Expanding tonal space/projections
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](\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.
Sequences of intervals with common numerator
In Part II: Planar extensions, we mentioned in the explanation of Fig.4 that there the
- “…slanted fine blue lines connect intervals that share a common numerator, since Harry Partch also known as utonalities”.
Here, in a projection with slanted mode-lines, interval sequences with a common numerator appear horizontally. Fig.1 shows an example at mode [math]n_0 [/math]= 6, [math](\frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3})[/math], printed in blue.
Polar projection using slanted mode-lines
In polar projection the slanted mode-lines form a separate spiral for each odd mode. Odd modes can never be an octave of a lower mode. Fig.3 shows an artistic interpretation of the polar projection of tonal space with slanted mode-lines.
- The center is the location of the fundamental, where mode [math]n_0[/math]=1 and [math]m[/math]=0. This corresponds to the origin of the Cartesian coordinate system as in Fig.1. The mode axis runs from the center upward. A clockwise angle of 2π (in radians) represents one octave up. Each yellow ball represents a pitch.
Logarithmic mode-lines
For the sake of completeness, another projection is shown here. This view may be of less practical importance to the musician, but from an aesthetic point of view it has a particularly well-balanced shape, especially in polar projection.
We can find a continuous function that binds the right boundary of each mode [math]n[/math] of the overtone scale to the left side of the adjacent mode [math]n[/math]+1-overtone scale. Measured vertically we recognize a set of lines of constant frequency difference (Fig.4).
Projecting our familiar pitch markers vertically onto the graph, this logarithmic function continuously connects all markers.
Polar projection with logarithmic spiral
When drawn in polar coordinates, the projection in Fig.4 shows a beautiful logarithmic spiral (Fig.5):
