The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the harmonic series.
42.
He had to build his own instruments to make music with pure frequency ratios ( not the complex fractions represented by intervals on the equal-tempered piano ), and to leave behind the Western tradition.
43.
The size of an interval ( also known as its width or height ) can be represented using two alternative and equivalently valid methods, each appropriate to a different context : frequency ratios or cents.
44.
In the years that followed, the frequency ratios of these intervals on keyboards and other fixed-tuning instruments would change slightly as different systems of tuning, such as meantone temperament, well temperament, and equal temperament were developed.
45.
:As equal temperament is " . . . a system of tuning in which every pair of adjacent notes has an identical frequency ratio . ", I would expect unequal to not include identical ratios between notes.
46.
In twelve-tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone, i . e . the frequency ratio of the interval between two adjacent notes, is the twelfth root of two:
47.
You want the notes to be approximately equally spaced, which you get exactly with equal tempering, but you also want harmonic intervals such as the fifth, with a frequency ratio of 3 : 2, to be represented.
48.
An equal-tempered fifth, at a frequency ratio of 2 7 / 12 : 1 ( or about 1.498307077 : 1 ) is approximately two cents narrower than a justly tuned fifth at a ratio of 3 : 2.
49.
In equal temperament tunings, the generating interval is often found by dividing some larger desired interval, often the octave ( ratio 2 : 1 ), into a number of smaller equal steps ( equal frequency ratios between successive notes ).
50.
However, because just intonation does not allow the irrational ratio of " 5 / 2, two different frequency ratios are used : the major tone ( 9 / 8 ) and the minor tone ( 10 / 9 ).
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