Section 3.1
Section 3.2
If we start at a frequency, f, go up by an interval of a fifth and then up by an interval of a fourth, what is our new frequency?
If we start at a frequency of 200 Hz, go up by an interval of a fifth and then down by an interval of a fourth, what is our new frequency?
a) Starting from a frequency of 520Hz, go up by three intervals of a fifth (3/2). What frequency are you at?
b) Now bring this note down by octaves until it is in the octave starting on the note with the frequency of 520 Hz. What frequency are you at, now?
A natural trumpet can only play harmonics of its fundamental freaquency: f, 2f, 3f, 4f, 5f, 6f, 7f,... How many notes are in the first octave of the trumpet. How many notes are in the second octave?
What octave must the trumpeter work in to have at least seven notes?
Section 3.3
Can the following series of frequencies correspond to a harmonic series, an overtone series, or both?
a) 500 Hz, 1020 Hz, 1600 Hz, 2000 Hz, ?
b) 400 Hz, 800 Hz, 1200 Hz, 1600 Hz, ?
c) 400 Hz, 1200 Hz, 2000 Hz, 2800 Hz, ?
A string is vibrating in the mode shown here:
a) If the length of the string is 37 cm, what is the wavelength of this mode of vibration.
b) If the fundamental mode has a frequency of 250 Hz, what is the frequency of this mode?
Consider a 1 meter length of string. Sketch the first four modes of vibration. Mark the nodes and antinodes, and find the wavelength, in each case.
Plot the first three modes of a vibrating string, fixed a both ends. If the length of the string is 2 meters, and the velocity of the wave on the string is 200 m/sec, what is the wavelength and frequency of each mode?
If you lightly touch the string of a violin or cello, the string vibrates is such a way as to have a node at the point that you touch the string.
a) If you touch the string right in the middle, sketch the first three modes that can vibrate.
b) What kind of series does the frequencies of these modes form with respect to the lowest frequency in part a)? I.e. is it a full harmonic series, a partial harmonic series or neither?
The strings on a violin, viola, or cello are tuned a fifth apart.
a) On the figure, indicate with an (x) where you would press your finger the fundamental string to bring its pitch up a fifth to create a unison between the strings. This technique requires you to place you finger precisely in the correct location.
b) Alternatively, you can place a finger lightly on each string to produce a harmonic. Indicate with a dot where you would place your finger on each string to produce a common harmonic.
c) Why is this technique more accurate than the first?
Consider a string on a cello or violin. If you press firmly on the string, you effectively change the length of the string without changing the tension.
a) Mark on the string with an X the position where you should press to raise the pitch of the string by a Perfect Fourth.
b) Now, if you lightly touch the string, you don't change the length, but you create a node in the vibrational mode at that point. Sketch the mode if you lightly touch the string at the same position as in part a).
c) What interval is the note generated in part b) above the original frequency?
The vibrational frequency of a string is given by the formula:
.In the next laboratory, we will measure the vibrational frequencies of a stretched string. We will stretch the string by hanging a weight from one end of the string:
L is the length of the string in meters, U is the mass of one meter of the string, T is the tension on the string and is given by the mass of the weight, M, in kg times the acceleration of gravity g = 9.8 m/sec2. So T = Mg. N can be any integer greater than or equal to 1 gives the different overtones.
a) If M = 0.4 kg, L = 0.7 m, U = 0.0003 kg/m, find the vibrational frequency of the first 4 overtones of the string.
f1 =
f2 =
f3 =
f4 =
b) How much must we increase or decrease M by to double the frequency of the string?
c) How much must we increase or decrease M to reduce the frequency by a fifth (3/2)?
Section 3.4
Starting from f = 1, find the Pythagorean scale formed by going down by fifths (3/2) and up octaves (2) to stay between 1 and 2.
Starting from f = 1, find the Pythagorean scale formed by going up by Perfect Fourths.
The pentatonic scale is consists of the following intervals:
a) Scales are generally formed by large steps and small steps that connect the adjacent notes in the scale. For example, the step from 1 to 9/8 is simply the interval 9/8. Find the next 4 steps of the pentatonic scale.
b) In part a), you should end up with only 2 intervals, a small step and a large step. Find the nearest Just interval to each of these steps. Remember, the Just intervals are ratios of integer less than 10. Note, one or both steps might already be a Just interval.
Section 3.5
Section 3.6
Which of the following is a possible mode for an air column open at one end and closed at the other? For the correct one, mark the nodes and antinodes and find the wavelength, if the length is 1 meter.
For each set of modes mark the ones that are proper modes and identify the nodes and antinodes for the proper ones.
VIBRATING STRING:
CLOSED-CLOSED AIR COLUMN:
a) Plot the first three modes of a 26 cm tube with two closed ends. Find the wavelength and corresponding frequencies. Use v = 343 m/sec.
b) Plot the first three modes of a tube with one closed end and one open end. Find the wavelength and corresponding frequencies.
Plot the first three modes of a 1 meter tube with two OPEN ends. Find the wavelength and corresponding frequencies. The velocity of sound is 343 meters/second.
a)Sketch the first two modes of an open-open air column with a length of 0.35 meters. Indicate the nodes with dots and antinodes with x's and find the wavelength and frequency.
b) Sketch the first two modes of an open-closed air column with a length of 0.35 meters. Indicate the nodes with dots and antinodes with x's and find the wavelength and frequency.
Sketch the second mode of vibration for the following tubes and find the wavelength and frequency of the mode if the tube is 0.5 m long:
a) An open-open tube.
b) An open-closed tube.
c) What is the interval between these two frequencies?
A finger hole in a woodwind instrument creates a node in the mode of vibration at that point. This is just like gently touching a string when plucked to produce an overtone. A finger hole is 10 cm from one end of a 30 cm tube. In which kind(s) of tube will this produce an overtone: closed-closed, open-closed, open-open. This means that you can find a proper mode with a node at the specified point. Sketch the case(s) that work.
Which overtones of a closed-closed air column have nodes in the middle of the tube?
Which overtones of an open-open air column have nodes in the middle of the tube?
With real instruments, it is sometimes hard to tell if the instrument acts like an open-open or an open-closed air column. A flute, for example, clearly has one open end, while the other end is closed. However, the blow hole may act like an open end. So, we would like another way to figure this out.
a) The lowest note on a flute is a "C" at 262 Hz. The length of the flute is about 24 inches. From this information, determine whether the flute acts like an open-open or open-closed air column. Does your answer predict the correct overtone series for a flute?
b) The length of a clarinet is also about 24 inches, but the lowest note on a clarinet is 147 Hz. Does the clarinet act like an open-open or open-closed air column and why? Does this agree with the overtone series?
Suppose you measure the following overtones of an air column: 375 Hz, 675 Hz, 875 Hz, 1175 Hz. However, you also realize that you probably did not record the fundamental frequency. Is the air column open-closed, open-open, or is the series wrong? (I.e. does not correspond to either case.) Give your reasoning. If it is a proper series, what is the fundamental frequency?
The following series is missing one or more of the lowest frequencies: 2500, 3500, 4500, 5500 Hz ?
a) This series can be part of the overtone series for an air column with which of the following boundary conditions (there may be more than one): closed-closed ends, closed-open ends, open-open ends?
b) What are the missing low frequencies?
The following series is missing one or more of the lowest frequencies: 5000, 7000, 9000, 11000 Hz ?
a) This series can be part of the overtone series for an air column with which of the following boundary conditions (there may be more than one): closed-closed ends, closed-open ends, open-open ends?
b) What are the missing low frequencies?
State whether the following series correspond to an open-open tube, an open-closed tube or neither. One or more of the low frequencies may be missing. If a series does correspond to an open-open or an open-closed tube, give the fundamental frequency of the tube.
|
|
Type of tube |
Fundamental |
|
300, 400, 500, 600 Hz ? |
|
|
|
600, 1000, 1400, 1800 Hz ? |
|
|
|
100, 200, 400, 800 Hz ? |
|
|
|
120, 220, 320, 420 Hz ? |
|
|
Each of these overtone series corresponds to an open-open or open-closed air column. However, some low frequencies may not have been recorded. Also, there is exactly one extra or one missing frequency. In each case, give that extra frequency, determine whether the tube is open-open or open-closed and give the fundamental.
|
|
Extra frequency |
Type of tube |
Fundamental |
|
375, 625, 700, 875, 1125 Hz ? |
|
|
|
|
300, 450, 750, 900, 1050 Hz ? |
|
|
|
|
450, 675, 900, 1125, 1300 Hz ? |
|
|
|
|
800, 960, 1600, 2240, 2880 Hz ? |
|
|
|
You have a tube that is 0.63 meters long. A resonance of the tube was measured for each of three different cases: both ends closed, one end open and one end closed, both ends open. Also, the frequency is affected by the end correction. For each frequency, state whether the tube is open-open, open-closed or closed-closed and what mode it is in. You don?t have to calculate the end correction, itself.
a) 402.0 Hz
b) 1088.9 Hz
c) 527.7 Hz
You start with a closed-closed tube with a fundamental frequency of 571.7 Hz. Assume that the speed of sound is 343 m/sec.
a) How long is the tube?
b) Now, you unblock the ends, so that you have an open-open tube with the same length as before. What should the radius of the tube be so that the open-open tube has a frequency one half step lower than the closed-closed tube? (Use 16/15 for the half step).
One of the projects demonstrated in a previous class was a pan flute and several interesting comments were made about it. Each note was made out of a open-closed tube. The radius of the tubes was approximately 1 cm.
a) First, consider just a regular open-closed tube with no end correction. How long should we make the tube so that it plays an A = 440 Hz?
b) Now, it was stated that inserting the stopper raised the pitch by about a whole step. How far into the tube must the stopper be pushed to raise the pitch by exactly a Just whole step (9/8)? Is you answer reasonable?
c) Then, it was stated that the end correction was not significant enough to worry about. Assume that we have an open closed tube that plays an A = 440 Hz if we don't consider the end correction, as in part a). Now, find out what the pitch would be if we did consider the end correction. Can you hear the difference between these two pitches?
One of the projects from a previous year was a wooden whistle as shown in the diagram:
a) One question that arose was whether the whistle acts as a closed-closed or open-closed air column. There is an opening, but it is off to the side, so the answer is not obvious. However, I measured the pitch of the whistle to be 816.7 Hz. What would the fundamental frequency of the whistle be for each of the two cases?
a) What is the whistle, closed-closed or open-closed? Is there an end correction?
b) Normally, the end correction means that the tube seems longer by an amount 0.6(radius of tube). If the whistle was open-closed, what would the fundamental frequency be with and without the end correction?
When you measure the resonances of an air column, you find the following series of frequencies: 310, 620, 930, 1240, ... Hz. The tube has a radius of 2 cm.
a) Is the tube open-open or open-closed?
b) Find the actual length of the tube, taking into account the end correction. Remember that an open-open tube has an end correction at each end.
This is my data for the same laboratory that you did (the length of the tube is 24 cm):
|
Closed Tube |
Frequency |
Wavelength |
|
1 |
771 Hz |
|
|
2 |
1468 Hz |
|
|
3 |
2184 Hz |
|
The speed of sound really must be the same in all cases, so assume that it is 347.5 m/sec. Calculate the wavelength of each mode. Now, make a CAREFUL sketch of the modes on the following graphs using the calculated wavelength (place the nodes and antinodes first):
What you can see is that the speaker is not a perfectly closed end, so the node is not exactly at the end of the tube. This is a big error for the fundamental. By the third node, things look O.K.
Assume an open-closed tube has a physical length of 24 cm and a diameter 3 cm. The end correction says that the pressure node at the open end is not right at the end but 0.6xradius of the tube beyond the end. Calculate the wavelength of the first three modes and make a careful sketch of the modes on the following graphs (place the nodes and antinodes first):
What is the frequency of the third mode with and without the end correction?
Section 3.7
Section 3.8
We have discussed intervals of an octave, a fifth, and a fourth. Of course, there are more intervals in music. In particular, the major third. The major third corresponds to going up 2 whole steps.
a) In the Pythagorian scale, what is the ratio of frequencies corresponding to a major third?
b) The answer to part a) is not a very simple fraction and some philosophers did not like this. What simple ratio of integers less than 10 is the closest to the Pythagorian major third. (This is called the Just scale.)
c) On a piano, a half step corresponds to a frequency ratio of 1.0595 and two half steps exactly equals a whole step. On a piano, what would be a major third?
a) Why does 1.0595 correspond to a half step on a piano?
b) Pythagoras would create a major sixth by going up a fifth twice and coming down by a fourth. What is this ratio?
c) What would a major sixth (nine half steps) correspond to on a piano?
d) The Just interval would be the closest ratio of small integers to the piano interval. What would a major sixth be in the Just scale?
a) Why does 1.0595 correspond to a half step on a piano?
b) Pythagoras would create a minor third by going up a fourth twice and coming down by a fifth. What is this ratio?
c) What would a minor third (three half steps) correspond to on a piano?
d) The Just interval would be the closest ratio of small integers to the piano interval. What would a minor third be in the Just scale?
a) The minor third on the piano corresponds to 3 half-steps. What is the equal temper minor third?
b) Pythagoras would create a minor third by going up a fifth and coming down by a Pythagorean third. What is the ratio for the Pythagorean minor third? (Give answer as a ratio of two numbers).
c) Pythagoras could also have formed the minor third by going up a fourth twice and coming down by a fifth. Does this give the same ratio as part a)? (Show your work!)
a) The minor sixth on the piano corresponds to 8 half-steps. What is the equal temper minor sixth?
b) Pythagoras would create a minor sixth by going DOWN by a Perfect Fifth FOUR times and coming up by octaves. What is the ratio for the Pythagorean minor sixth? (This ratio should be very close to the answer from part a).
c) The Just minor sixth is the ratio of two numbers, each less than 10, that is the closest to the answer in part a). What is the Just minor sixth?
The overtone series of a brass instrument is a harmonic series. If the fundamental frequency is f, the overtone series is 1f, 2f, 3f, 4f, 5f, 6f, 7f, 8f?. What combination of intervals from the fundamental does each of these tones correspond to? For example, 4f is two octaves above the fundamental.
We considered the intervals formed by the first seven overtones of a brass instrument, 1f, 2f, 3f, 4f, 5f, 6f, 7f. Now consider the next few: 8f, 9f, 10f, 11f, 12f. What interval does each overtone form with respect to the 8f overtone? If the interval is not exactly one of the intervals you know, which one is it closest to?
There are several intervals that we have not considered, yet. For each interval, find the ratio corresponding to that interval in the equal temper scale. Then find the ratio of integers, which is close to this value - this will determine the interval in the Just scale:
Minor third = 3 half steps
Minor sixth = 8 half steps
Major sixth = 9 half steps
Seventh = 11 half steps
Show that certain pairs of intervals combine to give an octave. How many pairs can you find? Some pairs may not come out perfectly in the Just scale or the Pythagorean scale. Give the pairs that come close and state whether they give a perfect Octave or not in each of the scales. Why will the pairs always come out perfectly in the Equal Temper scale?
Triads consist of three notes. They contain three different intervals: the first to second note, the first to third and the second to third.
a) The major triad has notes in the ratio 4:5:6. What are the three intervals in this triad? This is the triad that Ptolomy liked.
b) Notes in the ratio 3:4:5 also make up a triad. What are the three intervals in this triad?
c) (Not required, not for credit) What musical chord does this correspond to?
Look at the lower left-hand panel in the woodcut describing the Pythagorean ratios.
a) First assume that the numbers give the relative LENGTHS of the strings. Give the interval of the string (6) through (16) with respect to string (4).
b) Now interpret the drawing literally - that is, the numbers give the relative WEIGHTS on the strings. Again, give the interval that the strings (6) through (16) in that order make with respect to string (4). If the frequency is exactly equal to an interval that we know, give the name of the interval. Otherwise, give the ratio with three decimal points (for example 1.234).
Starting from f = 1, find the Pythagorean scale formed by going down by major thirds or minor thirds.