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Just intonation

22 July, 2019 - 10:18
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Just intonation is the system of tuning that is often used (sometimes unconsciously) by musicians who can make small tuning adjustments quickly. This includes vocalists, most wind instruments, and many string instruments. Look again at the harmonic series (Figure 6.1)

Figure 6.3 Just intonation
Both the 9:8 ratio and the 10:9 ratio in the harmonic series are written as whole notes. 9:8 is considered a major whole tone and 10:9 a minor whole tone. The difference between them is less than a quarter of a semitone. 

As the series goes on, the ratios get smaller and the notes closer together. Standard notation writes all of these "close together" intervals as whole steps (whole tones) or half steps (semitones), but they are of course all slightly different from each other. For example, the notes with frequency ratios of 9:8 and 10:9 and 11:10 are all written as whole steps. To compare how close (or far) they actually are, turn the ratios into decimals.

Whole Step Ratios Written as Decimals

  • 9/8 = 1.125
  • 10/9 = 1.111
  • 11/10 = 1.1
Note: In case you are curious, the size of the whole tone of the "mean tone" system is also the mean, or average, of the major and minor whole tones.

These are fairly small differences, but they can still be heard easily by the human ear. Just intonation uses both the 9:8 whole tone, which is called a major whole tone and the 10:9 whole tone, which is called a minor whole tone, in order to construct both pure thirds and pure fifths. Because chords are constructed of thirds and fifths (see Triads), this tuning makes typical Western harmonies particularly pleasing to the ear.

The problem with just intonation is that it matters which steps of the scale are major whole tones and which are minor whole tones, so an instrument tuned exactly to play with just intonation in the key of C major will have to retune to play in C sharp major or D major. For instruments that can tune almost instantly, like voices, violins, and trombones, this is not a problem; but it is unworkable for pianos, harps, and other instruments that cannot make small tuning adjustments quickly.

As of this writing, there was useful information about various tuning systems at several different websites, including The Development of Musical Tuning Systems, where one could hear what some intervals sound like in the different tuning systems, and Kyle Gann's Just Intonation Explained, which included some audio samples of works played in just intonation.