Can guitar frets perform multiplication?

Can Guitar Frets Perform Multiplication?

September 4, 2026
Roscoe, N.Y.

I’m judge that immoderate pictures are worthy a 1000 words, but others trigger a full batch of puzzlement. Such was the lawsuit pinch the screen of a book I precocious bought entitled Calculating pinch Tones: The Logarithmic Logic of Music:

This book was published by the Oughtred Society, an statement named successful grant of William Oughtred, the Anglican clergyman and mathematician who is credited pinch inventing the first logarithmic descent norm astir 1622. The Oughtred Society was founded successful 1991, “dedicated to the preservation and history of descent rules and different calculating instruments.” Their website contains a wealthiness of useful accusation connected the subject.

Although the Oughtred Society website still has a page dedicated to this book, they don’t look to beryllium trading it astatine this time, and a 2nd version is not disposable straight done the society.

This screen illustration intrigued maine because logarithms are progressive some successful our cognition of philharmonic transportation and successful the building of a descent rule. As I talk successful my web-book-in-progress The Lost Art of Logarithms, logarithms and the descent norm were primitively invented to easiness the tedious processes of multiplication and different mathematical tasks, but they proceed to beryllium captious for galore reasons, including the knowing of our perceptions of the earthy world.

But this screen illustration seems to connote a nonstop correspondence betwixt the spacing of tick marks connected a descent norm and the irregular spacing of frets connected a guitar, and moreover suggests that guitar frets are spaced successful specified a measurement that they could beryllium utilized to execute multiplications conscionable for illustration pinch a descent rule. Why other would the book beryllium entitled Calculating pinch Tones?

The illustration is reproduced connected page 34 of the book pinch a caption that includes a explanation of some halves of the graphic:

Guitar: Frets are positioned connected the cervix of the guitar to thief find the individual notes. The region from 1 to the adjacent becomes shorter arsenic the frets get person to the little bridge. The simplification is logarithmic. Slide Rule: The standard connected the descent norm corresponds to a logarithmic function. When utilizing the descent rule, 1 carries retired multiplication by adding distances. The progressive alteration successful region betwixt marks connected the descent norm corresponds to the aforesaid point connected the cervix of the guitar.

Despite really compelling that illustration and explanation are, thing bugged me. It seemed fundamentally wrong, and yet I couldn’t pinpoint the nonstop problem.

Multiplying pinch Guitar Frets

I knew what I had to do. First I had to get a guitar specified arsenic this one:

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

If your video show is not wide capable to show this full guitar, you tin usage touch aliases the rodent to resistance it horizontally into view.

If this guitar is genuinely tin of multiplying numbers, immoderate numeric labels request to beryllium added to it. Contrary to the terminology connected the screen of Calculating pinch Tones, the guitar drawstring is suspended betwixt the nut astatine the near of this illustration and the saddle astatine the right. Those mounts clasp the strings successful spot and govern their effective lengths erstwhile they are strummed aliases plucked erstwhile the strings are not pressed against immoderate frets. Midway betwixt the seed and saddle is the 12th fret, which efficaciously divides the drawstring successful half to play a transportation an octave higher than the drawstring alone.

It’s a small difficult to see, but successful the screen illustration, the seed is lined up pinch the 1 connected the all-important C and D scales of the descent rule, and the 12th fret is lined up pinch the 2. Let’s transportation these numbers to the guitar:

Between these 2 points are 11 frets, which must evidently beryllium branded arsenic fractions of 12 betwixt 1 and 2:

Many of these fractions tin beryllium expressed successful reduced forms, and that’s what I’ve done here:

The adjacent measurement is to saw the guitar successful half:

Sorry, but it has to beryllium done. Make a bully cleanable trim from apical to bottom.

Now the 2 pieces tin beryllium put backmost together but pinch the state to descent 1 portion comparative to the other:

Use touch aliases the rodent to resistance the apical half of the guitar to the correct to multiply 2 numbers. (If the bottommost half of the guitar can’t fresh connected your screen, you tin proceed to horizontally scroll it.)

For example, suppose you want to multiply 7/6 by 3/2. Move the apical half of the guitar truthful that the 1 connected the apical is aligned pinch the 7/6 connected the bottom:

Now find 3/2 connected the apical half. Opposite that connected the bottommost is the product: 7/4, which is so 7/6 times 3/2.

Most of the different number combinations require immoderate interpolation betwixt the frets, and that’s not ever easy, peculiarly because I’ve branded the frets pinch fractions alternatively than decimals. But a mates different combinations activity well, specified arsenic 5/4 times 4/3 equaling 5/3, and 4/3 times 3/2 equaling 2.

Just disconnected hand, it seems arsenic if this research is simply a success: Guitar frets tin decidedly multiply!

No, Guitar Frets Can Not Multiply.

Are you fresh for my celebratory triumphalism to beryllium brutally mocked?

Most acoustic guitars person 18 aliases 19 frets, truthful I can’t usage those guitars to research pinch fret-based multiplications overmuch beyond products of 2. But electrical guitars often person much frets, sometimes arsenic galore arsenic 24, which let for each drawstring to person a two-octave range.

I recovered a bully image of a two-octave electrical guitar connected the website of the philharmonic instrumentality shaper Donner. This is the inexpensive Donner DMT-100:

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

Notice the 2 sets of double dots connected the neck. These people the one-octave fret and the two-octave fret. The one-octave fret efficaciously divides the drawstring successful half, while the two-octave fret divides the remaining magnitude successful half again.

The seed and the frets tin beryllium branded likewise to the acoustic guitar, but really should I explanation the two-octave fret?

On the screen illustration of Calculating pinch Tones the seed is aligned pinch 1 connected the C and D scales of the descent rule, the 12th fret is aligned pinch 2, but the saddle is aligned pinch 4:

You tin click this to spot a larger version.

Although the corresponce betwixt the guitar frets and the descent norm seems to activity good for the first octave, the 24th fret is midway betwixt the 12th fret and the saddle, but that astir corresponds pinch 2.8 connected the descent rule. What does that mean? And what does it mean that the 4 connected the descent norm corresponds pinch the guitar saddle? The descent norm continues pinch 5, 6, and truthful distant up to 10, but guitar frets can’t spell beyond the saddle.

Despite my concerns, I person nary prime but to proceed labeling the electrical guitar frets up to 3:

Now it’s clip to saw this guitar successful half:

Don’t weep. That’s the guitar’s job.

The 2 halves tin beryllium put backmost together for different fret-based multiplication tool:

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

When the 1 connected the apical half is aligned pinch the 3/2 connected the bottom, the 2 connected the apical is immoderate region beyond the 3 connected the bottom.

Why does this fret-based descent norm look to activity for 1 octave but not for 2 octaves?

I’m acrophobic this taxable warrants a deeper dive.

How a Real Slide Rule Works (Briefly)

It seems arsenic if guitar frets are partially mimicking a descent norm but not rather nailing it.

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

y=10x

past the decimal (base-10) logarithm of y is defined for illustration this:

log(y)=x

It’s good known that if 2 powers of 10 are multiplied, past the exponents tin beryllium added:

10N×10M=10N+M

It tin past beryllium shown (as I laboriously show successful Chapter 3 of The Lost Art of Logarithms) that the sum of the logarithms of 2 numbers is the aforesaid arsenic the logarithm of the merchandise of those 2 numbers:

log(N×M)=log(N)+log(M)

The descent norm efficaciously implements this calculation successful a brace of sliding rulers. How these rulers are constructed is the occupation of Chapters 6 and 7 successful The Lost Art of Logarithms.

Here is the opening of the making of a 10-inch logarithmic scale. Each number connected the standard is positioned based connected the full magnitude of the standard (10 inches successful this case) multiplied by the number’s decimal logarithm. The standard starts pinch 1 because the logarithm of 1 is zero:

If this ruler is excessively wide to fresh connected your browser page, you tin usage touch aliases the rodent to scroll it horizontally into view.

In the aged days, these logarithms would beryllium obtained from a book. You tin alternatively usage a calculator. Most computer-based calculators must beryllium switched into Scientific mode to get entree to the log key. Phone-based calculators often person to beryllium turned sideways.

You could proceed pinch immoderate fractional numbers:

Eventually you extremity up pinch thing for illustration this:

The region of each of those tick marks from the opening of the logarithmic standard is adjacent to 10 inches times the logarithm of the number represented by that tick mark.

Put 2 of these logarithmic rulers look to look (as William Oughtred did successful 1622) and you person a descent rule:

I’ve besides added a covenient hairline that you tin resistance pinch the bluish circle. I’ve initialized this descent norm to show the multiplication of 1.56 and 2.72, but you tin resistance the apical ruler and the bluish circle pinch the hairline to research pinch different multiplications. You want to align the 1 connected the apical standard pinch the first number that you’re multiplying connected the bottommost scale. Then align the hairline pinch the 2nd number that you’re multiplying connected the apical scale. The hairline shows the merchandise connected the bottommost scale, successful this lawsuit 4.24.

But it’s overmuch much versatile than multiplying mini numbers. Multiplying 1.56 and 2.72 is fundamentally the aforesaid arsenic multiplying 15,600 by 27.2, aliases 0.0156 by 0.000272. The only quality is the decimal constituent successful the result.

If you can’t multiply the 2 numbers by shifting the apical standard to the right, displacement it to the left. Move the 10 connected the apical standard to align pinch the first number connected the bottommost scale. Here’s 8 times 7:

In this lawsuit you person to displacement a decimal constituent to the correct to get the merchandise of 56.

You tin besides usage this descent norm to divide: Align the dividend connected the bottommost standard pinch the divisor connected the apical scale. The quotient is connected the bottommost standard aligned pinch the opening aliases extremity of the apical scale.

There’s nary magic here. All you’re doing is adding aliases subtracting ruler lengths. But because the numbers connected the rulers are logarithms, past the numbers are efficaciously being multiplied. Here’s a small objection showing the ruler lengths pinch the aforesaid 2 numbers I utilized successful my first example:

For each position of the apical standard and the hairline, you tin spot that the 2 widths connected the apical adhd up to the width connected the bottom. But these widths correspond to logarithms of numbers connected the scale, truthful these numbers are efficaciously being multiplied.

Are guitar frets positioned similarly? And tin those positions beryllium quantified successful a akin measurement to uncover why they tin sometimes multiply and sometimes not?

Our Logarithmic Perception of Pitch

It mightiness look reasonable that guitar frets are spaced logarithmically because our cognition of philharmonic transportation is logarithmic.

What do I mean by that?

Sound is vibration. The wave of a vibration — really galore times the vibrating point goes backmost and distant each 2nd — is specified successful hertz (abbreviated Hz), named aft Heinrich Rudolf Hertz, the first personification to beryllium the beingness of electromagnetic waves. (Before the take of hertz successful the 1960s, the complaint of vibration was specified successful the self-explanatory units of cycles per second aliases cps.)

Humans tin mostly perceive sounds pinch frequencies betwixt 20 Hz and 20,000 Hz, but this varies by personification and age. As we get older, our expertise to perceive higher frequencies often deteriorates.

However, we don’t comprehend these frequencies linearly. In different words, we do not comprehend the quality betwixt 1000 Hz and 2000 Hz to beryllium the aforesaid arsenic the quality betwixt 2000 and 3000 Hz. Instead, we comprehend the quality betwixt 1000 and 2000 Hz to beryllium the aforesaid arsenic the quality betwixt 2000 and 4000 Hz.

This doubling of wave is perceived arsenic a alteration successful transportation called the octave, which forms the ground of euphony passim the world. As an easy reference, singing the first 2 notes of the verse “Somewhere complete the rainbow.” That’s an octave leap. Whether the opus is being sung by Björk aliases Barry White, the wave doubles from the “some” to the “where.”

As I talk successful excruciating item successful the as-yet-unfinished Chapter 13 of The Lost Art of Logarithms, euphony wrong the Western contented evolved to person 12 steps to the octave. The branded keys connected this soft exemplify a progression of octaves complete overmuch of the scope of the piano:

Each brace of consecutive branded keys is simply a doubling of wave and hence an interval of an octave. Middle C is the cardinal marked pinch a mini dot, and each the branded keys are A’s. I utilized A because modular tuning sets the A supra mediate C to beryllium 440 Hz. The frequencies of each the different A keys tin past beryllium calculated by progressively doubling aliases halving 440. These various A keys tin beryllium differentiated pinch alleged technological transportation names utilizing a subscripted number for the octave. These 8 branded keys are denoted arsenic A0, A1, A2, A3, A4, A5, A6, and A7,

Each of the 12 steps successful an octave is known arsenic a semitone.

And yes, these words make nary sense. The connection “octave” seems to connote 8 of something, and the connection “semitone” seems to propose half of something. A batch of this terminology originated galore hundreds of years ago, and it made consciousness astatine the time, but coming we’re simply stuck pinch it.

Over galore centuries, location evolved various ways that these 12 notes successful the octave would beryllium tuned. (Again, Chapter 13 of The Lost Art of Logarithms has overmuch much detail.) But opening successful the precocious 16th century, immoderate musicians and theorists began advocating that each semitone measurement beryllium the same. This is simply a type of tuning known arsenic equal temperament and it dominates Western euphony today, though not without immoderate dissension.

What does it mean that location are 12 adjacent semitones betwixt 220 and 440 Hz, and besides betwixt 440 and 880 Hz? It intends that the semitone cannot beryllium a fixed number of hertz but must alternatively beryllium a multiplication. The semitone has to beryllium a number than erstwhile multiplied by itself 12 times equals 2, the doubling of wave associated pinch the octave. This number is commonly expressed arsenic the twelfth guidelines of 2:

Semitone multiplier=212

aliases arsenic 2 to the 1/12th power:

Semitone multiplier=2112

For america today, it is easy to cipher that number. Actually, that’s not true. What’s easy for america is to punch that look into a calculator. In the Windows Calculator, for example, it’s accomplished by switching to Scientific mode, past punching successful 2, the xy key, the near parentheses, the 1 key, the disagreement (÷) key, past 12, the correct parentheses, and the adjacent (=) key.

The 12th guidelines of 2 is an irrational number; location is nary nonstop value. The calculator tin only springiness an approximate value, which present I’ve approximated moreover more:

Semitone multiplier=212 ≅1.059463...

If you multiply that number by itself 12 times, you get a consequence of astir 2. To support things simple, you tin usage 1.059 aliases 1.06 successful a pinch arsenic an moreover grosser approximation.

Suppose you number each the achromatic and achromatic keys connected the soft keyboard from near to correct opening astatine 1 (the statement A0) and ending pinch 88 (the statement C8). The statement A4 has a cardinal number of 49, and erstwhile the soft is tuned conventionally, a wave of 440 Hz. From that starting point, you tin past cipher the wave of immoderate cardinal connected the soft utilizing this formula:

frequency= 440×2 (key number−4912)

Or you tin spell the different measurement by expressing this narration pinch a logarithm. But because we’re dealing pinch powers of 2, it has to beryllium a binary (base-2) logarithm:

cardinal number= 49+ 12× log2 (frequency440)

Those 2 statements are equivalent.

We comprehend the 88 keys of the soft to beryllium adjacent steps successful pitch. Yet those cardinal numbers are logarithms of the frequency. This is what I mean erstwhile I asseverate that our cognition of transportation is logarithmic.

Frequency and String Length

The piano, harp, violin, and guitar each make sounds pinch vibrating strings. The wave of a vibrating drawstring depends connected 4 factors:

• the worldly that it’s made of,

• its thickness,

• its tension, and

• its length.

Musical instruments often incorporate strings of a assortment of materials, thicknesses, and lengths, and the strings are tuned by adjusting the tension. But erstwhile a peculiar drawstring is tuned, everything remains changeless isolated from (potentially) its length.

This is surely the lawsuit for a guitar. For immoderate peculiar string, the wave is governed wholly by the magnitude of the information of the drawstring that’s vibrating, which depends connected which fret the drawstring has been pressed against. (Well, that’s not wholly true. The hostility of the drawstring somewhat increases erstwhile it’s pressed against the fret, but let’s effort to clasp our sanity by ignoring that.)

Fortunately, the narration betwixt a vibrating string’s wave and its magnitude is very straightforward: They are inversely proportional. The longer the string, the little the frequency. Divide the magnitude successful half, and the wave is doubled for a leap of 1 octave.

Fretting the Guitar

The frets of a guitar must truthful beryllium positioned truthful that each successive fret increases the string’s wave by a facet of 1.059. Because the wave is inversely proportional to the length, each fret must efficaciously shorten the magnitude of the drawstring by the inverse of the semitone multiplier. One measurement to definitive this is arsenic 1/2 to the 1/12th power:

drawstring simplification factor = (12) 112 ≅ 0.943874...

I’ll usage the grosser approximation of 0.944 to make these sample calculations a small easier. Furthermore, let’s presume that the magnitude of the guitar drawstring from the seed to the saddle is 100 centimeters:

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

The 2nd fret gets messier. It must beryllium positioned truthful the drawstring has an effective magnitude of 94.4 times 0.944, aliases 89.1 centimeters.

You tin besides cipher 89.1 centimeters by multiplying 100 centimeters by 0.944 squared. That 2nd fret is 10.9 centimeters from the nut, aliases 5.3 centimeters beneath the first fret. The spacing betwixt the frets is decreasing already.

This process continues. Each fret must trim the magnitude of the drawstring to 0.944 times the magnitude associated pinch the erstwhile fret:

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

The abstraction betwixt the frets continues to progressively decrease. The region from the 4th fret to the 5th is 4.7 centimeters.

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

The twelfth fret is positioned astatine 100 centimeters times 0.944 to the twelfth power, aliases 50 centimeters from the saddle, efficaciously reducing the drawstring to half its length.

You tin summation the scope of the guitar by different octave pinch different 12 frets.

The 24th fret divides the drawstring successful half again.

The magnitude of the drawstring from a fret to the saddle tin beryllium calculated for illustration this:

drawstring magnitude from fret= drawstring length× (12) fret number12

The region of immoderate fret from the seed tin beryllium calculated for illustration this:

fret region from nut= drawstring length× (1− (12) fret number12 )

The seed is assumed to person a fret number of 0, truthful the fret region for the seed is besides 0. The 12th fret has a fret number of 12, truthful the fret region is half the drawstring length.

Building a Fret-Based Slide Rule

Notice that nary logarithms were progressive successful the improvement of the look for computing fret positions. Nevertheless, now that I person specified a formula, I tin make my ain fret-based descent norm without sawing immoderate much guitars successful half. I deliberation I established astatine the outset that only the first octave of frets seems to usability for illustration a accepted logarithmic descent rule. Including the 2nd octave of frets throws the full point off.

So let’s return that first octave of frets and explanation it precisely arsenic I branded the acoustic guitar pinch fractions betwixt 1 and 2:

Now let’s make different ruler the aforesaid measurement but pinch the tick marks connected the apical alternatively than the bottom:

You tin spot these 2 rulers together and usage touch aliases the rodent to move the apical ruler comparative to the bottom:

You tin effort retired a mates of the multiplications not requiring interpolation, specified arsenic 7/6 times 3/2 aliases 5/4 times 4/3, and now you mightiness announcement that it’s not exact. The tick marks don’t statement up precisely. This mightiness person been forgivable erstwhile awkwardly grappling pinch 2 pieces of a guitar sawed successful half, but if the guitar frets were spaced logarithmically, that would not happen.

I can’t make these rulers longer by including the 2nd octave of guitar frets. We already cognize that’s a disaster. However, logarithmic descent rules are self-similar. The portion of the descent norm from 2 to 4 is the aforesaid arsenic from 4 to 8 isolated from that the numbering is double:

I tin moreover crank the scales up to 10:

Try 5/2 times 8/3, which should beryllium 20/3, but it conscionable doesn’t statement up right.

So is this existent aliases is it fantasy?

A Tale of Two Ruler Scales

It’s annoying that a descent norm constructed from the mathematics of guitar frets seems to activity almost arsenic good arsenic a descent norm constructed from logarithms, but not exactly.

The pursuing schematic is not a descent rule. It is not interactive. Don’t effort to descent anything. The apical ruler standard is constructed based connected the look for positioning guitar frets, and the bottommost is constructed from logarithms:

I’ve stretched it retired truthful you don’t person to look intimately to spot that these are 2 different scales that only coincide astatine 1 and 2.

It’s besides revealing to chart the 2 functions beyond the scope of these scales. Earlier I showed a look for calculating the region of each fret from the nut:

fret region from nut= drawstring length× ( 1− (12) fret number12 )

I want to accommodate this look for a descent rule: The region of each tick people is based connected the number associated pinch that tick mark. That number n relates to the fret number for illustration this:

n=1+fret number12

Or:

fret number=12×(n−1)

Also, because the standard from 1 to 2 is half the drawstring length, let’s besides normalize the drawstring magnitude to 2 truthful the look for positioning a tick people becomes:

tick distance= 2− (12) n−2

This will beryllium the reddish statement successful the graph. When n equals 1 (the opening of the scale), the tick region is 0; erstwhile n equals 2 astatine the extremity of the scale, the tick region is 1.

The tick region successful the logarithmic standard from 1 to 2 is simply:

tick distance= log2 (n)

That’s the bluish statement successful the graph. Again, erstwhile n equals 1 astatine the opening of the scale, the tick region is 0, and erstwhile n equals 2, the tick region is 1.

Although the reddish and bluish lines look to coincide betwixt n adjacent to 1 and 2, these are decidedly not balanced functions!

Here’s what happened: Deceived arsenic I was by the cruelly deceptive illustration connected the screen of Calculating pinch Tones, I contorted my first labeling of the guitar frets to scope from 1 to 2. That implied a definite usability that of people coincided astatine those 2 points pinch the logarithm, but was different rather different. The remainder is fundamentally coincidence.

One of the themes of The Lost Art of Logarithms is that galore phenomena successful our earthy and human-made world tin beryllium interpreted logarithmically. That includes euphony but it does not see guitar frets.

Addendum: The Geometric Approach to Fret Positioning

Earlier I described the theoretical mathematics down positioning the frets connected a guitar. But the guitar has been astir for hundreds of years. How did group fig retired the positions of the frets without calculators and precision measuring instruments? For example, really were the frets positioned connected the guitar successful 1 of Johannes Vermeer’s past paintings, The Guitar Player, making love from 1672?

Public domain image from Wikipedia via Wikimedia Commons

One celebrated attack was to usage a elemental ratio from a book published successful Florence successful 1581 entitled Dialogo della musica antica et della moderna aliases “Dialogue of Ancient and Modern Music.”

The writer was Vincenzo Galilei, calved adjacent Florence astir apt sometime successful the 1520s. He studied lute astatine a young property and became a singer, teacher, and composer. Some of his euphony has been recorded: The medium The Well-Tempered Lute has a action of his compositions. He studied euphony mentation pinch Gioseffo Zarlino, and past refuted immoderate of Zarlino’s theories. His eldest son, Galileo, who was astir 17 erstwhile his begetter published the Dialogo, was later to execute immoderate renown successful the fields of astronomy and earthy philosophy, and besides wrote astir euphony mentation successful 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 pinch a complete English translator and overmuch commentary.

Here’s wherever Galilei discusses really to cipher fret positions for a lute:

I truthful disagreement the full statement A B into eighteen parts, and toward the precocious portion (moving distant from the debased part) wherever that first portion ends, I spot the first fret. I disagreement erstwhile much the full remainder of the aforesaid number of parts [i.e. eighteen] and from the very aforesaid band, I spot the first fret nether the second. I now proceed to administer the abstraction which remains beneath the frets, ever successful that very aforesaid order, up to [the number of] twelve, which brings maine precisely to [the point] wherever half of the full drawstring terminates. (pp. 299–300, bracketed phrases added by translator; corresponds to page 49 successful 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= 1817 ≅1.058824...

If you multiply 18/17 by itself 12 times, you won’t travel retired to 2. Instead, you get 1.985560, which is rather close.

Similarly, the drawstring simplification facet for calculating the position of frets is:

String simplification factor= 1718 = 0.944444...

If you return this ratio to the 12th power, it’s a small much than 1/2:

(1718) 12 ≅ 0.50364...

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

If you analyse immoderate treatise of euphony mentation from the clip of the Ancient Greeks done the Renaissance, you’ll spot a batch of geometrical drawings. When mixing arithmetic and geometry, ratios are perfect because you don’t person to measurement thing aliases get progressive pinch fractions. It’s each geometrical.

Vincenzo Galilei doesn’t explicate very good really the 18:17 ratio is utilized successful practice, but it’s easy to create a technique. Suppose you were building a lute aliases guitar, aliases possibly you’ve obtained a fretless guitar to which you want to adhd immoderate frets.

The first measurement is to get a ample expanse of insubstantial and tie a statement conception corresponding to the region betwixt the seed and the saddle:

Let’s telephone this the string line. Put the guitar aside, and tie a agelong horizontal statement perpendicular to the drawstring statement and aligned pinch its bottom:

Somewhere on that horizontal line, tie different shorter vertical line:

On that shorter vertical line, moving from the bottommost up, tie 18 evenly spaced horizontal tick marks:

This is thing that tin beryllium done pinch the accepted geometric drafting devices of compass and straightedge. The region betwixt these horizontal tick marks doesn’t matter, conscionable arsenic agelong arsenic they’re evenly spaced. If you’re lazy, you tin moreover usage an 18-inch ruler for marking this line.

Now tie a consecutive statement from the apical of the drawstring statement to the apical tick of the shorter vertical statement and proceed down to the base:

I’ve drawn a small dot wherever it crosses that bottommost horizontal. From that small dot, tie different diagonal that passes done the 2nd tick people and intersects the drawstring line:

Now get free of that smaller vertical statement pinch the tick marks. You don’t request it immoderate more:

I contend that immoderate vertical statement drawn from the apical diagonal to the guidelines will beryllium divided into 2 parts: the apical portion (between the 2 diagonals) is 1/18th of the full length. The bottommost portion (under the bottommost diagonal) is 17/18th of the full length.

The drawstring statement is 1 specified vertical line. The 2 diagonals disagreement that drawstring statement into 1/18th betwixt the 2 diagonals and 17/18th beneath the bottommost diagonal. The first fret goes wherever the bottommost diagonal intersects the string:

To find wherever the 2nd fret goes, first, widen a horizontal statement from the fret to the apical diagonal. This is shown present arsenic a dotted line:

You tin ideate a vertical statement drawn from the constituent wherever the dotted statement meets the apical diagonal down to the base. That statement is besides divided successful 2 parts. Draw conscionable a portion of that vertical statement down to the 2nd diagonal. This is besides a dotted line:

Now tie a horzizontal statement backmost to the drawstring line:

That is the position of the 2nd fret:

As you support going, you’ll spot that the region betwixt the frets is shrinking. That’s because each successive fret is 17/18 of the remaining magnitude of the drawstring aft the erstwhile fret. Here’s really to find the 3rd fret:

And here’s the fourth:

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

Keep going and you tin adhd different 12 frets culminating pinch the two-octave fret:

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

Vincenzo Galilei’s defense of the 18:17 ratio was apparently rather persuasive. In his basal book Tuning and Temperament: A Historical Survey (Michigan State College Press, 1951, 1952; Dover Publications, 2004), musicologist James Murray Barbour reports uncovering that “references to the 18:17 semitone screen 2 and a half centuries.” (p. 59) In different words, group were utilizing this ratio good into the 19th century. That’s why I presume that the guitar successful Vermeer’s coating was fretted pinch this technique.

Let’s spot really it useful successful practice, aliases astatine slightest successful a virtual guitar.

Here are 2 “guitars” pinch each of the six strings branded astatine the apical pinch their pitches. The guitar connected the near uses Vincenzo Galilei’s 18:17 ratio for determining the fret positions. The 1 connected the correct uses the twelfth guidelines of 2 for the calculations.

On a touch screen, you tin pluck a drawstring connected these 2 guitars pinch your finger. Otherwise you’ll request to resistance the rodent pointer crossed a drawstring pinch the fastener pressed. To play the statement associated pinch the unfastened string, pluck the drawstring nether the seed adjacent the top. For notes requiring frets, pluck the drawstring correct nether the fret.

A brace of dots are drawn correct supra the octave and two-octave frets. These dots are recovered connected immoderate guitars to thief orient the musician. The slider successful the lower-left area is simply a measurement control.

If you person a multitouch display, you tin pluck 2 strings simultaneously, possibly the aforesaid drawstring connected the 2 different guitars. At the very top, the frequencies will beryllium identical, but arsenic you get little down the frets, the frequencies deviate and you mightiness commencement to perceive beats.

It is apt that guitar builders who utilized Vincenzo Galilei’s 18:17 ratio made subtle adjustments to the frets truthful that a axenic octave interval could beryllium realized.