Tuning, Scales and Temperament Harmonic and Inharmonic ...

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scales http://www.anu.edu.au/ITA/ACAT/drw/PPofM/scales/scales6.html The Just Chromatic Scale A chromatic Just scale can be constructed by a procedure analogous to that used for the Pythagorean chromatic scale. In this case, the ambiguities are more numerous for the Just scale than for the Pythagorean. For example the tritone has two different values as does the raised 1st degree ( or lowered 2nd degree). The Just scale provides "cleaner" or "simpler" harmonies than does the Pythagorean scale. However, when the scale is played simultaneously with the original, as happens for a transposing instrument for example, there results notes which have two quite different ratios. Take for example a comparison of C major with D major. All the difficulties so far mentioned were known by the 17th Century and an attempt was made to resolve them by "tempering" the scale. BACK NEXT INDEX 1 of 1 2/4/2001 12:21 PM
scales http://www.anu.edu.au/ITA/ACAT/drw/PPofM/scales/scales7.html The Tempered Scale If you start from a particular note and ascend or descend in either perfect fifths or thirds, one never arrives at a note that is a whole number of octaves from the starting note. It is partly this reason that both the Pythagorean and Just scales have posed problems for recent Western musicians (in the last few hundred years). Adjustments have been made to these scales - in the spirit of acoustical compromise - to satisfy three incompatible requirements: 1. True intonation - that is exact thirds, fourths, fifths, etc 2. Complete freedom of modulation - that is from any one key to any other. 3. Convenience in the practical use of keyboard instruments - that is to make it possible for instruments which have been tuned differently to play together without retuning. Equal Temperament In this form of compromise, the first of these requirements is sacrificed for the good of the latter two requirements. In the equally tempered scale all semitone intervals are exactly the same ratio, irrespective of frequency location. This implies that all whole-tone intervals are likewise equal. In order to accomplish this the octave is divided into twelve equal semitone intervals. The two criteria then, are 1. That all semitone intervals be identical; i.e. 2. That the octave maintain its integrity; i.e. f(note 13) = 2 x f(note 1) The first criterion means that the ratio of all the semitone intervals is the same. Call this ratio a. The ratio of any two semitones is thus the same: 1 of 3 2/4/2001 12:22 PM
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