Suppose the length of the string is L, then it will vibrates with wave length 2L, 2L/2, 2L/3, 2L/4, ...

The longest wave length (with the lowest frequency f0) is 2L. Its vibration can be described like:

A sin(kx)cos(wt)

with k = 2*Pi/(wave length),

w = 2*Pi*frequency = 2*Pi*f.

Hence w/k = f*(wave length) = wave speed = v.

By just using dimensional analysis, we can find that:

sqrt[(string tension)/(linear mass density)] = sqrt(T/D) has the unit:

sqrt[(kg*m/s*s)/(kg/m)] = m/s, the unit (dimension) of v, so

v is proportional to sqrt[T/D]

Now we see that the frequency is given by:

f = v/(wave Length) ~ sqrt(T/D)/L.

which agrees with our observation of playing Er-Hu.

Let us denote the frequaency of key-note c by f0 (f0 is approximately 131 Hz to 132 Hz), then in the unit of f0, we have the following note-string length-frequency relations;

KEY-NOTE:	DO	RE	Mi	Fa	So	La	Si	Do
LABEL:	c	d	e	f	g	a	b	c'
Length in L	1	8/9	4/5	3/4	2/3	3/5	8/15	1/2
Freq. in f0	1	9/8	5/4	4/3	3/2	5/3	15/8	2/1