Expanding tonal space/projections
** THIS PAGE IS UNDER CONSTRUCTION **
This is Part III of a small series of articles discussing the model of tonal space.
On this subpage, we will explore additional views of tonal space (see Part I) as we create alternative projections for graphical representation.
Projections
Just as a natural environment can be visualized on a variety of maps with different geometries, the abstract model of tonal space can also be presented in different projections. Different projections of the model on paper or on screen are different views of the same abstract model.
Standard projection
Most illustrations of tonal space that we have seen in Part I (i.e., Fig.3) and Part II correspond to a particular projection of an abstract plane of tonal space.
With respect to frequency, both axes of this projection are logarithmically scaled. Although the horizontal interval axis has a visible linear scale, it indicates 1200 logarithmic cents per doubling of frequency (octave).
In the standard projection intervals with a common denominator are aligned horizontally. This makes it easy to find otonal chords whose intervals share a common denominator (i.e., [math]\displaystyle{ (\frac{4}{4}, \frac{5}{4}, \frac{6}{4}, \frac{7}{4}) }[/math]).
The vertical axis indicates the mode of an overtone scale. The mode is equal to the common denominator of the fractions describing the intervals in that row. Since the relationships between modes can themselves be interpreted as musical intervals, the mode axis has a logarithmic scale, which makes octaves (between modes) appear equally spaced.
Slanted mode-lines
A disadvantage of the standard projection is that the chain of intervals is discontinuous when an octave boundary has to be crossed. We have to look one octave higher on the mode axis to find the next intervals behind the octave boundary. This disadvantage can be overcome by a projection that implements slanted mode-lines (Fig.1).
All of the displayed intervals remain unchanged. However, they are projected vertically (in mode direction) onto slanted lines connecting the corresponding octaves.
Cylindrical projection
If we bend the projection plane around the vertical axis and glue the lower end of the octave to its upper end, we can observe the match of the slanted mode-lines at the octave boundary (Fig.2).
On the surface of this cylinder, a multi-start thread is created. All modes of the overtone scale that can be strung together in octaves form a common continuous thread on the surface. Modes 1, 3, 5, 7, 9... and all other odd-numbered modes each have their own thread.