Charts and the sky from computed positions: natal, transits, progressions, Hellenistic and Vedic techniques, the Chinese four pillars, fixed stars, mansions and hours read in period sources, and world maps.
Harmony of the Spheres
Kepler’s harmonies of the planets, from Harmonices mundi (Linz, 1619), Book V: his table of each planet’s daily motion seen from the Sun at aphelion and perihelion (p. 198, the errata applied), the interval in each ratio, the harmonies between neighbours, the Moon’s fourth, and the seven melodies of p. 207 as printed. The same motions come from the app’s ephemeris for comparison; the summaries are Unveil’s.
About this tool
History
Kepler printed Harmonices mundi at Linz in 1619. Its fifth book sets out the harmonic law he found on 15 May 1618 — the squares of the periods as the cubes of the mean distances — and then looks for music in the planets’ motions. He found it in their speeds as seen from the Sun: each planet moves slowest at aphelion and fastest at perihelion, and the ratio of the two is a musical interval. Saturn’s is a major third, Mars’s a fifth, the Earth’s a semitone. In chapter 6 he printed each planet’s song on a staff, and in the margin wrote that the Earth sings mi, fa, mi.
How it works
Kepler’s table of page 198 stands as printed, with its letters and the errata leaf’s corrections beside the misprints. For the orbit that holds the chosen date, the app finds each planet’s nearest and farthest points from the Sun in the VSOP87 positions, takes the motion at each, and names the nearest of Kepler’s intervals, with the difference in cents. The Moon is tried the same way, from the Earth, by the hour. The seven songs are drawn with the woodcut’s pitches and played, and the planets sound their pitches for the day by Kepler’s rule: the aphelion note at the aphelion motion, higher as the planet moves faster.
How it was checked
The tables and texts were read from the Smithsonian Libraries’ copy of 1619, checked against the Royal Danish Library’s copy and Frisch’s edition of 1864, and read again letter by letter by a second reader. The tests recompute the extreme motions for Kepler’s own year and find them within ten seconds of arc a day of his table for the five planets; only Mercury’s perihelion, 384′, is off by more, and he says himself that he rounded it. The same six intervals come out today as in 1619; the Earth’s semitone is 15 : 16 within thirteen cents, and the Moon’s fourth holds.
Limits
No English translation of Book V is in the public domain, so the Latin is given with summaries of Unveil’s own. The songs are drawn in treble and bass clefs, not the C clefs of the woodcut, and sounded in equal temperament by the browser. Kepler’s Moon motions are about three minutes an hour slower than the Moon’s true motion; his ratio stands beside the recomputed one. The universal harmonies of chapter 7 onward are not quoted.