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The phyics of music and the relationship of frequencies

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Right now in physics we are studying Rotational motion and torque, but since I already made a blog on that last week, I decided that this week I would create a post about music.

A few months ago my girlfriend asked me why certain harmonies sounded good to your ear and why some of them sounded bad. I just assumed that it had something to do with the fact that we grow up hearing certain harmonies in our music, and in other cultures where they might not listen to the same music as us, the harmonies that were pleasing to their ears would be different then ours.

Well, I decided to do a little internet research on the topic, and I found that the harmonies that common harmonies and chord progressions have small whole number ratios between their frequencies.

A is the number of cycles of the sound wave per second. It is measured in Hertz (Hz). The higher the frequency of the sound wave, the higher the pitch of the sound, and vice versa. Since the speed of sound remains at a constant 343.2 m/s in 20 degree Celsius air, the wavelength is inversly proportional to the frequency as shown by the equation:

velocitypng.latex?v=f\lambda

Now that we have some background in waves, we will jump into to these harmonies in music that sound nice.

The Perfect Octave and Perfect Fifth are two of the best consonances. If you play them together on an instrument, there is absolutely no dissonance. Pythagoras discovered that the ratio between a perfect octave was 2:1 and that the ratio between a perfect fifth was 3:2. With this knowledge, he began to mathematically build a scale.

If you want to build a scale between the octaves, you start with a note. The ratio of frequencies from that note to itself is 1:1, so we will call the starting pitch 1. The ratio of frequencies from that note to the octave above it is 2:1, so we will call the finishing pitch of the scale, 2. If we take the first note 1 and multiply it by the 3:2 ratio of the perfect fifth we get 3:2, which is the perfect fifth of the scale obviously. If we take 1 and divide it by 3:2, we get 2:3. However, 2/3 is not between 1 and 2 so it is out of the range of the scale. If we multiply it by the 2:1 ratio to get the next octave of the note, we get 4:3 which is an octave about 2:3. Interestingly enough, 4:3 is the perfect fourth of the scale!

If we continue using this method of multiplying or dividing by 3:2 and then doubling or halving the ratio to get between 1 and 2, we can build the Pythagorean scale, which is very close to the major scale, give or take a few Hz.

The ratio of the frequencies from the Pythagorean scale, with respect to the first note, is as follows:

1, 9:8, 81:64, 4:3, 3:2, 27:16, 243:128, 2

If you know the frequency of the starting pitch, you can multiply it by any of these ratios to find the new pitch. If you know that the frequency of A4 is 440 HZ, then you can find the frequency of the E above that (E5) by multiplying 440 by 3:2, since A to the E above it is a perfect Fifth. Using this we know that E5 has a frequency of 660 Hz.

The major scale that we use today is a little different then the Pythagorean scale, but only by a few Hz between some of the intervals. The Major scale's frequency ratio, with respects to the starting pitch is as follows:

1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2

The frequency ratio for the perfect fourth, perfect fifth, perfect octave, and whole step remain the same as the Pythagorean scale, but the thirds and sixths change. In medieval times, we saw a lot of music based around perfect fourths and fifths, but in the common practice period of music, music is based on tertian harmonies, or harmonies of thirds and sixths. Our major, minor, augmented, and diminished chords are all built around major and minor thirds/sixths. Major thirds and minor thirds have frequency ratios of 5:4 and 6:5 respectively, and major sixths and minor sixths have frequency ratios of 5:3 and 8:5. With these new frequency relationships, it is easy to see why thirds, fourths, fifths, and sixths sound so pleasing to our ears. The integer ratio of the frequencies is a small number! In fact, the most dissonant interval in the major scale is probably the Major Seventh, which has a frequency ratio of 15:8, which is the ratio with the largest, non-reducing, integers

The most common chord progressions in the world are based around the chords I, IV, and V. The I chord being a chord of thirds with starting pitch at the first pitch of the scale, The IV chord being a chord of thirds with starting pitch on the perfect 4th of the scale, and the V chord being a chord of thirds with starting pitch on the perfect 5th of the scale. When you realize that the roots of the chord are in ratios of 1:1, 4:3, and 3:2, and that the major and minor thirds are in frequency ratios of 5:4 and 6:5 respectively, it makes sense why this chord progression is so common and sounds pleasant to the ear.

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waves are one of my favorite topics (mostly because of their relationship to music). I think that overtones are another aspect that could be considered in why some harmonies sound different than others (another blog topic! :D)

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Crazy Business! Craziness! If you do 2:1|3:2|4:3|5:4|6:5 starting on C2 you get C2|C3|G3|C4|E4|G4. That is quite the epic major chord! 7:6 is probably where the niceness starts to dwindle. Do you know the ratio of the tritone?

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moe.ron, I appreciate your interest in the physics of music. I still have a copy of a paper I wrote in pre-calc in high school (a loooong time ago) where I looked into this, too. I was looking at timbre, and how overtones in the harmonic series create a more complex sound wave in different instruments. Shortly thereafter I visited the Boston Science Museum where they had a light organ--a wall-sized spectrum analyzer--that displayed the frequency and amplitude of music that was playing, and I was able to visualize what I had recently written about. I could have spent hours watching it.

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moe.ron, I appreciate your interest in the physics of music. I still have a copy of a paper I wrote in pre-calc in high school (a loooong time ago) where I looked into this, too. I was looking at timbre, and how overtones in the harmonic series create a more complex sound wave in different instruments. Shortly thereafter I visited the Boston Science Museum where they had a light organ--a wall-sized spectrum analyzer--that displayed the frequency and amplitude of music that was playing, and I was able to visualize what I had recently written about. I could have spent hours watching it.

philandfriends, that sounds like a very interesting paper and a fascinating instrument! Would you happen to know if the spectrum analyzer is still around?

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As I said earlier today, while the nice ratios are close the real ratios are 2^(n/12):1 where n is the number of halfsteps in the interval.

Odd that the nicest looking ratio 2^(6/12) or 2^(1/2):1 is the most dissonant tritone while the ugly looking 2^(5/12):1 and the 2^(7/12):1 of the "Perfect" Fourth and "Perfect" Fifth respectively are the most consonant.

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