthesis". Computer Music Journal. 12(2) pp. 11-13, 1988.  Hegarty, P. "Noise threshold: Merzbow and the end of natural sound". Organised Sound 7(1) pp. 193-200, 2002. Pure white noise. 2004. http://www.purewhite noise.comn', Visited 5/3-2005. Pierce, J. "Hearing in time and space", in Cook, P. R. (editor), Music, Cognition, and Computerized Sound. An Introduction to Psychoacoustics, Mit Press, pp 89-103, 2001. Warren, R. M. 1999. Auditory Perception, Cambridge University press.  Sekey A. & B. A. Hanson. "Improved 1-bark bandwidth auditory filter". J. Acoust. Soc. Am. 75(6), pp. 151-168, 1984.  Terhard, E. "Calculating virtual pitch", Hearing Research, pp. 155-182, 1979. Hirmi A., Karjalainen M., Savioja L.,. Vilimaiki V, Laine U. K., and Huopaniemi J. "Frequency-warped signal processing for audio applications," J. Aud. Eng. Soc., 48(11) pp. 1011-1031. November 2000.