Applying Principles in Musical Composition
Ever since music has been made, composers have wondered about the principles that created such melodic and flowing pieces. Composers have tried different techniques throughout the history of Western music, which is classified into different musical eras. One major concept in composition was mathematics. Some bright musicians were up to the challenge of creating music that intertwined with all sorts of math including numerical sequences, patterns within phrases, and simple ratios that created great consonance. A handful of these composers were Bach, Mozart, Stravinsky, and Debussy.
Bach
Johann Sebastian Bach was from the Baroque era, which was between the years 1600-1750. The end of the Baroque era was marked by the death of Bach, when other composers at the time were transitioning to the Classical era. Bach has been noted of using the Fibonacci sequence in his organ pieces.
Mozart
Mozart came from the Classical period. Like Haydn, he was one of the most influential composers of the era. Mozart implemented balance and symmetry in his pieces to make them extremely well crafted. He also created lots of patterns. He started with a four bar phrase. This was the foundation. Throughout the piece, he changed the shape of the foundation to create something new yet familiar. This way, the listener can still understand the context and the piece can move forward. Mozart had cleverly disguised these ideas so greatly, that some musicologists deny the possibility of Mozart creating all these links on purpose. Of course, there are too many connections within his pieces that it almost could not have all been coincidental.
Stravinsky
Stravinsky also had a big influence towards his era, the 20th century. He influenced many composers before and after his death, including Copland, a famous American composer. His music also used patterns called motifs. A motif in music is a group of 
notes that reappear throughout a piece to create the themes and melodies. Stravinsky used a technique in which he added or subtracted notes in his motifs without consideration of the changes in the meter. He also used a technique called dodecaphony, which was a twelve-tone technique in which all twelve notes have equal importance while not emphasizing any. Diverse rhythm was another key concept in the compositions Stravinsky created, especially in The Rite of Spring. The offbeat rhythm would make the piece eccentric and flowing.
Debussy
Debussy lived between the late-Romantic into the 20th century. In his music, mathematical structures are apparent, especially usage of the golden ratio and Fibonacci sequences. He may use these numbers to divide sections or show important themes. In one of his images, Reflections in Water, he sequences the keys of the piano with the numbers 34, 21, 13, and 8. The main climax rests at the φ position, or golden ratio.