a common space & database for harmonic overtones
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Well, it's been 8+ years since you posted this. I thought that I'd answer it anyway.
It isn't the logarithmic scale, but the natural scale/harmonic or overtone series. If you measure it, it isn't logarithmic, it's linear. Really. It just seems logarithmic because that's how our senses work as animals. Our bodies are sensitive to things based upon our size and therefore what threatens us most. Mice hear high-pitched sounds; whales hear low-pitched sounds; we're in between. So you'll notice that the decibel scale is written along the same lines.
So your body interprets the natural scale/harmonic/overtone series as being logarithmic when in fact if you measure it, it's linear. If you measure the diatonic scale, which is octave equivalent, you'll see that it's exponential. Since our senses are logarithmic, we perceive it as being linear. However, the spacing of the notes within the octaves is not consistent. It also explains why the cycle-per-second (Hertz) between the notes increases as you get higher.
The two systems combined around c.1260+ (look at the Rutland Psalter p.98r where a king is tuning his harp to a natural trumpet playing the natural scale). From the natural scale, we get the major and minor modes, triadic harmony and V-I cadence. With the diatonic scale, we have a scale that seems linear and where you can have instruments with different ranges playing together in different octaves. The natural scale is constrained to a gamut of notes. So with the acceptance of the natural scale, you have 10-stringed frame harps quickly evolving into 30-stringed Gothic harps.