''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...

: ''The Harmony of the World'', 1619) is a book by Johannes Kepler. In the work, written entirely in Latin, Kepler discusses

harmony In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. Howev ...

and

congruence Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...

in geometrical forms and physical phenomena. The final section of the work relates his discovery of the so-called " third law of planetary motion".

Background and history

Kepler began working on ''Harmonice Mundi'' sometime near 1599, which was the year Kepler sent a letter to Michael Maestlin detailing the mathematical data and proofs that he intended to use for his upcoming text, which he originally planned to name ''De harmonia mundi.'' Kepler was aware that the content of ''Harmonice Mundi'' closely resembled that of the subject matter for

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of import ...

's ''Harmonica,'' but was not concerned. The new

astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

Kepler would use-most notably the adoption of

elliptic orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, ...

s in the Copernican system-allowed him to explore new theorems. Another important development that allowed Kepler to establish his celestial-harmonic relationships, was the abandonment of the

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: M ...

as the basis for musical consonance and the adoption of geometrically supported musical ratios; this would eventually be what allowed Kepler to relate musical consonance and the angular velocities of the planets. Thus, Kepler could reason that his relationships gave evidence for God acting as a grand geometer, rather than a Pythagorean numerologist. Field, J. V. (1984). A Lutheran astrologer: Johannes Kepler. Archive for History of Exact Sciences, Vol. 31, No. 3, pp. 207–219. The concept of musical harmonies intrinsically existing within the spacing of the planets existed in medieval philosophy prior to Kepler.

Musica universalis The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as ...

was a traditional philosophical metaphor that was taught in the

quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...

, and was often called the "music of the spheres." Kepler was intrigued by this idea while he sought explanation for a rational arrangement of the heavenly bodies.Voelkel, J. R. (1995). The music of the heavens: Kepler's harmonic astronomy. 1994. Physics Today, 48(6), 59–60. When Kepler uses the term "harmony" it is not strictly referring to the musical definition, but rather a broader definition encompassing congruence in

Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...

and the workings of both the celestial and

terrestrial Terrestrial refers to things related to land or the planet Earth. Terrestrial may also refer to: * Terrestrial animal, an animal that lives on land opposed to living in water, or sometimes an animal that lives on or near the ground, as opposed to ...

bodies. He notes musical harmony as being a product of man, derived from angles, in contrast to a harmony that he refers to as being a phenomenon that interacts with the human

soul In many religious and philosophical traditions, there is a belief that a soul is "the immaterial aspect or essence of a human being". Etymology The Modern English noun '':wikt:soul, soul'' is derived from Old English ''sāwol, sāwel''. The ea ...

. In turn, this allowed Kepler to claim the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surf ...

has a soul because it is subjected to astrological harmony. While writing the book, Kepler had to defend

his mother ''His Mother'' is a 1912 American silent film produced by Kalem Company. It was directed by Sidney Olcott with Gene Gauntier and Jack J. Clark in the leading roles. It was one of more than a dozen films produced by the Kalem Company filmed in Ire ...

in court after she had been accused of

witchcraft Witchcraft traditionally means the use of magic or supernatural powers to harm others. A practitioner is a witch. In medieval and early modern Europe, where the term originated, accused witches were usually women who were believed to have u ...

Content

Kepler divides ''The Harmony of the World'' into five long chapters: the first is on regular polygons; the second is on the congruence of figures; the third is on the origin of harmonic proportions in music; the fourth is on harmonic configurations in astrology; the fifth is on the harmony of the motions of the planets.Brackenridge, J. (1982). Kepler, elliptical orbits, and celestial circularity: A study in the persistence of metaphysical commitment part II. Annals of Science, 39(3), 265. SmallStellatedDodecahedron

Chapters 1 and 2 of ''The Harmony of the World'' contain most of Kepler's contributions concerning

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

. He is primarily interested with how polygons, which he defines as either regular or semiregular, can come to be fixed together around a central point on a plane to form congruence. His primary objective was to be able to rank polygons based on a measure of sociability, or rather, their ability to form partial congruence when combined with other polyhedra. He returns to this concept later in ''Harmonice Mundi'' with relation to astronomical explanations. In the second chapter is the earliest mathematical understanding of two types of

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

star polyhedra, the

small Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text ...

and great stellated dodecahedron; they would later be called Kepler's solids or Kepler Polyhedra and, together with two regular polyhedra discovered by

Louis Poinsot Louis Poinsot (3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a co ...

, as the Kepler–Poinsot polyhedra. He describes polyhedra in terms of their faces, which is similar to the model used in

Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institutio ...

's ''

Timaeus Timaeus (or Timaios) is a Greek name. It may refer to: * ''Timaeus'' (dialogue), a Socratic dialogue by Plato *Timaeus of Locri, 5th-century BC Pythagorean philosopher, appearing in Plato's dialogue *Timaeus (historian) (c. 345 BC-c. 250 BC), Greek ...

'' to describe the formation of

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s in terms of basic triangles. The book features illustrations of solids and tiling patterns, some of which are related to the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. While medieval philosophers spoke metaphorically of the "music of the spheres", Kepler discovered physical harmonies in planetary motion. He found that the difference between the maximum and minimum angular speeds of a

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...

in its orbit approximates a harmonic proportion. For instance, the maximum angular speed of the Earth as measured from the Sun varies by a

semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

(a ratio of 16:15), from ''mi'' to ''fa'', between

aphelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any e ...

and

perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any e ...

. Venus only varies by a tiny 25:24 interval (called a diesis in musical terms). Kepler explains the reason for the Earth's small harmonic range: The celestial choir Kepler formed was made up of a tenor (

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin atmos ...

), two bass (

Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; ...

and

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousandt ...

), a soprano ( Mercury), and two altos (

Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...

and Earth). Mercury, with its large elliptical orbit, was determined to be able to produce the greatest number of notes, while Venus was found to be capable of only a single note because its orbit is nearly a circle. At very rare intervals all of the planets would sing together in "perfect concord": Kepler proposed that this may have happened only once in history, perhaps at the time of creation. Kepler reminds us that harmonic order is only mimicked by man, but has origin in the alignment of the heavenly bodies: Kepler discovers that all but one of the ratios of the maximum and minimum speeds of planets on neighboring

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

s approximate musical harmonies within a margin of error of less than a diesis (a 25:24 interval). The orbits of Mars and Jupiter produce the one exception to this rule, creating the inharmonic ratio of 18:19. Chapter 3 o be written Chapter 4 o be written Chapter 5 includes a long digression on astrology. This is immediately followed by Kepler's third law of planetary motion, which shows a constant proportionality between the cube of the semi-major axis of a planet's orbit and the square of the time of its orbital period. Kepler's previous book, ''

Astronomia nova ''Astronomia nova'' ( English: ''New Astronomy'', full title in original Latin: ) is a book, published in 1609, that contains the results of the astronomer Johannes Kepler's ten-year-long investigation of the motion of Mars. One of the most ...

'', related the discovery of the first two principles now known as Kepler's laws.

Recent history

A copy of the 1619 edition was stolen from the

National Library of Sweden The National Library of Sweden ( sv, Kungliga biblioteket, ''KB'', meaning "the Royal Library") is Sweden's national library. It collects and preserves all domestic printed and audio-visual materials in Swedish, as well as content with Swedish ...

in the 1990s.

Use in recent music

A small number of recent compositions either make reference to or are based on the concepts of Harmonice Mundi or Harmony of the Spheres. The most notable of these are: * Laurie Spiegel: ''Kepler's Harmony of the Worlds'' (1977). An excerpt of the piece was selected by

Carl Sagan Carl Edward Sagan (; ; November 9, 1934December 20, 1996) was an American astronomer, planetary scientist, cosmologist, astrophysicist, astrobiologist, author, and science communicator. His best known scientific contribution is research on ...

for inclusion on the

Voyager Golden Record The Voyager Golden Records are two phonograph records that were included aboard both Voyager spacecraft launched in 1977. The records contain sounds and images selected to portray the diversity of life and culture on Earth, and are intended fo ...

, launched aboard the Voyager spacecraft. *

Mike Oldfield Mike may refer to: Animals * Mike (cat), cat and guardian of the British Museum * Mike the Headless Chicken, chicken that lived for 18 months after his head had been cut off * Mike (chimpanzee), a chimpanzee featured in several books and document ...

, (English musician and composer, born 1953), ''Music of the Spheres'' (album released in 2008 by

Mercury Records Mercury Records is an American record label owned by Universal Music Group. It had significant success as an independent operation in the 1940s and 1950s. Smash Records and Fontana Records were sub labels of Mercury. In the United States, it is ...

). * Joep Franssens (Dutch composer, born 1955), ''Harmony of the Spheres'' (cycle in five movements for mixed choir and string orchestra), composed 2001. *

Philip Glass Philip Glass (born January 31, 1937) is an American composer and pianist. He is widely regarded as one of the most influential composers of the late 20th century. Glass's work has been associated with minimalism, being built up from repetitive ...

, American composer, ''Kepler'' (opera) (2009), homage to Johannes Kepler, commissioned by the city of

Linz Linz ( , ; cs, Linec) is the capital of Upper Austria and third-largest city in Austria. In the north of the country, it is on the Danube south of the Czech border. In 2018, the population was 204,846. In 2009, it was a European Capital ...

, where the astronomer lived. * Tim Watts, (English composer, born 1979), ''Kepler's Trial'' (2016–2017), premiered at St John's College, Cambridge (2016); revised version performed at the

Victoria and Albert Museum The Victoria and Albert Museum (often abbreviated as the V&A) in London is the world's largest museum of applied arts, decorative arts and design, housing a permanent collection of over 2.27 million objects. It was founded in 1852 and ...

, 9 November 2017 *

Paul Hindemith Paul Hindemith (; 16 November 189528 December 1963) was a German composer, music theorist, teacher, violist and conductor. He founded the Amar Quartet in 1921, touring extensively in Europe. As a composer, he became a major advocate of the '' ...

, German composer Die Harmonie der Welt Symphony (originally entitled Symphonie „Die Harmonie der Welt“ in German), IPH 50, is a symphony composed in 1951, and which served as the basis for the 1957 opera Die Harmonie der Welt. * Miriam Monaghan (British recorder player and composer) ''Kepler’s Planets'' (2019), written fo
Palisander Recorder Quartet
Extracts were premiered live on

BBC Radio 3 BBC Radio 3 is a British national radio station owned and operated by the BBC. It replaced the BBC Third Programme in 1967 and broadcasts classical music and opera, with jazz, world music, drama, culture and the arts also featuring. The st ...

In Tune (October 2019) with full concert premiere at

London International Festival of Early Music The London International Festival of Early Music (LIFEM) is an English music festival which is devoted mainly to baroque and renaissance music. It takes place each November in Blackheath, London, at Blackheath Halls. History The festival was fo ...

(November 2019). * Dave Soldier, American composer, wrote Motet: Harmony of the World (2022), closely hewing to Kepler's instructions in the book for a future composer to write a motet, including the use a specific 6 voice choir (recorded by the microtonal choir, Ekmeles), the particular just intonation intervals, and the harmonies allowed in Kepler's diagrams. The text sets the Prayer to the Sun by

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophe ...

in ancient Greek, a poet heavily quoted in Kepler's text.

References

External links

''Harmonice mundi''
("The Harmony of the Worlds") in fulltext facsimile; Carnegie-Mellon University
''Harmonice Mundi''
at

Archive.org The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...

Harmonies of the World
excerpt from ' translated by Charles Glenn Wallis {{Authority control 17th-century Latin books Harmonice Mundi 1619 in science Astronomy books Astrological texts Physics books Pythagorean philosophy Astrological aspects Platonic solids Mathematics books Dynamics of the Solar System Harmony Works by Johannes Kepler Music in space