In the discussion of 15 octave species (chapter VI),See F-VAL ms. 148, fol. 67r-68r). Aurelian ascends from the lowest octave of Hypodorian (A—g—a) up to the highest octave a ninth higher which he called "Hyperlydian" (b—aa—bb). Including this conclusion at the end, the whole passage was compiled literally from Aurelius Cassiodorus."De musica" as the chapter V, §8 of "De artibus ac disciplinis liberalium litterarum" (Gerbert 1784, i:17-18). This only reference to a triphonic tone system within Carolingian chant theory, also known as the "lesser perfect system" within the Aristoxenian school,See Mathiesen's article about Ancient Greece (table 1). was probably the reason for Aurelian's own name as author and compiler of music theory. It proves that the early use of b flat was more influenced by triphonia. The tone system behind the 15 octaves is built in three conjunct ("synemmena") tetrachords (ypodorius—dorius—hyperdorius: A—D—G—c), with another tetrachord below (hypo) and a third one above (hyper).In general, scholars of Western chant tend to underestimate this oriental influence, because they find no practical use for it and already Cassiodorus' interpretation is based on a misunderstanding—probably of another Latin source, which was nearly as popular as Boethius' music treatise: Martianus Capella's De nuptiis Philologiae et Mercurii (Liber IX De harmonia). Even the reproduction of the wing diagram (taken from Munich, Bayerische Staatsbibliothek, Ms. clm 14629, fol. 220r) could not convince Charles Atkinson (2008, 72), that Cassiodorus' diatonic tropes in chromatic order were based on a diatonic interpretation of the chromatic Iastian and enharmonic Eolian division, so that the Lydian tonus became based on F#. Commentaries by John Scotus Eriugena and Remigius of Auxerre testified the difficulties, that Carolingian philosophers had, when they tried to understand the practical use of temporary changes from the diatonic to the enharmonic or the chromatic genus (Atkinson 2008, 65-84). Within triphonia, three species like Dorian (D—c—d defined by the tetrachord tone—half tone—tone), Phrygian (E—d—e defined by the tetrachord half tone—tone—tone), and Lydian were already enough to represent the tone system, as it was needed during the use of b flat. But Cassiodorus' simplification corrupted the Lydian tonus, so that the tetrachords were based on C#—F#—b—e as a kind of transposed Phrygian species, and the Eolian tonus (C—F—♭—e♭) replaced it.
