๐ต String Harmonics Calculator
The nth harmonic's frequency = fundamental ร n. Nodes sit at k/n of the string length (k = 1 to n-1).
Enter a string's fundamental (open-string) frequency and a harmonic number (2nd, 3rd harmonic, etc.) to calculate the pitch produced by that harmonic and the positions of its "nodes" โ the points along the string that don't vibrate. Useful for understanding harmonics (artificial/natural harmonics) on stringed instruments.
How to use
- Enter the string's fundamental (open-string) frequency (Hz).
- Select a harmonic number (2nd through 8th harmonic).
- The resulting frequency and node positions are shown in the diagram.
How the calculation works
The sound of a string or wind instrument contains, besides the lowest note (the fundamental), whole-number multiples of its frequency called harmonics (overtones). Enter the fundamental and a harmonic number n to see the frequency of the nth harmonic and a diagram of where the nodes (points that do not move) fall on a string vibrating in that harmonic. Frequency of harmonic n = fundamental ร n Node positions = k/n of the string length (k = 1 to nโ1) The mix of harmonics is why instruments sound different even when playing the same pitch.
Worked example
Fundamental 110 Hz (guitar 5th string, A2) 2nd harmonic 220 Hz (an octave up): node at the middle (12th fret) 3rd harmonic 330 Hz (an octave and a fifth up): nodes at 1/3 and 2/3 (near the 7th and 19th frets) 4th harmonic 440 Hz (two octaves up): nodes at 1/4, 1/2 and 3/4 (such as the 5th fret) 5th harmonic 550 Hz (two octaves and a major third up)
Things to be aware of
- Guitar harmonics are played by lightly touching a node and plucking, so only that harmonic sounds.
- Some harmonics do not match equal temperament: the 7th harmonic is about 31 cents flatter than an equal-tempered minor 7th.
- Brass players change pitch largely by switching between harmonics with their lips.
FAQ
What is a "node"?
A node is a point on a vibrating string where the amplitude stays at zero. For the nth harmonic, nodes appear at points dividing the string into n equal parts (excluding the two ends). The red dots in the diagram mark the node positions.
What's the formula?
The nth harmonic's frequency is n times the fundamental (f = n ร fโ). Node positions sit at k/n of the string's length (k = 1, 2, ..., n-1). For example, the 3rd harmonic has nodes at 1/3 and 2/3 of the string.
How does this relate to playing harmonics on a guitar?
Lightly touching a string at a node position while plucking produces just that harmonic's pitch โ this is how "harmonics" are played. Touching at the 12th fret (the string's midpoint, a node of the 2nd harmonic) sounds an octave up; touching near the 7th or 19th fret (nodes of the 3rd harmonic) sounds an octave plus a perfect fifth up.