Every string has specific frequencies at which it naturally vibrates based on its:

1. Length: The longer the string, the lower the fundamental frequency. f = 1/length
2. Tension: The tighter the string, the higher the frequencies. F = √T
3. Density: Frequency decreases as density of the material increases. F = 1/√μ

Timbre is a consequence of the overtones. The string can also vibrate in more complex modes by dividing the string into sections. Each section vibrates independently at a higher frequency. These modes of vibration occur at specific frequencies where the string fits an integer number of half-wavelengths along its length. An overtone is any frequency higher than the fundamental in the harmonic series. The fundamental and the overtones are called partials. Harmonic partials are the frequency which are integer multiple of the fundamental including the fundamental frequency where the relationship is by 1 times itself.

Overtones:

• any frequency component above the fundamental
• cab be harmonic or inharmonic
• includes all the frequencies generated by the vibrating system

Harmonics:

• all the frequencies including the fundamental
• frequencies that are integer multiple of the fundamental
• follow a particular mathematical relationship eg. 2x, 3x, 4x of the fundamental

All harmonics are overtones but not all overtones are harmonics

Harmonic note: is a musical note that is played by amplifying or preventing vibration of certain overtones in a stringed instrument hence has higher pitch than the base note and is pleasant to hear. The player avoids the other frequencies by touching the string on certain points called nodal points of the overtones. The partials have a higher pitch because they are a multiple of the base note. Frequency is on a linear scale while pitch is on a logarithmic scale. The first overtone is one octave higher than the base. If we consider a Sitar or Tambura with a base frequency 200Hz considering it to be षड्ज (Shadaj) the following notes will be generated by the harmonics.

Nodal point: The nodes of the natural harmonic are located at whole number fraction of a string. For an instrument the nodal points are located across the length of the string from bridge to bridge.

nodal points: L/n
where n is the sequence of the harmonic and L is the length
If we consider a string of 100cm at 220Hz

How to play the harmonic note: Each harmonic has a node where there is no vibration and antinode where there is maximum vibration. When you put a finger on any of the nodes you're preventing the string to vibrate in the segments that it would normally move (the antinodes). This forces the string to produce harmonics for the point in string you touched. The touch on the string damps out unwanted vibrations that don't correspond to that harmonic. So for example if you slightly touch the point of second harmonic i.e. in the middle you're preventing the string to vibrate in its natural mode. It now must vibrate in thr second harmonic with the string divided into two parts. The isolation of vibration produces pure harmonic tone

Most Sitars have a bridge to bridge length of around 90cm. When measuring the length of the bridge is taken into account for the calculation. In Sitar one bridge has negligible touch point but the other bridge called Jawari has significant amount of length along the contact point. So to accurately measure the freely suspended length of the string we have to subtract the length of the Jawari from the final value.