Tempering as the
symmetrization of scales

A solution to these problems is the
introduction of tempered systems - an even distribution of 12
identical subintervals, half-tones, within the octave (Fig.
3.5). Since the octave is an interval with a value of 2, the
value of each subinterval is 2^{[1/12]}. The
corresponding string lengths form a geometric progression : (^{1}/_{2})^{0}=1,
(^{1}/_{2})^{[1/12]}, (^{1}/_{2})^{[2/12]},...,
(^{1}/_{2})^{[12/12]}=^{1}/_{2}.
The ratio of its consecutive terms is the constant value of the
halftones 2^{[1/12]}. In this manner a complete
symmetrization of the system has taken place and deviations have
been eliminated, as well as the non-equivalence between sharps
and flats and the difference between chromatic and diatonic
half-tones. Because exponents add up (or are subtracted) when
multiplying or dividing two degrees of the same number (a^{m}a^{n}=a^{m+n}),
the complex multiplicative interval structure of the Pythagorean
or harmonic system has been substituted by a simple additive
structure made up of exponents. For example, the sum of
intervals 2^{[7/12]} and 2^{[5/12]} is the
product of their values 2^{[7/12]}×2^{[5/12]}=2^{([7/12]+[5/12])}=2^{[12/12]}=2
(octave), and the difference is the ratio of their values 2^{[7/12]}:2^{[5/12]}=2^{([7/12]-[5/12])}=2^{[2/12]}.
Note that by tempering we achieve an approximation of the string
lengths and, consequently, the corresponding pitch, where none
of the intervals (except for the octave) coincide with the
"perfect" harmonic intervals [2/3], [3/4], [4/5] (i.e., 2^{[7/12]}
» ^{2}/_{3}, 2^{[5/12]} » ^{3}/_{4},
2^{[4/12]} » ^{4}/_{5}).

**
Figure 3.5 The tempered system
and its interval structure. **