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The most important scale in the Western tradition is the Diatonic Scale , but the scales used and proposed in various historical eras and parts of the world have been many and varied. Scales may broadly be classed as scales of Just Intonation , Tempered Scales , and practice-based scales. A scale is in just intonation if the ratios between the frequencies for all degrees of the scale are either ratios of small integers, or obtained by a succession of such ratios. It is tempered if it represents an adjustment, or tempering, of just intonation. It is practice-based if it simply reflects musical practice, as for instance various measurements of the Tuning of a Gamelan might do. PYTHAGOREAN TUNING See Also: Pythagorean tuning Pythagorean tuning is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is 81:64 = (9:8)2, rather than the independent and harmonic just 5:4, directly below. A whole tone is a secondary interval, being derived from two perfect fifths, (3:2)2/2 = 9:8. JUST INTONATION See Also: Just intonation If we take the ratios constituting a scale in just intonation, there will be a largest Prime Number to be found among their prime factorizations. This is called the prime limit of the scale; a scale which uses only the primes 2, 3 and 5 is called a 5-limit scale. Below is a typical example of a 5-limit justly tuned scale, one of the scales Johannes Kepler presents in his Harmonice Mundi or Harmonics of the World of 1619, in connection with planetary motion. The same scale was given in transposed form by Alexander Malcolm in 1721 and theorist Jose Wuerschmidt in the last century and is used in an inverted form in the music of northern India. American composer Terry Riley also made use of the inverted form of it in his ‘‘Harp of New Albion’’. Despite this impressive pedigree, it is only one out of large number of somewhat similar scales. (In theory unisons and octaves and their multiples are also "perfect" but this terminology is rarely used.) To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the frequency we associate to the unison, which will often be the tonic frequency. For instance, with a tonic of A4 (A natural above middle C), the frequency is 440 Hz , and a justly tuned fifth above it (E5) is simply
The just major third, 5:4 and minor third, 6:5, are a syntonic comma, 81:80, apart from their pythagorean equivalents 81:64 and 32:27 respectively. According to Carl Dahlhaus (1990, p.187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals." TEMPERAMENT Western Common Practice Music usually cannot be played in just intonation, even when it is confined to a single key. This is because the Supertonic chord, or ii-chord, which is the most important of the minor triads in a major key, serves to bridge between the Dominant and Subdominant , having a root at once a minor third below the root of the subdominant triad, and hence sharing two of its notes, and a fifth above the root of the dominant triad or dominant seventh chord. The problem becomes still worse when Modulation , the key changes so important to common practice music, comes into play. The scale of the Western tradition is by its very nature neither one of just intonation nor one defined only in practice, but is a systematically tempered scale. The tempering can involve either the irregularities of Well Temperament or be constructed as a Regular Temperament , either some form of Equal Temperament or some other regular meantone, but in all cases will involve the fundamental features of Meantone Temperament . Meantone, however, is not the only worthwhile temperament nor is the equal division of the octave into twelve parts the only reasonable way to so divide it. Many other systems of temperament are possible, leading to a variety of harmonic relationships characteristic to them. These characteristics depend on what just intervals, called commas, which differ slightly from the unison become a unison when tempered. In meantone, for example, the root of a ii-chord regarded as being a fifth above the dominant would be a major whole tone of 9/8 if the fifths were tuned justly, but would be a minor whole tone of 10/9 if it is taken to be a just minor third of 6/5 below a just subdominant degree of 4/3. These are being equated, so meantone temperament is tempering out the difference between 9/8 and 10/9. This means their ratio, (9/8)/(10/9) = 81/80, is tempered to a unison. The interval 81/80, called the Syntonic Comma or comma of Didymus, is the key comma of meantone temperament, and the fact that it becomes a unison in meantone temperament is a key fact of Western music. Equal temperament See Also: Equal temperament In equal temperament the half step, rather than the fifth or third, is the basis of tuning. Each half step is the interval of the twelfth root of two, so that twelve of these equal half steps add up to exactly an octave. This is the 12-step, equal tempered scale. In non-equal tempered scales, it is necessary to retune whenever changing keys. Also, fretted instrument must use an equal tempered scale, or else the frets would not align evenly across the strings. Equal tempered scales have been built using 19 equally spaced tones, and also 24 equally spaced tones. These scale have their uses, but the 12 tone scale does the best job approximating the perfect fifth, perfect fourth, minor third, major third, minor sixth, and major sixth. The 12 tone, equal tempered scale does compromise in approximating these tones. But, this tonal compromise is normally forgiven in light of the advantage of quick key changes and uniformly tuned instruments. Sound samples Below are Ogg Vorbis files demonstrating the difference between just intonation and equal temperament. You may need to play the samples several times before you can pick the difference.
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