If you’re a string instrument player who enjoys figuring out tunes by ear, you may have encountered a strange, subtle dissonance that doesn’t appear to make sense when comparing what you hear in a recording to what you hear when you to try to play that same song yourself. Sure, you could be playing the wrong notes altogether, but it may very well be the way you tuned your instrument.
Back in the early days of music, musicians tuned pianos, violins, harps, and other stringed instruments according to harmonic tuning. This tuning method uses – yes you guessed it – harmonics! Each note creates a unique sound wave and when these waves create consistent, repeating patterns, they’re referred to as harmonics.
Every harmonic wave is related to other harmonic waves by whole number ratios (Henderson, 2019). This can be seen on one particular string when the vibration is in motion. In the chart below, each dotted line represents a string, and the number of crests and troughs (high and low points) in each wave represents the name of the harmonic:
This continues on and on, though you can’t easily hear harmonics as they get higher.
On a stringed instrument, let’s say you want to play a harmonic on a certain string, you would lightly touch the string at one of the nodes which are “located at 1/2, 1/3, 1/4, etc. the length of the string” (Zukofsky, 1968). In the chart above, the nodes on each dotted line are located where the wave intersects it (excluding the two outer ends). There are zero nodes in the 1st harmonic, one in the 2nd harmonic, two in the 3rd harmonic, and so on.
Since the waves are related in whole number ratios, so are their frequencies (the rate at which they vibrate). When a player plays harmonic notes together or in sequence, the notes are literally in “harmony” and sound pleasing.
This tuning method works perfectly when you play a song in one key. However, if you play a song that changes key, or want to play multiple songs that are in different keys without having to retune your instrument every time you switch songs, a problem arises…
To get from one note to the next octave, let’s say middle C to the next C up, you have to travel 12 notes total. In this example it would be: C#/Db, D, D#/Eb, E, F, F#/Gb, G, G#/Ab, A, A#/Bb, B, and C. The frequency of that next C should be exactly 2 times the frequency of middle C (which is a ratio of 1:2). But, no matter which harmonic interval ratio you tried to tune all the notes to, be it perfect fifths, perfect fourths, major thirds, etc., the numbers never even out perfectly to a whole number ratio (Hudson, 2015).
Equal Tempered Tuning
To get around this problem, “equal tempered tuning” was developed. Proposed ever since the late 1590s and largely adopted from the 18th century on (“Equal Temperament”, 2019), this tuning method uses the concept of dividing all the notes evenly.
In equaled tempered tuning, the frequency of each note is the 12th root of 2 multiplied by the frequency of the note just before it (Hudson, 2015). This enables the frequency of the C above middle C to be exactly twice the frequency of the middle C – an exact ratio of 1:2. Voilà!
If you’ve heard of “musical cents”, each equally tempered note is worth 100 cents and an octave is equal to 1200 cents (Davids, 2019).
Unfortunately, this whole number ratio only works out for octaves, so all the other harmonic ratios are slightly off (a little sharp or flat), but for almost all intents and purposes, they’re close enough where you don’t have to keep retuning.
Today, you will scarcely hear anything other than equal tempered tuning as all mainstream tuners and just about every music education program and professional performance group uses this method. Yet when you do happen to come across harmonic tuning, you’ll be sure to notice it, and honestly, it sounds a little nicer!
How to Tune Using Harmonic Intervals
When you want to tune harmonically, you can still make some use of your tuner, especially if it displays exact frequencies, but it’s probably easier to use your ears.
If you’re playing a violin, viola, cello, bass, or mandolin, tune using perfect fifths. Similarly, a mandolin player should tune each pair of strings in perfect fifths. Please note that this won’t always work for international or Medieval instruments in the violin family (which can have more than 4 main strings and/or drone strings).
6-string guitar players can tune using all perfect fourths except for a major third between the G and B strings. 7-string guitars follow the same pattern, adding in one more string that is tuned down a perfect fourth. 12-string guitars are tuned identically to their 6-string counterparts, but each string pair is an octave apart (thin string is high, thick string is low), with the B and E strings pairs sharing identical pitches. 4-string bass guitar players can also tune using all perfect fourths.
6-String Guitar: EADGBe
7-String Guitar: BEADGBe
12-String Guitar: EEAADDGGBBee
4-String Bass Guitar: EADG
Soprano ukuleles, concert ukuleles, and tenor ukuleles all follow the same pattern of down a perfect fifth, up a major third, and up a perfect fourth. Baritone ukuleles however go up a perfect fourth, up a major third, and up a perfect fourth. (All ukes share those last two string interval relations.)
Soprano, Concert, and Tenor Ukulele: GCEA
Baritone Ukulele: DGBE
The traditional 5-string banjo is tuned first down a perfect eleventh, then up a perfect fourth, up a major third, and up a minor third. The 6-string banjo is tuned just as a standard guitar, but produces that wonderful, twangy banjo tonal quality, and the tenor banjo is tuned with all perfect fifths.
5-String Banjo: GDGBD
6-String Banjo: EADGBE
Tenor Banjo: CGDA
When tuning a piano, pick a key you like and tune according to that particular key alone. (Just remember you’ll have to change the key again if your song does.)
Harps vary in tuning with most folk and Celtic harps being tuned to a particular key. However, they skip notes so not all notes in the key will be present. Pedal and lever harps also skip notes, but the pedals/levers enable you reach the notes that are missing by altering the pitch of the strings.
Alternate Tunings and Choices
There are a bunch of alternate tunings for various music styles, also called “cross tuning” or “scordatura” (Italian for “detune”). Alternate tunings can use either harmonic or equal tempered tuning styles, but you’d tune the strings in a slightly different order, or to completely different notes than they are normally tuned to. These exist for just about every string instrument, even the traditional orchestral instruments.
Some guitar and bass players prefer multi-scale guitars/basses with fanned frets, giving them a different scale range altogether, while others use fretless guitars/basses to further push the intonation boundaries of their sound. Once in a while, players create their own tunings, like famous artist Jewel.
With so many tunings to choose from, you’ll never run out of options. Now when you encounter a musician or music group who decided to incorporate one of these tuning styles, you’ll be able to play their music the way they do, or reconfigure it in your own unique style. So have fun and tune on to your liking!
Davids, Paul. “Why didn’t Frusciante Tune his Guitar?” YouTube, uploaded by Paul Davids, 13 May 2019, www.youtube.com/watch?v=Daw93bRHe4Y.
The Editors of Encyclopaedia Britannica. “Equal Temperament.” Encyclopaedia Britannica, 2019, www.britannica.com/art/equal-temperament.
Henderson, Tom. “Fundamental Frequency and Harmonics.” the Physics Classroom, Sound Waves and Music-Lesson-4 – Resonance and Standing Waves, 2019, www.physicsclassroom.com/class/sound/Lesson-4/Fundamental-Frequency-and-Harmonics.
Hudson, Alex. “Why it’s impossible to Tune a Piano.” YouTube, uploaded by MinutePhysics, 17 September 2015, metro.co.uk/2015/09/18/its-impossible-to-tune-a-piano-5398180.
Zukofsky, Paul. “On Violin Harmonics.” Perspectives of New Music, Vol. 6, No. 2, Princeton University Press, 1968, www.musicalobservations.com/publications/harmonics.html.