A little Wave theory
The tone (frquency) made by the string, is determined by its length (l), its mass (m) and the force of tension (F). The speed (v) of the sound in the material of the string can be determined with use of ths formula:
Now the speed (v) is known, the keynote can be determined. The keynote is created by a the half of the wavelenght ((Lambda)). As l = /2 the following formula can be used for determining the keynote:
All stringed instruments work this way.
Notice: The units are: v [m/s] m = [kg] l and = [m] F = [N] (Newton) f = [Hz]
Vibrating air columns:
Resonce boxes - Air oscillations in a tube closed in one end.
The classic example is organ pipes. There are open pipes and closed pipes (called "gedackt"). The length of the pipe determines the keynote it gives. The speed of sound in air (at 20°C) is 343 m/s. As the organ pipe is supposed to give a certain tone frequency (f), and knowing the speed of sound in air (343 m/s), the wavelength can be determined with this formula:
The wavelength is not necessarily the same as the length of the pipe. In a "gedackt" that is closed at one end, the waves will look like this:
The lenght of the pipe will be a fourth part of the keynotes wavelength as the closed end creates a zero point. The maximum will be at the open end, even a little further from the opening. Thus organ pipes and wind instruments are actually shorter than the calculated values. When working with brasswind instruments I usually make the pipe ca. 10% shorter than the calculated value.
Besides the keynote, the pipe will create some overtones, here the first overtone is shown:
The frequency of the first overtone is calculated with this formula:
Also panflutes work this way
Organ pipes closed at one end will have weaker overtones than the open ones.
Open pipes - Waves in open tubes.
Some organ pipes are open at both ends. Here the maximum of the waves will lay at the open ends, and a zero will lay in the middle of the pipe. Thus a half wavelength lays in the tube. Its length (l) will be the half wavelenght
The calculating of goes the same way as earlier with this formula:
An open pipe must tus be double as long as a closed to create the same keynote. It will create more rich overtones than the closed one, this is further enhanced by a wide tube. Overtones in open tubes look like this (1. overtone):
More about Overtones
In the days of silent movies a cinema organ used to play during the movies. In the USA they made an experiment: The open pipes in the lowest octave were losed at the top, causing them to play one octave deeper. This intrasound (ca 7-16 Hz) can not be herd by humans, but our body does feel those tones, and they cause us to be uncomfortable and nervous. This was uses while playing thrillers, nearly causing a panic in the audience. The change was simple: A stopper in the upper opening of the pipe, removing the stopper would restore the normal function of the organ.
Recorders, transverse flutes, and brasswind instruments work as open pipes.
The tone scale is senn here, the octaves are numbered from 1-8, see the frequencies in the link below. First note in each time is a C, followed by D, E, F, G, A, H then a niw time begins.
List with tone names and frequencies (Opens in new window)