In the history, the European musical theory
and aesthetics constituted by an empirical-rational
method the system of musical rules: the laws
for the construction of musical works, based on
seven-tone tonality.
The
major -
minor tonal system
with a tonic as the tonal center dominated in the
whole European musical tradition for the centuries.
From the basic statement that the laws of mathematics
and proportions (arithmetical, geometrical and harmonical)
could describe the nature, Pythagoreans derived the theory
of musical harmonies, identifying it with the harmony of
the well-organized universe. Beginning from
the tetrakis – 1, 2, 3, 4 – the ordered sequence
of small natural numbers, and forming their
proportions 1:2, 2:3, 3:4, the three basic proportional vibrations -
(2:3) and perfect fourth (3:4),
and the natural overtone series are
derived. Pythagoras (or the Pythagoreans)
made two important statements with regard to them, one an aesthetic claim, the
other a mathematical observation. The aesthetic claim was that they believed that
each of the planets emitted a tone, and that the universe thus resonates with
a "fifthmusic of the spheres" of which the natural overtone series
on our earth is only an image. The mathematical observation regarded the
circle that can be built from the interval represented in the second proportional
vibration 2:3 (the perfect fifth), in relation to the sequence built from the first,
1:2 (the octave). In solving the problem of the Pythagorean comma
(the difference between the tone obtained as a result of stacking twelve consecutive
perfect fifths from that reached in seven octaves), as the result of
(the tuning based on perfect fifths and octaves)
it was obtained the chromatic scale with twelve equal semitones. (Well-tempered tuning
was devised by Werckmeister, a contemporary of J.S. Bach, hence the "temperingWell-tempered Clavier").
The tones of that scale may be simply denoted as 0,1,...,11 (mod 12),
where the enharmonic tones are treated as the same, and where the unit is a semitone.
In fact, the result was a symmetrization: the division of octave
in twelve equal semitone intervals. During the history, from all
two of them are distinguished:
the major (Ionian) and
minor (Aeolian).
In the sense of their meaning and
emotional-symbolical role they are a well-known example
of antisymmetry
.
modes
Continuing with this approach, we may prove the remarkable
fact: the tonal system
major -
minor is completely invariant
with regard to the
0 2 4 5 5 3 1 that could be, according 0 2 4 5 5 3 1
If we denote the intervals 6, 5,
4, 3, 2, 1
by the corresponding colored lines, as the result we obtain the following circle diagram (Fig. 2).
From that diagram, we can notice two important properties: (1) the singularity
of the interval
Because the tritone triangle is the center of instability in every tonality,
for any triad (or chord) we may define its coefficient of stability
Different symmetries, resulting from
the basic symmetrical organization of the tonal system
occur in the case of all triads and chords, and together with
their individual symmetries represent the standardized
patterns of tonal music.
Figure 3 shows all the seven possible four-tone chords (given by the intervals)
and their symmetries. The mirror-axes are denoted by the white lines,
antisymmetry axes by the red broken lines, and the centers of
antisymmetry by the red dots.
This way, after the mutual identification of
complementary intervals (e.g. third –
sixth,
forth -
fifth...),
(i.e. by reducing all the
intervals between two tones to the minimal ones),
we succeeded to discover the basic symmetry of the
major - minor tonal system:
its complete
interval tritonei=6.
From this basic symmetries resulted a large
number of symmetries in the complete harmony level
of the tonal music. So, in the beginning it was
the .
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