ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

How Guitar Fret Distances are Designed

Updated on October 27, 2009

This study is surely one that a person wanting to learn about the properties of vibrating strings and its application to string instruments should take up, since it includes so much of value, especially as a concrete example of vibrating strings fixed at both ends , its properties and application to the guitar, a very familiar musical instrument. Whether you are a musician, a physicist or anyone who enjoys guitar music, have you ever stopped to think about how to compute the correct fret positions on the guitar fingerboard?

Peré Mersenne (1588-1648) asserted that the pitch of a string is inversely proportional to the length and the square root of the linear mass density, and directly proportional to the square root of the tension:

These mathematical laws obeyed by vibrating strings as outlined by Mersenne can be learned from experiment and this is sufficient for practical purposes.

In this hub, the properties of vibrating strings fixed at two ends is derived by reasoning from some general principles of wide application and the result includes Mersenne’s’ rules, not only would this manner strengthen the validity of the principles, but would give practice in applying them.

Vibrating Strings Fixed at Both Ends

A guitar is a common example of a string fixed at both ends which can vibrate. When we pluck it, the string starts to vibrate in one or more loops. The vibrations of such a string are called standing waves and they satisfy the relationship between wavelength and frequency that comes from the definition of waves:

where v is the speed of the wave, f is the frequency (measured in Hertz, Hz) and λ is the wavelength.

The velocity of a wave depends on the properties of the medium it travels. The velocity of a wave on a stretched string depends on the tension ‘T’ in the string and on the string’s linear mass density ‘µ

These standing waves may be thought of as the result of waves that travel down the string in both directions, reflected at the end and re-reflected at the other, and so on. The frequencies at which the standing waves are produced are the natural frequency or resonance frequency of the string.

                The fundamental frequency, the main pitch you hear, is the lowest tone and it comes from the string vibrating with one big arc from bottom to top. The fundamental satisfies the condition

where λ is the wavelength of the fundamental and L is the length of the freely vibrating portion of the string, it correspond to one antinode (or loop). Since the string is fixed at both ends, so any vibration of the string must have nodes at each end. That limits the possible vibrations of the string (‘mode of vibration’ in simple terms just means style or way of vibrating). The corresponding frequencies at which the standing waves are produced are the natural or resonance frequencies of the string.

                The other natural frequencies are called overtones; when they are integral multiples of the fundamental, they are also called harmonics. In general we can write

In order to find the fundamental frequency f of the vibration, the equations

Thus, the formula for the fundamental frequency of a string fixed at both ends (Eqn 1).

Equation 1 is the basis for the design of stringed instruments such as piano, violins and guitars. To ‘tune’ a string-instrument, the tensions in the various strings are adjusted until their fundamental frequencies are correct. This equation shows the relationship between the fundamental frequency, length and linear mass density of a vibrating string fixed at both ends.

                 It includes Mersenne’s rules and accounts for the different masses and tensions of strings of the same length to create different pitches of guitar strings. Individual pitches on the strings are created by pressing down beneath a fret, thus effectively shortening the length of the string.

The lowest note on the piano vibrates at 27.5 Hz and the highest vibrates at 4,186 Hz. In guitar, the open 6th string is the lowest note which vibrates at 82.4 Hz (Table 1). This note is called E note denoted by E0. If you double the frequency, you get the note E1, one octave higher than E0 which has a frequency of 164.8 Hz. ‘Octaves’ of a note are just multiples of the original frequency. In general

where F0 is any given note’s frequency and Fn is the frequency of the note n octaves above F0 where n=1,2,3…

                ­The highest note on a guitar of twelve frets is the note E3 (that is three octaves up from E0). From one note up to its octave, the average human ear can distinguished only 12 different tones or notes. So when a chromatic scale (comprising all twelve notes in an octave) is played, we perceived that all sounds between the first note and the last note (the first note’s frequency doubled) are included.

The twelve notes in a chromatic scale are A, Bb, B, C, Db, D, Eb, E, F, F#, G, G#. The equal-tempered tuning is the standard system in music and it assigns the corresponding frequencies to the notes. Tuning is not linear. One has to double the frequency to get the next octave up; one has to quadruple it to get the next octave after that. Consequently, the notes within the chromatic scale are not equally distributed in frequency (or in length). So in general practice, musicians universally assigned A3 to a frequency of

FA3=440 Hz and therefore A4 will be at FA4=880 Hz. We then take logarithms of A3 and A4 frequencies. Next, we mark in 11 equally spaced points between

We then apply arc-logarithms to those points and arrive at the half-step ratio r of notes in the equal tempered tuning. Therefore each successive note in the chromatic scale is r higher in frequency.

On a guitar, different strings have different linear mass density. The first string is like a thread, and the sixth string is wound much thicker and heavier. The tension in the strings is controlled by the tuning pegs. The length of the open string (also called the scale length) is the distance from the nut to the saddle. When you press down on a string at a fret (assuming tension is constant), you change the length of the string, and therefore its frequency when vibrating. The frets are spaced out so that proper frequencies are produced when the string is held down at each fret. Fig 1 shows the guitar fingerboard.

Table 2 shows the frequencies of the notes in the first string and the twelve fret bar positions on the fingerboard based on scale length of 26 inches.


    0 of 8192 characters used
    Post Comment
    • profile image


      6 years ago

      This is an inneffective table as the string length is not always 26". My equation d=L(1-2^-n/12) is much more usefull as d represents the distance from to nut to whatever fret no matter what the length of the string. It is only derived from combining f=f0*2^n/12 and knowing that frequency is inversly proportional to Ld. the part in brackets also tells us the protion of the string at each fret and when 5 and 7 is plugged into n. the portion of the string is 0.33 and 0.25. which explains why there are such strong harmonics as you are forcing a node on relevant points.

    • profile image

      9 years ago

      that's pretty cool

    • Bnji profile imageAUTHOR


      9 years ago from Iligan City, Philippines

      Thanks william for reading this hub. This is actually a research paper i created for our physics conference when i was still in college. I love classical guitars.

    • William Tan Seng profile image

      William Tan Seng 

      9 years ago

      Wow, I did not know about the complexity of design based on computation until reading this post.

      Thanks for the information.


    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at:

    Show Details
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the or domains, for performance and efficiency reasons. (Privacy Policy)
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
    ClickscoThis is a data management platform studying reader behavior (Privacy Policy)