Alex,

Ah, yes, that's making sense now.

A spiral is a good analogue to pitch because both are periodic, pitch at the octave, spirals at 360 degrees; when you go up an octave or once around a spiral, you end up at a place which is both the same and different as where you were. One question is: what kind of spiral to use?

A lot of people have picked the helix to model pitch space (e.g. Wikipedia says

spiral pitch music

will give you
gobs of references.

If you
consider not just pitch but the position of harmonics, the analogy is a little
trickier. Your solution, to use a logarithmic spiral so that linear frequency is
expressed visually, has some nice properties, but is problematic because of the
range of sizes involved. If you make each octave twice as big as its lower
neighbor, the difference in size between the low octaves and the high ones is
big; for example, here's what the range of about six octaves looks like:

If the center of this spiral
represents the lowest frequency we typically use musically as a pitch (about 60
Hz), then here's what the first twelve harmonics of middle C would look like:

Okay, that's good ... you can see
that the harmonics are equally spaced (as you can in a linear spectrogram). The fundamental
and the 2nd, 4th, 8th harmonics are all lined up (at 3 o'clock) as are the 3rd,
6th and 12th (at about 8 o'clock), which allows you to see that they go
together, but as soon as you add another pitch, for example, G a fifth higher ...

... it starts to get harder to
scan visually. Extending it into three dimensions makes it yet harder, because
points that were in a straight line are now skewed, and distances around the
circle are changed by foreshortening:

For me, there are also connotive problems. *Bigger* suggests lower and bigger: longer strings,
longer resonators, big instruments, big animals with low voices, etc. So having
higher-pitched parts of the sound represented by bigger physical things seems a
little counter-intuitive. There's also the question of the size of intervals;
here's what semitones look like in the logarithmic spirals:

For me, semitones sound roughly
the same size from octave to octave, so the contrast of tightly-spaced semitones
at the bottom and far-spaced ones at the top doesn't match what I hear.

In hearing, equally-spaced
harmonics group together into a single "sound object" and it would be great to
have a visual representation of harmonics that had the same property. The
standard approach to this is called the *harmonic
sieve*, which goes like this: put the spectral energy on a logarithmic scale
so that intervals are the same size at all frequencies, make a template that has
the spacing of harmonics (octave, fifth, fourth, major third, minor third,
etc.), and look for places where the template lines up with the energy. This
works pretty well for finding fundamentals, but there are problems in making a
visualization of it: if you show all the places where the sieve can exist, you
get a lot of visual clutter, but if you only show the places where it matches,
you have to have some criteria for what constitutes a match, and these tend to
be either arbitrary, wrong, or very complex. Another problem is that the
symmetry of the sieve is not visible in a logarithmic scale, so it looks
arbitrary (even though it's not).

There's an idea for a display that's been in the back of my
mind for a while; I spent most of today trying to make a prototype of it, but
wasn't able to get it working well enough to demonstrate a nice result. The
basic idea is this: for each fundamental frequency, there's a constant
inter-harmonic distance; if you wrap the spectrum around a circle with that
inter-harmonic distance as its circumference, the harmonics of that fundamental
frequency would line up. At frequencies that weren't harmonics of the
fundamental frequency, the energy would still be present, just not lined up. My
thought was that the places where the harmonics lined up would have a kind of
visual integrity. For example, here's the data for a sawtooth wave, with one
turn of the spiral equal to the inter-harmonic distance:

or

That was promising, but then the
question is: what happens if the inter-harmonic distance doesn't match a single
turn of the spiral? For example, if it's equal to a half-turn, you get this:

Pretty, but not what I wanted.
Anyway, the idea was that you'd arrange a set of these spirals so that the
"line-up" rows would be prominent, and it would be visually obvious which spiral
corresponded to a frequency with harmonic energy.

S.