Music brings joy to those who play, write, or listen. Bring up how it is connected to mathematics, and many people will be puzzled. However, once the shock recedes and they think about it, they realize that there are striking similarities. Not only the obvious ones, like beats in a measure, but the wave lengths and ratios between notes.
image sourceStudies have shown that babies who listen the classical music can grasp mathematical concepts quicker than those who did not. In essence, math is creative and beautiful, which is why such elegant connections are made between them. This paper will attempt to discuss the simple and complex connections between the two studies as well as attempt to uncover new ideas in the fields of music, such as a wind instrument with a piano's range.
Music is made up of beats. Beats are pulses in which time is marked. The most common measure has four beats in it, which means no matter the combination of notes, they must add up to four beats. For example
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- Level one is called a whole note, meaning that this note will get four beats. 1note @ 4 beats
- Level two is composed of two half notes, which are two beats each. 1 note @ 2 beats
- Level three is 4 quarter notes representing one beat each. 1 note @ 1 beat.
- Level four is 8 8th notes, which equal one half a beat each. 1 note @ .5 beats
- Level five shows 16 16th notes, which would each receive one quarter of a beat. 1 note @ .25 beats.
The same holds true for rests: one whole rest equals two half rests equals four quarter rests, and so on and so forth. The following picture shows the relationships between rests and beats.
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Basics first. Before attempting to play music, any musician will look at the time signature. The top number tells a musician how many beats are in one measure and the bottom one reveals which note will get one beat. In the above example, there will be four beats in one measure, and a quarter note (1/4 a measure) will get one beat. But this does not mean that only quarter notes can take be in that measure. Take a look at the first measure above
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Another simple aspect of music is the tempo. Tempo is recorded in beats per minute and tells the conductor and musicians how fast a piece of music should go. This is marked at the beginning of a piece above the staff (see above). A tempo of 60 is very slow at one beat per second while a tempo of 180 is very fast at three beats per second. The formula to figure out how long a song will be is:
60S=MB
T
S equals time in seconds, m equals the number of measures, b equals the beats in one and t equals the tempo
The key signature tells the musician which keys to play in. Different keys are made up of different notes even though the ratios to the first note remain the same.
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Take any note, and the transition to the note directly to the left or right of it is called a half step. The note two notes to the left or right is called a whole step. For example, take “F”. An “F#” and an “E” are considered half steps, while “G” and “D#” are considered whole steps. To create a major key we used this pattern. Start with any note and then:
+1 +1 +1/2 +1 +1 +1 +1/2
which will give you the key of that note. This will also give you a major scale. According to Answer, majors scales are defined as: “an ascending or descending collection of pitches proceeding by a specified scheme of intervals”. This interval is: x+1+1+1+(1/2)+1+1+1+(1/2), with x representing any note, which was discussed earlier. There are thirteen major scales that are distinct, C, C#, D, D#, E, E#, F, F#, G, G#, A, A#, B. Any scales after these would be defined as octaves, one whole pattern above the others. To find octaves we use the formula (with x representing the first note of any scale):
x=x+13
Now this pattern is the same for all notes and scales. To transpose a scale or change it to a different key, you use a method similar to a shift cipher. For instance; take the notes C, D, E, F, and G in the key of C (no sharps of flats). If one wants to transpose these notes, one first attaches numerical values to the notes based on their scale. C is the first note of the scale so it gets a value of one. D is the third note so it gets a value of 3:
