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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
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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;font!
-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>&nbs!
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
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this list see list information at <a
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