In this magazine we are introduced to musical score - a graph whose vertical axis represents pitch and whose horizontal axis represents time. We are brought back again to the time of Pythagoras where mathematical principles underlie many musical phenomena. In this article it goes on to say that it is even more surprising "that our understanding of the mathematical structure of the spaces in which musical phenomena operate remains fragmentary." I would never have thought how far back the roots of the connections between music and math were. This article points out that, music theorists classify chords in categories such as major triads, grouping chords together with their translations in some appropriate space. Western musicians are accustomed to a discrete view of pitch space, corresponding to the chromatic scale playable on the piano, but the general problem requires consideration of a larger space in which continuous pitch variation is possible. The various equivalence relations give rise to an assortment of quotient spaces.
In this article we are introduced to Tymoczko's work. He developed a project that characterizes these spaces in great generality and relates the geometry of the spaces to the musical behavior of the chords that inhabit them. Tymoczko's mathematical music theory stated that, "although Terra incognito to practicing musicians and even to many professional music theorists, has in recent years blossomed into a sizable and multifaceted industry." Tymoczko talks about the pitch-class set theory which is the study of a discrete 12-note quotient space, developed as a means of confronting the analytical challenges posed by "post-tonal" music of the 20th century. He included in his study the diatonic set theory which investigates the subtle and beautiful relationship between the 12-note chromatic scale and diatonic scales such as the C major scale, with seven unequally spaced notes per octave.
References:
Hook, Julian. "MATHEMATICS:Exploring Musical Space." Science Magazine Science 7 July 2006:Vol. 313. no. 5783, p7 July 2006: pp. 49 - 50.
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