One of the first interdisciplinary approaches to a holistic understanding of our world was that of Pythagoras and his disciples. They created the theory of the fundamental significance of numbers in the objective world and in music. This theory reduced all existence to number, meaning that all entities are ultimately reducible to numerical relationships that link not only mathematics to music but also to acoustics, geometry, and astronomy. Even the dependence of the dynamics of world structure on the interaction of pairs of opposites—of which the even–odd polarity essential to numbers is primary—emerges from these numerical relationships. Pythagoras would have been pleased to learn of attractors opposing in character, created by simple feedback loops of numbers, and forming tenuous boundaries—dynamic sites of instability and creativity.
Pythagorean thinking deeply influenced the development of classical Greek philosophy and medieval European thought, especially the astrological belief that the planetary harmony of the universe affects everything, including terrestrial affairs, through space–time configurations of cosmic bodies. People were intrigued by the precision of numerical relationships between musical harmonies, which deeply touch the human soul, and the prosaic arithmetical ratios of integers. This connection was first demonstrated by Pythagoras himself in the sixth century B.C. In his famous experiment, a stretched string on a monochord was divided by simple arithmetical ratios—1:2, 2:3, 3:4, 4:5, and 5:6—and plucked. It was a Eureka moment when he discovered that these respective partitions of the string create the consonant intervals of harmony.
One tone is not yet music. One might say it is only a promise of music. The promise is fulfilled, and music comes into being, only when one tone follows another. Strictly speaking, therefore, the basic elements of music are not individual tones but the movements between tones. Each of these movements spans a certain pitch distance. The pitch distance between two tones is called an interval. It is the basic element of melody and of individual musical motion. Melody is a succession of intervals rather than of tones. Intervals can be consonant or dissonant.
[ Nodes of a vibrating string are harmonics. Conversely, antinodes
—points of maximum amplitude—occur midway between nodes. ]
It was Pythagoras’ great discovery to see that the ratios of the first small integers up to six give rise to consonant intervals; the smaller these integers, the more complete the resonance. A string divided in the ratio 1:2 yields the octave (C–C), an equisonance of the fundamental tone. The ratio 2:3 yields the fifth (C–G); 3:4 the fourth (C–F); 4:5 the major third (C–E); and 5:6 the minor third. These correspond to the consonant intervals of octave, fifth, fourth, major third, minor third, and the sixth. The pairs of notes given in brackets are examples of the respective consonances.
The minor sixth, created by the ratio 5:8, seems to go beyond the limit of six. Yet eight—the only integer greater than six involved here—is the third power of two and thus a member of the series of consonant numbers. Eight is created by an octave operation, which produces absolutely equisonant tones. All authorities agree that, besides the equisonant octave, there are no consonant intervals other than the third, the fourth, the fifth, and the sixth. If more than two notes are to be consonant, each pair of them must also be consonant.
As mentioned already, the most complete consonance within the range of an octave is the major perfect chord C–E–G (4:5:6), which unites the major third and the fifth with the fundamental note. These concepts of harmony and consonant intervals are formed by the first terms in the series of overtones, or harmonics, produced by a vibrating string. [...] Whenever there is a musical sound, there is an addition of harmonics that relate the fundamental tone to an infinity of overtones, which influence the quality of the consonant fundamental. The overtones up to the sixth harmonic represent the consonant intervals: the octave, the fifth, the fourth, the major third, the minor third, and the sixth.
Figure 19: Smoothed time series of consecutive impulses of the torque (IOT), with epochs indicated by dots. The resulting wave pattern corresponds to the secular cycle of sunspot activity. The average wavelength is 166 years, with each extremum occurring at mean intervals of 83 years, aligned with a maximum in the secular sunspot cycle. These maxima, as identified by Wolfgang Gleissberg, are marked by bold arrows. Minima occur when the wave approaches zero. This wave pattern reflects the influence of solar system configurations that generate impulses of the torque.
Figure 34 shows the combination of the consonant intervals known as the major sixth (3:5) and the minor sixth (5:8) as they emerge in solar-system processes over thousands of years. These intervals are marked by vertical triangles and large numbers. The curve depicts the supersecular variation of energy in the secular torque wave, part of which was shown in points along the curve represent epochs of extrema, labeled by Aₛ numbers from −64 to +28, corresponding to the period from 5259 BC to AD 2347. The mean cycle length is 391 years. Black triangles indicate maxima in the corresponding supersecular sunspot cycle, while open triangles indicate minima. When the energy exceeds certain quantitative thresholds, shown by hatched horizontal lines, a phase jump occurs in the correlated supersecular sunspot cycle. These critical phases are marked by vertical dotted lines. A new phase jump is expected around 2030.
It points toward a supersecular minimum comparable to the Egyptian minimum (E) around 1369 BC, a prolonged period marked by notable cooling and glacier advance. The ratio 3:5:8, representing the major and minor sixth, marks the intervals that separate these rare phase jumps indicated by the vertical dotted lines. The 317.7-year period of the triple conjunction of Jupiter, Saturn, and Uranus is also involved in this relationship, as shown by the small numbers beneath the large numbers at the top of the figure.
[...] Another confirmation of the hypothesis that consonant intervals play an important role with respect to the Sun's eruptional activity are the connections presented in Figure 34 that cover thousands of years. It has been shown in Figure 19 that consecutive impulses of the torque (IOT) in the Sun’s motion about the center of mass (CM) of the solar system, when taken to constitute a smoothed time series, form a wave-pattern the positive and negative extrema (±As) of which coincide with maxima in the secular sunspot cycle. This Gleissberg cycle, with a mean period of 83 years, which modulates the intensity of the 11-year sunspot cycle, is in turn modulated by a supersecular sunspot cycle with a mean period of about 400 years. The Maunder Minimum of sunspot activity in the 17th century and a supersecular maximum in the 12th century are features of this supersecular cycle. It seems to be related to the energy in the secular wave presented in Figure 19.
This energy may be measured by squared values of the secular extrema ±As. When these values are taken to form another smoothed time series, a supersecular wave emerges as plotted in Figure 34. It runs parallel with the supersecular sunspot cycle. Its mean period is 391 years, but it varies from 166 to 665 years. Each dot in the plot indicates the epoch of a secular extremum (±As). These epochs are numbered from -64 to +28 and range from 5259 B.C. to 2347 A.D. Black triangles indicate maxima in the correlated supersecular sunspot curve and white triangles minima. The medieval maximum, which was together a climate optimum (O), the Spoerer Minimum (S), and the Maunder Minimum (M) are marked by respective abbreviations. The extrema in the supersecular wave properly reflect all marked peaks and troughs in the supersecular sunspot curve derived from radiocarbon data.
This energy may be measured by squared values of the secular extrema ±As. When these values are taken to form another smoothed time series, a supersecular wave emerges as plotted in Figure 34. It runs parallel with the supersecular sunspot cycle. Its mean period is 391 years, but it varies from 166 to 665 years. Each dot in the plot indicates the epoch of a secular extremum (±As). These epochs are numbered from -64 to +28 and range from 5259 B.C. to 2347 A.D. Black triangles indicate maxima in the correlated supersecular sunspot curve and white triangles minima. The medieval maximum, which was together a climate optimum (O), the Spoerer Minimum (S), and the Maunder Minimum (M) are marked by respective abbreviations. The extrema in the supersecular wave properly reflect all marked peaks and troughs in the supersecular sunspot curve derived from radiocarbon data.
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