Explain why placing your finiger exactly in the middle of a guitar or violin string and pressing it against the neck should give a tone about an octave higher than the open (full length) string.
Explain ( in terms of the tension in the string) why you may in reality get a difference slightly greater than an octave, especially if the bridges are high and hold the string very far above the fingerboard.
When you place your finger exactly in the middle of a guitar or violin string and press it against the neck, you are essentially dividing the string into two equal parts. This produces a higher pitch because dividing the string in half shortens the vibrating length, resulting in an increase in frequency and thus a higher tone.
To understand why this happens in terms of string tension, we need to consider the physics behind it. When you pluck an open string, it vibrates at its natural frequency, which depends on various factors such as the length, tension, and mass of the string. When you place your finger on the string and press it against the neck, you effectively change the length of the vibrating section.
In the case of dividing the string in half by placing your finger exactly in the middle, the tension in each half of the divided string becomes twice that of the original open string. This increased tension causes the frequency of the vibrating length to double, resulting in a pitch that is approximately one octave higher than the open string.
However, in reality, there might be a slight difference in the pitch, especially if the bridges are high and hold the string far above the fingerboard. This is due to a couple of factors. First, the increased tension caused by the higher bridge can further raise the pitch of the divided string. Second, the distance between the bridge and your finger placement can affect the overall tension and vibration of the string, potentially causing a greater difference in pitch than just an octave.
The specific amount by which the pitch exceeds an octave depends on the individual instrument and the specific setup, including factors such as the height of the bridge, the fingerboard, and the type and gauge of the strings.