After a brief survey of the use of alternative tuning systems in twentieth-century music, Chalmers begins his elucidation and expansion of tetrachordal theories with the arithmetic approach of Pythagoras and Ptolemy
, in which intervals are expressed as numerical ratios, or as harmonic divisions - 1/2, 1/3, 1/4, and so on - of a string, and the complementary approach of Aristoxenus and his followers, who thought of musical intervals as spatial distances divisible into parts in the same way that a line can be divided with a ruler (and much as on a piano keyboard, where equal intervals span constant distances).
721 BC: The first recorded eclipse was observed by the Babylonians, according to the Greek astronomer Ptolemy
As it happened, Ptolemy
was also the greatest of ancient geographers.
She worked on an instrument on Philae called Ptolemy
that sniffs and analyses gas and dust.
After his death Berenice took her children to Egypt and married the first Greek Pharoah, Ptolemy
His first articles on the eastern place-names of Ptolemy
appeared more than fifty years ago (e.
The text of the Geographia was a verse reimagining of Ptolemy
's second-century text, written in the Tuscan vernacular, which described the journey of its author around the known world in seven days, guided by Ptolemy
, as Dante had been by Virgil.
The presumed author, Ptolemy
Lagides, was one of Alexander's "closest friends and most trusted generals" in the Macedonian army, and if it had survived, many historians believe his book would have been the most accurate firsthand account of the conquests of Alexander the Great.
ON THIS DAY 721 BC: The first recorded eclipse was observed by the Babylonians, according to the Greek astronomer Ptolemy
II leverages Coverity Scan to highlight the importance of software quality
Allen wrote that for Ptolemy
the group was [tau][epsilon][tau][rho][alpha][pi][lambda][epsilon][upsilon][rho]ov (tetrapleuron), which translates to "quadrilateral.
He explained the passive reception by the eyes of light rays that are reflected from objects, not an active emanation of light rays from the eyes, which was what Ptolemy
Arsinoe IV was Cleopatra's younger half-sister or sister, both of them fathered by Ptolemy
XII Auletes, though whether they shared a mother is not clear.
Her father was the pharaoh Ptolemy
XII Auletes, who was married to the Egyptian queen Cleopatra V.
Then they turn to the description of the Central Asiatic countries by second-century CE Alexandrian astronomer Ptolemy
in his "Geography," which dates to about 150 CE, so some six centuries after Herodotus.