<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div>On Oct 14, 2008, at 12:19 PM, Porritt, David wrote:</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; ">Fred:<br><br>The 12th root of 2.0012 is the equivalent of the 19th root of 3 to 4 decimal places (1.059516053 rather than 1.059526065).<br><br>dp<br><br>David M. Porritt, RPT<br><a href="mailto:dporritt@smu.edu">dporritt@smu.edu</a></span></blockquote><br></div>Thanks, Dave! Considerably closer than I thought. Multiplied out over the full range of a piano, these differences matter, but not as much as inharmonicity produces. It's interesting to remember that stacking 12 "just, theoretical" 5ths, from A0 to A7, produces a theoretical excess of 24 cents over a "just, theoretical" seven octaves. Quite a bit less than standard tuning "stretch" of even the most conservative among us.<div><div><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><br class="Apple-interchange-newline">Regards,</div><div>Fred Sturm</div><div>University of New Mexico</div><div><a href="mailto:fssturm@unm.edu">fssturm@unm.edu</a></div><div><br class="webkit-block-placeholder"></div></div></div>PS to Ric: re-read what Dave wrote. He's coming up with an increase of 2 ( to 2.0012) where the 12th root is quite close to the 19th root of 3. Obviously one can carry this process out as many decimal places as one wants, and one will come to practical convergence at some point.<div></div></div></body></html>