Activity: Piano Construction We are going to design a piano. Before starting, we need to consider what characteristics we will plan for our piano. Most pianos have 88 keys and therefore 88 different pitches or tones. Each string produces a different tone, so the average piano has 88 strings. We will design for a minimum of 85 tones and therefore, 85 strings. The frame of the piano must fit into the living room of a typical family home. The total tension (the sum of all the string tensions) that a grand piano frame can withstand is about 30 tons. Some music theory: Pianos are designed according to the western musical scale which contains 12 pitches per octave. If we start with A, the pitches would be named: A, A#, B, C, C#, D, D#, E, F, F#, G, G# This series would then be repeated beginning again with A. The second (higher) A is considered to be an octave higher than the first (lower) A. If the lowest note is an A, then what note will the 85th tone be? A is 1st, 13th, 25th, 37th, 49th, 61st, 73rd, 85th How many octaves will the range of this piano span? 7 octaves If the frequency of the lowest tone is 32 Hz. What will be the frequency of the tone, one octave higher? 64 Hz What will be the frequency of the highest tone? 32 Hz (2^7) = 4096 Hz General Physics theory: What is the equation for the velocity of a wave on a string in terms of frequency and length? (Hint: What is the relationship between string Length and Wavelength for the fundamental tone on a string?) Since only ½ wavelength is on a string in the fundamental, then λ=2L What is the equation for the velocity of a wave on a string in terms of the tension and type of material? 𝑣 𝜋 𝑇 𝜇 Combine these equations into one and solve for frequency. This equation will be the basis for our piano design. 𝐿𝑓 𝑓 𝜋 𝑇 𝜇 𝜋 𝑇 𝐿 𝜇 Now, before we start changing things, lets establish some values to start with. Let’s start with a steel string. Design Strategies: I. Changing length while keeping everything else constant. If the highest tone is produced with a string of length 3 inches, how long will the string need to be to produce a tone one octave below this? 6 inches What will be the length of the longest string? 3 in (2^7) = 384 in =32 feet. Will this piano fit in the living room? Not in any room I’ve seen in a house. II. Changing Tension while keeping everything else constant. If the lowest tone is produced on a string with 10 N of force, how much force will be needed to produce a tone one octave above this? For twice the freq we need 4 times the force. Notice the square root. What tension will produce the highest pitch? 10 N (4^7)=163840 newtons This is over 36800 pounds! And this is only for the highest tension string. Will this piano’s total tension work for the piano frame specified? Lets add up all the A string tensions: 10 + 40 +160 + 640+2560+10240+40960+163840 =218450 (this is over 49000 pounds or almost 25 tons). We haven’t even done any of the other notes and we are almost to 30 tons. III. Changing the linear density while keeping everything else constant. We will change the linear density by making the wire thicker (changing the diameter). If we start with a steel string of diameter 1/32”, what is the linear density? The density of steel is 7800 kg/m3. You need to decide whether to use metric or inches. I’ll use metric. 1/64 in = 0.0396875 cm. 𝑚 𝜌 𝑉 So 𝜌𝑉 𝑚 𝑉𝑜𝑙𝑢𝑚𝑒 𝜋𝑟 2 𝐿 𝑚 𝜌𝑉 𝜌𝜋𝑟 2 𝐿 𝜇 𝜌𝜋𝑟 2 𝐿 𝐿 𝐿 𝑘𝑔 𝑘𝑔 𝜇 7800𝑚3 𝜋 0.000396875𝑚 2 0.0038597 𝑚 If the highest tone is produced on a string with a 1/32 inch diameter, what diameter string will produce a tone one octave below? From the equation above you can see that the linear density is a function of radius squared. When the radius is doubled, the linear density is increased by 4. 𝑓 𝜋 𝑇 𝐿 𝜇 By plugging into the equation you can see that increasing r by 2 causes the frequency to go down by 2. 𝑓 𝜋 𝑇 𝐿 𝜇 𝑓 𝜋 𝑇 𝐿𝑟 𝜇 2 What diameter will produce the lowest pitch? 3 By doubling the radius 7 times, the diameter is doubled as many times. This means that the diameter will be 4 inches to make the lowest A tone! Will these string diameters result in a useful piano? Explain your reasoning. Think about making a nice tiny wooden hammer to strike a 4 inch diameter string to make the tone!