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Intonation: Why what when how


By Bayla Keyes



  • Acoustic reality based on the overtone series and the system discovered by the Greek mathematician Pythagoras in around 530 B.C.E. 

  • Fifth based, therefore ideal for violins

  • To use, tune with as many perfect fifths and their inverse, perfect fourths, as possible

  • Whole steps and all major intervals will be wide, half steps and all minor intervals narrow

  • Used by string soloists when playing with orchestra because it brings them out of texture

  • Will make violin ring and project because of maximum sympathetic vibration of open strings

  • Will increase emotional tension

To use expressive intonation, slightly raise the third and seventh tones in major scales (most often the sixth as well); lower the third tones in minor scales. Slightly exaggerate all accidentals-- put sharps higher and flats lower. (In this system, an Eb will be lower than a D#!) Tune all notes possible to open strings using pure octaves and fourths and when necessary, combinations of these intervals. Half steps should be as close as possible. This will bring an expressive, intense quality to your intervals, and will be particularly helpful in fast passagework, where the ear perceives half-steps differently. Note that this intonation is most frequently used in single lines in concertos and solo works. Sprinkle it sparingly, like a spice, when playing with piano or in chamber ensemble!

Equal Tempered

  • Compromise system devised in the sixteenth century

  • Divides octave into twelve equal half-steps, ideal for keyboard instruments and multiple keys

  • Only intervals which ring are the octaves. No perfect fifths or fourths because fifths are too small, fourths are too wide

  • Whole steps and all major intervals will be narrower than in Pythagorean system, half steps and all minor intervals wider

Using equal temperament is advisable in many if not most situations, e.g. orchestral playing, matching the piano or in combination with winds.  However you can expect to be using Pythagorean intervals as well, in order to make perfect ringing fourths and fifths with other musicians. Equal tempered intervals will not ring because, except for octaves, they do not create overtones. 

Just – for Thirds and Sixths Only!

  • Acoustic reality based on subtones. Correct size of thirds and sixths will produce a barely audible subtone which is actually  a harmonic of the root of a triad containing them

  • Described in 1482 by the Spanish theorist Bartolomeo Ramos de Pareja 

  • Major thirds and sixths will be narrow, minor thirds and sixths wide

  • To use, listen for the “third tone” while playing 3rds and 6ths; adjust until it coalesces

  • Exclusively used while playing thirds and sixths, either by themselves or when tuning inside a chord


This is a specialized intonation, used when playing thirds or sixths alone or when tuning chords in a string or wind ensemble. Do not use it when playing with piano!

Comparison of Interval Sizes

Major  Expressive: Widest Equal-Tempered: Narrower Just: Narrowest (3rds & 6ths only)

Minor Expressive: Narrowest Equal-Tempered: Wider Just: Widest (3rds & 6ths only)


Simple Way to Organize and Practice Your Intonation Systems


To use expressive intonation, divide your pitches into three families:

  • The BRIGHT family contains those pitches which relate up to the open E string, e.g. E, B, F#, C#, G#, D#, A# and all other sharps.

  • The DARK family contains those pitches which relate down to the open G string, e.g. G, C, F, Bb, Eb, Ab, Db, Gb, and all other flats.

  • The CROSSOVER pitches are the A and D strings.


To practice expressive intonation, tune your violin to these pitches:

E at A = 442 or 443; move drone to E

A at 441

D at A = 441; move drone to D

G at A = 440; move drone to G

Put your drone on A = 441 to practice Ds and As.

Put your drone on A = 440 and then move the pitch to whichever member of the dark family agrees with the most pitches in a section of your piece.

Put your drone on A = 442 or 443 and then move the pitch to whichever member of the bright family agrees with the most pitches in a section of your piece.



The above example shows how this classification of pitches will result in the wide whole steps and narrow half steps which are a feature of expressive or Pythagorean intonation. I have marked the members of the bright family.


To use equal tempered intonation, simply keep your drone at A = 441 and move to a pitch which will agree with the most pitches in a section of your piece.


To tone your ear, practice your scales using both kinds of intonation!


Examples of Tuning with Open Strings to Create Pythagorean Intervals







Examples of Creation of Subtones when Using Just Intonation

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