Paste your essay in here…INTRODUCTION
Mathematics can be applied to everything in the world. It is one of the most arduous fro many students, but is definitely one of the most beneficial subjects. From the obvious personal applications of math to its wide ranging strategy uses, math is ubiquitous. From calculating tips, determining gas mileage, budgeting finance to landscaping buildings, designing animations and games, mathematics is needed in all situations.
Most people do not realize classical music heavily relies on math. Many composers loved to involve math within their work. Wolfgang Amadeus Mozart is one of the most famous composers who implemented math into their work. He often used patterns and sequences in this music. As seen in the beginning of the Marriage of Figaro, the singer recites a list of numbers, which is a mathematical pattern Mozart created, which was that his number sequences all added up to 144. Other composers followed his lead and used math through other methods to improve their work.
By studying music and its sound, mathematical patterns can be discovered. Composers have used symmetry in their work, which can be seen in the sheet music but also when the song is heard. In symphonies, when the music rises and falls equally symmetry in these scales is shown. Time signature heavily use fractions, since specific beats and notes are divided up to fit per measure. Research has shown a relation between octaves with wavelength and frequency, which can be calculated with math. As illustrated, math can be applied to every concept in our world, even classical music through patterns sequences, two-piece relationships, fractions, and symmetry.
SYMMETRY
Symmetry is when two shapes or parts of a shape are identical. This is a common characteristic in math, most often seen in graphs and in geometry. The most common form of symmetry is reflectional symmetry, in which one part reflected over a specified line creates an identical part. This can be seen in graphs, for example when a function is reflected over the y-axis to display identical images in both quadrants. Another example is when a shape is cut down the middle by a line of symmetry and both sides are identical.
In classical music reflectional symmetry is musical inversion, in which one section of the measure is inverted. The most well-known example is rising and falling scales. This shows symmetry because if the notes were split through the middle, it would be seen that there are an equal amount of notes on both sides of the split, and that they are in the same rhythm. Mozart often used inversion in his songs. For example, in his song, Piano Sonata No. 16 (Sonata Facile), he uses a scale technique at least four times, in which the notes rise and fall identically and symmetrically as seen in Figure 1.1.
Another type is translation symmetry, in which an image is copied and translated to make an identical copy. For example, on a graph, the function can be copied and translated up x units of the axises in a proportional matter for each copy to make a chain of translated symmetrical shapes. In music this is called repetition or a sequence of notes, which is when a section of the song is repeated. For example, in Ludwig Van Beethoven’s Moonlight Sonata, First Movement, he introduced the song by having the right hand using this technique for the first two measures as seen in Figure 1.2.
FRACTIONS
All musical pieces use fractions to organize the beats in each song. When looking sheet music, the notes are similar to mathematical symbols. All songs are divided into measures, which means for every specified number of beats, a new measure begins. The number of beats per measure is indicated with a time signature, which is at the beginning of the song and is signified with two numbers on top of each other. The number on the top tells the musician how many beats there are per measure. For example, if the top number were a 3, there would be three beats per measure. The number on the bottom indicates the rhythm and pace, by signifying how many “beats” equals a whole.
Each number corresponds to a number of notes, for example, 1 is a whole note, which is a white note that plays for four beats. 2 is a half note which would be a white note with one tail that plays for two beats. 4 is a quarter note, which would be a filled in black note with one tail which would play for ¼ of a beat. 8 is an eighth note, which would a black note with a curved out tail or two black notes with connecting tails, which would play for an ⅛ of a beat. 16 is a sixteenth note, which would be a black note with two curved out tails or four black notes connected with two black lines, which would play for 1/16 of a beat. All of these notes also have rest or silence notes, which are notes that show a pause in the sound, but last for the same amount of beats as the note they correspond to. For example, a whole rest note looks like an upside down top hat, and lasts for four beats but the musician does not play anything during this count. This applies to all the rest notes for all the other notes. A musical note diagram can be seen in Figure 2.1
An example of a common time signature is 4 over 4, which means the four beats of that measure need to add up to 4 quarter notes. Another example of a common signature is 3 over 4, which means three beats needs to add up the 4 quarter notes. The most common time signature is 4 over 4, because four beats is much easier to fit into a rhythm that requires four beats per measure.
While this may be confusing, different rest symbols or types of notes are used to keep up with these rhythms. For example, for a time signature of 4 over 4 could have two quarter notes, followed by a half note, which would equal 4 quarters, as seen in Figure 2.2.
Therefore, when composers write their music, they need to keep in mind that in each measure, the notes need to add up to the time signatures. For example, in a 3 over 4 time signature, if the composer wrote one half note and 4 eighth notes, this would add up to 4 beats, which would be 4 over 4, instead of 3 over 4. The composer could write one half note and 2 eighth notes instead to fit this time signature.
FREQUENCY AND WAVELENGTH:
Frequency is a vibration that is measured per second, and wavelength is the length between the highest points of a wave, also known as crests. Mathematics are directly related to the frequency in musical notes. This observation began with Ancient Greek philosopher, Pythagoras. He concluded that octaves of notes were related by factors of 2 or ½. Pythagoras came to this conclusion by measuring a musical string. He took a stretched string and folded it in half and held the opposite sides closed so it would form an ellipse-like shape. He plucked the string and recorded the sound of it for his research. He then twisted the shape in half, and it played the same sound, but in a lower tone.
Pythagoras’ observations can be confirmed with the measured frequencies of musical notes in Hertz (Hz). The equation to find a note that is an octave lower, meaning a lower pitch, is 0.5f or f/2, in which f is the frequency of the note given. For a note of an octave higher or higher pitch, the equation would be 2f, with f as the frequency of the note given.
For example, a the frequency of a middle G note is 392 Hz, and a G note one octave lower is half of the middle frequency, so the lower G is 196 Hz. The result is the G note an octave higher, which would be the middle G multiplied by 2, to get 784 Hz.
While these equations work for most frequencies, it is the most basic equation to find different octaves of notes. These equations would only be used for a perfectly tuned instrument, which is nearly impossible. More complex equation is used to find the frequencies and wavelengths of different notes in different octaves. The equation is fn = f0 * (a)n, in which “f0” s the frequency known, “fn” is the frequency to be solved for, “n” is the number of notes the two notes are from each other, and “a” is 21/12 of 1.0595.
For example, if f0 is 392 Hz for a middle G note, and I want to find a middle A note, the equation would be f2=392*(21/12)2, which would equal to 440.04 Hz, which is the most accurate frequency for a middle A when tuning instruments. This equation is more precise for exact frequencies, while the other equations are used to find a more basic relationship between octaves.
A similar is seen with wavelength in centimeters (cm), however, the equations are reverse of the ones for frequency. To find the wavelength of an octave lower, the equation would be 2w, in which w is the wavelength given. The equation to find the wavelength of a note an octave above is 0.5w or w/2, in which w represents the wavelength given.
For example, the wavelength of a middle B note is 69.85 centimeters. The wavelength of a B note that is an octave lower, the middle B’s length multiplied by 2, which would result in a wavelength of 139.7 centimeters. For a B note that is an octave lower, the middle B would be divided by 2, or multiplied by ½, which would result in a wavelength of 34.925 centimeters.
CONCLUSION:
All elements of the world consist of mathematics. The world is surrounded by bits and pieces constructed by trigonometry, algebra, geometry and more. Music especially is heavily influenced by mathematics. The time signatures and measured are divided by fractions using mathematical proportions. Frequency and wavelength have been studied and show a relationship with note octaves. Musicians have also used symmetry to make music that falls, rises, shifts, and reflects proportionally to make music that is more appealing to the ear. Without mathematics, classical music would not be as beautiful and organized as it is.