Sorry about that. Outlook strikes again and does not transform equations to HTML format. If you use outlook all the equations show up. I will resend in the morning. David
Sent from my iPad > On Dec 27, 2013, at 12:34 AM, "David Smith" <[email protected]> wrote: > > > ------=_NextPart_001_0004_01CF029B.5B4EA1A0 > Content-Type: text/plain; > charset="us-ascii" > Content-Transfer-Encoding: 7bit > > Merry Post Christmas. I finally had some time to work through the math (beat > Mathematica into submission) for the string tension and do a plot. Quite > instructive. > > > > For the conditions of my 11th course gut C string: > > > > Diameter = 0.00203 > > Density = 1360 > > Len = 0.685 > > Tension = 28.0 > > > > Which produces a pitch at 58.2 HZ. > > For the Frequency as a function of Tension the equation is: > > > > > > For the Cents as a function of Tension around the C string frequency the > equation is: > > > > or > > Cents[T_] > using the values for the C string. > > > > For an increase in Tension of 1.0 newton (0.1 kg) the difference in pitch is > 30.3744 Cents > > For an decrease in Tension of 1.0 newtom (0.1kg) the different in pitch is > -31.4817 Cents > > > > Pretty sensitive to small changes in tension. > > > > The partial derivative of this with T_ is: > > CentsPerNewton[T_] > > > This is a pretty straightforward equation. It states that the sensitivity of > the string (in cents) for the C string to Tension is inversely proportional > to the Tension. That means that if we increase the tension the change in > pitch (in cents) goes down). > > > > The chart is: > > > > > > > > So, higher tension strings will reduce the sensitivity. But not by a lot (if > we keep to a reasonable range). The bottom line is that the 11th course of a > baroque lute at this string length using gut is just a pain to tune. The > only reasonable choice is to provide a better tuning mechanism such as the > planetary gear tuners. > > > > Anyway, thanks for your patience as I work through this. It has been fun and > now I think I understand what is happening. > > > > Regards > > David > > > > -----Original Message----- > From: [email protected] [mailto:[email protected]] On Behalf > Of David van Ooijen > Sent: Friday, December 20, 2013 9:03 AM > Cc: Lute List > Subject: [LUTE] Re: Question on String Tension > > > >>> When I plot the partial derivative of F'(T) using the values for > > this string I find that the sensitivity is actually quite small; less > > than 1/10th of a hertz per Newton > > << > > Don't think in Hertz. The difference between 440 and 441Hz is a smaller > > difference in pitch than between 40 and 41 Hz. Think in cents, it makes > > calculating easier, and the results are easer to understand too. > > David > > > > -- > > > > > > To get on or off this list see list information at > <http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html> > http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html > > > ------=_NextPart_001_0004_01CF029B.5B4EA1A0 > Content-Type: text/html; > charset="us-ascii" > Content-Transfer-Encoding: quoted-printable > > <html xmlns:v="urn:schemas-microsoft-com:vml" > xmlns:o="urn:schemas-microsoft-com:office:office" > xmlns:w="urn:schemas-microsoft-com:office:word" > xmlns:m="http://schemas.microsoft.com/office/2004/12/omml"; > xmlns="http://www.w3.org/TR/REC-html40";><head><meta http-equiv=Content-Type > content="text/html; charset=us-ascii"><meta name=Generator content="Microsoft > Word 15 (filtered medium)"><!--[if !mso]><style>v\:* > {behavior:url(#default#VML);} > o\:* {behavior:url(#default#VML);} > w\:* {behavior:url(#default#VML);} > .shape {behavior:url(#default#VML);} > </style><![endif]--><style><!-- > /* Font Definitions */ > @font-face > {font-family:Courier; > panose-1:2 7 4 9 2 2 5 2 4 4;} > @font-face > {font-family:"Cambria Math"; > panose-1:2 4 5 3 5 4 6 3 2 4;} > @font-face > {font-family:Calibri; > panose-1:2 15 5 2 2 2 4 3 2 4;} > /* Style Definitions */ > p.MsoNormal, li.MsoNormal, div.MsoNormal > {margin:0in; > margin-bottom:.0001pt; > font-size:11.0pt; > font-family:"Calibri","sans-serif";} > a:link, span.MsoHyperlink > {mso-style-priority:99; > color:#0563C1; > text-decoration:underline;} > a:visited, span.MsoHyperlinkFollowed > {mso-style-priority:99; > color:#954F72; > text-decoration:underline;} > p.MsoPlainText, li.MsoPlainText, div.MsoPlainText > {mso-style-priority:99; > mso-style-link:"Plain Text Char"; > margin:0in; > margin-bottom:.0001pt; > font-size:11.0pt; > font-family:"Calibri","sans-serif";} > span.PlainTextChar > {mso-style-name:"Plain Text Char"; > mso-style-priority:99; > mso-style-link:"Plain Text"; > font-family:"Calibri","sans-serif";} > span.MathematicaFormatStandardForm > {mso-style-name:MathematicaFormatStandardForm; > mso-style-priority:99; > font-family:Courier;} > .MsoChpDefault > {mso-style-type:export-only;} > @page WordSection1 > {size:8.5in 11.0in; > margin:1.0in 1.0in 1.0in 1.0in;} > div.WordSection1 > {page:WordSection1;} > --></style><!--[if gte mso 9]><xml> > <o:shapedefaults v:ext="edit" spidmax="1026" /> > </xml><![endif]--><!--[if gte mso 9]><xml> > <o:shapelayout v:ext="edit"> > <o:idmap v:ext="edit" data="1" /> > </o:shapelayout></xml><![endif]--></head><body lang=EN-US link="#0563C1" > vlink="#954F72"><div class=WordSection1><p class=MsoPlainText>Merry Post > Christmas. I finally had some time to work through the math (beat Mathematica > into submission) for the string tension and do a plot. Quite > instructive.<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText>For the conditions of my 11th course gut C > string:<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText style='margin-left:.5in'>Diameter = > 0.00203<o:p></o:p></p><p class=MsoPlainText style='margin-left:.5in'>Density > = 1360<o:p></o:p></p><p class=MsoPlainText style='margin-left:.5in'>Len = > 0.685<o:p></o:p></p><p class=MsoPlainText style='margin-left:.5in'>Tension = > 28.0<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText>Which produces a pitch at 58.2 HZ.<o:p></o:p></p><p > class=MsoPlainText>For the Frequency as a function of Tension the equation > is:<o! :p! >> </o:p></p><p class=MsoPlainText style='margin-left:.5in'><!--[if gte >> msEquation 12]><m:oMathPara><m:oMathParaPr><m:jc >> m:val="left"/></m:oMathParaPr><m:oMath><span style='font-family:"Cambria >> Math","serif"'> <m:r><m:rPr><m:scr m:val="roman"/><m:sty >> m:val="p"/></m:rPr>Frequency</m:r><m:r><i>[</i></m:r><m:r><m:rPr><m:scr >> m:val="roman"/><m:sty >> m:val="p"/></m:rPr>T_</m:r><m:r><i>]=</i></m:r></span><m:f><m:fPr><span >> style='font-family:"Cambria >> Math","serif"'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSup><m:sSupPr><span >> style='font-family:"Cambria >> Math","serif"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span >> style='font-family:"Cambria >> Math","serif"'><m:r>(</m:r></span></i><m:f><m:fPr><span >> style='font-family:"Cambria >> Math","serif"'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i><span >> style='font-family:"Cambria >> Math","serif"'><m:r>T</m:r></span></i></m:num><m:den><i><span >> style='font-family:"Cambria Math","serif"'><m:r>π</m:r></span></i><span >> style='font-family:"! C! > ambria Math","serif"'><m:r><m:rPr><m:scr m:val="roman"/><m:sty! > m:val="p"/></m:rPr>Density</m:r></span></m:den></m:f><i><span > style='font-family:"Cambria > Math","serif"'><m:r>)</m:r></span></i></m:e><m:sup><i><span > style='font-family:"Cambria > Math","serif"'><m:r>0.5</m:r></span></i></m:sup></m:sSup></m:num><m:den><span > style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Diameter</m:r><m:r><i>*</i></m:r><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Len</m:r></span></m:den></m:f></m:oMath></m:oMathPara><![endif]--><![if > !msEquation]><span > style='font-size:11.0pt;font-family:"Calibri","sans-serif";mso-fareast-language:EN-US'><img > width=205 height=47 id="_x0000_i1025" > src="cid:[email protected]"></span><![endif]><o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>For the Cents > as a function of Tension around the C string frequency the equation > is:<o:p></o:p></p><p class=MsoPlainText style='margin-left:.5in'><!--[if gte > msEquation 12]><! m! > :oMathPara><m:oMathParaPr><m:jc m:val="left"/></m:oMathParaPr><m:oMath><span > style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Cents</m:r><m:r><i>[</i></m:r><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>T_</m:r><m:r><i>]=1200.*</i></m:r><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Log</m:r><m:r><i>[2,</i></m:r></span><m:f><m:fPr><m:type > m:val="lin"/><span style='font-family:"Cambria > Math","serif"'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><span > style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Frequency</m:r><m:r><i>[</i></m:r><m:r><i>T</i></m:r><m:r><i>]</i></m:r></span></m:num><m:den><i><span > style='font-family:"Cambria > Math","serif"'><m:r>58.2168</m:r></span></i></m:den></m:f><i><span > style='font-family:"Cambria > Math","serif"'><m:r>]</m:r></span></i></m:oMath></m:oMathPara><![endif]--><![if > !msEquation]><span style='font-size:11.0pt;fon! t! > -family:"Calibri","sans-serif";mso-fareast-language:EN-US'><img width=3! > 17 height=17 id="_x0000_i1025" > src="cid:[email protected]"></span><![endif]><o:p></o:p></p><p > class=MsoPlainText>or<o:p></o:p></p><p > class=MsoPlainText> > Cents[T_] = <!--[if gte msEquation 12]><m:oMath><i><span > style='font-family:"Cambria Math","serif"'><m:r>1731.23</m:r></span></i><span > style='font-family:"Cambria Math","serif"'><m:r><m:rPr><m:scr > m:val="roman"/><m:sty > m:val="p"/></m:rPr>Log</m:r><m:r><i>[0.188982</i></m:r></span><m:sSup><m:sSupPr><span > style='font-family:"Cambria > Math","serif"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i><span > style='font-family:"Cambria > Math","serif"'><m:r>T</m:r></span></i></m:e><m:sup><i><span > style='font-family:"Cambria > Math","serif"'><m:r>0.5</m:r></span></i></m:sup></m:sSup><i><span > style='font-family:"Cambria > Math","serif"'><m:r>]</m:r></span></i></m:oMath><![endif]--><![if > !msEquation]><span style='font-size:11.0pt;font-family! :! > "Calibri","sans-serif";position:relative;top:3.0pt;mso-text-raise:-3.0pt;mso-fareast-language:EN-US'><img > width=169 height=18 id="_x0000_i1025" > src="cid:[email protected]"></span><![endif]><o:p></o:p></p><p > class=MsoPlainText>using the values for the C string.<o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>For an increase > in Tension of 1.0 newton (0.1 kg) the difference in pitch is 30.3744 > Cents<o:p></o:p></p><p class=MsoPlainText>For an decrease in Tension of 1.0 > newtom (0.1kg) the different in pitch is -31.4817 Cents<o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>Pretty > sensitive to small changes in tension.<o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>The partial > derivative of this with T_ is:<o:p></o:p></p><p > class=MsoPlainText> > CentsPerNewton[T_] = <!--[if gte msEquation! ! > 12]><m:oMath><m:f><m:fPr><span style='font-family:"Cambria Math","serif! > "'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i><span > style='font-family:"Cambria > Math","serif"'><m:r>865.617</m:r></span></i></m:num><m:den><i><span > style='font-family:"Cambria > Math","serif"'><m:r>T</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if > !msEquation]><span > style='font-size:11.0pt;font-family:"Calibri","sans-serif";position:relative;top:6.0pt;mso-text-raise:-6.0pt;mso-fareast-language:EN-US'><img > width=39 height=25 id="_x0000_i1025" > src="cid:[email protected]"></span><![endif]><o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>This is a > pretty straightforward equation. It states that the sensitivity of the string > (in cents) for the C string to Tension is inversely proportional to the > Tension. That means that if we increase the tension the change in pitch (in > cents) goes down).<o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>The chart > is:<o:p></o:p></p><p class=MsoPlainText><o:p>&nb! s! > p;</o:p></p><p class=MsoPlainText><span > class=MathematicaFormatStandardForm><img width=360 height=228 > id="Picture_x0020_3" > src="cid:[email protected]"></span><o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText><a > name="_MailEndCompose">So, higher tension strings will reduce the > sensitivity. But not by a lot (if we keep to a reasonable range). The bottom > line is that the 11<sup>th</sup> course of a baroque lute at this string > length using gut is just a pain to tune. The only reasonable choice is to > provide a better tuning mechanism such as the planetary gear > tuners.<o:p></o:p></a></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText>Anyway, thanks for your patience as I work through this. > It has been fun and now I think I understand what is > happening.<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText>Regards<o:p></o:p></p><p > class=MsoPlainText>David<o:p></o:p></p><p class=MsoPlainText><o:p>! &! > nbsp;</o:p></p><p class=MsoPlainText>-----Original Message-----<br>From! > : [email protected] [mailto:[email protected]] On Behalf Of > David van Ooijen<br>Sent: Friday, December 20, 2013 9:03 AM<br>Cc: Lute > List<br>Subject: [LUTE] Re: Question on String Tension</p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText> > >> When I plot the partial derivative of F'(T) using the values > for<o:p></o:p></p><p class=MsoPlainText> this string I find that > the sensitivity is actually quite small; less<o:p></o:p></p><p > class=MsoPlainText> than 1/10th of a hertz per > Newton<o:p></o:p></p><p class=MsoPlainText> > <<<o:p></o:p></p><p class=MsoPlainText> Don't think in > Hertz. The difference between 440 and 441Hz is a smaller<o:p></o:p></p><p > class=MsoPlainText> difference in pitch than between 40 and 41 > Hz. Think in cents, it makes<o:p></o:p></p><p class=MsoPlainText> > calculating easier, and the results are easer to understand > too.<o:p></o:p></p! >! > <p class=MsoPlainText> David<o:p></o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText> > --<o:p></o:p></p><p class=MsoPlainText><o:p> </o:p></p><p > class=MsoPlainText><o:p> </o:p></p><p class=MsoPlainText>To get on or > off this list see list information at <a > href="http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html";><span > style='color:windowtext;text-decoration:none'>http://www.cs.dartmouth.edu/~wbc/lute-admin/index.html</span></a><o:p></o:p></p></div></body></html> > ------=_NextPart_001_0004_01CF029B.5B4EA1A0-- > > --
