For a vibrating string, the frequency is inversely proportional to the length, directly proportional to the square root of its tension, and inversely proportional to the square root of its weight per unit length.
Pitch does vary with temperature, the higher the temperature, the higher the velocity, and therefore the higher the frequency for the same wavelength. But with string instruments, this difference is very small.
|c e||Major 3rd||4:5|
|c e-flat||minor 3rd||5:6|
|c a||Major 6th||3:5|
|c a-flat||minor 6th||5:8|
The notes are those that you would find on a piano. A "c" is a basic note. That is why this table revolves around it. The interval is how far the second note in the table is from the "c". The frequency ratio is the ratio of the frequency of the second note to the "c".
To add intervals, you must multiply the frequency ratios.
Absolute pitch is where a pitch can be referenced to another note that is absolute. In other words, a C is always a C and has the same number of cycles per second (which is the frequency of the note and always remains constant). So if another pitch is compared to this note it will give us the actual pitch of that note and we can give that pitch a new name, like "A" (depending on what the ratio is between the two pitches). Absolute pitch is usually found by tuning everything to a tuning fork (the pitch of a tuning fork does not change).
Relative pitch is only a comparison of two pitches to each other. There is no way of knowing what the actual note is without comparing to a tuning fork, or an in-tune instrument, or with someone who has perfect pitch (someone who knows exactly what a "C" is and can sing it and are correct).
For our experiment, we used relative pitch, but since we compared it with Allison's flute (which is fairly in-tune), the notes we got are very close to the actual pitches on the absolute scale.
Stillman Drake believes that Galileo actually did conduct experiments in which he measured the weights tied onto similar strings of the same length and measure the musical interval between the two strings. He observed that the interval between the two notes was related to the inverse squares of the length of the string when the same weight was attached and the same interval was observed.
Thus music played a key role in Galileo's experiments because of the physics, pitch, and the timing considerations. Since Galileo couldn't measure time with a watch(since there were no watches back then), he had to measure time with a water clock and using internal rhythm (like a musical conductor can split time up into smaller intervals in his mind) he was able to time a ball rolling down an incline plane, or a ball rolling down a plane into a parabola curve.
Galileo probably also used frets (the ridges on a stringed musical instrument, like the violin or lute) in his inclined plane experiments because when the ball hit the fret it would make a noise and he could time how long it took the ball to get from one fret to the next. He could then adjust his frets so that the time interval between each one was the same.