# Finger Piano Physics- Calculating Length and Building a Prototype.

1. Frequencies of desired notes and corresponding calculated lengths of keys:

 Note f (Hz) L (m) F2 87.31 0.061345708 F2# 92.5 0.05959987 G2 98 0.057903277 G2# 103.83 0.056254174 A2 110 0.054653734 A2# 116.54 0.053098062 B2 123.47 0.051586428 C3 130.81 0.050118228 C3# 138.59 0.048691172 D3 146.83 0.047305189 D3# 155.56 0.045958644 E3 164.81 0.044650302
1. Provide a reasoning on why you selected this particular octave (middle versus higher or lower).  Perhaps due to length restrictions etc.  Show calculations showing what your minimum and maximum lengths are.

I selected this octave of a middle frequency based on the calculated lengths necessary to produce the frequencies of these notes. I have a length limit to the keys of my finger piano, with the length of my longest key with the shortest possible section held under the bridge representing the maximum length (9 cm) and the length of my shortest key with possible section held under the bridge representing the minimum (4 cm). These restrictions are simply a function of the material from which the keys were cut: the putty knives provide inherent limits on the length of the keys, because unlike wood, they are not sold in a variety of lengths. Notes of higher frequencies are more audible on these keys. Therefore, within my length limits, the 2nd and 3rd harmonics are the most audible and possible to produce.

Calculations: See attached documents.

1. Prototype:
1. How did you prototype your instrument without building an exact replica?
1. Using a box of similar dimensions and of identical wood to the body of my final instrument, I attached two bridges made of maple to the side of the box adjacent to the hinges. These bridges were made of a lighter, more flexible wood than the bridges of my final instrument. The smaller of the two bridges was 1.5 cm. wide by 1 cm. tall. The larger bridge was 2 cm. wide by 2.5 cm. tall. These bridges were divided into two sections and attached to the box with three hexagonal bolts, with the head of each bolt in the interior of the box and the opposite end holding the bridge in place by a washer and a wingnut. The bridges held in place the twelve spring steel keys, with five keys per section. The keys lengths were adjusted by loosening the winguts and lifting the bridges to extend or shorten the length of the key that was free to vibrate. The space between keys and the space between bolts was greater than those on the final instrument.
2. What design considerations were you hoping to validate with your prototype?
1. I did not cut a hole in the resonator box in my prototype to test whether the sound would be sufficiently amplified without the hole. I also tested the importance of the pliability of the wooden bridge, and I tested whether placing the keys perpendicular to the resonating surface would sufficiently amplify the notes.
1. What did you learn from this experience and how do you intend to change the actual design or building process?
1. The wooden bridge on my prototype was very pliable, and the sections of bridge between the bolts bent in the center, reducing the pressure on the keys in the center of each section and allowing both ends to vibrate slightly, producing an unpleasant sound. In my final instrument, used  basswood bridges, which are much stronger than my previous maple bridge and apply pressure evenly to the ends of the keys.
2. I initially used hexagonal bolts to fasten the bridge and apply pressure to the keys, but the hexagonal bolts do not grip the wood. Once the bolts were tightened to a certain point with the wingnuts, the bolt would turn in its hole and no more pressure could be applied. In my final instrument, I fastened the bridges with carriage bolts, which dig into the wood and hold themselves in place.
3. The frequencies that are amplified by the resonator box without the hole are D2, D2#, and E2, which are the highest frequencies on the finger piano. I am considering whether the final instrument requires a resonator hole to amplify all notes. One possible change would involve removing the back of the box and extending the keys perpendicular to the opening to amplify all notes and leave the un-plucked ends of the keys to vibrate freely.
4. 4. Calculation explanations:
 Note f (Hz) π v (m/s) K m L (m) F2 87.31 3.141592654 5790 0.000144509 1 0.061345708 F2# 92.5 3.141592654 5790 0.000144509 1 0.05959987 G2 98 3.141592654 5790 0.000144509 1 0.057903277 G2# 103.83 3.141592654 5790 0.000144509 1 0.056254174 A2 110 3.141592654 5790 0.000144509 1 0.054653734 A2# 116.54 3.141592654 5790 0.000144509 1 0.053098062 B2 123.47 3.141592654 5790 0.000144509 1 0.051586428 C3 130.81 3.141592654 5790 0.000144509 1 0.050118228 C3# 138.59 3.141592654 5790 0.000144509 1 0.048691172 D3 146.83 3.141592654 5790 0.000144509 1 0.047305189 D3# 155.56 3.141592654 5790 0.000144509 1 0.045958644 E3 164.81 3.141592654 5790 0.000144509 1 0.044650302

Calculations/Explanations/Nodes and Anti-nodes: See attached document.

Calculations of Minimum and Maximum Lengths