Expanding tonal space/planar extensions

This is Part II of a small series of articles discussing the model of tonal space.
On this subpage, we explore what happens beyond the represented boundaries of tonal space (from Part I) as we extend the plane in different directions.
** THIS IS A WORK IN PROGRESS 1.Feb 2025 **

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

Fig.1: The Horizon Chart illustrating overtone scales up to Mode 32

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).

Fig.2: Mapping Harry Partch's Genesis scale into tonal space

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 (Fig.2) is a small plus sign (+). The ♦-markers indicate Partch’s multiple-number ratios beyond the 11-limit, x-markers indicate unimplemented intervals.