And thank god for equal temperament, because the last thing you want is perfect intervals that don't leave any space for proper resonance.
«In general, the interference equation can be used to measure resonant amplitudes for any musical interval under any temperament or octave division. This equation tells us that minimum resonance occurs at the fourth root of an octave (or square root of twelve) while maximum resonance occurs at the cube root of half an octave. Taken together, these results offer clear evidence that harmonic interference balances naturally around 12 as the most rational and harmonic number possible.»
«We find here the most amazing thing. The arithmetic mean converges toward PI, or mathematical constant π ≈ 3.14159, located in the middle of the curve. We further find this point in the distribution curve to be equal to Unity (or 1) when the domain value X = 12. This is significant because twelve is the square root of 144, the value shared by both harmonic and Fibonacci series in a 12-step octave. Squaring each of the table values and dividing by twelve confirms that 12.02383 ≈ 12 is the point of balance between foreground and background.
The significance of twelve as a point of balance in the octave interference pattern is proven further by plugging it into the equation, confirming the curve height equal to Unity at the octave. But even more significant than this is the fact that plugging the square root of twelve into the equation results in the amplitude y = 5.0666. Care to guess what this number represents?
It is none other than the y-axis amplitude for the golden ratio in an octave. Yes, the square root of twelve in the Gaussian interference pattern occurs precisely at Φ, right in the “cracks between the keys” of a major 3rd and minor 3rd in an octave. Just like the dense lattice region between a major 6th and minor 6th, the infinite golden ratio also provides an anti-harmonic proportion in the lower half of an octave. This occurs naturally at the square root of 12 (or fourth root of 144) in a 12-step octave.
No matter how you do the math, both harmonic and Fibonacci series reach a harmonic balance with one another at n=12 and an anti-harmonic dead zone at n=√12. Division of the octave by twelve (not eleven, nineteen or any other number) is revealed here as a completely natural pattern produced by linear harmonics that are curved in pitch space by Fibonacci proportions as they converge to Φ. Could Gioseffo Zarlino’s decision to divide the octave into twelve steps have involved some knowledge of this simple relation between harmonics and the Fibonacci series?»
«As a surprising correspondence between music and math, this little trick reveals the Pythagorean comma accurate to 3 decimal places. More amazing still, if we recalculate using the un-rounded arithmetic mean 12.02383 found earlier in place of 12, we obtain a slightly better estimate for the Pythagorean comma good to 4 decimal places. This bizarre associative property in the interference equation using the anti-harmonic golden ratio location of n=√12 proves the golden ratio is a physical property in the natural harmonic series and not some kind of error or “evil” in nature as portrayed by the Church. Vibration needs room to resonate in space and the Pythagorean comma created by the golden ratio appears to be just the right amount of room needed.»