September 4, 2026
Roscoe, N.Y.

I’m sure that some pictures are worth a thousand words, but others trigger a whole lot of puzzlement. Such was the case with the cover of a book I recently bought entitled Calculating with Tones: The Logarithmic Logic of Music:

This book was published by the Oughtred Society, an organization named in honor of William Oughtred, the Anglican clergyman and mathematician who is credited with inventing the first logarithmic slide rule around 1622. The Oughtred Society was founded in 1991, “dedicated to the preservation and history of slide rules and other calculating instruments.” Their website contains a wealth of useful information on the subject.

Although the Oughtred Society website still has a page dedicated to this book, they don’t seem to be selling it at this time, and a second edition is not available directly through the society.

This cover illustration intrigued me because logarithms are involved both in our perception of musical pitch and in the construction of a slide rule. As I discuss in my web-book-in-progress The Lost Art of Logarithms, logarithms and the slide rule were originally invented to ease the tedious processes of multiplication and other mathematical tasks, but they continue to be vital for many reasons, including the understanding of our perceptions of the natural world.

But this cover illustration seems to imply a direct correspondence between the spacing of tick marks on a slide rule and the irregular spacing of frets on a guitar, and even suggests that guitar frets are spaced in such a way that they could be used to perform multiplications just like with a slide rule. Why else would the book be entitled Calculating with Tones?

The illustration is reproduced on page 34 of the book with a caption that includes a description of both halves of the graphic:

Guitar: Frets are positioned on the neck of the guitar to help locate the individual notes. The distance from one to the next becomes shorter as the frets get closer to the lower bridge. The reduction is logarithmic.

Slide Rule: The scale on the slide rule corresponds to a logarithmic function. When using the slide rule, one carries out multiplication by adding distances. The progressive decrease in distance between marks on the slide rule corresponds to the same thing on the neck of the guitar.

Despite how compelling that illustration and description are, something bugged me. It seemed fundamentally wrong, and yet I couldn’t pinpoint the exact problem.

Multiplying with Guitar Frets

I knew what I had to do. First I had to get a guitar such as this one:

Guitar image from Wikipedia by Martin Möller, CC BY-SA 2.0 DE via Wikimedia Commons; modifications by me

If your video display is not wide enough to show this entire guitar, you can use touch or the mouse to drag it horizontally into view.

If this guitar is truly capable of multiplying numbers, some numeric labels need to be added to it. Contrary to the terminology on the cover of Calculating with Tones, the guitar string is suspended between the nut at the left of this illustration and the saddle at the right. Those mounts hold the strings in place and govern their effective lengths when they are strummed or plucked when the strings are not pressed against any frets. Midway between the nut and saddle is the 12th fret, which effectively divides the string in half to play a pitch an octave higher than the string alone.

It’s a little hard to see, but in the cover illustration, the nut is lined up with the 1 on the all-important C and D scales of the slide rule, and the 12th fret is lined up with the 2. Let’s transfer these numbers to the guitar:

Between these two points are 11 frets, which must obviously be labeled as fractions of 12 between 1 and 2:

Many of these fractions can be expressed in reduced forms, and that’s what I’ve done here:

The next step is to saw the guitar in half:

Sorry, but it has to be done. Make a nice clean cut from top to bottom.

Now the two pieces can be put back together but with the freedom to slide one part relative to the other:

Use touch or the mouse to drag the top half of the guitar to the right to multiply two numbers. (If the bottom half of the guitar can’t fit on your screen, you can continue to horizontally scroll it.)

For example, suppose you want to multiply 7/6 by 3/2. Move the top half of the guitar so that the 1 on the top is aligned with the 7/6 on the bottom:

Now find 3/2 on the top half. Opposite that on the bottom is the product: 7/4, which is indeed 7/6 times 3/2.

Most of the other number combinations require some interpolation between the frets, and that’s not always easy, particularly because I’ve labeled the frets with fractions rather than decimals. But a couple other combinations work well, such as 5/4 times 4/3 equaling 5/3, and 4/3 times 3/2 equaling 2.

Just off hand, it seems as if this experiment is a success: Guitar frets can definitely multiply!

No, Guitar Frets Can Not Multiply.

Are you ready for my celebratory triumphalism to be brutally mocked?

Most acoustic guitars have 18 or 19 frets, so I can’t use those guitars to experiment with fret-based multiplications much beyond products of 2. But electric guitars often have more frets, sometimes as many as 24, which allow for each string to have a two-octave range.

I found a good image of a two-octave electric guitar on the website of the musical instrument manufacturer Donner. This is the inexpensive Donner DMT-100:

Donner DMT-100: from their website; modifications by me

Notice the two sets of double dots on the neck. These mark the one-octave fret and the two-octave fret. The one-octave fret effectively divides the string in half, while the two-octave fret divides the remaining length in half again.

The nut and the frets can be labeled similarly to the acoustic guitar, but how should I label the two-octave fret?

On the cover illustration of Calculating with Tones the nut is aligned with 1 on the C and D scales of the slide rule, the 12th fret is aligned with 2, but the saddle is aligned with 4:

You can click this to see a larger version.

Although the corresponce between the guitar frets and the slide rule seems to work well for the first octave, the 24th fret is midway between the 12th fret and the saddle, but that approximately corresponds with 2.8 on the slide rule. What does that mean? And what does it mean that the 4 on the slide rule corresponds with the guitar saddle? The slide rule continues with 5, 6, and so forth up to 10, but guitar frets can’t go beyond the saddle.

Despite my concerns, I have no choice but to continue labeling the electric guitar frets up to 3:

Now it’s time to saw this guitar in half:

Don’t weep. That’s the guitar’s job.

The two halves can be put back together for another fret-based multiplication tool:

Slide the top half to the right to multiply. It’s now easy to find pairs of numbers that do not work right. For example, here’s a multiplication of 3/2 and 2, which should equal 3:

When the 1 on the top half is aligned with the 3/2 on the bottom, the 2 on the top is some distance beyond the 3 on the bottom.

Why does this fret-based slide rule seem to work for one octave but not for two octaves?

I’m afraid this subject warrants a deeper dive.

How a Real Slide Rule Works (Briefly)

It seems as if guitar frets are partially mimicking a slide rule but not quite nailing it.

Logarithms were invented to simplify the multiplication of multi-digit numbers. Today we understand logarithms as the inverse of exponentiation. If

$y={10}^x$

then the decimal (base-10) logarithm of y is defined like this:

$log\left(y\right)=x$

It’s well known that if two powers of 10 are multiplied, then the exponents can be added:

${10}^N\times {10}^M={10}^{N+M}$

It can then be shown (as I laboriously demonstrate in Chapter 3 of The Lost Art of Logarithms) that the sum of the logarithms of two numbers is the same as the logarithm of the product of those two numbers:

$log(N\times M)=log\left(N\right)+log\left(M\right)$

The slide rule effectively implements this calculation in a pair of sliding rulers. How these rulers are constructed is the job of Chapters 6 and 7 in The Lost Art of Logarithms.

Here is the beginning of the making of a 10-inch logarithmic scale. Each number on the scale is positioned based on the total length of the scale (10 inches in this case) multiplied by the number’s decimal logarithm. The scale starts with 1 because the logarithm of 1 is zero:

If this ruler is too wide to fit on your browser page, you can use touch or the mouse to scroll it horizontally into view.

In the old days, these logarithms would be obtained from a book. You can alternatively use a calculator. Most computer-based calculators must be switched into Scientific mode to get access to the log key. Phone-based calculators often have to be turned sideways.

You could continue with some fractional numbers:

Eventually you end up with something like this:

The distance of each of those tick marks from the beginning of the logarithmic scale is equal to 10 inches times the logarithm of the number represented by that tick mark.

Put two of these logarithmic rulers face to face (as William Oughtred did in 1622) and you have a slide rule:

I’ve also added a covenient hairline that you can drag with the blue circle. I’ve initialized this slide rule to show the multiplication of 1.56 and 2.72, but you can drag the top ruler and the blue circle with the hairline to experiment with other multiplications. You want to align the 1 on the top scale with the first number that you’re multiplying on the bottom scale. Then align the hairline with the second number that you’re multiplying on the top scale. The hairline shows the product on the bottom scale, in this case 4.24.

But it’s much more versatile than multiplying small numbers. Multiplying 1.56 and 2.72 is basically the same as multiplying 15,600 by 27.2, or 0.0156 by 0.000272. The only difference is the decimal point in the result.

If you can’t multiply the two numbers by shifting the top scale to the right, shift it to the left. Move the 10 on the top scale to align with the first number on the bottom scale. Here’s 8 times 7:

In this case you have to shift a decimal point to the right to get the product of 56.

You can also use this slide rule to divide: Align the dividend on the bottom scale with the divisor on the top scale. The quotient is on the bottom scale aligned with the beginning or end of the top scale.

There’s no magic here. All you’re doing is adding or subtracting ruler lengths. But because the numbers on the rulers are logarithms, then the numbers are effectively being multiplied. Here’s a little demonstration showing the ruler lengths with the same two numbers I used in my first example:

For each position of the top scale and the hairline, you can see that the two widths on the top add up to the width on the bottom. But these widths correspond to logarithms of numbers on the scale, so these numbers are effectively being multiplied.

Are guitar frets positioned similarly? And can those positions be quantified in a similar way to reveal why they can sometimes multiply and sometimes not?

Our Logarithmic Perception of Pitch

It might seem reasonable that guitar frets are spaced logarithmically because our perception of musical pitch is logarithmic.

What do I mean by that?

Sound is vibration. The frequency of a vibration — how many times the vibrating thing goes back and forth every second — is specified in hertz (abbreviated Hz), named after Heinrich Rudolf Hertz, the first person to prove the existence of electromagnetic waves. (Before the adoption of hertz in the 1960s, the rate of vibration was specified in the self-explanatory units of cycles per second or cps.)

Humans can generally hear sounds with frequencies between 20 Hz and 20,000 Hz, but this varies by person and age. As we get older, our ability to hear higher frequencies often deteriorates.

However, we don’t perceive these frequencies linearly. In other words, we do not perceive the difference between 1000 Hz and 2000 Hz to be the same as the difference between 2000 and 3000 Hz. Instead, we perceive the difference between 1000 and 2000 Hz to be the same as the difference between 2000 and 4000 Hz.

This doubling of frequency is perceived as a change in pitch called the octave, which forms the basis of music throughout the world. As an easy reference, sing the first two notes of the verse “Somewhere over the rainbow.” That’s an octave leap. Whether the song is being sung by Björk or Barry White, the frequency doubles from the “some” to the “where.”

As I discuss in excruciating detail in the as-yet-unfinished Chapter 13 of The Lost Art of Logarithms, music within the Western tradition evolved to have 12 steps to the octave. The labeled keys on this piano illustrate a progression of octaves over much of the range of the piano:

Each pair of consecutive labeled keys is a doubling of frequency and hence an interval of an octave. Middle C is the key marked with a tiny dot, and all the labeled keys are A’s. I used A because standard tuning sets the A above middle C to be 440 Hz. The frequencies of all the other A keys can then be calculated by progressively doubling or halving 440. These various A keys can be differentiated with so-called scientific pitch names using a subscripted number for the octave. These eight labeled keys are denoted as A₀, A₁, A₂, A₃, A₄, A₅, A₆, and A₇,

Each of the 12 steps in an octave is known as a semitone.

And yes, these words make no sense. The word “octave” seems to imply eight of something, and the word “semitone” seems to suggest half of something. A lot of this terminology originated many centuries ago, and it made sense at the time, but today we’re simply stuck with it.

Over many centuries, there evolved various ways that these 12 notes in the octave would be tuned. (Again, Chapter 13 of The Lost Art of Logarithms has much more detail.) But beginning in the late 16th century, some musicians and theorists began advocating that each semitone step be the same. This is a type of tuning known as equal temperament and it dominates Western music today, although not without some dissension.

What does it mean that there are 12 equal semitones between 220 and 440 Hz, and also between 440 and 880 Hz? It means that the semitone cannot be a fixed number of hertz but must instead be a multiplication. The semitone has to be a number than when multiplied by itself 12 times equals 2, the doubling of frequency associated with the octave. This number is commonly expressed as the twelfth root of 2:

$''Semitone\; multiplier''=\sqrt[12]{2}$

or as 2 to the 1/12th power:

$''Semitone\; multiplier''=2^{\frac{1}{12}}$

For us today, it is easy to calculate that number. Actually, that’s not true. What’s easy for us is to punch that expression into a calculator. In the Windows Calculator, for example, it’s accomplished by switching to Scientific mode, then punching in 2, the x^y key, the left parentheses, the 1 key, the divide (÷) key, then 12, the right parentheses, and the equal (=) key.

The 12th root of 2 is an irrational number; there is no exact value. The calculator can only give an approximate value, which here I’ve approximated even more:

$''Semitone\; multiplier''=\sqrt[12]{2}\cong \mathrm{1.059463...}$

If you multiply that number by itself 12 times, you get a result of about 2. To keep things simple, you can use 1.059 or 1.06 in a pinch as an even grosser approximation.

Suppose you number all the black and white keys on the piano keyboard from left to right beginning at 1 (the note A₀) and ending with 88 (the note C₈). The note A₄ has a key number of 49, and when the piano is tuned conventionally, a frequency of 440 Hz. From that starting point, you can then calculate the frequency of any key on the piano using this formula:

$''frequency''=440\times 2^{\left(\frac{''key\; number''-49}{12}\right)}$

Or you can go the other way by expressing this relationship with a logarithm. But because we’re dealing with powers of 2, it has to be a binary (base-2) logarithm:

$''key\; number''=49+12\times {log}_2\left(\frac{''frequency''}{440}\right)$

Those two statements are equivalent.

We perceive the 88 keys of the piano to be equal steps in pitch. Yet those key numbers are logarithms of the frequency. This is what I mean when I assert that our perception of pitch is logarithmic.

Frequency and String Length

The piano, harp, violin, and guitar all make sounds with vibrating strings. The frequency of a vibrating string depends on four factors:

• the material that it’s made of,

Musical instruments often contain strings of a variety of materials, thicknesses, and lengths, and the strings are tuned by adjusting the tension. But once a particular string is tuned, everything remains constant except (potentially) its length.

This is certainly the case for a guitar. For any particular string, the frequency is governed entirely by the length of the portion of the string that’s vibrating, which depends on which fret the string has been pressed against. (Well, that’s not entirely true. The tension of the string slightly increases when it’s pressed against the fret, but let’s try to retain our sanity by ignoring that.)

Fortunately, the relationship between a vibrating string’s frequency and its length is very straightforward: They are inversely proportional. The longer the string, the lower the frequency. Divide the length in half, and the frequency is doubled for a leap of one octave.

Fretting the Guitar

The frets of a guitar must therefore be positioned so that each successive fret increases the string’s frequency by a factor of 1.059. Because the frequency is inversely proportional to the length, each fret must effectively shorten the length of the string by the inverse of the semitone multiplier. One way to express this is as 1/2 to the 1/12th power:

$''string\; reduction\; factor''={\left(\frac{1}{2}\right)}^{\frac{1}{12}}\cong \mathrm{0.943874...}$

I’ll use the grosser approximation of 0.944 to make these sample calculations a little easier. Furthermore, let’s assume that the length of the guitar string from the nut to the saddle is 100 centimeters:

The first fret must be 0.944 times that length, or 94.4 centimeters from the saddle and hence 5.6 centimeters from the nut:

The second fret gets messier. It must be positioned so the string has an effective length of 94.4 times 0.944, or 89.1 centimeters.

You can also calculate 89.1 centimeters by multiplying 100 centimeters by 0.944 squared. That second fret is 10.9 centimeters from the nut, or 5.3 centimeters below the first fret. The spacing between the frets is decreasing already.

This process continues. Each fret must reduce the length of the string to 0.944 times the length associated with the previous fret:

That 84.1 length is 100 times 0.944 cubed. And again:

The space between the frets continues to progressively decrease. The distance from the 4th fret to the 5th is 4.7 centimeters.

Eventually you get to the 12th fret, which reduces the string to half its original length:

The twelfth fret is positioned at 100 centimeters times 0.944 to the twelfth power, or 50 centimeters from the saddle, effectively reducing the string to half its length.

You can increase the range of the guitar by another octave with another 12 frets.

The 24th fret divides the string in half again.

The length of the string from a fret to the saddle can be calculated like this:

$''string\; length\; from\; fret''=''string\; length''\times {\left(\frac{1}{2}\right)}^{\frac{\mathrm{fret\; number}}{12}}$

The distance of any fret from the nut can be calculated like this:

$''fret\; distance\; from\; nut''=''string\; length''\times (1-{\left(\frac{1}{2}\right)}^{\frac{\mathrm{fret\; number}}{12}})$

The nut is assumed to have a fret number of 0, so the fret distance for the nut is also 0. The 12th fret has a fret number of 12, so the fret distance is half the string length.

Building a Fret-Based Slide Rule

Notice that no logarithms were involved in the development of the formula for computing fret positions. Nevertheless, now that I have such a formula, I can make my own fret-based slide rule without sawing any more guitars in half. I think I established at the outset that only the first octave of frets seems to function like a conventional logarithmic slide rule. Including the second octave of frets throws the whole thing off.

So let’s take that first octave of frets and label it exactly as I labeled the acoustic guitar with fractions between 1 and 2:

Now let’s make another ruler the same way but with the tick marks on the top rather than the bottom:

You can place these two rulers together and use touch or the mouse to move the top ruler relative to the bottom:

You can try out a couple of the multiplications not requiring interpolation, such as 7/6 times 3/2 or 5/4 times 4/3, and now you might notice that it’s not exact. The tick marks don’t line up precisely. This might have been forgivable when awkwardly grappling with two pieces of a guitar sawed in half, but if the guitar frets were spaced logarithmically, that would not happen.

I can’t make these rulers longer by including the second octave of guitar frets. We already know that’s a disaster. However, logarithmic slide rules are self-similar. The part of the slide rule from 2 to 4 is the same as from 4 to 8 except that the numbering is double:

I can even crank the scales up to 10:

Try 5/2 times 8/3, which should be 20/3, but it just doesn’t line up right.

So is this real or is it fantasy?

A Tale of Two Ruler Scales

It’s annoying that a slide rule constructed from the mathematics of guitar frets seems to work almost as well as a slide rule constructed from logarithms, but not exactly.

The following graphic is not a slide rule. It is not interactive. Don’t try to slide anything. The top ruler scale is constructed based on the formula for positioning guitar frets, and the bottom is constructed from logarithms:

I’ve stretched it out so you don’t have to look closely to see that these are two different scales that only coincide at 1 and 2.

It’s also revealing to graph the two functions beyond the range of these scales. Earlier I showed a formula for calculating the distance of each fret from the nut:

$''fret\; distance\; from\; nut''=''string\; length''\times (1-{\left(\frac{1}{2}\right)}^{\frac{\mathrm{fret\; number}}{12}})$

I want to adapt this formula for a slide rule: The distance of each tick mark is based on the number associated with that tick mark. That number n relates to the fret number like this:

$n=1+\frac{\mathrm{fret\; number}}{12}$

Or:

$\mathrm{fret\; number}=12\times (n-1)$

Also, because the scale from 1 to 2 is half the string length, let’s also normalize the string length to 2 so the formula for positioning a tick mark becomes:

$\mathrm{tick\; distance}=2-{\left(\frac{1}{2}\right)}^{n-2}$

This will be the red line in the graph. When n equals 1 (the beginning of the scale), the tick distance is 0; when n equals 2 at the end of the scale, the tick distance is 1.

The tick distance in the logarithmic scale from 1 to 2 is simply:

$\mathrm{tick\; distance}={log}_2\left(n\right)$

That’s the blue line in the graph. Again, when n equals 1 at the beginning of the scale, the tick distance is 0, and when n equals 2, the tick distance is 1.

Although the red and blue lines seem to coincide between n equal to 1 and 2, these are definitely not equivalent functions!

Here’s what happened: Deceived as I was by the cruelly deceptive illustration on the cover of Calculating with Tones, I contorted my initial labeling of the guitar frets to range from 1 to 2. That implied a certain function that of course coincided at those two points with the logarithm, but was otherwise quite different. The rest is basically coincidence.

One of the themes of The Lost Art of Logarithms is that many phenomena in our natural and human-made world can be interpreted logarithmically. That includes music but it does not include guitar frets.

Addendum: The Geometric Approach to Fret Positioning

Earlier I described the theoretical mathematics behind positioning the frets on a guitar. But the guitar has been around for hundreds of years. How did people figure out the positions of the frets without calculators and precision measuring instruments? For example, how were the frets positioned on the guitar in one of Johannes Vermeer’s last paintings, The Guitar Player, dating from 1672?

Public domain image from Wikipedia via Wikimedia Commons

One popular approach was to use a simple ratio from a book published in Florence in 1581 entitled Dialogo della musica antica et della moderna or “Dialogue of Ancient and Modern Music.”

The author was Vincenzo Galilei, born near Florence probably sometime in the 1520s. He studied lute at a young age and became a singer, teacher, and composer. Some of his music has been recorded: The album The Well-Tempered Lute has a selection of his compositions. He studied music theory with Gioseffo Zarlino, and then refuted some of Zarlino’s theories. His eldest son, Galileo, who was about 17 when his father published the Dialogo, was later to achieve some renown in the fields of astronomy and natural philosophy, and also wrote about music theory in his Two New Sciences of 1638.

The Internet Archive has the original 1581 edition of Papa Galilei’s Dialogo; the International Music Score Library Project (IMSLP) has a 1967 reprint), and the University of North Texas Digital Libary has a doctoral thesis by Robert H. Herman with a complete English translation and much commentary.

Here’s where Galilei discusses how to calculate fret positions for a lute:

I therefore divide the whole line A B into eighteen parts, and toward the high part (moving away from the low part) where that first part ends, I place the first fret. I divide once more the whole remainder of the same number of parts [i.e. eighteen] and from the very same band, I place the first fret under the second. I now proceed to distribute the space which remains beneath the frets, always in that very same order, up to [the number of] twelve, which brings me exactly to [the point] where half of the entire string terminates. (pp. 299–300, bracketed phrases added by translator; corresponds to page 49 in the original Italian edition)

Vincenzo Galilei is suggesting a ratio of 18:17 for positioning the frets. This implies that the semitone multiplier is approximated by:

$''Semitone\; multiplier''=\frac{18}{17}\cong \mathrm{1.058824...}$

If you multiply 18/17 by itself 12 times, you won’t come out to 2. Instead, you get 1.985560, which is quite close.

Similarly, the string reduction factor for calculating the position of frets is:

$''String\; reduction\; factor''=\frac{17}{18}=\mathrm{0.944444...}$

If you take this ratio to the 12th power, it’s a little more than 1/2:

${\left(\frac{17}{18}\right)}^{12}\cong \mathrm{0.50364...}$

But the advantage of 17/18 is that it’s a ratio and ratios are extemely useful.

If you examine any treatise of music theory from the time of the Ancient Greeks through the Renaissance, you’ll see a lot of geometrical drawings. When mixing arithmetic and geometry, ratios are ideal because you don’t have to measure anything or get involved with fractions. It’s all geometrical.

Vincenzo Galilei doesn’t explain very well how the 18:17 ratio is used in practice, but it’s easy to develop a technique. Suppose you were building a lute or guitar, or perhaps you’ve obtained a fretless guitar to which you want to add some frets.

The first step is to get a large sheet of paper and draw a line segment corresponding to the distance between the nut and the saddle:

Let’s call this the string line. Put the guitar aside, and draw a long horizontal line perpendicular to the string line and aligned with its bottom:

Somewhere along that horizontal line, draw another shorter vertical line:

On that shorter vertical line, working from the bottom up, draw 18 evenly spaced horizontal tick marks:

This is something that can be done with the traditional geometric drawing tools of compass and straightedge. The distance between these horizontal tick marks doesn’t matter, just as long as they’re evenly spaced. If you’re lazy, you can even use an 18-inch ruler for marking this line.

Now draw a straight line from the top of the string line to the top tick of the shorter vertical line and continue down to the base:

I’ve drawn a little dot where it crosses that bottom horizontal. From that little dot, draw another diagonal that passes through the second tick mark and intersects the string line:

Now get rid of that smaller vertical line with the tick marks. You don’t need it any more:

I contend that any vertical line drawn from the top diagonal to the base will be divided into two parts: the top part (between the two diagonals) is 1/18th of the total length. The bottom part (under the bottom diagonal) is 17/18th of the total length.

The string line is one such vertical line. The two diagonals divide that string line into 1/18th between the two diagonals and 17/18th below the bottom diagonal. The first fret goes where the bottom diagonal intersects the string:

To determine where the second fret goes, first, extend a horizontal line from the fret to the top diagonal. This is shown here as a dotted line:

You can imagine a vertical line drawn from the point where the dotted line meets the top diagonal down to the base. That line is also divided in two parts. Draw just a part of that vertical line down to the second diagonal. This is also a dotted line:

Now draw a horzizontal line back to the string line:

That is the position of the second fret:

As you keep going, you’ll see that the distance between the frets is shrinking. That’s because each successive fret is 17/18 of the remaining length of the string after the previous fret. Here’s how to find the third fret:

And here’s the fourth:

Just keep going and you get to the octave fret that I’ve made a little wider for emphasis:

Keep going and you can add another twelve frets culminating with the two-octave fret:

And now you have a template for fretting the guitar up to a two-octave range:

Vincenzo Galilei’s advocacy of the 18:17 ratio was apparently quite persuasive. In his essential book Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951, 1952; Dover Publications, 2004), musicologist James Murray Barbour reports finding that “references to the 18:17 semitone cover two and a half centuries.” (p. 59) In other words, people were using this ratio well into the 19th century. That’s why I assume that the guitar in Vermeer’s painting was fretted with this technique.

Let’s see how it works in practice, or at least in a virtual guitar.

Here are two “guitars” with each of the six strings labeled at the top with their pitches. The guitar on the left uses Vincenzo Galilei’s 18:17 ratio for determining the fret positions. The one on the right uses the twelfth root of 2 for the calculations.

On a touch screen, you can pluck a string on these two guitars with your finger. Otherwise you’ll need to drag the mouse pointer across a string with the button pressed. To play the note associated with the open string, pluck the string under the nut near the top. For notes requiring frets, pluck the string right under the fret.

A pair of dots are drawn right above the octave and two-octave frets. These dots are found on some guitars to help orient the musician. The slider in the lower-left corner is a volume control.

If you have a multitouch display, you can pluck two strings simultaneously, perhaps the same string on the two different guitars. At the very top, the frequencies will be identical, but as you get lower down the frets, the frequencies deviate and you might start to hear beats.

It is likely that guitar builders who used Vincenzo Galilei’s 18:17 ratio made subtle adjustments to the frets so that a pure octave interval could be realized.