Music

 Mathematical harmony has, over the years, been used deliberately and unknowingly in musical composition. Mathematician Gottfried Leibnez wrote that "music is the pleasure the human mind experiences from counting without being aware that it is counting."

Italian composer and lutenist, Vincenzo Galelie also wrote a discourse on string theory, which states that "the universe is constructed by tiny vibrating particles", which are assumed to be even smaller than subatomic particles, and these interact with each other by twisting, folding and vibrating. They create phenomena like electromagnetism and gravity. This string theory may also be linked to sound and hence music.  

Composers often rely on the golden ratio and Fibonacci sequence to enhance their compositions, and construct balanced, melodic, dynamic movements and rhythms.

Claude Debussy's music, for example, use the golden ratio, prominently the "dramatic" climax of "Cloches a travers les feuilles", as the ratio of number of bars to climax bar is 1.618.

Composers tend to depend on symmetry for construction and development of progressions in themes and variation. They utilize mathematical structures, including prime numbers to evoke a feeling of unease. Moreover, they introduce hints of dissonance through nontraditional rhythms and meter. Its infamous example is found in Messiaen's "Quartet for the End of Time".

According to Du Sautoy, in two notes an octave apart, the frequencies of the two notes are in an exact 1:2 ratio, so we tend to assume we're hearing the same name. 

Researcher Natalie Hojier labelled the Golden Ratio and Fibonacci Sequence as the most effective compositional tools. 

Beethoven, a renowned composer and pianist, has work that reflects the Fibonacci Sequence and Golden Ratio. Piano Sonata No. 14 in C# Minor, op. 27, no. 2 represents "a model of tonal clarity", and in its earliest progression, there are regularly expanding tonal blocks. "Each new point of arrival develops from the preceding tonal area and simultaneously prepares for the next". Since each new tempo is temporally connected both to preceding events and those yet to come, the sensation of gathering strength prevails. The intervals are spaced further and further apart, and their durational ratios remain numerically constant.

The first movement of Beethoven's Fifth Symphony also has occurrences of Golden Ratio by some, such as Haylock.

Mozart, arguably the most renowned composer is said to have had a fascination and preference for mathematical and symbolic number compositions in music. Some believe that in his Piano Sonata #1 in C major K279, Mozart seems to have used the Sequence.

Chopin's Prelude in A Minor is a harmonic example of the Golden Ratio. Rogers has called it strategically placed and deliberately paced", and said the timing works as a series of signals. These elements govern the harmonious ambiguities and contribute to predicting the eventual establishment of tonal stability, which is to arrive at the end.

Even in relatively modern music, artists like Vijay Iyer, Michael Jackson (Billie Jean), Tool (Lateralus) have shown understanding and use of the Fibonacci sequence and Golden Ratio.

The arrangement of the first few keys of the piano represent Fibonacci numbers. The 13 keys in groups of threes and twos complete a full octave.