The (sinusoidal) output of a signal generator drives a speaker placed over one end of a pipe, both of whose ends are open. As you vary the frequency of the tone from the signal generator, you (and the class) notice that at certain frequencies, the sound becomes very much louder than it is when the generator is tuned anywhere between them. The four pipes of different lengths allow you to explore the relationship between these frequencies and the length of the pipe. By firmly holding the rubber stopper against the bottom end of the pipe, you can show how these frequencies change when the pipe is closed at one end, versus when it is open at both ends.

When you send a wave through a particular medium, if that medium has well-defined ends, as a vibrating string or an air column contained within a pipe, when the wave reaches the opposite end of the medium, it is reflected back on itself. Whether it is reflected in phase or out of phase with itself depends on whether that particular end of the medium is “closed” or “open.” (See demonstration 40.66 -- Longitudinal wave machine with free or fixed end. This phenomenon is perhaps easier to see with demonstration 40.60 -- Transverse wave machine with free or fixed end.) With an organ pipe, at an open end, the air is not axially constrained by the pipe, so the motion of the air is greatest there, and there is an antinode (the reflection is in phase). At a closed end, the air is constrained, the motion is least, and there is a node (the reflection is out of phase). With both ends open, then, there is an antinode at each end of the pipe, and resonance – a maximum in intensity of the sound – occurs at any frequency for which the length of the pipe corresponds to an integral number of half wavelengths. When this condition is satisfied, the superposition of the wave with its reflections at the ends of the tube forms a pattern of alternating antinodes and nodes, with an antinode at each end of the tube, and the number of nodes and antinodes between them depending on how many half wavelengths fit within the length of the tube. Such a pattern is called a standing wave. The lowest frequency for which this occurs is called the fundamental, or the first harmonic. If we call the fundamental f, we should observe harmonics at f, 2f, 3f, 4f, 5f, etc. With one end open and the other end closed, as when you firmly press the rubber stopper to the bottom of the pipe, there is an antinode at the open end and a node at the closed end, and resonance occurs at any frequency for which the length of the pipe corresponds to an odd integral number of quarter wavelengths. If we call the fundamental f, we observe harmonics at f, 3f, 5f, 7f, 9f, etc. The frequency of the fundamental is half that of the fundamental for the open pipe; it is an octave lower. For both sets of harmonics, each harmonic above the fundamental has one more node and one more antinode that the one below it. For example, the second harmonic for the pipe with both ends open has an antinode at each end and one in the middle, and a node centered between each end and the middle. The third harmonic for the pipe with one end closed and one end open, if we call the pipe length l, has an antinode at one end, a node (1/3)l from that end, an antinode (1/3)l from the node, and a node at the other end of the pipe. Note that these nodes and antinodes are displacement nodes and antinodes. Where the motion of the air is least, the pressure is greatest, and where the motion of the air is greatest, the pressure is least. Thus, a displacement node corresponds to a pressure antinode, and a displacement antinode corresponds to a pressure node.

The four organ pipes are pitched, according to the now-obsolete “physics” standard, to middle C (256 Hz), E a major third above that (320 Hz, shown in the clamp in the photograph above), G a perfect fifth above middle C (384 Hz) and C the octave above middle C (512 Hz), and are stamped accordingly. They actually resonate at slightly higher frequencies than these, probably because they are designed to fit an inlet that, with the knife edge at one end of the pipe, forms the whistle that sets the air column vibrating. This attachment probably increases the effective length of the pipe somewhat, so without it, the pipe is a bit shorter, and the frequency is a bit higher than what is stamped on the pipe. For the pipe shown in the photograph, with both ends open, you should observe resonances at approximately 338 Hz, 679 Hz, 1,018 Hz, 1,354 Hz, 1,698 Hz, 2,038 Hz and 2,357 Hz. These are all quite close to multiples of 338 Hz. (You can probably go higher, as well.) With one end closed, you should observe resonances at about 174 Hz, 525 Hz, 874 Hz, 1,221 Hz, 1,570 Hz, 1,910 Hz, 2,266 Hz, 2,634 Hz and 2,964 Hz. These are all quite close to odd multiples of 174 Hz. (Again, you can probably go higher, if you wish.)

The overtone series

Perhaps the most famous experiments that led to our understanding of the harmonic series, especially its importance in building musical scales and intervals, were those that Pythagoras performed on a monochord. (See demonstrations 44.15 – Weighted string sound box, and 44.18 – Sonometer.) This series is often called the overtone series. The fundamental is the lowest-frequency member of the series. It is common to refer to the second harmonic as the first overtone, the third harmonic as the second overtone, etc. As noted on the page for demonstration 44.12 – Ear model, we hear differences between pitches according to the ratios of their frequencies. That is, we hear any pair of pitches whose frequencies are in a particular ratio as being the same musical interval apart. As noted above, for a pipe with both ends open, the harmonics are successive integer multiples of the frequency of the fundamental. The interval between any two members of a harmonic series – two pitches whose frequencies are in whole number ratios – will sound pleasant. If the frequencies of two pitches cannot be expressed as a whole-number ratio, then the pitches are not harmonically related, and their interval sounds out of tune. As we go up in the harmonic series, the frequency ratios of neighboring harmonics become successively smaller, and thus, so do the musical intervals between them. A frequency ratio of 2:1, which we have between the second harmonic and the fundamental, corresponds to a musical interval of an octave. For the first three octaves in the harmonic series produced by a pipe that is open at both ends, the frequencies and intervals are as in the following table.

Pipe Open at Both Ends (All Half-Wavelength Harmonics)
n	ν	Interval from Previous Harmonic		Interval from Fundamental
1	ν0		Unison (perfect prime)		Unison (perfect prime)
2	2ν0		Octave		Octave
3	3ν0		Perfect fifth		Octave + One perfect fifth
4	4ν0		Perfect fourth		Two octaves
5	5ν0		Major third		Two octaves + One major third
6	6ν0		Minor third		Two octaves + One perfect fifth
7	7ν0		Minor third*		Two octaves + One minor seventh*
8	8ν0		Major second*		Three octaves

Intervals are named according to the number of scale degrees that separate the two pitches that make them. Going up from the first step of a major scale, the tonic, the rest of the scale steps are, in succession, a major second, major third, perfect fourth, perfect fifth, major sixth, major seventh and an octave above the tonic. The intervals between neighboring scale degrees are major and minor seconds – whole steps and half steps, respectively.Within the first three octaves of the harmonic series, we have the major third, which corresponds to a frequecy ratio of 5:4, the perfect fourth, whose ratio is 4:3, the perfect fifth, whose ratio is 3:2, and the octave, at 2:1. The asterisks on the intervals for the seventh and eighth harmonics are there because though they are close, they do not equal the ratios by which we normally define these intervals. The building of scales and intervals from the overtone series is a rich and fascinating subject. Suffice it to say here, that frequency ratios for the various intervals are commonly based on just intonation, a system in which all intervals are derived from the natural fifth, natural major third and natural octave, i.e., those shown in the table above. This system is considered to provide the most pleasant-sounding major triads (built on the first, fourth and fifth steps of the scale), because it puts them all in 4:5:6 ratios. (Note that the fourth, fifth and sixth harmonics together make up a major triad. This property is probably why the German name for just intonation is reine or natürliche Stimmung, for “pure” or “natural voicing.”) Unfortunately, this system has serious disadvantages that prevent its use for tuning instruments, but it does provide a useful reference for calculating frequency ratios for intervals that we hear as being “pure” or “natural.” Intervals having these frequency ratios are called just-tuned or just. The ratios given above for the major third (5:4), perfect fourth (4:3), perfect fifth (3:2) and octave (2:1) correspond to those for just-tuned intervals. A just minor second has a frequency ratio of 16:15, major second 9:8, minor third 6:5, minor sixth 8:5, major sixth 5:3, minor seventh 16:9, and major seventh 15:8.

Most woodwind and brass instruments, and, of course, open organ pipes, allow all half-wavelength harmonics, and produce the overtone series shown above.

As noted above, if we close one end of the pipe, we obtain harmonics for which the length of the pipe corresponds to odd-integer numbers of quarter wavelengths. The frequencies of the harmonics are odd multiples of the frequency of the fundamental. Because the length of the pipe is one quarter wavelength, the fundamental frequency is half that of the fundamental for the pipe with both ends open, or one octave lower. The table below lists the first eight members of this series, with their intervals.

Pipe Open at One End and Closed at the Other End (Odd Quarter-Wavelength Harmonics)
n	ν	Interval from Previous Harmonic		Interval from Fundamental
1	ν0		Unison (perfect prime)		Unison (perfect prime)
3	3ν0		Octave + One perfect fifth		Octave + One perfect fifth
5	5ν0		Major sixth		Two octaves + One major third
7	7ν0		Augmented fourth/Diminished fifth*		Two octaves + One minor seventh*
9	9ν0		Major third*		Three octaves + One major second
11	11ν0		Minor third*		Three octaves + One perfect fourth*
13	13ν0		Minor third*		Three octaves + One minor sixth*
15	15ν0		Major second*		Three octaves + One major seventh

As in the first table, asterisks indicate intervals whose frequency ratios differ from those of just-tuned intervals. For these I have chosen the names of the just-tuned intervals closest to which these intervals lie.

The clarinet and, of course, closed organ pipes, allow only odd quarter-wavelength harmonics, and produce the overtone series shown above.

References:

1) Resnick, Robert and Halliday, David. Physics, Part One, Third Edition (New York: John Wiley and Sons, 1977), pp. 412-413.
2) Berg, Richard E. and Stork, David G. The Physics of Sound (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1982), pp. 67-75, 319-320, 349-351.
3) Apel, Willi. Harvard Dictionary of Music (Cambridge, Massachusetts: Harvard University Press, 1955) pp. 359-363 (Intervals, and Intervals, Calculation of), pp. 384-385 (Just intonation).