Extension 1 for this module (8 points):
Creating an Instrument
We have seen how sound can be represented as a summation of sine waves. In this extension, you will process a description of an instrument, which specifies the relative intensities of that instruments overtones. You will modify a copy of PlayThatTune so that the sound emitted for each note has the profile of the specified instrument.Warm Up
- Update your repository and locate the instrument package in the extensions folder.
- You will find a copy of PlayThatTune in the instrument package, credited to Sedgewick's book, as this code is adpated from Sedgewick's PlayThatTune.
- Run PlayThatTune and verify that it plays the music files correctly.
Overview
- For the purposes of this work, let's define a frequency factor as a multiplicative factor to be applied
to a frequency f:
In the example below, let's assume that f is 440 Hz (cycles per second), which is concert A. On a piano, this is the A above middle C:
- a factor of 1 results in the same frequency as f, because 1*f==f.
- a factor of 2 yields a frequency an octave higher than f.
For this example, we obtain 880 Hz, which is the A above high C on a piano:
- a factor of 4 yields a frequency two octaves higher than f.
Here we obtain 1760 Hz, an octave above the A above high C:
- a factor of 3 is between one and two octaves higher, and it turns out to be one octave plus
a just (or, perfect) fifth
above
the pitch corresponding to f.
Here we obtain the frequency 1320 Hz, which sounds like the E above high C.
- a factor of 0.75 (ratio of 3:4) yields a note that sounds like an E, but dividing by 4 lowers its
pitch by two octaves. With f at concert A, a frequency factor of 3:4 yields a frequency of 330, whichsounds like the E above middle C:
- It is thus convenient to specify a frequency factor as the ratio of two integers. In the above examples we see 1:1, 2:1, 4:1, 3:1, and 3:4.
- To build an actual instrument, we must pick the frequency factors we
wish to hear, and specify how strong each should be. The table below
specifies such an instrument:
The first row is the fundamental frequency. The second is an octave higher, and is 3/4 as strong as the fundamental frequency. The third row specifies a pitch that is a fifth above the fundamental freqnecy, and quieter still at half the strength of the fundamental frequency.Frequency Factor Relative Numerator Denominator Strength 1 1 1 2 1 0.75 3 2 0.50 The waveform that results from such an instrument specification is shown below in red, with the three frequency factors shown in black:
- The red solid waveform is literally the sum of the black dotted waveforms. The sound you produce is sampled from the red waveform, and that is the subject of this extension.
- The plot shown above is not the subject of this extension, but is shown
to help explain how complex sounds are built from simple ones,
as follows:
- The black waveform with the largest amplitude (the tallest black waveform) is the fundamental frequency at relative strength 1. The sample shown is for 1/440 of a second of a 440 Hz concert-A. This is sufficient ot show one full cycle of the fundamental frequency.
- The black waveform with the next largest amplitude has relative strength 0.75. It is an octave higher, so it oscillates twice in the timespan of the fundamental pitch's waveform.
- The black waveform of smallest amplitude (at half the strength of the fundamental frequency) is the 3:2 frequency factor, which sounds like a fifth above the fundamental frequency. As expected, it exhibits 1.5 cycles in the timespan of the fundamental pitch's waveform.
- The sound of the red waveform, which is what you will produce below, is similar to an oboe or a clarinet (in the opinion of my son).
Procedure
- First prompt the user for how many frequency factors the user wishes to specify. In the above example, 3 such frequencies were used.
- For each frequency factor, prompt the user for its
- Numerator (an int)
- Denominator (an int)
- Relative strength (a double)
The information about the frequency factors must be saved (in arrays) for future use.
- The code in PlayThatTune, copied with attribution
from PlayThatTune, reads in a file and computes a value
in a variable called hz that is the fundamental frequency
of the tune being played.
The hz value is computed from concert A (440 Hz), taking the specified number of equally spaced chromatic steps above (or below) concert A.
The relevant details, explained in the lecture slides for this module, are not necessary to complete this extension, but please ask if you would like clarification.
- Your task is to modify the assignment to the sample a[i] in
the provided code:
- the fundamental frequency (hz, shown in the red box) of the desired pitch
- relative strength 1.0 (the result of the sine function call is implicitly multiplied by 1, shown at the red dot).
In place of that value, you must compute the sum of sine wave samples, one for each frequency factor, as follows:
- a[i] is initially zero.
- For each frequency factor
fk, add to a[i] a sample obtained
as follows:
- The value passed to the sine function must be multiplied by the frequency factor. This means multiplying the red-boxed hz by the ratio of the numerator and denominator of the frequency factor.
- The value returned from the sine function must be multiplied by the relative strength of the frequency factor. This happens at the red dot.
- Test your program on the A.txt file first, which is a single note. Then try Ascale.txt and some other songs.
When you done with this extension, you must be cleared by the TA to receive credit.
- Commit all your work to your repository
- Fill in the form below with the relevant information
- Have a TA check your work
- The TA should check your work and then fill in his or her name
- Click OK while the TA watches
End of extension 1
Extension 2 for this module (12 points):
Warm Up
To understand this extension, you should first be familiar with the extension in which an instrument's sound is produced as the sum of sine waves.You will modify your program further in this extension to produce the sine-wave plots that depict how sine-wave addition occurs.
The details of this assignment are not completely specified so that you must think through what is needed to produce meaningful plots. Ask for help as needed!
Procedure
- An example of a plot is shown below:
- The black waveform with the largest amplitude (the tallest black waveform) is the fundamental frequency at relative strength 1. The sample shown is for 1/440 of a second of a 440 Hz concert-A. This is sufficient ot show one full cycle of that fundamental frequency.
- The black waveform with the next largest amplitude has relative strength 0.75. It is an octave higher, so it oscillates twice in the timespan of the fundamental pitch's waveform, at 880 Hz.
- The black waveform of smallest amplitude (at half the strength of the fundamental frequency) is 660 Hz: the 3:2 frequency factor, which sounds like a fifth above the fundamental frequency. As expected, it exhibits 1.5 cycles in the timespan of the fundamental pitch's waveform.
- The red waveform is the sum of the black ones.
- Each of the frequency factors incorporated into an instrument is shown separately by plotting dots of the curve as the values are computed.
- With frequencies on the order of hundereds or thousands of
cycles per second, we must limit the time of the plot so that we
can see one or two cycles clearly.
Think about how many samples you need to capture to show one complete cycle at 440 Hz.
One way to reason about this is to use the units of the various computations and multiply or divide them to obtain the property you seek.
- Plot the individual waveforms using black dots, as shown above.
- Plot the summation waveform using red connected line segments, as shown above.
When you done with this extension, you must be cleared by the TA to receive credit.
- Commit all your work to your repository
- Fill in the form below with the relevant information
- Have a TA check your work
- The TA should check your work and then fill in his or her name
- Click OK while the TA watches
End of extension 2
Extension 3 for this module (2 points):
Extension points available for Lab 4
If you have added sound and pictures to your solution for Lab 4, demo those to a TA and receive points for this extension.
When you done with this extension, you must be cleared by the TA to receive credit.
- Commit all your work to your repository
- Fill in the form below with the relevant information
- Have a TA check your work
- The TA should check your work and then fill in his or her name
- Click OK while the TA watches
End of extension 3