Just intonation

Just intonation is any musical tuning in which the frequencies of notes are related by whole number ratios. Any interval tuned in this way is called a just interval; in other words, the two notes are members of the same harmonic series. Although in theory two notes tuned in the frequency ratio 1024:927 might be said to be justly tuned, in practice only ratios using quite small numbers tend to be called just. Intervals used are then capable of being more consonant.

Why isn't just intonation used much?

Some fixed just intonation scales and systems, such as the diatonic scale above, produce wolf intervals. The above scale allows a minor tone to occur next to a semitone which produces the awkward ratio 32:27 for C:A, and still worse, a minor tone next to a fourth giving 40:27 for E:A. Moving A down to 10/9 alleviates these difficulties but creates new ones: D:A becomes 27:20, and A:F# becomes 32:27.

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You can have more frets on a guitar to handle both A's, 9/8 with G and 10/9 with G so that C:A can be played as 6:5 while D:A can still be played as 3:2. 9/8 and 10/9 are less than 1/53 octave apart, so mechanical and performance considerations have made this approach extremely rare. And the problem of how to tune chords such as C-E-G-A-D is left unresolved (for instance, A could be 4:3 below D (making it 9/8, if G is 1) or 4:3 above E (making it 10/9, if G is 1) but not both at the same time, so one of the fourths in the chord will have to be an out-of-tune wolf interval). However the frets may be removed entirely -- this, unfortunately, makes in-tune fingering of many chords exceedingly difficult, due to the construction and mechanics of the human hand -- and the problem of how to tune chords such as C-E-G-A-D in just intonation remains unresolved.

Related Topics:
Fret - Guitar

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For many instruments tuned in just intonation, you can't change keys without retuning your instrument. For instance, if you tune a piano to just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys impractical to impossible.

Related Topics:
Keys - Modulation

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Synthesizers have proven a valuable tool for composers wanting to experiment with just intonation. Many commercial synthesizers provide the ability to use built-in just intonation scales or to program your own. Wendy Carlos used a system on her 1986 album Beauty in the Beast, where one electronic keyboard was used to play the notes, and another used to instantly set the root note to which all intervals were tuned, which allowed for modulation. On her 1987 lecture album Secrets of Synthesis there are audible examples of the difference in sound between traditional equal temperament and just intonation.

Related Topics:
Synthesizer - Wendy Carlos - 1986 - 1987

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If the value of the major and minor tones are adjusted so that they are both equal, one gets a meantone temperament. This eliminates the wolf intervals from the diatonic scale without too much damage to the pure consonances; hence meantone tuning was the primary keyboard tuning used in Western music from about 1480 to 1780. A 12-note keyboard in meantone is limited to six major keys (typically B-flat, F, C, G, D, and A) with no wolf intervals; it was not uncommon during this era to find keyboard with split keys so that more than 12 notes, and thus more than six wolfless major keys, could be accessed.

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If in addition the semitone is altered so that an interval of two semitones is equal to one tone, you get the 12 notes used in modern Western music (see equal temperament), which allows one to travel through twelve equally consonant keys with only 12 notes on an instrument. The adventurous harmonic paths blazed by composers such as Beethoven and Schubert require such a closed system, though in their day keyboard tunings were probably somewhat unequal as in a well temperament. Regardless, the influence of such composers was such as to lead to the total abandonment of meantone temperament, which held out longest (until the 1850s) only in England and Spain. 12-note equal temperament does quite a bit more damage to the consonances than meantone temperament; today, many musicians, after exposure to tunings with purer harmonies, refuse to return to 12-note equal temperament due to its noticeably less-pure consonances.

Related Topics:
Tone - Equal temperament - Beethoven - Schubert - Well temperament

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