''Harmonice Mundi'' (

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

: ''The Harmony of the World'', 1619) is a book by

Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...

. In the work, written entirely in Latin, Kepler discusses

harmony In music, harmony is the concept of combining different sounds in order to create new, distinct musical ideas. Theories of harmony seek to describe or explain the effects created by distinct pitches or tones coinciding with one another; harm ...

and congruence in geometrical forms and physical phenomena. The final section of the work relates his discovery of the so-called third law of planetary motion. The full title is ''Harmonices mundi libri V'' (''The Five Books of The Harmony of the World''), which is commonly but ungrammatically shortened to ''Harmonices mundi.''

Background and history

Kepler began working on ''Harmonice Mundi'' around 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 (; , ; ; – 160s/170s AD) was a Greco-Roman mathematician, astronomer, astrologer, geographer, and music theorist who wrote about a dozen scientific treatises, three of which were important to later Byzantine science, Byzant ...

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

astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

Kepler would use (most notably the adoption of

elliptic orbit In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some orbits have been referre ...

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 determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...

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 a form ...

'' 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 th ...

, 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 is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...

and the workings of both the celestial and terrestrial 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 The soul is the purported Mind–body dualism, immaterial aspect or essence of a Outline of life forms, living being. It is typically believed to be Immortality, immortal and to exist apart from the material world. The three main theories that ...

. In turn, this allowed Kepler to claim the

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

has a soul because it is subjected to astrological harmony. While writing the book, Kepler had to defend his mother in court after she had been accused of

witchcraft Witchcraft is the use of Magic (supernatural), magic by a person called a witch. Traditionally, "witchcraft" means the use of magic to inflict supernatural harm or misfortune on others, and this remains the most common and widespread meanin ...

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. Ioanniskepplerih00kepl 0081

Chapter 1 and 2

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

polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

. 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 star polyhedra, the

small Small means of insignificant size Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to three geometrical measures: length, area, or ...

and

great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at eac ...

; they would later be called Kepler's solids or Kepler Polyhedra and, together with two regular polyhedra discovered by Louis Poinsot, as the Kepler–Poinsot polyhedra. He describes polyhedra in terms of their faces, which is similar to the model used in

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

's '' Timaeus'' to describe the formation of

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

. 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, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...

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 minor second, 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 ...

(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. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...

and

perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. The line of apsides (also called apse line, or major axis of the orbit) is the line connecting the two extreme values. Apsides perta ...

Venus Venus is the second planet from the Sun. It is often called Earth's "twin" or "sister" planet for having almost the same size and mass, and the closest orbit to Earth's. While both are rocky planets, Venus has an atmosphere much thicker ...

only varies by a tiny 25:24 interval (called a

diesis In classical music from Western culture, a diesis ( or enharmonic diesis, plural dieses ( , or "difference"; Greek: "leak" or "escape" is either an accidental (see sharp), or a very small musical interval, usually defined as the differe ...

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. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...

), 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 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...

and

Jupiter Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...

), a soprano ( Mercury), and two altos (Venus 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 (also known as orbital revolution) 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 ...

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 5

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'', 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 (, ''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 association published ab ...

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 Laurie Spiegel (born September 20, 1945) is an American composer. She has worked at Bell Labs, Bell Laboratories, in computer graphics, and is known primarily for her electronic music electronic music, compositions and her algorithmic compositio ...

: ''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 and science communicator. His best known scientific contribution is his research on the possibility of extraterrestrial life, including e ...

for inclusion on the

Voyager Golden Record The Voyager Golden Records are two identical phonograph records, one of each which were included aboard the two Voyager spacecraft launched in 1977. The records contain sounds and data to reconstruct raster scan images selected to portray the di ...

, launched aboard the Voyager spacecraft. *

Mike Oldfield Michael Gordon Oldfield (born 15 May 1953) is an English retired musician, songwriter and producer best known for his debut studio album ''Tubular Bells'' (1973), which became an unexpected critical and commercial success. Though primarily a gu ...

, (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. Mercury Records released ...

). * 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 minimal music, minimalism, being built up fr ...

, American composer, ''Kepler'' opera (2009), homage to

, commissioned by the city of

Linz Linz (Pronunciation: , ; ) is the capital of Upper Austria and List of cities and towns in Austria, third-largest city in Austria. Located on the river Danube, the city is in the far north of Austria, south of the border with the Czech Repub ...

, 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 (abbreviated V&A) in London is the world's largest museum of applied arts, decorative arts and design, housing a permanent collection of over 2.8 million objects. It was founded in 1852 and named after Queen ...

, 9 November 2017 *

Paul Hindemith Paul Hindemith ( ; ; 16 November 189528 December 1963) was a German and American 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 advo ...

, 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, Radio drama, drama, High culture, culture and the arts ...

In Tune (October 2019) with full concert premiere at London International Festival of Early Music (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 In Western classical music, a motet is mainly a vocal musical composition, of highly diverse form and style, from high medieval music to the present. The motet was one of the preeminent polyphonic forms of Renaissance music. According to the Eng ...

, including the use of a specific six-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 (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

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
Harmonies of the World
excerpt from ' translated by Charles Glenn Wallis {{Authority control 17th-century books in Latin 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