Different ways of expressing the same tension equation

Mr. Cris Forster, I have forwarded your message to "Hamlet's Ghost", and he had the following to say, which I hope will be conclusive on the subject of tensions.

Cordially,
Mr. Ozan Yarman

-----------------------------------------------------------------------------------------

My Dear Ozan,
His Equations (1) and (2) are identical, since obviously, the diameter is twice the size of the radius. For this reason, I would say, they are redundant.

However, these equations are correct for the "first harmonic" only. Else, one should multiply the tension (or the "weight" that will create the tension) by n^2, where n, being an integer, points to the nth harmonic.

His equation (3) too is identical to his equation (1); recall that I have previously explained that the mass per unit length is nothing else but D^2 x pi x rho.

Anyway, I am glad he finally agrees with you about the fact that equation (3) (the one you had furnished sometime ago), is not an approximate relationship, but is as rigorous as his first equation; they are in fact, as I keep on saying, identical relationships...

His last equation too is correct, but, although he defined it, I would have used another symbol for the total mass so as not to confuse it with `mass per unit length`. Nevertheless, after writing eqation (3), his equation (4) is again a triviality, and for this reason, redundant.

On the whole, I am glad to see that he has tacitly agreed with you in the end...

My best wishes to him...

Pap T.

----- Original Message -----
From: Cris Forster
To: Tuning
Sent: 27 Mayıs 2005 Cuma 22:51
Subject: [tuning] Four String Tension Equations

Here is a complete set of four string tension equations, where
each equation gives dimensionally consistent results.

T = tension, in Newtons

F = frequency, in Hertz, or cycles per second

L = length, in meters

D = diameter, in meters

r = radius, in meters

rho = mass density of the stringing material, in kg/m^3

M/u.l. = mass per unit length = Pi * r^2 * rho, in kg/m

M = total mass, in kg

When the string diameter is given:

((1)) T = F^2 * L^2 * D^2 * Pi * rho

((2)) T = 4 * F^2 * L^2 * Pi * r^2 * rho

When the string diameter is not known, one must determine the
mass per unit length ( M/u.l. ) by weighing one unit length (one
meter) of the string on a scale:

((3)) T = 4 * F^2 * L^2 * M/u.l.

When the string diameter is not known, one must determine the
total mass ( M ) by weighing the vibrating string length ( L ) on a
scale:

((4)) T = 4 * F^2 * L * M

Cris Forster, Music Director
www.Chrysalis-Foundation.org

>My Dear Ozan,
>His Equations (1) and (2) are identical, since obviously, the
>diameter is twice the size of the radius. For this reason, I would
>say, they are redundant.

Dear Ozan's Father,

As a builder of traditional and original acoustic musical
instruments, I find little redundancy in the mathematics of
vibrating systems. I am passionate about the subject of _musical
mathematics_ because it is my life's work. This commitment
leaves very little on my redundancy plate. I have simply
attempted to be thorough.

I have communicated with builders of acoustic musical
instruments for 30 years. Some builders prefer to measure
tension with a diameter variable, others prefer to measure tension
with a radius variable. I know no better way to dispel confusion
than to be thorough.

>However, these equations are correct for the "first harmonic"
>only. Else, one should multiply the tension (or the "weight" that
>will create the tension) by n^2, where n, being an integer, points
>to the nth harmonic.

Builders of musical instruments only consider the tension of the
fundamental mode of vibration, or of the first harmonic. Here's
why: most strings are tensioned between 40% and 50% of their
_break strength_. If the tension falls too much below the lower
limit, the strings are too loose and will sound weak. If the
tension rises too much above the upper limit, the strings are too
tight and will break prematurely. To calculate a string's break
strength, we must know the _tensile strength_ of the stringing
material. Tensile strength refers to the maximum tension a material
can withstand without tearing. I won't digress with detailed break
strength calculations. Suffice it to say, the tensile strength of
spring steel music wire varies with diameter.

>His equation (3) too is identical to his equation (1); recall that
>I have previously explained that the mass per unit length is
>nothing else but D^2 x pi x rho.
>Anyway, I am glad he finally agrees with you about the fact that
>equation (3) (the one you had furnished sometime ago), is not an
>approximate relationship, but is as rigorous as his first equation;
>they are in fact, as I keep on saying, identical relationships...

In Ozan's original text, and I quote,

******************************

The middle strings are manufactured from stainless steel such
that each weigh about 20 grams per unit meter, or 8 grams (0.008
kgs) per 0.4 meters.

******************************

the diameter of the sample string is not included.
Hence, no one is able to verify the accuracy of this physical
property of the wire.

>His last equation too is correct, but, although he defined it, I
>would have used another symbol for the total mass so as not
>to confuse it with `mass per unit length`. Nevertheless, after
>writing eqation (3), his equation (4) is again a triviality, and
>for this reason, redundant.

Again, I believe it is better to be thorough than to be terse.
Also, Equations 3 and 4 are highly interesting to me because they
would enable a prospective instrument builder to contemplate
these kinds of calculations without owning an expensive
micrometer. An accurate postage scale could be used to measure
either the mass per unit length (M/u.l.) or the total mass (M) of a
string. Personally, I find this kind of approach, especially in a
classroom where for financial reasons a precision micrometer may
not exist, highly interesting and most relevant.

>On the whole, I am glad to see that he has tacitly agreed with
you in the end...
>My best wishes to him...
>
>Pap T.

Dear Distinguished Professor at MIT,

My best wishes to you as well.

Cris Forster, Music Director
www.Chrysalis-Foundation.org

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Mr. Cris Forster, I have forwarded your message to "Hamlet's
Ghost", and he had the following to say, which I hope will be
conclusive on the subject of tensions.
>
> Cordially,
> Mr. Ozan Yarman
>
> -------------------------------------------------------------------
----------------------
>
> My Dear Ozan,
> His Equations (1) and (2) are identical, since obviously, the
diameter is twice the size of the radius. For this reason, I would
say, they are redundant.
>
> However, these equations are correct for the "first harmonic"
only. Else, one should multiply the tension (or the "weight" that
will create the tension) by n^2, where n, being an integer, points
to the nth harmonic.
>
> His equation (3) too is identical to his equation (1); recall that
I have previously explained that the mass per unit length is nothing
else but D^2 x pi x rho.
>
> Anyway, I am glad he finally agrees with you about the fact that
equation (3) (the one you had furnished sometime ago), is not an
approximate relationship, but is as rigorous as his first equation;
they are in fact, as I keep on saying, identical relationships...
>
> His last equation too is correct, but, although he defined it, I
would have used another symbol for the total mass so as not to
confuse it with `mass per unit length`. Nevertheless, after writing
eqation (3), his equation (4) is again a triviality, and for this
reason, redundant.
>
> On the whole, I am glad to see that he has tacitly agreed with you
in the end...
>
> My best wishes to him...
>
>
> Pap T.
>
>
>
> ----- Original Message -----
> From: Cris Forster
> To: Tuning
> Sent: 27 Mayýs 2005 Cuma 22:51
> Subject: [tuning] Four String Tension Equations
>
>
> Here is a complete set of four string tension equations, where
> each equation gives dimensionally consistent results.
>
> T = tension, in Newtons
>
> F = frequency, in Hertz, or cycles per second
>
> L = length, in meters
>
> D = diameter, in meters
>
> r = radius, in meters
>
> rho = mass density of the stringing material, in kg/m^3
>
> M/u.l. = mass per unit length = Pi * r^2 * rho, in kg/m
>
> M = total mass, in kg
>
>
> When the string diameter is given:
>
>
> ((1)) T = F^2 * L^2 * D^2 * Pi * rho
>
> ((2)) T = 4 * F^2 * L^2 * Pi * r^2 * rho
>
>
> When the string diameter is not known, one must determine the
> mass per unit length ( M/u.l. ) by weighing one unit length (one
> meter) of the string on a scale:
>
> ((3)) T = 4 * F^2 * L^2 * M/u.l.
>
>
> When the string diameter is not known, one must determine the
> total mass ( M ) by weighing the vibrating string length ( L )
on a
> scale:
>
> ((4)) T = 4 * F^2 * L * M
>
>
> Cris Forster, Music Director
> www.Chrysalis-Foundation.org

Mr. Cris Forster,

It is with utter regret that I observe you have shown not the least bit of decency in addressing my person, though you yourself have complained most provocatively about my lack thereof only recently over something frivolous at best. Ignoring the presence of your intermediary and consigning to treat him as an errand-boy is an act of despicable disrespect in my sight. Given your maturity and position, conduct so unbecoming of a man of your stature is unfortunate to say the least.

The only reason I asked my father's assistance (who is not a part of this list, as you should have realized well by now, and who probably has much more important things to do then pursuing endless arguments over the same material), is evidently my ignorance in the simplest equations of physics. Not only is it ethically permissable, but also inevitable, that I would require to ask his opinions on the matter and ask that he elucidate your input.

However, for you to exploit this oppurtunity to best me since the beginning, is quite shameful. Equally shameful is your bringing about an argument on `mass per unit lenght` error in my previous reasoning which I assumed to have corrected on short notice and which you seemed to have tacitly acknowledged at last.

I will nevertheless forward your response to him, and deliver his answer in case he considers it worthwhile to reply to you.

Cordially,
Mr. Ozan Yarman

----- Original Message -----
From: Cris Forster
To: tuning@yahoogroups.com
Sent: 28 Mayıs 2005 Cumartesi 19:25
Subject: [tuning] Re: Different ways of expressing the same tension equation

>My Dear Ozan,
>His Equations (1) and (2) are identical, since obviously, the
>diameter is twice the size of the radius. For this reason, I would
>say, they are redundant.

Dear Ozan's Father,

As a builder of traditional and original acoustic musical
instruments, I find little redundancy in the mathematics of
vibrating systems. I am passionate about the subject of _musical
mathematics_ because it is my life's work. This commitment
leaves very little on my redundancy plate. I have simply
attempted to be thorough.

I have communicated with builders of acoustic musical
instruments for 30 years. Some builders prefer to measure
tension with a diameter variable, others prefer to measure tension
with a radius variable. I know no better way to dispel confusion
than to be thorough.

>However, these equations are correct for the "first harmonic"
>only. Else, one should multiply the tension (or the "weight" that
>will create the tension) by n^2, where n, being an integer, points
>to the nth harmonic.

Builders of musical instruments only consider the tension of the
fundamental mode of vibration, or of the first harmonic. Here's
why: most strings are tensioned between 40% and 50% of their
_break strength_. If the tension falls too much below the lower
limit, the strings are too loose and will sound weak. If the
tension rises too much above the upper limit, the strings are too
tight and will break prematurely. To calculate a string's break
strength, we must know the _tensile strength_ of the stringing
material. Tensile strength refers to the maximum tension a material
can withstand without tearing. I won't digress with detailed break
strength calculations. Suffice it to say, the tensile strength of
spring steel music wire varies with diameter.

>His equation (3) too is identical to his equation (1); recall that
>I have previously explained that the mass per unit length is
>nothing else but D^2 x pi x rho.
>Anyway, I am glad he finally agrees with you about the fact that
>equation (3) (the one you had furnished sometime ago), is not an
>approximate relationship, but is as rigorous as his first equation;
>they are in fact, as I keep on saying, identical relationships...

In Ozan's original text, and I quote,

******************************

The middle strings are manufactured from stainless steel such
that each weigh about 20 grams per unit meter, or 8 grams (0.008
kgs) per 0.4 meters.

******************************

the diameter of the sample string is not included.
Hence, no one is able to verify the accuracy of this physical
property of the wire.

>His last equation too is correct, but, although he defined it, I
>would have used another symbol for the total mass so as not
>to confuse it with `mass per unit length`. Nevertheless, after
>writing eqation (3), his equation (4) is again a triviality, and
>for this reason, redundant.

Again, I believe it is better to be thorough than to be terse.
Also, Equations 3 and 4 are highly interesting to me because they
would enable a prospective instrument builder to contemplate
these kinds of calculations without owning an expensive
micrometer. An accurate postage scale could be used to measure
either the mass per unit length (M/u.l.) or the total mass (M) of a
string. Personally, I find this kind of approach, especially in a
classroom where for financial reasons a precision micrometer may
not exist, highly interesting and most relevant.

>On the whole, I am glad to see that he has tacitly agreed with
you in the end...
>My best wishes to him...
>
>Pap T.

Dear Distinguished Professor at MIT,

My best wishes to you as well.

Cris Forster, Music Director
www.Chrysalis-Foundation.org

Dear Mr. Ozan Yarman,

Given the context of Equation 2,
the physical properties of your string indicate:

20 grams / unit meter = Pi * r^2 * rho

However, I cannot solve for r^2 because I don't know your value of
rho for stainless steel wire.

Thank you for pointing out the Catch-22, double bind, or dilemma
of my predicament. In an email, I don't know how to address two
people at the same time, when one person is clearly speaking and
the other person is clearly not speaking. I could have potentially
accomplish this feat by addressing your father through you, but,
instead, I chose to address your father directly and with respect.
I would have preferred, of course, that he respond to me in kind,
but on hindsight, I have concluded it's all for the best.

Again, I take full responsibility for all my actions and writings.

In the event that I unintentionally offended you and/or your father,
I resolutely and happily yield the floor to you, to him, or to both
of you for the last time.

Cris Forster, Music Director
www.Chrysalis-Foundation.org

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Mr. Cris Forster,
>
> It is with utter regret that I observe you have shown not the
least bit of decency in addressing my person, though you yourself
have complained most provocatively about my lack thereof only
recently over something frivolous at best. Ignoring the presence of
your intermediary and consigning to treat him as an errand-boy is an
act of despicable disrespect in my sight. Given your maturity and
position, conduct so unbecoming of a man of your stature is
unfortunate to say the least.
>
> The only reason I asked my father's assistance (who is not a part
of this list, as you should have realized well by now, and who
probably has much more important things to do then pursuing endless
arguments over the same material), is evidently my ignorance in the
simplest equations of physics. Not only is it ethically permissable,
but also inevitable, that I would require to ask his opinions on the
matter and ask that he elucidate your input.
>
> However, for you to exploit this oppurtunity to best me since the
beginning, is quite shameful. Equally shameful is your bringing
about an argument on `mass per unit lenght` error in my previous
reasoning which I assumed to have corrected on short notice and
which you seemed to have tacitly acknowledged at last.
>
> I will nevertheless forward your response to him, and deliver his
answer in case he considers it worthwhile to reply to you.
>
> Cordially,
> Mr. Ozan Yarman
>
>
>
>
>
> ----- Original Message -----
> From: Cris Forster
> To: tuning@yahoogroups.com
> Sent: 28 Mayýs 2005 Cumartesi 19:25
> Subject: [tuning] Re: Different ways of expressing the same
tension equation
>
>
> >My Dear Ozan,
> >His Equations (1) and (2) are identical, since obviously, the
> >diameter is twice the size of the radius. For this reason, I
would
> >say, they are redundant.
>
>
> Dear Ozan's Father,
>
> As a builder of traditional and original acoustic musical
> instruments, I find little redundancy in the mathematics of
> vibrating systems. I am passionate about the subject of
_musical
> mathematics_ because it is my life's work. This commitment
> leaves very little on my redundancy plate. I have simply
> attempted to be thorough.
>
> I have communicated with builders of acoustic musical
> instruments for 30 years. Some builders prefer to measure
> tension with a diameter variable, others prefer to measure
tension
> with a radius variable. I know no better way to dispel
confusion
> than to be thorough.
>
> >However, these equations are correct for the "first harmonic"
> >only. Else, one should multiply the tension (or the "weight"
that
> >will create the tension) by n^2, where n, being an integer,
points
> >to the nth harmonic.
>
> Builders of musical instruments only consider the tension of the
> fundamental mode of vibration, or of the first harmonic. Here's
> why: most strings are tensioned between 40% and 50% of their
> _break strength_. If the tension falls too much below the lower
> limit, the strings are too loose and will sound weak. If the
> tension rises too much above the upper limit, the strings are
too
> tight and will break prematurely. To calculate a string's break
> strength, we must know the _tensile strength_ of the stringing
> material. Tensile strength refers to the maximum tension a
material
> can withstand without tearing. I won't digress with detailed
break
> strength calculations. Suffice it to say, the tensile strength
of
> spring steel music wire varies with diameter.
>
>
> >His equation (3) too is identical to his equation (1); recall
that
> >I have previously explained that the mass per unit length is
> >nothing else but D^2 x pi x rho.
> >Anyway, I am glad he finally agrees with you about the fact
that
> >equation (3) (the one you had furnished sometime ago), is not
an
> >approximate relationship, but is as rigorous as his first
equation;
> >they are in fact, as I keep on saying, identical
relationships...
>
>
> In Ozan's original text, and I quote,
>
>
> ******************************
>
> The middle strings are manufactured from stainless steel such
> that each weigh about 20 grams per unit meter, or 8 grams (0.008
> kgs) per 0.4 meters.
>
> ******************************
>
>
> the diameter of the sample string is not included.
> Hence, no one is able to verify the accuracy of this physical
> property of the wire.
>
>
> >His last equation too is correct, but, although he defined it,
I
> >would have used another symbol for the total mass so as not
> >to confuse it with `mass per unit length`. Nevertheless, after
> >writing eqation (3), his equation (4) is again a triviality,
and
> >for this reason, redundant.
>
>
> Again, I believe it is better to be thorough than to be terse.
> Also, Equations 3 and 4 are highly interesting to me because
they
> would enable a prospective instrument builder to contemplate
> these kinds of calculations without owning an expensive
> micrometer. An accurate postage scale could be used to measure
> either the mass per unit length (M/u.l.) or the total mass (M)
of a
> string. Personally, I find this kind of approach, especially in
a
> classroom where for financial reasons a precision micrometer may
> not exist, highly interesting and most relevant.
>
>
> >On the whole, I am glad to see that he has tacitly agreed with
> you in the end...
> >My best wishes to him...
> >
> >Pap T.
>
>
> Dear Distinguished Professor at MIT,
>
> My best wishes to you as well.
>
>
> Cris Forster, Music Director
> www.Chrysalis-Foundation.org