<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">What does "X is Dominant of Y" mean (in the paradigmatic sense)? Is X a chord and Y a region? Under such an assumption it would be impossible to speak of secondary dominants without an additional operation of "typecasting" wherein chords are turned into regions. What is the basis for such a "typecasting" operation: Something like the recursivity of the Tonnetz - as mentioned by Nicolas Meełs? It seems that the acceptance of secondary dominants as instances of recursion has strong consequences for the entire theory.<div>A typical syntagmatic trace for the chord-as-region-casting is a ii - V/V progression. If we assume recursion we assume that behind the scene the "V" in "V/I" is casted as the "V" in "I/V". </div><div>Analogously we have the reverse direction, the V - ii/IV progression (think of Chopin's E-minor prelude). Here the "IV" in "IV/I" is casted as "IV" in "I/IV" behind the scene. </div><div>What are other typical syntagmatic traces for such chord-as-region-castings? </div><div><div><br class="webkit-block-placeholder"></div><div>Sincerely </div><div>Thomas Noll <br><div><br class="Apple-interchange-newline"><blockquote type="cite"><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; ">The<span class="Apple-converted-space"> </span><i>Tonnetz</i><span class="Apple-converted-space"> </span>would appear to me as an example of recursion, in that it can be taken to represent pitches (as in Euler), or chords (as at times in Riemann), or tonalities (Schoenberg's regions). This exemplifies the assumption that functions are the same or similar at these three embedded levels. Some may remember that at one of the early OxMac conferences (in the late '80s, I think), Leonard Meyer forcefully questioned this assumption.</span></blockquote></div><br><div> <span class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><span class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><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>Tel (priv.): +34 93 268 75 19</div><div>Tel (mobil): +34 66 368 12 02</div><div><br class="khtml-block-placeholder"></div><div>*********************************************************</div><div><br class="khtml-block-placeholder"></div><div><br><br class="khtml-block-placeholder"></div><br class="Apple-interchange-newline"></span></span> </div><br></div></div></body></html>