Musical pitches increase in frequency exponentially. All octaves are exactly half the distance on the fretboard of the next lowest octave. Naturally, the higher up the neck we go, the shorter the fret spacing.
For a more detailed answer continue reading.
Understanding The Distance Between Frets On A Guitar
For the curious among you, you may have just asked “but why is each octave half the distance?” To explain that we need to go a little more in-depth into cycles, frequency, and pitch.
The fundamentals of sound
If you are interested in learning more, I’ve already covered the fundamentals of sound in detail here, so we’ll just stick to the basics.
As human beings, our ears detect vibrations that have been produced from around us, whether it be vibrations generated from the air conditioner, the wife asking you to do the dishes for the third time that day, or vibrations from a musical instrument.
Those vibrations stimulate our eardrums, which in turn transfer the vibrations to 3 tiny bones in our ear, which then send those vibrations to our inner ear where nerve endings are stimulated. This generates electrical signals that are then sent and interpreted by our brains.
Great, but then what is the difference between say the sound of a car engine and a clean middle C played on the piano? How can our brains differentiate the two?
This brings us to the next topic, pitch.
So we’ve established that we hear by interpreting vibrations. We measure these vibrations using a unit called hertz (commonly written as Hz).
It’s a measurement of how many times something vibrates in 1 second. So for example, the middle C on the piano is 262hz, which means when the string of the piano is struck it vibrates 262 times per second.
If this was to vibrate faster, let’s say at 293 times per second, it would produce a higher note, in this case, D3. Likewise, if it was to vibrate slower, let’s say at 246hz it would produce a lower note, B3.
So, fast vibrations produce high notes, slow vibrations produce low notes, and any difference in pitch is due to the frequency, or the number of completed vibrations that occur per second. Easy!
A note on tone, timbre, and the fundamental
This very scientific method of saying ‘C4 is 262hz’ only really applies when we’re talking about a pure sine wave.
When we play something like a distorted electric guitar or sing a note, we’re throwing out tons of different frequencies.
It’s perhaps easier to visualize this using a spectrum analyzer. Here’s a picture of a guitar after playing a C3 note, with pitch laid across the bottom extending from 0hz to 20,000hz, and the volume (or amplitude) of each frequency it’s outputting along the Y-axis.
As you might have expected, there are tons of frequencies there.
This is the culmination of everything that went into producing that note, from the guitar pick material, how hard you plucked, the string tension, and the fret position on the string you plucked, you get the idea.
So for all the frequencies (we can also call these harmonics) this guitar output, you’ll see that the loudest frequency that was output was 130hz, which happens to be our C3 note.
The loudest frequency output is called ‘the fundamental’ and is what makes us hear this as a C3 note, even though there are many other frequencies present.
All the other quieter harmonics are what give us our ‘tone’. Each instrument outputs its own kind of harmonics which is why a C3 on a piano doesn’t sound like a C3 on a guitar, even though that 130hz will be the loudest frequency on both.
Cycles and exponential growth
Now we’ve established what the fundamental is when a note is produced, from here on out we are going to only be referring to that fundamental.
All the harmonics are simply creating the character and timbre of the instrument, but they are not loud enough to make us ‘hear’ their notes.
Ok so we’ve established that the fundamental is the frequency we hear the loudest, and we measure it using hertz, or cycles. Which simply refers to how many times that frequency cycles (or vibrates) in 1 second.
So the more cycles per second, the higher the frequency and therefore the higher the pitch.
Let’s take our C note across a range of octaves and see how many cycles or times they vibrate per second.
Let’s stop there as we’ll be outside of the human hearing range for C11.
What you’ll notice about all these C notes is each octave vibrates at double the speed of the one before it.
To produce a C5 note our fundamental needs to vibrate 523 times a second while a C6 note needs to vibrate 1046 times per second.
Transferring this knowledge over to the guitar
Let’s now take our guitar in standard tuning, when we play our lowest string without holding any frets down we are playing an E2 note, in which the fundamental (loudest frequency) is 82hz.
Or another way to put it is that the string is vibrating 82 times per second, or that it cycles 82 times per second.
To play 1 octave higher from that we need to fret the 12th fret note, which is exactly at the middle point between the nut and bridge of the guitar, which increases in frequency due to the differences between the two fret locations in string tension.
Which makes sense, 82hz is E2, so doubling that, or holding the string down on the 12th fret, effectively halving the length of the string will make it vibrate twice as fast at 164hz, producing an E3 note.
On a 24’’ scale length guitar that E3 is at the 12’’ mark, the halfway point.
Now where we start to experience that exponential increase is when we want to produce an E4 note. To play E4 we would have to play the 24th fret. (The only acoustic I’m aware of that has 24 frets is the Guild F45 but I digress).
In any case, wouldn’t the 24th fret be another 12 inches up the neck?
No, it’s only 6’’ higher. Because those vibrations are increasing exponentially, so by just moving up 6’’ the frequency is doubled, whereas previously we needed to move up 12’’.
Now let’s pretend for a second we had a 60 fret guitar where the upper frets went all the way to the bridge. We would only need to move a further 3’’ above the 24th fret to double the frequencies again and reach the next octave to hit an E5.
Then to get to E6 you’d only need to climb a further 1.5’’ up the neck.
This is why the higher we climb in pitch, the space between frets is reduced, because moving up a small distance increases the frequency + pitch more and more, requiring less and less distance to be covered before we’ve reached the desired vibration or cycle speed to produce the next note.
The Rule of 18
So we’ve talked a bit about octaves, and how if you divide a string in half you achieve the same note, but it vibrates at double the frequency making it an octave higher.
But what about all the other additional frets? Is there some kind of method behind them?
Absolutely! And we refer to this as the rule of 18. In fact, it’s the rule of 17.81672, but that doesn’t sound quite as glamorous.
This is a fret placement formula we can use for calculating fret placement. All we need to figure it out is the scale length of the guitar and our magic number, which we’ll just call 17.82 for simplicity (the guitar doesn’t have perfect temperament anyway).
So very quickly, the scale length of a guitar is the distance between the bridge and the nut, this is the area in which, when a string is plucked, it will vibrate within. While shorter scale guitars are increasing in popularity, the most common scale length on a guitar is 25.5’’ so we’ll use that as our example.
So we now take our scale length of 25.5 and divide it by 17.82, which gives us 1.431.
That now tells us we need to place our first fret 1.431 inches in from the nut.
To work out the formula of the next fret we do the same thing, but then add the sum of every fret before it.
So the second fret is: 25.5 divided by 17.82 + 1.431 = 2.782
That tells us the distance between the guitar nut and the second fret is 2.782 inches or 1.351 inches away from the first fret.
Notice how the distance between frets 0 and 1 was 1.431, while the difference between frets 1 and 2 was 1.351.
That’s the distance between each fret getting smaller, by the time you get to the 24th fret it will only be 0.379 inches away from the 23rd fret.
Here is a really handy fret position calculator you can play with to learn a little bit more about fret distance and spacing which helps visualize that ‘exponential’ quality of pitch and distance.
Discussions about fret placement accuracy and many of the concepts mentioned above can be tricky to understand. But if there’s one thing you can take away from this to start letting this concept sink in it’s that pitch increases exponentially, not linearly.
So each time you climb an octave, it’s only half the distance up the fretboard when compared to the octave previous to it.