Exploring Music's Complexities

Music and Mathematics

Study of Sound

need for clefs

Music and mathematics, can they be classified into the same category?  To make music, you must know how to break “sound” into elements of pitch, rhythm and tempo (1). Science teaches us that sound is vibration, and the frequency of vibration is what makes different sounds. Music then is the study of the sound created by those vibrations, and puts them into patterns that elicit emotion. Therefore we can say music is based on mathematics. We know that sound is vibration, and the frequency of vibration is what makes different sounds. Music then is the study of the sound created by those vibrations, and puts them into patterns that elicit emotion.  Music also demonstrates an assortment of number properties (2).

History

Ancient cultures exhibited an interest in investigating the expression of musical scales as numerical ratios (1) and that “all nature consists of harmony arising out of numbers”. In Pythagorean tuning, the unison, fourth, and fifth intervals have ratios 1/1, 4/3, and 3/2 respectively, corresponding to the generators 1, 5, and 7 in our set.  Musical use of words such as meter and measure suggest the importance of music in ancient times.  Today composers use set theory, abstract algebra and number theory to create new ways of creating and hearing music.

Reading Music Notes and Fractions

Music teachers also use numbers and math to describe and teach music (4).  For example, music pieces consist of measures or bars representing time.  These are mathematical divisions of time.  Beats provide a framework for combining notes of different duration.  Beats give music its regular rhythmic patterns. The groupings of strong and weak beats are called meter (2).  A time signature provides information about rhythm.

Expressed mathematically 4 x 1/4=1

 It is usually written in two numbers.  Each note has a different shape to tell its time.  There are whole notes, half notes, quarter notes, 1/8th notes and 16th notes.  The number on the bottom indicates which note receives a single beat.  The one on the top tell the performer how many of these notes receive a single beat.  The shapes of the notes provide information about beat length of time.  A whole note would use up the entire measure, a quarter note would use up one fourth of the measure.  A note with a dot after it lengthens the note by half.  The relationship and values of fractions spell out how long to hold a note. The prime articulates of form are rhythm and texture.  Rhythm, meter and the tempo of a composition’s pulse are related to the measurement of time and frequency. These provide an analogy in geometry (3).

Other Connections to Mathematics

Different tuning systems create variations in frequency ratios between intervals. It is said that Pythagoras and/or his followers were the first to draw attention to the fact that musical intervals could be expressed as numerical ratios and that more consonant intervals had ratios of small integers. In Pythagorean tuning, the unison, fourth, and fifth intervals have ratios 1/1, 4/3, and 3/2 respectively, corresponding to the generators 1, 5, and 7 in our set.

Set theory is used to organize musical objects and to describe their relationships.  It can be used to discover deep structures through the use of transposition and inversion.

Abstract algebra has been used to describe just intonation using a free aeolian group.

Theory of regular temperaments emphasizes transformations between musical objects rather than the musical objects themselves.

The chromatic scale can be thought of as a torsor for the group Z/12Z.

“Real and complex analysis have also been made use of the Riemann zeta functions to study of equal division of the octave.” (1).

 

References

  1. Music and mathematics.   https://en.wikipedia.org/wiki/Music_and_mathemarics
  2. Reginald Smith Brindle. The New Music. Oxford University Press.    Pp 42-3
  3. Glydon, Natasha. Music, Math, and Patterns. http://mathcentral.uregina.ca/beyond/araticals/Music/musico.html

 

 

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