User:Holger Stoltenberg/planar extensions
Expanding tonal space/planar extensions
In this section, we will explore what happens beyond the represented boundaries of tonal space as we extend the plane in different directions.
Extending tonal space upward (mode axis)
If we go up the vertical axis from Mode 16 of the overtone scale to the 6th octave, we pass Modes 17 through 31 to reach Mode 32.
The Mode 32 Horizon Chart
There is an exponential growth in the number of intervals per row as we step up the logarithmic mode axis in octaves. Each additional octave in mode direction doubles the number of intervals per row.
The actual top row (Mode 32) is equivalent to 32afdo (Fig.1).
The Genesis scale
When we try to visualize Harry Partch's 43 note Genesis scale by mapping it into tonal space, we even have to resort to a Mode 81 overtone scale (Fig.2).
It is worth noting that the Genesis scale covers all intervals of tonal space from Mode 1 to Mode 11 without a gap (with the exception of intervals containing prime factors of 13 or greater, which was a design choice). The marker for these 11-limit intervals is a small plus sign (+). The ♦-markers indicate Partch’s multiple-number ratios beyond the 11-limit, x-markers indicate unimplemented intervals.
Horizontal extension of tonal space (interval axis)
Horizontal tiling
Since overtone scales are defined as octave repeating, identical tiles placed along the horizontal interval axis can illustrate the use of cross-octave intervals or the possibility of rendering intervals in the next higher (blue frame) or lower (green frame) octave (Fig.3).
Fig.3: Three copies of a Mode 16 Horizon Chart along the interval axis
Extending tonal space to the right
As we know, interval spacing actually gets narrower in the next higher octave of an overtone scale. For a more natural appearance of adjacent octaves, we can shift the two corresponding frames (e.g. orange and blue) vertically by one octave. This even leaves room for the more dense intervals of the 6th octave in the upper right corner.
Fig.4: Two octaves view of tonal space
The slanted fine blue and orange lines connect intervals that share a common numerator, since Harry Partch also known as utonalities (for example starting at harmonic h6: [math]( \frac{6}{6}, \frac{6}{5}, \frac{6}{4}, \frac{6}{3}, \frac{6}{2})[/math] ).
To the editor's surprise, Partch's 11-limit tonality diamond can be mapped seamlessly to tonal space right at the boundary between two octaves (Fig.5).
Extending tonal space downwards
Extending tonal space downwards by fractional modes smaller than 1 means to enter a restricted area. We enter the field of tritaves, doubled octaves and pentaves...