![]() ![]() The result is that the diagonals are made up of pitch classes separated by minor thirds (in the southeast direction) and major thirds (northeast). In the tonnetz shown in Figure 2, the rows are simply the entries of the circle of fifths, while the columns are the twelve diatonic pitch classes arranged chromatically (by half steps). Beginning with the musical writings of the 18 th century mathematician Leonhard Euler, and continuing at least through the work of the influential music theorist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) in the 19 th century, the representation of harmonic concepts in a two dimensional array called a Tonnetz (Tonal Network) has guided the understanding of tonal harmony. ![]()
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