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This paper investigates sparse noise sequences, including the previously proposed velvet noise and its novel variants defined here. All sequences consist of sample values minus one, zero, and plus one only, and the location and the sign of each impulse is randomly chosen. Two of the proposed algorithms are direct variants of the original velvet noise requiring two random number sequences for determining the impulse locations and signs. In one of the proposed algorithms the impulse locations and signs are drawn from the same random number sequence, which is advantageous in terms of implementation. Moreover, two of the new sequences include known regions of zeros. The perceived smoothness of the proposed sequences was studied with a listening test in which test subjects compared the noise sequences against a reference signal that was a Gaussian white noise. The results show that the original velvet noise sounds smoother than the reference at 2000 impulses per second. At 4000 impulses per second, also three of the proposed algorithms are perceived smoother than the Gaussian noise sequence. These observations can be exploited in the synthesis of noisy sounds and in artificial reverberation.