__Special Points Of Harmonic Balance On A String__

__( For the case of fundamental & dividing frequencies sounded in a trinity)__Some points of harmonic balance on a string can be heard when sounding fundamental open, and a contact point divides the sting into a ratio of parts, both sides of the contact point strings sounded.

The formula used for one side of the dividing contact point string sounded is:

frequency, f1 = f0*(Te/(Te-t) , Where fo = fundamental open string frequency, Te = time it takes for the contact point travelling at constant velocity to move from start to end. t = time passed

For the other length of the string on the other side of the point of contact, frequency, f2 = fo*(Te/t)

Letting the fundamental open string = 1 for the frequency for convenience

if a ratio, r1 is sounded for another frequency

Then the length of string corresponding to r1 = 1/r1 ( letting open string length be = 1)

The length of string on the other side of the contact point dividing the string into a particular ratio, will = 1/r2

So 1 = (1/r1) + (1/r2)

Rearranging gives:

r1 = r2/(r2-1)

Designating diff1 = (r1 - 1)

diff2 = (r2 -1)

Sum1 = (r1 +1)

Sum2 = (r2 +1)

By sound experimentation of moving a contact point which divides the string into a ratio of r2/r1 or r1/r2 depending on which side you want the ratio to be, some special points of balance can be heard due to the coupled ratios of r1, r2, diff1,diff2,sum1,sum2, relative to each other in different formats being equal, or a multiple of 1 or 2 for the more pronounced points of balance heard along the string.

The more intangibly perceived points of balance on the string correspond to relationships like 3/2,4/3 5/4etc...

When diff2/sum1 = 3

(r1)^2 = 4/3

r1 = 2/sqroot(3) = 249.022cents r2= 3479.962c

When 3*sum1 = sum2

then 3(r1)^2 - 2(r1) - 2 = 0

r1 = (1+sqroot(7))/3 = 337.504cents r2= 2996.598

When diff2/sum1 = 2

then (r1)^2 = 3/2

r1 = sqroot(3/2) = 350.977cents r2=2935.345

When 3*(r1) = 1/diff1

then 3*(r1)^2 - 3*(r1) -1 = 0

r1 = (1+sqrt(7/3))/2 = 405.271cents r2=2712.496

When 2*diff1 = (r1)/sum1

then 2*(r1)^2 - (r1) - 2 = 0

r1 = (1+sqroot(17)/4 = 428.422cents r2=2627.43

When 2*(r1) = 1/(diff1)

then 2*(r1)^2 - 2*(r1) - 1 = 0

r1 = (1+sqrt(3))/2= 539.981cents r2=2279.962

When (1/diff1) - (r1)/sum1 = 2

the same as saying r2/sum1 = 3/2

then 3*(r1)^2 - 2(r1) - 3 = 0

r1 = (1+sqroot(10))/3 = 566.893cents r2=2208.502

When 1/sum1 = diff1

then (r1)^2 = 2

r1 = sqroot(2) = 600.000cents r2= 2125.864

When 3*r1 = sum2

then 3(r1)^2 - 5(r1) + 1 = 0

r1 = (5+sqroot(13))/6 = 624.366cents r2/r1 = 1444.049cents r2= 2068.415

When r1 = 1/diff1 {Golden ratio phi}

r1 = (1+sqroot(5))/2 = 833.090cents r2/r1 = 833.090cents r2= 1666.180

When diff2 / diff1 = 2

then 2*(r1)^2 - 4*(r1) + 1 = 0

r1 = 1+1/sqroot(2) = 925.864cents r2/r1 = 600cents r2= 1525.864

When sum1 = 2*diff2

r1 = sqroot(3) = 950.978cents r2/r1 = 539.981cents r2= 1490.959

When diff1/diff2 = 2/sum2

then 2*(r1)^2 - 3*r1 - 1 = 0

r1 = (3+sqr(17))/4 = 999.008cents r2/r1 = 428.422cents r2= 1427.430

When 3*diff1 = 2*diff2

3(r1)^2 - 6(r1) + 1 = 0

r1 = 1+sqroot(2/3) = 1033.390cents r2/r1 = 350.977cents r2=1384.367

When 3*diff2 = 4*diff1

then 4(r1)^2 - 8(r1) + 1 = 0

r1 = 1+sqroot(3)/2 = 1079.962cents r2/r1 = 249.022cents r2=1328.984

Note that when the ratio r2/r1 is stated for the higher range intervals, these ratios of r2/r1 already exist for lower range values of r1

These points of harmonic balance can only be strongly perceived for this special case of sounding both sides of the contact point strings, which divide an open string of the fundamental frequency. Experimental results seem to clarify this

But for the case of only one side of the contact point string frequency sounded with the fundamental, these frequencies of harmonic balance in this report can not be perceived. Only Just intonation whole number relationships for the ratio of frequencies seem to be the only points of harmonic balance.

**Here is a video of the synchronicity of harmonic geometry of a string with the sound of these trinity of frequencies**