Dynamical Systems, Celestial Mechanics, and Music: Pythagoras Revisited

Cartwright, Julyan H. E.; González, Diego; Piro, Oreste

doi:10.1007/s00283-020-10025-x

Cited by 5 publications

(6 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, the four main tones of the Pythagorean musical set describe a Keplerian orbit. In fact, if a is the semi-major axis, b the semi-minor axis and l = a(1 − e 2 ), it is found that a is the arithmetic mean, b the geometric mean and l the harmonic mean of r min and r max (Cartwright et al, 2021).…”

Section: The 12-tji and 12-tet Tuning Systems And Their Consonancesmentioning

confidence: 99%

“…Thus, we searched alternative orbital metrics and checked whether they could produce a musical correspondence for all planetary pairs. The assumption is that ratios adopted in the traditional musical tuning systems and, in particular, those that form consonances, are peculiar because they are harmoniously interrelated and, therefore, may unearth important physical relations (Cartwright et al, 2021).…”

Section: Introductionmentioning

confidence: 99%

“…These ratios can also be derived as harmonic and arithmetic means of 1/1 and 3/2 in the case of thirds and means between 4/3 and 2/1 in the case of sixths (Forster, 2010). The proposed system was defined as just because all notes are related by intervals that are defined by rational numbers (Cartwright et al, 2021). In the tuning systems employing just-intonation, exact harmonic consonance ratios are maximized.…”

Section: Introductionmentioning

confidence: 99%

“…Herein we adopt a similar transdisciplinary approach and show that music theory can still be useful to explore some possible unknown features that characterize the interplanetary gravitational organization of our solar system. Indeed, Kepler himself was inspired by musical principles (Cartwright et al, 2021).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System

Bank

Scafetta

2022

Front. Astron. Space Sci.

View full text Add to dashboard Cite

Orbital systems are often self-organized and/or characterized by harmonic relations. Inspired by music theory, we rewrite the Geddes and King-Hele (QJRAS, 24, 10–13, 1983) equations for mirror symmetries among the distances of the planets of the Solar System in an elegant and compact form by using the 2/3rd power of the ratios of the semi-major axis lengths of two neighboring planets (eight pairs, including the belt of the asteroids). This metric suggests that the Solar System could be characterized by a scaling and mirror-like structure relative to the asteroid belt that relates together the terrestrial and Jovian planets. These relations are based on a 9/8 ratio multiplied by powers of 2, which correspond musically to the interval of the Pythagorean epogdoon (a Major Second) and its addition with one or more octaves. Extensions of the same model are discussed and found compatible also with the still hypothetical vulcanoid asteroids versus the transneptunian objects. The found relation also suggests that the planetary self-organization of our system could be generated by the 3:1 and 7:3 resonances of Jupiter, which are already known to have shaped the asteroid belt. The proposed model predicts the main Kirkwood asteroid gaps and the ratio among the planetary orbital parameters with a 99% accuracy, which is three times better than an alternative, recently proposed harmonic-resonance model for the Solar System. Furthermore, the ratios of neighboring planetary pairs correspond to four musical “consonances” having frequency ratios of 5/4 (Major Third), 4/3 (Perfect Fourth), 3/2 (Perfect Fifth) and 8/5 (Minor Sixth); the probability of obtaining this result randomly has a p < 0.001. Musical consonances are “pleasing” tones that harmoniously interrelate when sounded together, which suggests that the orbits of the planets of our Solar System could form some kind of gravitationally optimized and coordinated structure. Physical modeling indicates that energy non-conserving perturbations could drive a planetary system into a self-organized periodic state with characteristics vaguely similar of those found in our Solar System. However, our specific finding suggests that the planetary organization of our Solar System could be rather peculiar and based on more complex and unknown dynamical structures.

show abstract

Section: The 12-tji and 12-tet Tuning Systems And Their Consonancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System

Bank

Scafetta

2022

Front. Astron. Space Sci.

View full text Add to dashboard Cite

show abstract

“…10 Geodesy also has represented an important practical aspect in early civilizations that led rapidly into geometric developments. We find the apex of number in the Pythagorean doctrine that the entire universe is governed by numbers; for the Pythagoreans, that meant integer and rational numbers [49] .…”

Section: Methodsmentioning

confidence: 86%

Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero

Rojo-Garibaldi

Rangoni

González

et al. 2021

Heliyon

Self Cite

View full text Add to dashboard Cite

Pre-Columbian Mesoamerica was a fertile crescent for the development of number systems. A form of vigesimal system seems to have been present from the first Olmec civilization onwards, to which succeeding peoples made contributions. We discuss the Maya use of the representational redundancy present in their Long Count calendar, a non-power positional number representation system with multipliers 1, 20, 18 × 20, …, 18 × 20 n . We demonstrate that the Mesoamericans did not need to invent positional notation and discover zero at the same time because they were not afraid of using a number system in which the same number can be written in different ways. A Long Count number system with digits from 0 to 20 is seen later to pass to one using digits 0 to 19, which leads us to propose that even earlier there may have been an initial zeroless bijective numeration system whose digits ran from 1 to 20. Mesoamerica was able to make this conceptual leap to the concept of a cardinal zero to perform arithmetic owing to a familiarity with multiple and redundant number representation systems.

show abstract

2500 years ago scientific theories of the origin of life arose in ancient Greece

Cartwright

2024

Discov Life

View full text Add to dashboard Cite

What have the Greeks ever done for us? Everything. This short commentary celebrating the earliest history of origin of life research seeks to bring to the attention of the community of origin of life researchers that there is a much longer tradition of scientific theories in this subject than some people are aware of; there are 2500 years of thinking that we should not forget about. In Ionian Greece, Thales initiated ideas that we can see as the beginnings of science itself. Anaximander, his pupil, was particularly interested in the origins of the world, including how life had arisen, and proposed a species of proto-evolutionary theory of the origin of life. 2500 years ago, Anaxagoras left Ionia for Athens taking ideas of scientific thinking with him. Anaxagoras is credited with being the originator of a further theory of the origin of life, that of panspermia. The scientific conception of knowledge was attacked almost immediately, and Anaxagoras ended up exiled, but that idea of the knowability of things survived and led, many centuries later, to the scientific revolution and to our current human civilization.

show abstract

Dynamical Systems, Celestial Mechanics, and Music: Pythagoras Revisited

Cited by 5 publications

References 34 publications

Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System

Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System

Non-power positional number representation systems, bijective numeration, and the Mesoamerican discovery of zero

2500 years ago scientific theories of the origin of life arose in ancient Greece

Contact Info

Product

Resources

About