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Conversation between Alex and Stephen (prev top next)

2007mar09

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 In music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency). Googling for
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.

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