In ancient Greece, (c. 500 B.C.E) the Ionian philosopher/mathematician/scientist Pythagoras (c. 570 - 475 BCE) developed the first theory of music. He recognized that musical notes and their natural harmonies are based on mathematical ratios. (3:2 for the 5th harmony of any note, 2:1 for an octave). Pythagoras' system was used throughout the Middle Ages.
But natural harmonies created a problem for composers starting about 1500 A.D. The problem was resolved in the Late Baroque era by J.S. Bach and others.
(The 12 notes of the scale in the cycle of 5ths: E♭–B♭–F–C–G–D–A–E–B–F♯–C♯–G♯)
The Greek tragedies of the great playwrights Sophocles and Euripides were sung. They had no idea what the music sounded like (only recently have there been attempts to recreate ancient Greek music). In the early Baroque Era, the first operas were attempts to recreate what they believed experiencing the Greek plays was like. Greek tragedy will greatly influence the history of opera.
With the fall of the Western Roman Empire (c. 400) ancient Greek culture is lost in Italy, France, and Spain. (It had never reached England or Germany in any significant way). It would be kept alive in the Eastern Roman Empire (the Byzantine Empire) based in Constantinople. Ancient Greek thinking would find its way back into western Europe in the High Middle Ages followed by Roman thought by the late 15th century. The revival of ancient thought would spark changes in religion, politics, science and the arts in western Europe. The results would be Renaissance humanism followed by the invention of opera,
video - Rediscovering Ancient Greek Music (2017) 15:40
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