Pythagorean TemperamentA pentatonic musical scale can be devised with the use of only the octave, fifth and fourth. It produces three intervals with ratio 9/8 and two larger intervals. If a 9/8 (whole tone) interval is carved out of the larger ones, a smaller (semitone) interval is left: BC and EF. This creates a Pythagorean diatonic scale. If the semitone thus created is taken from the whole tone, a chromatic semitone of different size is left over. This leads to some of the difficulties of Pythagorean temperament and other temperaments  such difficulties ultimately led to the development of equal temperament. The use of the octave and the fifth (ratio 3:2) is attributed to Pythagorus in the sixth century BC.

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Pythagorean IntervalsThe Pythagorean temperament produces two different semitones. When expressed in cents, it becomes evident that their differences are well above the just noticeable difference in pitch. This leads to certain difficulties which ultimately led to the development of equal temperament.

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Temperament ProblemsFor over two centuries the predominant musical scale used, at least for western music, has been the equal tempered scale. The ability to freely modulate between musical keys and the equivalence of all musical keys were strong enough features to overcome reservations about the compromises made with the smallintegerratio rule. The kinds of problems which led to this compromise scale were:

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The Whole ToneIn the buildup of a pentatonic scale using the musical intervals which have been found to be universally consonant ( octave, fifth, and fourth), an interval of ratio 9/8 naturally emerges. This interval satisfies the basic condition for consonance and it occurs in the basic pentatonic scale. This suggests the carving of another such whole tone from the larger interval remaining, leaving the smaller 256/243 interval. This whole tone is used in the Just and Pythagorean temperaments. In devising the equal tempered scale it was important to maintain not only the octave, fifth, and fourth at close to their just values, but also to maintain the whole tone. Expressed in cents notation, the natural whole tone is 204¢, compared to 200¢ for the equal tempered whole tone, just within the accepted 5¢ just noticeable difference. 
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