Homelab 2 [Solutions] In this homelab we will build a monochord and measure the fundamental and harmonic frequencies of a steel string. The materials you will need will be handed out in class. They are: a piece of wood with two holes in it, two bent nails, and a steel guitar string. The string we will give you has a diameter of 0.010 inch. You will also find it helpful to have some kind of adhesive tape handy when you put the string on the monochord. As soon as you can, you should put a piece of tape on the end of the string. The end is sharp and the tape will keep you from hurting your fingers. Step 1 Push the nails into the holes as shown above. They should go almost, but not quite, all the way through the board. If you push them too far in they will stick out the bottom, the board will not rest flat, and you might scratch yourself on them. You won't need a hammer to put the nails in because the holes are already big enough. You might need to use a book or some other solid object to push them in, or it might help to twist them while you push. The nails we are using are called 'coated sinkers.' They have a sticky coating that will keep them from turning in the holes when you don't want them to. It cannot be iterated enough to be careful with the nails. Refer to the diagram above if you are unsure about how the final product of this step looks like. Step 2 Now put the wire on the monochord. First, put two pieces of tape within reach so you will be able to get them while you are trying to keep the wire from unwinding. Loop the end of your string with the little brass ring around one of the nails, close to where it goes into the board. Slip the other end of the string through the brass ring and pull it tight (click for photo). This will hold the brass ring end on the first nail. Now, pull the other end of the wire down to the other nail until there is a straight section of wire running from one nail to the other, about 1/4" or less above the board. Wrap the wire about 10 times around vertical part of the second nail, ending with a few turns around the horizontal part of the second nail, and tape it down. You will probably still have a foot or so of wire left. Just coil it up and tape it out of the way. If you turn the second nail you will now see that you can tighten and tune the string. Before you bring it up to pitch it is important to arrange the coils on the vertical parts of the second nail so that the last few turns of the wire (before it heads for the other nail) are on top of a few other turns (click for photo). This way, as you increase the tension on the wire it will pinch the turns underneath and this will keep it from slipping. If your wire slips, the pitch will not hold steady and you will not be able to make good frequency measurements. Again, these instructions should be straightforward. This is just setting up for the experiment. You do want to keep in mind that you must make full use of the string and tape to ensure the wire is secure for accuracy. The links to the photos should be clear enough about how you would go about this. Step 3 It’s time to tune your monochord. Your goal is to tune the string up to a frequency of about 400 Hz, or around G4. The exact pitch is not important. You should bring the pitch up slowly from below to avoid breaking the string. If you don't have an instrument to tune to, or don't know how to match pitch by ear, you can use Raven Lite to measure the frequency. (You could use a guitar tuner too, but be careful that you are tuning to the right octave. If you try to tune it a whole octave too high it will break.) When you look at the spectrogram with Raven Lite you will see a whole series of frequencies (the harmonics of the string), but the frequency you are trying to tune is just the lowest one, which corresponds to the musical pitch. Once the string is tuned, check and see if the pitch is stable. It may drift downwards for the first few minutes, but then it should soon become stable, as long as you pluck it gently. If the pitch keeps drifting downwards the string may be slipping on one or both of the nails. Check to see that the turns overlap, as discussed above. If you are still unsure about exactly what your note should sound like, it should be around a “G”, if not slightly higher. Raven Lite: To start a new recording, click on the small microphone button to start a new recording. It looks something like this: . There is no need to change anything with regards to the settings, just make sure you choose the appropriate audio recording device only if you are using another external microphone or headphones. Then, click on the small green arrow near the bottom of the screen, which looks like this: . It may ask you for permission to use the microphone, which you must obviously accept. The software will start plotting two graphs: a blue one on the top, and a more vibrantly colored one on the bottom. After making sure there is minimal background noise, pluck the string near the microphone and observe the graph draw a spike. Hover your mouse around the top of the colored graph at the spike, and you will see a reading for the frequency. Step 4 The main goal of this homelab is to measure the frequency of vibration of at least the first three natural modes (a.k.a “harmonics” or “partials”) of the string. (Sometimes the lowest mode is called the “fundamental,” and the ones above it are called “overtones.” In this language we want to measure the frequency of the fundamental and the first two overtones.) The first three modes we are looking for are pictured in Figs. 7.7-7.9 of Hartmann. To excite the first mode, you just gently pluck the string near the center. For the second mode, pluck the string near one end while lightly touching it with a fingertip exactly at the mid-point. This creates a node at the center of the string. You will have to experiment to find the best spot to touch the string. Once you find it, use a pencil to mark the spot on the board so you can easily find it again. To excite the next mode, you touch the string 1/3 of the way from one nail while plucking it near the other nail. In general, to excite mode N, you touch the string at point that is 1/N of the total length from one end. You can see from Figs. 7.7-7.9 that this point is a node for the Nth mode. Experiment until you have all of these points marked for the first three modes and you can clearly hear the sound of each mode. It may take you a while to develop the skill to do this. Once you get the hang of it, see if you can excite even higher modes, like the fourth or fifth. Exciting the higher nodes may not be as easy simply because of the limitations of the string – its length, mass, etc. However, using the Raven Lite Software, you will notice that the frequency increases by a factor of N depending on the fraction 1/N of the string that you touch it at to create the first node. Step 5 Make a recording of the sound of each of the first three modes with Raven Lite. It will be easiest to identify them in the spectrogram if you put them all on a single recording, one after another. Notice that, for each mode, besides the lowest frequency, the spectrogram will also contain higher frequencies. This is unavoidable because more than one mode has a node at the position where you put your finger. (For example, the second and fourth modes both have nodes at the center of the string. So, when you are exciting the second mode you will probably excite some fourth mode as well.) You can save recordings by clicking on “File” at the top of the window, then “Save As”, and then enter the filename and location to save it. Step 6 Using your recording, carefully measure the frequencies of the first three modes. Since your tones will last at least a second, you can set the 'focus' control to a large number of samples for better frequency resolution. With care, you can measure each of the three frequencies with an accuracy of about 1 Hz. The resulting frequencies should be 400Hz, 800Hz and 1200Hz for nodes 1, 2 and 3 respectively. This is because the frequency is directly proportional to the number number of the fundamental. Step 7 Finally, make a graph of your results on this form. Use the y-axis for frequency and the x- axis for the mode number. Start the y-axis at zero frequency, and evenly space the mode numbers along the x-axis.