This is the first of a series of threads devoted to various approaches to music theory and harmony.
Tonality
Rameau probably wrote the first book completely devoted to tonal harmony, in 1722. Since then not much has changed. There are some new innovations, though, which have appeared recently.
Leonhard Euler (1707-1783) proposed his "Euler plane" about the same time as Rameau was writing his treatise. He made a grid, and placed fifths on the horizontal axis, and major thirds on the vertical axis.
Guerino Mazzola used the Euler plane in 2002 to propose his "altered scales" of C major. He keeps C and G at the center, and adds any other five adjacent notes to create a 7-note scale. As you can see, the "C-G" at the start is what assures stability of the scale. A total of 32 scales can be generated this way.
It's also possible, on the Euler graph, to circle the notes of a Cmajor and D major scale. This way, one can visualize easily which notes are common to both, and which differ, in order to make modulations.
Mazzola also put the Euler grid on a torus (a doughnut shape). This replaces fifths with minor seconds, but now the intervals circle around the doughnut. Other intervals also circle around if followed at different angles.
There are other systems of graphing. Dmitri Tymoczko does this in his Geometry of Music (2011), and Franck Jedrzejewski (2006) also, by constructing a hexagonal grid (like interconnected stop signs), which graphs the 24 major and minor triads. This reveals connections of relative minors, parallel minors, and other "function" areas.
IRCAM has done similar research, and some have made grids using 24 notes per octave.
Tonality
Rameau probably wrote the first book completely devoted to tonal harmony, in 1722. Since then not much has changed. There are some new innovations, though, which have appeared recently.
Leonhard Euler (1707-1783) proposed his "Euler plane" about the same time as Rameau was writing his treatise. He made a grid, and placed fifths on the horizontal axis, and major thirds on the vertical axis.
Guerino Mazzola used the Euler plane in 2002 to propose his "altered scales" of C major. He keeps C and G at the center, and adds any other five adjacent notes to create a 7-note scale. As you can see, the "C-G" at the start is what assures stability of the scale. A total of 32 scales can be generated this way.
It's also possible, on the Euler graph, to circle the notes of a Cmajor and D major scale. This way, one can visualize easily which notes are common to both, and which differ, in order to make modulations.
Mazzola also put the Euler grid on a torus (a doughnut shape). This replaces fifths with minor seconds, but now the intervals circle around the doughnut. Other intervals also circle around if followed at different angles.
There are other systems of graphing. Dmitri Tymoczko does this in his Geometry of Music (2011), and Franck Jedrzejewski (2006) also, by constructing a hexagonal grid (like interconnected stop signs), which graphs the 24 major and minor triads. This reveals connections of relative minors, parallel minors, and other "function" areas.
IRCAM has done similar research, and some have made grids using 24 notes per octave.