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# Mathematics of Musical Frequency

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## Lara Bockenstedt

on 4 December 2013

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#### Transcript of Mathematics of Musical Frequency

Mathematics of Musical Frequency
The Science of Sound
Music, similar for many people, has always fascinated me. It connects people world wide and I have been playing instruments for over ten years.
In order to understand musical frequency, we must first understand the science of sound. Here is a clip from a youtuber (Charlie Mcdonnell's video "Fun Science: Sound") who creatively explains this difficult topic: (watch until the end of the song)
Understanding Musical Frequency
Do you recall Charlie singing about higher frequencies of sound and lower frequencies? He explained the fact that sounds actually come to us in the form of a wave. A sound wave "creates minute pockets of higher and lower air pressure, and all the sounds we hear are caused by these pressure changes." When we listen to music, the frequency of the wave actually determines the pitch when these air pockets have different levels of air pressure.
What?
Confusing, right? Well, consider the note "middle C". A "C major" scale or chord are the easiest to learn because there are no sharps or flats, and it is the note dead center in the middle of a piano. The "middle C" note has a frequency of about 262 Hertz . . . meaning 262 pockets of higher air pressure are hitting your ear every second (specifically every 0.00382 seconds.) A higher frequency sound at "Middle G" for example would have air pockets arriving more frequently at 392 Hertz, or air pockets being received by your ear every 0.00255 seconds. Lower frequency sounds would have air pockets arriving at your ear more slowly and at a lower Hertz number.
Octaves
In the musical world, there are instruments that generally play at higher octaves, (violins, flutes,) and instruments that play at lower octaves, (cellos, tubas.) It is very critical that they sound good when playing in an ensemble. Similarly, when singing, jumping octaves can be common; such as when we sing happy birthday and we jump from middle c to high c in the third line of the song.

What makes these octaves sound good together??

Well, there is a rule that applies to finding the frequency of these octave jumps. To go from "middle c" to "high c" the frequency would double for the air pockets arriving every 0.00191 seconds. Because of this basic rule, we know that to go from "middle c" to "lower c" the frequency would have to be divided by half. If two octaves are played at the same time, ("middle c" and "high c" for example, a pattern emerges in the arrival time of each pocket of sound from the two notes.

Chords
Now that we know about octaves, it's time to add some creativity- with chords! A "C major" chord includes three notes: C, E, and G, all separated by major thirds. What makes these three notes playing together sound so good to our ears? The trick of it isn't that the frequencies line up perfectly with each other, similar to octaves,rather they compliment each other. For every four frequencies of "Middle C" being received, your ear has also received 5 frequencies of "Middle E" and 6 frequencies of "Middle G" . . . . and then the cycle repeats itself.
Mathematicians and Music
early musicians were aware of the importance of "tuning" the instruments so that the frequencies would be correct. Mathematicians such as Pathagoras in the 6th century BCE were very precise in their method of tuning. For example, they checked to make sure that "Middle G"'s frequencies were always 3/2 times that of "Middle C"'s
Amplitude
Moving beyond the pitch of sounds we hear, there is a science and math to how loud the noises we hear are. Noise frequencies of varying length and energy are measured in decibels for loudness, and humans typically register sound at about 5 decibels. A decibel of about 130 would be considered very painful. In fact, some Police Stations in Texas carry around decibel meters to judge loudness for giving tickets based on noise complaints.
More Math!!!
Most of our major and minor scales come from ancient Greece where math was a crucial part of the culture. Plato believed music to be essential to education- seeing its ties to both math and inner harmony. While most of the music from ancient Greece has been lost, their meticulous work on music theory dating as early as 1200 BC has contributed immensely towards our current knowledge of music.

Another contribution to music from mathematical history is the Pythagorean scale. The Pythagorean scale allows us to take a note such as "Middle C" and to mathematically find the other notes on the scale using fractions!
Further Information:
If you wish to educate yourself on the topic more extensively, this video explains more of the intricacies of tuning & mathematics
Thank you!
Works Cited:

http://jackhdavid.thehouseofdavid.com/papers/math.html

http://plus.maths.org/content/magical-mathematics-music

The two videos used:

1. Charlie Mcdonnell's video "Fun Science: Sound"

2. Zacharyjrsman's video "The Math of Music"
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