I apologise for the mistake in my graph. The following is what I actually meant.
In your V-series, version 23, the "octave spiral harmonic energy with critical band; each turn of the spiral is an octave", is the closest to what I imagined.
In this case however, (excuse the pathetic drawing) the translation is into three dimensions:
In the image above, the distance between each consecutive octave doubles (imagine that this is an accurate representation) because of the increasing size of the spiral on the x and z axis. The ascent on the Y axis is constant, indicating our linear perception of intervals. Each frequency position along the spiral would have its corresponding octaves lined up above it along the diagonal axis of the spiral, (as shown with the example of 100Hz). So if you were to follow the sequence of intervals up the chromatic scale to the next octave, the spacings between the intervals increases, however their angular placement remains consistent over all octaves.
Although in the task of imagining the position of the harmonics starting at 100Hz, the placement along the spiral may not easily demonstrate the integer multiple relationship with the fundamental, but it would easily present it as a particular note of the scale. So it would not address the issue you raised:
|The problem with picking a single space (dimension) for frequency is that there is no space in which both equal frequency distances (the spacing between harmonics) and equal pitch distances (the distance spanned by a particular musical interval) are represented by equal visual distances.|
An interesting adaptation might be to bring the position of tones and semitones of the scale into the picture by slightly (but noticeably) deforming the distances to and from the centre of the spiral similar to the different lengths of the black and white keys of the piano.