Mathematics and its Relation with Music and its Harmonics

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Ever since c. 17th century, musical compositions have manipulated the standard aspects of music, which include rhythm and melody. More importantly, many musical compositions have incorporated complex math within, for examples again, melodies and rhythms that create a uniqueness that has yet to be perfectly matched by other composers. One other key aspect of music in general, harmony, is where a fairly complex mathematical formula is involved: the harmonic mean. Because harmony is a major component of music, it is no surprise that this harmonic mean can be applied to nearly all types of music. Basically, a harmonic mean-as it applies to music-~is any possible division between an original note and the octave of that note that produces a different note. With that, there is a sequence in the divisions between a note and its octave that is not very consistent in distance from the original note. One instrument in particular that can demonstrate this type of mean is the vioiin because of the harmonic tones produced whenever a finger is pressed on a string. At certain points on a string. different notes are produced by the harmonics of the upper and iower naives of the string, which would be examples of harmonic means.
The diagram above is of a D-string on a violin from the beginning of the neck where the string first crosses the neck to where the string touches the bridge. The halfway notation marks the most prominent harmonic mean on a violin: the octave. Its harmonic mean is 1/2
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