The fret spacing on a guitar can give us a a very important insight into how human volition (i.e., choice or free will) works biologically, at the deepest level of our brain neurons. The insight is that the biochemistry of our nerve cells has a natural logarithmic, rather than linear, mathematics built into it, which is crucial to how we hear, see, and measure probabilities in the world around us. It is why we survive as a species from “time t” to “time t+1”. In an earlier post, I noted how we think our number systems are nicely linear and evenly-spaced, but in many cases in nature, the distance between, say, a “one” and a “two” of a thing is greater than between an “eight” and a “nine” of a thing. You can see this in the gradually-decreasing spacing of the frets as they go up the keyboard.
In the case of our ears, we “think” that we hear linearly, say, in the twelve tones of the western chromatic scale, or the eight-note tone-semitone “do-re-mi-fa” scale (and noting that different cultures “hear music” differently), but these notes are not evenly spaced in frequency. The low “A” note on a guitar (the open fifth string) is typically tuned at 220 cycles per second, but same note one octave higher is 440 cycles per second. The next octave “A” (the fifth fret on the first string) is 880 cycles, and you can even reach to the next highest “A” at 1760 cycles on many guitars.
Each octave doubles the frequency, at 2 times, 4 times, 8 times, and 16 times the original frequency, or 2 to the first power, the second power, the third power, and the fourth power, respectively. These latter values (one through four) are called the “logarithms” of a base 2 numbering system. 
Too much math! What this means in real life is that that our inner ear’s specialized cell structures are responding to the logarithm of the external sound, and our brain neurons then clean that up into the perception of harmonious and disharmonious “music to our ears.” 
And the same goes for the loudness of the sound. The decibel (dB) scale used by sound engineers, and conceptually represented by the numbers one to ten on a traditional amplifier volume knob (or one to eleven if you are Spinal Tap – see the video below) is a logarithmic scale. The power difference between 10 dB and 20 dB is not double, rather it is ten times, but the human ear does not perceive that increase of power. Twenty decibels does not “sound” ten times louder than ten decibels. It “sounds” about twice as loud in our brain’s interpretation.
Two laws of physics, collectively called the Weber–Fechner law, quantify this phenomenon of the difference between the actual change in a physical stimulus versus the perceived change. Our cell biology responds to proportion, not absolute quantity, and when we translate that into the human expression of mathematics via logarithms. Indeed, one way to think of a “natural logarithm” is that it is quantifying the mathematical relationship between a natural rate of increase or growth and time itself.
And it is not just about our sense of hearing. Our senses of taste, touch and smell also follow this law. As does our vision, which is the subject of Part Two of this post.
Part Two of this series is now posted.
- Read more about “Ant choices and t+1” here.
- This guitar is left-handed, designed for those of us with some genetic/epigenitic-based mirror-imaged brain functions.
- The distances of the notes between the octaves are then theoretically 1/12th fractions of the logarithm value. Fret spacing is set to shorten the distance between the bridge and the fret accordingly, but it is also adjusted for physical factors like string type and height, and the “equal temperament” of our “western” musical scale, which deserves its own topic.
- The next level down, the actual biochemistry of that “logarithmic counting process,” is outside my area of expertise and I will mercifully avoid going down that road. The science of music and the brain was the subject of one of the last books by the great science writer Oliver Sacks: Sacks, Oliver. Musicophilia: Tales of Music and the Brain. Picador, 2012.