Scholars have attempted to understand the relationship between mathematics and music for many years. People in the field of music can readily understand the definite relationship between mathematics and music. In fact, some studies suggest that mathematics is in every field of study. Harmonic rations dominated music during the Pythagorean times. Studies have identified the relationship between tonal quality and mathematics.
Tonal quality is responsible for creating the audible impression of music. These are “sound waves vibrating at distinct frequencies” (McPhee 1). Musicians believe that tonal quality creates enjoyable music to mathematicians. In order to understand the relationship between tonal quality and mathematics, we can focus on the concept of dissonance and harmony in music. Harmony and dissonance require specific notes to blend in music.
Conversely, not all notes can blend in music. Ancient Greeks noted that mathematical integers relate to musical notes (Reid 1). In this context, Greeks observed that vibrating objects produced overtones. These overtones originate from a single object. As a result, the tones created harmonic series like 1/2, 1/3, 1/4, 1/5.
Tuning practices control pitches of music in order to create a common tone between intervals. Therefore, tuning must have fixed reference. Music systems may have 440 Hz as equivalent to A. Tuning must control out of tune tones using specific reference pitch.
Mathematical analysis indicates that integer multiples also generate their base frequencies, but with weak intensity. For instance, when a string with specific length generates “a frequency of 220 Hz, there are corresponding frequencies of 440 Hz, 660 Hz, and 880 Hz and others” (Hearle 1).
Octave also highlights the relationship between mathematics and music. This relationship is evident when creating high or low musical notes in vibration. Octave also creates both interval and chord depending on their numbers. Both musicians and mathematicians use the analogy of shortening or lengthening a string to explain the concept of octave. The idea shows that any change in the length of the string results into a new tone.
Music and mathematics also relate in Pythagorean tuning. This explains the scientific relationship between human ear and sound waves. This idea resulted in a mathematical-based system of tuning. This is the rule of perfect fifths in mathematics and music.
It shows that intervals between “the first and the fifth note of the proposed scale would be tuned to a ratio of 2:3” (McPhee 1). In the same case, it uses the ratio of 1:2 to indicate the relationship between the first and eight notes. These ratios indicate the relationship between wavelengths of different notes.
Tempering has mathematical explanations for creating harmonic and melodic pair of notes. These creations result into ratios and compounds. This is where musicians apply 12-tone scale through the octave and pure fifth with reference to rations of 2:1 and 3:2 (Hearle 1).
Musicians use the Pythagorean scale to build music systems using integer notes. Developers of musical keyboards apply the idea of integer notes to develop the system. They use a system of tuning known as “tempering when creating even-tempered or well-tempered systems” (Reid 1). In this process, mathematical rules apply. In an even-tempered system, the system creates all equal or slightly different notes of the same tune and then distributes the error on an equal basis among all scale notes to create compound musical chords.
There is also equal temperament mathematically defined as 1:12. This is the standard for European applicable in Western music. This style is simple and versatile making it popular among users. All the ratios between 12 correspond to two as applicable in modern pianos.
The development of meantone also has mathematical concepts. Meantone uses Just Intonation of the scale, but with pure thirds. The model has C, G, D, A followed by a pure major. Just Intonation has regular tones with harmonic numbers, which rely on a single primary frequency. This is the opposite of equal temperament.
Writers argue that people tend to prefer the Pythagorean intervals. Therefore, we have to learn how to listen to other sour tuning from well-tempered music. Further, some studies have shown that musicians tend to apply sweet notes of Pythagorean.
Mathematical aspects help in understanding the concept of wavelengths and musical notes. In this case, mathematical concepts help in discovery of various ratios, which can result into harmony in music. The musician must combine musical notes with complementary wavelengths to create both high and low waves, which must relate with each other for harmonic sound. This aims to avoid dissonance in music notes.
Mathematics and music also relate on the golden ration and Fibonacci sequence. The above principles mainly focus on the aesthetic nature of music. Fibonacci sequence has systematic sequence of numbers in which every two preceding numbers add up to the next number (1, 1, 2, 3, 5, and 8 etc).
These numbers help in comprehending musical patterns. On the other hand, the golden ratio is mysterious, scientific ratio responsible for aesthetic in music. These are mathematical concepts, which help people appreciate aesthetic nature of music. For instance, some groups of notes in a given scale show Fibonacci sequence as well as the golden ratio.
Hearle, Noah. The Relationship between Mathematics and Musical Temperament. (n.d). Web.
McPhee, Isaac. Music and Mathematics. 11 March 2008. Web.
Reid, Harvey. On Mathematics and Music. (n.d). Web. <http://www.woodpecker.com/writing/essays/math+music.html>.
Mathematics and Music