In this essay we will dedicate our attention to the foundation behind the art of music and determine the mathematical principles implicated in its’ creative practice. Mathematics and art have always been considered as two separate cultures. However, possible connections have been explored and the existence of a link has been verified. Proof of an abundance of mathematical principles implicated in music is visible through the use of pitches, scales, geometry, the Golden Ratio and the Fibonacci Sequence. This essay will play as a means of demonstrating the relationship that mathematics and art share in order to uncover and conceive distinguished discoveries and fabrications.
Art is a medium that is used by artists to express emotions of the heart that are given birth to by human desire. These desires are channelled in ways that are pleasing to people and music has integrated itself into this art class by adopting artistic features. Through proficiently expressing the human desire, artists appeal to the heart of the audience to create something beautiful. Mathematics is a process that aims to achieve the maximum potential of the human desire, fulfilling it entirely by means of fashioning fabrications for the purpose of humans. Mathematicians first locate evidences and truths to reach a conclusion that can be represented in the form of something beneficial in order to create something beautiful, a perfect world.
In order to comprehend the foundation of mathematics and art we have to explore the content of music and the depth of mathematics involved by investigating music in general. Pitch is defined as how sharp a produced note is through vibrations per second. In a typical 12-note scale octave if we look at the root note ‘C’ we can find that it vibrates at a rate of 261.63 vibrations per second or hertz. However, mathematically gauging this, by doubling or halving the rate of vibration an octave transition occurs. Where 523.26 Hertz is an octave higher and 130.815 Hertz is an octave lower. Therefore the higher a note is the greater the frequency differences while, the lower the note the smaller the frequency differences.
During the time of the Ancient Greeks there lived a renowned geometer, known as Pythagoras of Samos who played a leading role in the development of mathematics. A geometer is a mathematician whose area of study is geometry while Pythagoras in particular demonstrated his attraction to music being a musician himself playing the lyre. Through observing music he established a musical system contributing towards the mathematical theory of music. All notes became part of a series of fractions where by looking at root note ‘C’ again, we find that notes an octave higher or lower naturally demonstrated to be double or half the frequency. However, if we look at the following whole tone that is note ‘D’ we can see that it is 9/8 of note ‘C’. So by multiplying the frequency of ‘C’, 261.63 by 9/8 or 1.125 a result of 294.33 Hertz is produced.
By further investigating the concept of scales in music through the application of ratios, evidence dictates that a preset ratio could determine the frequency of every subsequent note throughout an entire piano of keys. A scale is typically comprised of 12 keys, 5 black and 7 white otherwise known as 12 semitones. Naturally to reach the 13th semitone, the Octave note, the frequency of the root note ‘C’ has to be doubled. Therefore, the only number that the Octave note can retain is 2, double the root note. Thus a pattern is adopted, in which the 12th root of 2 is 1.05946309. So in order to reach the Octave note, 1.05946309 is multiplied by itself 12 times resulting in 2. Therefore, to obtain the subsequent frequency value of every note, all notes are defined as multiples of the same basic interval that is 1.05946309. In the case of note ‘C’, 261.63 Hertz is multiplied by 1.05946309 to generate 277.18 Hertz, ‘C Sharp’, the following black key semitone note. By persisting with this pattern the frequency of the Octave note ‘C’, 523.25 Hertz can be attained. This became to be known as Equal Temperament, a musical system of tuning known as a music temperament. A temperament is achieved through compromising pure intervals of just intonation to satisfy the prerequisites of the system. Just intonation is any musical tuning where the frequencies of notes are associated through ratios of small whole numbers. Intervals tuned in this fashion are known as just intervals. This combines together to craft beautiful compositions. Johann Sebastian Bach once called this scale the Well-Tempered Scale that was employed within the Well-Tempered Clavier, a collection of solo keyboard music composed by Johann Sebastian Bach himself.
By approaching music at a geometrical level however, affine transformation, the act of preserving co-linearity even after transformation exists in Fugues. A Fugue is a composition making use of counterpoint, the relationship between two or more voices. It builds on the ‘subject’, the theme of the piece that is primarily introduced and persists with numerous appearances throughout the composition. Music is written onto paper following a simple set of rules. Notes are read from left to right following time-translations and pitch is governed from high to low, or at intervals following frequency-translations. The subject may be scaled appropriately through inversions (flipped upside-down) or reversed to demonstrate retrograde motion. Retrograde motion depicts opposing bodies of movement much like the orbit of one body about another. Note shape determines the note length where open notes are longest and filled notes with the most bars attached are shortest. By understanding a Fugue we can comprehend the intentions behind Johann Sebastian Bach’s first book, titled “Twenty-four Preludes and Fugues” containing Fugues of all 24 major and minor keys. 4 for each note of the scale composed "for the profit and use of musical youth desirous of learning, and especially for the pastime of those already skilled in this study," (Johann Sebastian Bach, 1722).
Through understanding the basics of music we can understand mathematics’ true extent of involvement. The Golden Ratio has made numerous appearances to dictate perfect scenarios throughout history and many art pieces. It is an infinite equation often referred to as the Greek Letter Phi. The exact value is found by dividing the sum of the square root of 5 and 1 by 2 resulting in 1.61803399.
The Fibonacci Sequence maintains an infinite pattern where the following number assumes the role of the sum of the two preceding numbers. Therefore, the initial numbers are 0, 1, 1, 2, 3, 5, 8, 13 while, as progression occurs it gradually replicates the Golden Ratio itself. This sequence was developed by Leonardo of Pisa but more commonly referred to as Fibonacci.
The Fibonacci Sequence emerges as a basis for the construction of the music system.
Using the middle ‘C’ scale including the Octave note there are ‘13’ keys. In the octave there are ‘8’ white keys, or tones and ‘5’ black keys separated into groupings of ‘3’ and ‘2’. The ‘5th’ and ‘3rd’ notes are the root of all chords. Typically in a scale the ‘5th’ note becomes dominant which is ‘G’ in this case. The ‘8th’ note and semitone of the ‘13’ keys, utilizing fractions shows evidence which exhibits a case of the golden ratio as 8/13 which is 0.61538461. Although not precise it roughly demonstrates the ratio. Both the Fibonacci Sequence and the Golden Ratios characteristically illustrate inclusion within timing in musical compositions. This would be in the form of a climax during the 61.8% mark of the song. Therefore in a 100 bar song, the climax would transpire on the 62nd bar. It is through following this infinite pattern that music can become mathematically and artistically beautiful.
In conclusion, through recognizing the depth of mathematics in art, it becomes apparent that music can adopt mathematical properties. Music has always been a creative practice developed and perfected for the purpose of expressing desires to touch the hearts of audiences. While, in the shadows mathematics has always been incorporated into the very basis of music itself. By implicating mathematics into the equation of music, we find that it adopts a more pleasant atmosphere. Therefore, by adopting mathematical properties in a composition noticeable differences manifest from the proper application of varying pitches and intervals. However, the Fibonacci Sequence and the Golden Ratio prove to be dominant features and leading theories that support the actual construction of a composition with mathematics. With the use of water xylophones as a medium a clear demonstration of rates of vibration or frequencies are better heard and maintained. By incorporating fractions of water volume to determine a played note, a single scale can be replicated. Through the unison of mathematical features, music can be perceived as something truly beautiful.
