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Math and Music Essay

Math and music are connected in many ways. Math is seen to be as very precise. Music is often seen as a way to express emotion. They are actually both very closely related together. Music is an expression of scales and notes that are strung together to make sound. Math is the subject of numbers and symbols used to write formulas and equations. At its foundation, music and math are related. In this essay, you will show that math and music are related in many ways. They are more closely related then what they are seen to be. Numbers to beats. Pitch to rhythm.

Rhythm

Math and music’s connection begins with something called rhythm. Music is built on rhythm. Same as how mathematics is based on numbers. Rhythm is made whenever the time range is split into different pieces by some movement or sound. There are many everyday life examples of rhythm the beating of your heart, when waves hit the shore of a beach and the systematic way the traffic light blinks is rhythm. Rhythm measures time so the measure and time signature are created to make rules for a certain piece of music. A piece of music is divided into equal measures. Each measure represents the same amount of time. Each measure gets split into equal shares, or beats.

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A time signature has two parts. It resembles a fraction. The top number (numerator) is how many beats in each measure. The bottom number (denominator) indicates tells you which note to count. For example, 4/4 is the most common time signature. The four at the top represents how many beats in that measure (4). The four at the bottom indicates which note to count (in this case, a whole note). Beats are in notes. These represent how long to hold the note for. For example, a quarter note equals one beat.

How many beats in measure, four. (Numerator)
How many beats in measure, four. (Numerator)

Which note to count for, whole note. (Denominator)
Which note to count for, whole note. (Denominator)

Binary Number System

Music is related to math with the binary number system. By following this pattern, one can see how each succeeding power (of two) gives a new note to work with (ex: sixteenth notes, thirty-second notes, sixty-fourth notes, one hundred-twenty-eight notes, and so forth). This pattern is also used for rests. A rest that is a whole rest is equal to a whole note. A half rest is equal to a half note.

This pattern continues on. In 4/4 time there is one whole note in a measure, this equals 20=1. Two half notes go in a measure. The binary version of this is 21 = 2 half notes per measure. 4 quarter notes in a measure. The binary version of this is 22=4 quarter notes in a measure. 8 eighth notes go in a measure. The binary version of this is 23=8 eighth in each measure. 16 sixteenth notes fall in each measure. The binary version is 24=16 sixteenth notes in a measure.

Binary Number System is shown above

Adding a ‘dot’ after any note increases the value of the note by one half of the original note value. This also applies to rests.

All of these rests and notes can be a combination of many arrangements to make different rhythms. The only condition it has is that there must the same exact number of beats in every single measure.

A time signature of 4/4 says that every measure, no matter what notes they contain, must equal four beats. The fractional way of saying this is the sum of the fractions that every individualized note represents, must always equal one. This is because 4/4 simplified is “one.” Here are a few examples that will and will not work out.

Another very common time signature is 3/4. The fractional way of saying this 3/4. The quarter note would still get one beat (due to the fact a four is at the bottom) but this time there would only be three beats in a measure. This basically means the total number of beats must be three. These are some examples that will and won’t work.

Math can be used to determine where the second note of the two will fall in relation to the three-note rhythmic cycle. This concept is the least common denominator (LCM). Since the LCM of two and three is six, one would divide the measure into six equal counts to determine where each and every note would fall. The six count measure can be counted as “one and two and three and.” (In the time signature of 3/4, each and every one of these counts signifies an eighth note, because three quarter notes equal six eighth notes.)

In the measure below the first rhythmic cycle has three quarter notes in each measure. Each one is taking up exactly two counts. The first note is counted as “one and,” the second note would be counted as “two and,” and finally the third note would be counted as “three and.” The second rhythmic scale has two dotted quarter notes in every measure. The first dotted quarter note is counted as “one and two.” While the second dotted quarter notes starts on the ‘and’ of two, and is counted as “three and.”

Give one of these cycles to each of your hands and try to play them all at once, beating on a table or some other surface. It may even help to count aloud while doing this to make sure all the beats are falling on the right count. A much more complicated rhythm is three aainst four. The least common multiple of three and four is twelve so so the measure is divided amongst twelve equal parts. (In this case, each count signifies one sixteenth note, because three quarter notes equals twelve sixteenth notes.) This cycle can be counted as “one e and a, two e and a, three e and a, four e and a.”

While trying to beat out this rhythm as well, one may find that beating out a two against three is far easier then beating out a three against four, though it is quite possible to play both.

Every single thing surronding us has a rhythm. Ocean water has a rhythm. Protons and neutrons have rhythm. In every case, however, the rhythm moves the vibrations of the rhythm to the surronding material. Whether it be water, the ground, air, or something else, rhythm transfers vibrations. When rhythms distrupt the medium in a periodic way (repeating at equal times for equal amounts of time) they create something called wave motion. A wave has a high and low point just like an ocean wave has a high point and a low point. The high point in a wave is called the crest. The low point is called the trough. One wave equals one cycle.

The first wave is called a transeverse wave. A transverse wave is a wave that lets the particles in the medium vibrate perpendicular to the direction that the wave is traveling.

Particles in medium travel this way
Particles in medium travel this way

Wave travelsthis way!
Wave travelsthis way!

Attach a rope to something in front of you then give it a little slack. Imagine jerking the rope up and down really quick. Jerk the rope
Jerk the rope

Wave moves along rope
Wave moves along rope

The movement of one’s hand sends a wave going horizontally down the rope whilst the rope itself moved up and down. Crest

Trough
Crest

Trough

Particles in mediumtravel this way
Particles in mediumtravel this way

Wave travelsthis way!
Wave travelsthis way!

Crest

Trough
Crest

Trough

When a violin string gets plucked, it works exactly like the rope. The pluck, instead of a jerk, creates the wave. The wave travels along the string horizontally, thus, the air particles around it move ever so little vertically. Particles in mediumtravel this way

Particles in mediumtravel this way

Wave travelsthis way!
Wave travelsthis way!

An example of transverese waves are sine waves. Here a few examples.

2
Another type of wave is called a longitudinal wave. In this wave, the particles vibrate parallel to the direction the wave is traveling.

A longitudinal wave is sent when you knock over the first dominoe. This is because the dominoes fall in the direction of the wave.

Another example of a longitudanal wave is a Slinky ® toy. Hang a slinky from the ceiling, with a weight attached to it’s end, if you pull on the weight and then let go, the slinky goes up and down many times. The wave and the medium move parallel to each other.

Sound waves are also longitudinal.

The source of sound waves directs a vibration outwards in the air. At the points of compression, many air molecules crowd together and the pressure gets very high. At it’s point of refraction, the molecules are far apart and the air pressure is low. Sound waves create points of compression and refraction.

An example of a transverse wave is when one plucks a violin string. The wave that it produces however is longitudinal. The wave travels through the air, hits your eardrum and lets one hear the note.

A direct connection can be seen between two kinds of waves. The crest of a transverse wave has a direct relation to the point of compression in a longitudinal wave. The trough of the transverse wave corresponds to the point of rarefraction in the longitudinal wave.

Amplitiude, frequency and wavelengths are charecteristics of a wave.

Amplitude (A) is the distance from the top of the crest to where the wave originated from. The wavelegnth (λ) is any point on the vibrations to the corresponding next one. It is the distance a wave travels in one cycle. The frequency (f) is the number of waves per second. Frequency is measured in Hertz. One Hertz (Hz) = one vibration/seond. The period (T) is the amount of time it takes for one whole wave or cycle to complete fully. The period and frequency are recipricols of on another. (T=1/f). The loudness is how the listener measures amplitude. The larger the amplitude the louder the loudness. The smaller the amplitude the quieter the loudness.

The pitch is the listeners measuremet of frequency. It shows how high or low a sound is. The higher the frequency, the higher the pitch. The lower the frequency, the lower the pitch.

The water experiment can explain pitch. The more water in the glass the lower the pitch. The less water in the glass, the higher the pitch. In a complicated tone, there is something called a partial. The root tone with the smallest frequency is called the fundamental frequency. In most musical tones, the frequencies are integer multiples. The first one would be f. The second would be 2f. The third would be 3f. This pattern continues.

1st harmonic f=100 Hz
2nd harmonic2f=200 Hz
3rd harmonic3f=300 Hz
4th harmonic4f=400 Hz
1st harmonic f=100 Hz
2nd harmonic2f=200 Hz
3rd harmonic3f=300 Hz
4th harmonic4f=400 Hz

If the fundamental frequency is 100 Hz, these would be the frequencies of the first four harmonics:

1st harmonic f=220 Hz
2nd harmonic2f=440 Hz
3rd harmonic3f=660 Hz
4th harmonic4f=880 Hz
1st harmonic f=220 Hz
2nd harmonic2f=440 Hz
3rd harmonic3f=660 Hz
4th harmonic4f=880 Hz
If the fundamental frequency is 220 Hz, these would be the frequencies:

Handel (1685-1759) used a tuning fork for A with a frequency of 422.5 Hz. By the 1800’s the highest frequency was 461 Hz in America and 455 Hz in Great Britain. Since stringed instruments sound better when tuned higher, the frequency probably would have kept rising. However is 1953 the standard of 440 Hz was agreed tooo. Still, some people use a frequency of 442 or 444Hz.

The Piano
5 black keys

7 white keys
5 black keys

7 white keys

On the piano keyboard, there are 88 keys. It has a pattern that repeats every 12 keys. The pattern contains 7 white keys and 5 black keys.

The white keys are given a letter name A through G. The black keys also get letter names, just with either a flat or sharp symbol after it. For example, the black key between C and D is has two names, C# or D♭. The distance between two anearby keys on the piano is called a half step (for example, between C and C# or E and F). Two half steps make a whole step (for example between C and D or E and F#). A sharp raises it a half step meanwhile the flat lowers it half a step.

Geometry

Math is related to music by geometry. Geometric transformations are like musical transformations. A geometric transformation relocates the figure while keeping the size and shape. The original piece or geometric figure is not changed. The simplest geometric transformation is when the figure slides in a certain direction. The results are the same size, shape and angle measurement. This is called a translation in geometry

First place the music notes on the vertices of this triangle. Then move the notes are to the staff. The musical version of the geometric translation appears.

Geometric Translation – Repetition

The most simple of translation are in “When the Saints Go Marching In.” The repetitiveness is the theme of this song. The notes are played the same, just in different measures of the music. This means that different measure have the same notes.

Another example is in Row, Row, Row your Boat.

Geometric translations do not only have to be horizontal. They can be raised or lowered. It can be raised or lowered vertically which means the pitch can be higher or lower.

Transposition is a more sophisticated application of translation to music. It involves the movement of an exact sequence of notes to an

Geometric Translation – Transposition

Transposition is another application of translation in music. It involves the movement of an exact sequence of notes to another place on the scale. The notes are in another key. This is shown in the song “Yankee Doodle.”

Another example of this can be found in “O Christmas Tree.”

Geometric Transposition – Reflection

When the geometric figure is reflected across a line, the result is a mirror image of the original figure. The size, shape, angle and measurement remain unchanged. Another name for reflection is a “flip.” There are two types of reflections, one over the x-axis and one over the y-axis.

The musical version of this is called retrogression, is shown below.

An easy-to-see reflection is in the song “Raindrops Keep Falling on my Head.”

An additional example is shown in the Shaker tune “Simple Gifts”

A geometric reflection across the x-axis is the same except for the fact that the line of reflection is horizontal instead of than vertical. In music, it is called inversion and can take several different forms. One is in harmony:

The other form of inversion is in melody and can be shown in Greensleeves.

Transposition – Glide Reflection

This is the third form of geometric transformation, which is called glide reflection. It is a translation followed by a reflection or a slide and then a flip.

You can see inversion in Guantanamera, a popular Spanish song.

Rotation

A rotation occurs when a geometric figure is rotated 180 degrees around a point. The figure is moved to another location. It is also called a turn.

This can also be done by reflecting over both axes, in any order.

The Circle of Fifths and The Chromatic Circle

The circle of fifths can be plotted from the chromatic scale by using multiplication. The chromatic scale is based on 12 notes which cannot be repeated until all notes are played. Multiply the numbers by 7. The reason we are multiplying by 7 is that there are 7 whole tones. Number the 12 notes of the chromatioc scale from: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)

Showing all of the notes on the chromatic scale:
0=C, 2=D, 4=E, 5=F, 7=G, 9=A, 11=B, 1=C#, 3=D, 6=F#, 8=G#, 10=A# Now multiple the whole row by 7 (0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77)
Then subtract 12 from every number until the final number becomes less then 12: (0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5)
And this is equal too:
(C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, F)
Which is the circle of fifths (this is enharmonically related too): (C, G, D, A, E, B, G♭, D♭, A♭, E♭, B♭, F).
This is the chromatic circle with the circle of fifths inside. (Star dodecagarm) This is the chromatic circle with the circle of fifths inside. (Star dodecagarm)

Fibonacci Sequence

Mozart is thought to be one of the greatest musicians and composers in the world. He used Fibonacci Sequence in some of his piano concertos (a concerto is a musical composition normally composed in three parts or movements.) Fibonacci sequence is the sequence of numbers, in which the sum of the two previous numbers equals that number ex: 0, 1, 1, 2, 3, 5, 8, 13…). In the margins of some of his music, he wrote down equations.

For example, in Sonata No. 1 in C Major, there are 100 measures in the first movement (A movement is a self-contained part of a composition.) The first section, of the movement, along with the theme, has 32 measures. The last section of the movement has 68 measures. This is perfect division, using natural numbers. This formatting can be seen in the second movement, in turn. Although there is no actual evidence concerning this matter, the perfect divisions of this piece of musis is quite easy to see.

Fibonacci sequence goes on infinetly. The first number is 1. Every following number is the sum of the previous two. Adding 1 to nothing would give you 1. The third number would then be 2, the sum of 1 and 1. The fourth number would be 3 (to get this you would add 2+1) and the fifth number would be 5 (to get this you would add 3+2). These are some examples of Fibonn aci numbers:

Fibonacci Sequence is everywhere. For example, the Fibonacci sequence gets shown on the piano because of the way the keys are setup. An octave is made up of thirteen keys. Eight of the keys are white and five are black. The black keys are split into groups of two and three. Each scale has eight notes. The scale is based off of the third and fifth tones.

Both pitches are whole tones which are two steps away from the first note in the scale (also known as the root). There is also something called the Fibonacci Ratio. A Fibonacci ratio is any Fibonacci number divided by one adjacent in the series. For example, 2/3 is a Fibonacci ratio. So are 5/8 and 8/13. This pattern continues on. The farther along the ratios are placed, the more they have in common. They also become more and more exactly equal to 0.618.

The porportion that these ratios show is thought to be, by many, to look appealing to the eye. It is called it the golden porportion. A Hungarian composer named Béla Bartók often used this technique while creating his compositions.

The chart below is based on the Fibonacci ratios. The root tone A has a frequency of 440 Hertz. To find high A you multiply the Fibonacci ratio of 2/1 by 440 Hertz to get 880 Hertz. To get the frequency of note C, multiply 3/5 by 440 to get 264 Hertz. Harmonics are based off of Fibonacci ratios.

Bibliography
http://www.goldennumber.net/music/
http://www.sciencefairadventure.com/ProjectDetail.aspx?ProjectID=150 Math and
Music: Harmonious Connections by: Trudi Hammel Garland and Charity Vaughn Kahn

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