Scales with
different semi-tone intervals

The Pythagoreans derive their theory
of musical harmony, which they identify with the harmony of a
well-organized universe, based on the assumption that nature can
be described by mathematical laws which boil down to proportion
(arithmetic, geometric and harmonic) and are based on small
integers (Ghyka 1987, p.38; Ghyka
1977, p. 4). Beginning with a
sequence of small integers - *tetrakis* 1, 2, 3, 4 (Ghyka
1987, page 12), a sequence of their proportions (ratios) is
formed - [1/2], [2/3], [3/4]. Based on the division of a string
in these proportions, we get four basic tones: C^{1}=1,
C^{2}=^{1}/_{2}, G^{1}=^{2}/_{3}
and F^{1}=^{3}/_{4} and the unit
interval (unit measure): F^{1}-G^{1}. Knowing
that the frequency f(m) of a tone is the reciprocal value of the
corresponding length of string m, f(m)=^{1}/_{m},
and that the interval i between two tones is the ratio of their
frequencies, i=f(m):f(n)=^{n}/_{m}, then the
proportion i_{1}=^{2}/_{3}:^{3}/_{4}=^{8}/_{9}
gives the basic interval i_{1}=F^{1}-G^{1}
(Fig. 3.2).

**
Figure 3.2
The frequencies of the basic tones of the Pythagorean scale and
the unit interval F**^{1}-G^{1}.

In an attempt to measure all other
intervals using the unit interval i_{1}, with C^{1}=1
and G^{1}=^{1}/_{2} as the starting
points, the Pythagoreans determined that D^{1}=^{8}/_{9},
and E^{1}=^{8}/_{9}^{2}=[64/81],
i.e., that A_{1}=^{2}/_{3}×^{8}/_{9}=[16/27],
B^{1}=^{2}/_{3}×(^{8}/_{9})^{2}=[128/243].
However, due to the incommensurability of tetrachords C^{1}-F^{1}
and G^{1}-C^{2} with a unit interval i_{1}=^{8}/_{9}
chosen in this manner, the following intervals occur: i_{2}=^{3}/_{4}:[64/81]
= ^{1}/_{2}:[128/243] = [243/256] ~ 0.94922...
Note that the interval i_{2} does not represent the
exact value of the half tone - the length of the string which
corresponds to the value of the half tone is obtained from the
equation ^{2}/_{3}:x=x:^{3}/_{4},
i.e., x=[(Ö2)/2]. Consequently, the value of the half tone is
[(i_{1})/2]=^{2}/_{3}:[(Ö2)/2]=[(2Ö2)/3]
~ 0.94281... Generally speaking, the length of string x which
corresponds to the center of an interval is the geometric mean
of the lengths of string m and n which correspond to the ends of
the interval, i.e., x=Ö{mn}. The entire structure of such a
distribution of intervals is multiplicative, i.e., it is defined
by the products (ratios) of numbers which correspond to the
lengths of the string. The result obtained is a sequence
generated by fourths, fifths and octaves. If we take C^{1}
as the initial tone and C^{2} as the end tone, we obtain
the interval sequence i_{1} (C^{1}-D^{1}),
i_{1} (D^{1}-E^{1}), i_{2} (E^{1}-F^{1}),
i_{1} (F^{1}-G^{1}), i_{1} (G^{1}-A^{1}),
i_{1} (G^{1}-B^{1}), i_{2} (B^{1}-C^{2}).
Note that the this initial sequence is asymmetric, but that it
allows tetrachords (C^{1}-F^{1}), (G^{1}-C^{2})
which satisfy translational symmetry, i.e. transposition. If we
continue the tonic sequence in both directions, higher and
lower, we obtain a periodic translational structure with the
following period: 5i_{1}+2i_{2}=(C^{1}-C^{2})
which is a sequence of octaves of the tonal system. Since the
initial interval sequence is assymetric, all of its points are
non-equivalent and thus each one of them may serve as a starting
point. The result obtained in this manner are seven modes
(diatonic scales): Ionic, Doric, Phrygian, Lydian, Myxolydian,
Aeolian and Locrian (Despic, 1987, Holopov,
1974). The principle
of construction of a Pythagorean system based on perfect fifths
results in the occurrence of deviation - the Pythagorean comma.
This is the incongruence between a sequence consisting of twelve
fifths and a sequence of seven octaves, i.e., the difference
between the chromatic and diatonic half-tone (Despic,
1987).

**
Figure 3.3
The frequency of tones and the interval structure of the
Pythagorean scale. **

From the sequence [1/2], [2/3],
[3/4], [4/5] results a natural or harmonic division of a string:
C_{1}=1, C_{2}=^{1}/_{2}, G_{1}=^{2}/_{3},
F_{1}=^{3}/_{4}, E_{1}=^{4}/_{5}.
This sequence defines two intervals i_{1}=^{8}/_{9}
(F^{1}-G^{1}), i_{2}=[15/16] (E^{1}-F^{1}).
Based on this we obtain the interval sequence i_{1} (C^{1}-D^{1}),
i_{3} (D^{1}-E^{1}), i_{2} (E^{1}-F^{1}),
i_{1} (F^{1}-G^{1}), i_{3} (G^{1}-A^{1}),
i_{1} (A^{1}-B^{1}), i_{2} (B^{1}-C^{2}),
where i_{3}=[9/10] (Fig. 3.4).

**
Figure 3.4
The basic tones of a harmonic scale (C**^{1}, E^{1},
F^{1}, G^{1}, C^{2}), basic intervals (E^{1}-F^{1}),
(F^{1}-G^{1}) and the interval structure of a
harmonic scale.

Of course, the occurrence of three
different half-tone intervals caused deviations, particularly
noticeable with keyboard instruments, as well as a significant
increment of the complexity of the theory of intervals. The
introduction of sharp and flat systems within the mentioned
initial interval sequences, due to the difference between the
half-tones, requires the discernment of chromatic and diatonic
intervals and the corresponding scales. Thanks to the irregular
division of an octave into 12 subintervals (an irregular
half-tone sequence), the definition of the interval value is
based on the name of the interval, as a function of the place of
the particular tone in the scale, instead of as a real value of
the interval, i.e., the difference in tonal pitch. The asymmetry
of the initial system causes a high level of complexity of the
entire structure which is especially visible, for example, in
interval classification, their inversion rules and scale
classification. This brings about the creation of instruments
with keyboards (ënharmonic instruments") where we note a
difference between enharmonic tones (Encyclopedia of Music,
1963, p. 706).