Can Guitar Frets Perform Multiplication?

September 4, 2026Roscoe, 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=10x then the decimal (base-10) logarithm of y is defined like this: log(y)=x It’s well known that if two powers of 10 are multiplied, then the exponents can be added: 10N×10M=10N+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×M)=log(N)+log(M) 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