Proceedings of the 22 nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2–6, 2019 GENERALIZATIONS OF VELVET NOISE AND THEIR USE IN 1-BIT MUSIC Kurt James Werner The Sonic Arts Research Centre (SARC) Queen’s University Belfast (QUB) Belfast, United Kingdom of Great Britain and Northern Ireland kurt.james.werner@gmail.com ABSTRACT A family of spectrally-flat noise sequences called “Velvet Noise” have found use in reverb modeling, decorrelation, speech syn- thesis, and abstract sound synthesis. These noise sequences are ternary—they consist of only the values -1, 0, and +1. They are also sparse in time, with pulse density being their main design parameter, and at typical audio sampling rates need only several thousand non-zero samples per second to sound “smooth.” This paper proposes “Crushed Velvet Noise” (CVN) general- izations to the classic family of Velvet Noise sequences includ- ing “Original Velvet Noise” (OVN), “Additive Random Noise” (ARN), and “Totally Random Noise” (TRN). In these generaliza- tions, the probability of getting a positive or negative impulse is a free parameter. Manipulating this probability gives Crushed OVN and ARN low-shelf spectra rather than the flat spectra of standard Velvet Noise, while the spectrum of Crushed TRN is still flat. This new family of noise sequences is still ternary and sparse in time. However, pulse density now controls the shelf cutoff frequency, and the distribution of polarities controls the shelf depth. Crushed Velvet Noise sequences with pulses of only a single polarity are particularly useful in a niche style of music called “1- bit music”: music with a binary waveform consisting of only 0s and 1s. We propose Crushed Velvet Noise as a valuable tool in 1- bit music composition, where its sparsity allows for good approx- imations to operations, such as addition, which are impossible for signals in general in the 1-bit domain. 1. INTRODUCTION In 2007, Karjalainen and Järveläinen defined a new type of sparse noise sequences which they called “Velvet Noise” [1]. This was later more specifically termed “Original Velvet Noise” (OVN) by Välimäki et al. [2]. These noise sequences have some similarities to sparse noise investigated by Schreiber in 1960 [3] and have a few peculiar qualities. First, they are sparse in time—most of their samples are actually zero. Second, the non-zero samples only take the values -1 and +1. Specifically, OVN is produced by defin- ing a pulse density, splitting time into equal-length windows, and distributing a single impulse into each window, with both its ex- act position within the window and its sign (±) randomized. OVN is clearly not i.i.d. (independent and identically distributed), since the value of each sample within a window is related to the values of all other samples in the window. Despite this, OVN remark- ably has a flat magnitude spectrum and an autocorrelation which Copyright: c  2019 Kurt James Werner et al. This is an open-access article dis- tributed under the terms of the Creative Commons Attribution 3.0 Unported License, which permits unrestricted use, distribution, and reproduction in any medium, pro- vided the original author and source are credited. is nearly zero everywhere except at zero-lag, making it very similar to Gaussian white noise (an i.i.d. process which is neither sparse in time nor limited to particular values). With a sufficiently high pulse density, Velvet Noise can even sound just as “smooth” as Gaussian noise [2]. One fascinating property of Velvet Noise sequences is that they are very efficient to convolve by, since they are very sparse (mostly 0s) and the non-zero values (±1) don’t require an actual multiplication during convolution [4, 5]. These properties have led Velvet Noise to be used in reverb modeling [1, 5, 6, 7, 8, 9], the design of decorrelation filters [10, 11], speech synthesis [12, 13], and abstract sound synthesis [14, 15]. Related to OVN, several other sparse ternary noise sequences have been proposed, including Additive Random Noise (ARN), Totally Random Noise (TRN), Extended Velvet Noise (EVN), Ran- dom Integer Noise (RIN) [2]. ARN randomizes the spacing be- tween pulses rather than distributing a single pulse per window. TRN has a random chance of generating a pulse of a random sign for every single sample. EVN takes OVN and restricts the sample location to only a portion of each window, enforcing a second level of sparsity. RIN is identical to ARN, although both the pulse off- sets and sign are read from a precomputed table of integer random numbers rather than independent random numbers [2]. In this paper, I propose generalizations to the family of Velvet Noise sequences which are called “Crushed Velvet Noise” (CVN). Specifically, I propose new variants of OVN, ARN, and TRN now called COVN, CARN, and CTRN (the “C” denoting “Crushed” for each). In Crushed Velvet Noise Sequences, the signs of each pulse is not assigned based on a 50% probability, but rather this probability is exposed as a free parameter that can be manipulated along with the pulse density. This small change allows the creation of a variety of spectra with different properties. COVN and CARN both have low-shelf-like Power Spectral Densities (PSDs), with their cuttoff frequency controlled by the pulse density and their shelf attenuation controlled by the free parameter determining the probability of a positive or negative sign for each pulse. Adjusting this probability to extreme settings gives sequences where either -1s or +1s do not appear 1 . Variants of CVN which only have 0s and 1s are of particular use to a niche approach to electronic music composition called “1-bit music,” where the only allowable signal levels are 0 and 1. In the end of this paper, I ex- plain how Velvet Noise and the proposed novel variants can be used in 1-bit composition, specifically highlighting their poten- 1 The case where there is a 100% chance of a +1 and no chance of a -1 has already been investigated briefly in [13], where a unipolar variant of OVN is called Unipolar Velvet Noise (UVN). The sparse noise sequence explored by Schreiber [3] was also unipolar. With these in mind, we can say that this paper fills in the gaps between the proposed unipolar variant and the bipolar OVN sequence. DAFX-1