Tag: relationship between string length and frequency

(Guitar) String Theory 2: Why Do Frets Get Closer Together? 

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

Frets on a guitar are placed 1 semitone apart. The 12th fret produces a note one octave above the open (full-length) string.

The Relationship Between Pitch And Frequency

The frequency of a note is the speed at which a sound wave vibrates in order to produce a given pitch. The lower the frequency, the lower the pitch.

The common factor between the pitch of a note and its frequency is the octave. One octave equals 12 semitones, where each semitone sounds the same distance apart as the next, like centimetre or inch markings on a ruler. 

An octave is also the frequency ratio of 2:1. Every 12 semitones higher, the frequency doubles. We can look at the relationship between sound waves and what we hear by creating a graph with pitch on one axis and frequency on the other. It would look something like this:

The above frequencies are based on a guitar A string, A = 110Hz.

  • One octave higher = double the frequency.
  • Double the frequency = half the wavelength and thus half the string length.
  • One octave higher than the open (full-length) string is half the string length, half-way from the nut to the saddle.
  • The next octave higher is half of the remaining string length = 3/4 of the string away from the nut.

In other words, the first half of the string has 12 frets and the next quarter of the string also has 12 frets.

The effect of this relationship is that for every semitone higher in pitch, the frequency increases by a little bit more than the last semitone.

The Relationship Between Frequency And String Length

Frequency and wavelength are inversely related: as one goes up, the other goes down. As the frequency increases, the wavelength, and thus the string length, becomes smaller, a little less so for each semitone. 

Strings are effectively half a wave. Higher notes are produced by making the playing part of the string, and thus the wave length, shorter. For each semitone higher, the adjustment is a little less than the previous semitone. The frets mark these positions.

Why do we care? Maybe we don’t need to, but isn’t it nice to know why frets are laid out differently from piano keys?

(Guitar) String Theory 1: Strings and Octaves

This post is one of a growing series of holistic investigations into various aspects of music theory. The full list can be found in the Posts page under the category Music Theory De-Mystified.

All comments are welcome. If you enjoy my post, please give it a like and share it or subscribe to my blog.

A plucked guitar string is a good physical representation of half a sound wave. 

Sound waves, like ripples in a pond, are wave shaped pulses that travel and spread away from the source. Single frequencies have an evenly-curved shape called a sine wave. A complete wave, from the start to where it begins to repeat, is called a cycle.

One Wave Cycle

Unlike ripples in a pond, a string on a guitar (or any string instrument) is fixed and doesn’t travel. A vibrating string produces half a sine wave at a time, moving gradually upward then downward for each wave cycle. (The full sine wave is twice the length of the string.)

A Guitar (or other stringed instrument) String Is Half A Sine Wave

When you lightly touch the string above the 12th fret (half-way along its length) and pluck the string, we hear a pure sound called a harmonic. By not pressing all the way down, both halves of the string are free to vibrate: only the middle is blocked, allowing a complete sine wave of half the string length.

Guitar String With Octave Harmonic

The sound we hear is exactly one octave above the sound of the open (whole) string.

  • One octave higher = half the string length.
  • In other words, one octave higher = half the wavelength.

By the way, you can check the accuracy of a guitar’s intonation by comparing just touching the string at the 12th fret to pressing all the way down at (behind) the 12th fret. The pitch should sound the same.

In Why Are Octaves Special? we saw that one octave higher = double the frequency, so:

  • double the frequency = half the wavelength. As the frequency goes higher, the sound wave becomes shorter.

You can also place a finger lightly over the 5th fret, 1/4 of the string length, and hear a note 2 octaves above the open string, at 4x the frequency.

This is just another way of demonstrating the close relationship that exists between notes one or more octaves apart. The octave is fundamental to how music behaves. It is a universal musical phenomenon, independent of genre or culture.

Even though we don’t think of sound waves when playing or listening, I suspect that we are innately aware of them. We tend to think of bass notes as big and piccolo or tin whistle notes as little…

Bear with me- there’s a little more in the next post, (Guitar) String Theory 2: Why Do Frets Get Closer Together?