The interesting connection between math and music
This essay examines the relationship between mathematics and music from three different points of view. The first describes some ideas about. 3. Music and Mathematics: An Introduction to their Relationship. 6. Historical Connections Between Mathematics and Music. 9. Music Theorists and. From the rich complexity of the Bach fugues to the catchy songs of the Beatles, music and mathematics overlap in all kinds of interesting ways.
Wave mechanics The brain turns out to be adept at decomposing sinusoids into their component frequencies. If two pitches share a lot of overtones, we tend to hear them as consonant, at least here in the western world.
The relationship between absolute pitches and pitch classes is an excellent doorway into logarithms generally.
You also need logarithms to understand decibels and loudness perception. Symmetry Music is really just a way of applying symmetry to events in time. See this delightful paper by Vi Hart about symmetry and transformations in the musical plane.
Mathematics & Music
Combinatorics and graph theory Generating diatonic chords from a scale is an exercise in combinatorics. Seventeenth-century European bellringing introduced one of the earliest nontrivial results in graph theory, change or method ringing. Discrete mathematics The pitch continuum is, well, continuous, but tuning systems and scales are discrete.
The voice, fretless stringed instruments and trombones produce continuous pitches. Keyboards, fretted string instruments and saxophones produce discrete pitches. While it is possible to construct equal temperament scale with any number of notes for example, the tone Arab tone systemthe most common number is 12, which makes up the equal-temperament chromatic scale.
In western music, a division into twelve intervals is commonly assumed unless it is specified otherwise.
AMS :: Mathematics and Music
For the chromatic scale, the octave is divided into twelve equal parts, each semitone half-step is an interval of the twelfth root of two so that twelve of these equal half steps add up to exactly an octave. With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings.
In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as musical keyboards. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world. Equally tempered scales have been used and instruments built using various other numbers of equal intervals.
The 19 equal temperamentfirst proposed and used by Guillaume Costeley in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal semitone equal temperament at the cost of a flatter fifth. The overall effect is one of greater consonance. Twenty-four equal temperamentwith twenty-four equally spaced tones, is widespread in the pedagogy and notation of Arabic music.
However, in theory and practice, the intonation of Arabic music conforms to rational ratiosas opposed to the irrational ratios of equally tempered systems. These neutral seconds, however, vary slightly in their ratios dependent on maqamas well as geography.
Indeed, Arabic music historian Habib Hassan Touma has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music.
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To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture. Set theory music Musical set theory uses the language of mathematical set theory in an elementary way to organize musical objects and describe their relationships.
To analyze the structure of a piece of typically atonal music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as transposition and inversionone can discover deep structures in the music.