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Hi Dmitri,<div><br></div><div><br></div><div>On Mar 25, 2009, at 10:20 AM, Dmitri Tymoczko wrote:</div><blockquote type="cite"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "></div></blockquote><div><font class="Apple-style-span" face="Helvetica" size="3"><span class="Apple-style-span" style="font-size: 12px; "><br></span></font></div><div></div><blockquote type="cite"><div><span class="Apple-style-span" style="font-family: Helvetica; font-size: 12px; ">I think discussions of the issue of recursion could stand to pay more attention to the distinction between intrinsic structure and our psychological organization of a structure.</span></div></blockquote><div><br></div><div>One important distinction must be made: recursive vs non- recursive is a property of the <b>grammar (i.e. the model)</b>, not of the <b>corpus</b>. <i>Any</i> finite corpus can be modeled with a non-recursive grammar, even if it contains strings like</div><div><br></div><div>ABCDCBA</div><div><br></div><div>This is because, if the corpus is finite, you'll never have to deal with infinite depth of recursion. Of course, if recursive structures keep occurring with high frequency and at deep levels of embedding, it will be rather forced and unnatural to insist on a non-recursive model. The real question therefore becomes one of the model's <b>economy, explanatory power, ability to relate to other disciplines </b>(e.g. cognition) etc.</div><div><br></div><div>To summarize, in my mind recursive vs non-recursive is a question of <b>model selection</b>, not a question of what the music <b>is<span class="Apple-style-span" style="font-weight: normal; "> </span>intrinsically</b>.</div><div><br></div><div>For example, if you claim that Roman numeral progressions can be modeled quite adequately using a non-recursive finite-state model, you are explicitly or implicitly making certain model-selection commitments. More specifically, you systematically avoid <b>interpretations</b> of the corpus that involve higher levels of embedding, such as</div><div><br></div><div>( ( ( ( (I ii V I) ii V I) ii V I) ii V I) ii V I) ii V I</div><div><br></div><div>and instead claim that an interpretation of </div><div><br></div><div>I (ii V I) (ii V I) (ii V I) (ii V I) (ii V I) (ii V I)</div><div><br></div><div>is just as good. In other words, you are making modeling choices by balancing qualities such as model simplicity vs goodness-of-fit to the data. ("If a complex model only gives us minor improvements in accuracy over a simpler one, we might as well just go with the simpler model.")</div><div><br></div><div>These are all modeling decisions which can be ultimately quite subjective. Even if not everybody agrees on a single set of model selection criteria, at least these criteria have to be made explicit, and it is precisely these criteria that we should be debating.</div><div><br></div><div><div><div><div>Best,</div><div>Panos</div><div><br></div></div></div></div><div><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Times; font-size: 18px; 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; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Times; font-size: 18px; font-style: normal; font-variant: normal; font-weight: normal; 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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; "><span class="Apple-style-span" style="border-collapse: separate; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; color: rgb(0, 0, 0); font-family: Times; font-size: 18px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; -webkit-text-decorations-in-effect: none; text-indent: 0px; -webkit-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; "><span class="Apple-style-span" style="font-size: 18px; "><div><br></div></span>% ===================================================</span></font></div><div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% Panayotis Mavromatis</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% Music Theory Director, Music and Performing Arts</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% The Steinhardt School, New York University</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% 35 West 4th Street, Room 777</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% New York, NY 10012</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; ">% Phone: (212) 998-5287</span></font></div><div><font class="Apple-style-span" size="4"><span class="Apple-style-span" style="font-size: 14px; 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