<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Dear colleages,<div>with students in theory class we have been working on the idea of an infinite accellerando (or ritardando) in analogy to the illusion of infinite ascend (or descend) in pitch height in the Shepard scale (or glissando). The solution we came up with is a combination of </div><div>(1) a non-periodic self-similar rhythm (such as investigated by Norman Carey and David Clampitt in their 1996 PNM article: "Self-similar pitch structures, their duals, and rhythmic analogues"), which is played simultaneously at several speeds, where "old" levels fade out and new levels fade in through a Gaussian envelope for the amplitude, and</div><div>(2) an exponential time flow (which is the physical time of performance as a function of symbolic score time). All of the (infinitely many) voices silently start simultaenously at minus infinity ;-)</div><div>One of the students, Christobal Massis Valencia (percussionist), realized a ritardando (in combination with an "infinite" descend) as a homework on that class:</div><div><a href="http://user.cs.tu-berlin.de/~noll/sqrt2">http://user.cs.tu-berlin.de/~noll/sqrt2</a></div><div>Thomas Noll</div><div><br></div><div><div><div><div><br></div></div><div apple-content-edited="true"><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 12px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div><div><div>*********************************************************</div><div>Thomas Noll</div><div><a href="http://flp.cs.tu-berlin.de/~noll">http://flp.cs.tu-berlin.de/~noll</a></div><div><a href="mailto:noll@cs.tu-berlin.de">noll@cs.tu-berlin.de</a></div><div>Escola Superior de Musica de Catalunya, Barcelona </div><div>Departament de Teoria i Composició </div><div><br class="khtml-block-placeholder"></div><div>*********************************************************</div></div></div></div><div><br></div></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"> </div><br></div></div></body></html>