remembered by name since the 5th century. Compositions had been related to them, but they must be reconstructed by notated sources which date centuries later. The melodic neume notation of Byzantine music developed late since the 10th century, with the exception of an earlier ekphonetic notation, interpunction signs used in lectionaries, but modal signatures for the eight echoi can already be found in fragments (papyri) of monastic hymn books (tropologia) dating back to the 6th century. Amid the rise of Christian civilization within Hellenism, many concepts of knowledge and education survived during the imperial age, when Christianity became the official religion. The Pythagorean sect and music as part of the four "cyclical exercises" (ἐγκύκλια μαθήματα) that preceded the Latin quadrivium and science today based on mathematics, established mainly among Greeks in southern Italy (at Taranto and Crotone). Greek anachoretes of the early Middle Ages did still follow this education. The Calabrian Cassiodorus founded Vivarium where he translated Greek texts (science, theology and the Bible), and John of Damascus who learnt Greek from a Calabrian monk Kosmas, a slave in the household of his privileged father at Damascus, mentioned mathematics as part of the speculative philosophy. According to him philosophy was divided into theory (theology, physics, mathematics) and practice (ethics, economy, politics), and the Pythagorean heritage was part of the former, while only the ethic effects of music were relevant in practice. The mathematic science harmonics was usually not mixed with the concrete topics of a chant manual. Nevertheless, Byzantine music is modal and entirely dependent on the Ancient Greek concept of harmonics. Its tonal system is based on a synthesis with ancient Greek models, but we have no sources left that explain to us how this synthesis was done. Carolingian cantors could mix the science of harmonics with a discussion of church tones, named after the ethnic names of the octave species and their transposition tropes, because they invented their own octoechos on the basis of the Byzantine one. But they made no use of earlier Pythagorean concepts that had been fundamental for Byzantine music, including: It is not evident by the sources, when exactly the position of the minor or half tone moved between the and . It seems that the fixed degrees (hestotes) became part of a new concept of the echos as melodic mode (not simply octave species), after the echoi had been called by the ethnic