Musical Scales Lab

Goal: 

To understand the mechanisms of the musical scale and find the frequencies of the notes between C and the C one octave higher. 

Background Information: 

Any note that is one full octave above another note also has twice the frequency of the lower note. Therefore, the note that is one full octave lower has half the frequency of the higher note. For example, a C that has a 100 Hz frequency has a C one octave higher than it with a 200 Hz frequency. According to the Greeks, a perfect 5th, or musical interval, can be composed by two notes that have a 3:2 ratio. 

Process: 

Since the Greeks claimed that a perfect fifth is a simple 3:2 ratio, we used this as our guide for establishing our scale. This means that we started with 100 Hz (C) and proceeded to multiply it by 1.5, and continue to multiply those results by 1.5 until the value exceeds 200 Hz. Once the value has reached or surpassed 200 Hz, we have passed the frequency value for the C one octave above the C at 100 Hz. After ordering them, we can see that this scale does not give us values of frequencies with constant ratios. Therefore, we are instructed to use a different method which will give us values of frequencies with correct ratios.
To find a more precise scale, also known as an even-tempered scale, we calculated the ratio needed to space the notes equally

Original 12 note scale:

C: 100 Hz
C#: 106.7871094 Hz
D: 112.5 Hz
D#: 126.5625 Hz
E: 142.3828125 Hz
F: 150 Hz
F#: 160.1806641
G: 168.75
G#: 189.84375 Hz
A: 213.5742188 Hz
A#: 225 Hz
B: 253.125 Hz
C: 284.765625 Hz


Even Tempered Scale:

100x^11=200
x^11=2
x= 1.059462094


C: 100 Hz
C#: 105.9463094 Hz
D: 112.2462048 Hz
D#: 118.9207114 Hz
E: 125.9921049 Hz
F: 133.4839852 Hz
F#: 141.421356 Hz
G: 149.8307074 Hz
G#: 158.7401048 Hz
A: 168.1792826 Hz
A#: 178.1797431 Hz
B: 188.7748619 Hz
C: 199.9999993 Hz

Conclusion:
This lab demonstrates to us the complex nature of musical scales and the use of ratios in creating them. Given the need for evenly spaced scales (with ratios not absolute differences), we must use exponents to find the values for our scales.