Whether the musical theory of the ecclesiastical chants prior to the fifth century--apart from the Ambrosian hymns--was influenced more by Roman or Oriental traditions is a moot point. The question, however, is not vital, as the real founder of the church system of modes, the Pythagorean philosopher Boethius (fifth century), was professedly an imitator of the Greek theorists. Boethius speaks of the ancients with something like veneration, and takes pride in writing like a Greek, after the fashion of Plato, Aristotle, Aristoxenus, Nichomachus, Philolaus, and Ptolemy. He is a mathematician rather than a musician. He congratulates Ptolemy on having ignored the testimony of the ear and condemns the practice of music as interfering with the just and logical consideration of theory. Cassiodorus and Isidore of Seville speak in somewhat the same fashion. Boethius is the authority for a long line of church musicians, including Hucbald, Guido, Englebert, Jean de Muris, Adam de Fulda, and Alcuin. The mathematical view of music fathered by him gained such prevalence that in the curriculum of mediæval universities music was placed among the mathematical sciences. We cannot do more here than briefly indicate the tone system used in the church after the liturgy had been definitely formulated, without going into the question of its earlier evolution. The system of modes was professedly founded on the tetrachordal species of the Greeks (see Chap. IV, p. 112), but with an obvious misunderstanding of the Greek system. At first only four forms were recognized, namely, the so-called authentic (from [Greek: ], to govern) modes of St. Ambrose. According to tradition St. Gregory added to these four 'plagal’ (from [Greek: plagios], oblique). At any rate before the eleventh century there were eight accepted church modes, four 'authentic’ modes, and four 'plagal’ or derived modes. Later theorists taught the existence of fourteen different scales. Two of these were rejected as 'impure’; the other twelve remained in use for centuries, and were known by the names of their Greek prototypes, but these names, too, were misapplied, as will be seen from the following table, where the octave D-d corresponding to the Phrygian species of the Greeks is named Dorian, and vice versa the octave F-f (Greek mixolydian) has become the Lydian, and so forth. The difference between authentic and plagal modes was chiefly one of range and emphasis. For example, in every mode two tones were considered, and were, as a matter of fact, of predominant importance; the _final_ or note on which the piece ended, somewhat analogous to our tonic or key-note, and the _dominant_, the note most frequently touched in the course of the melody, the centre of gravity, so to speak, about which the melody moved. In the authentic modes the melody never sank below the final except in cadence, where it might take the note immediately below the final and rise by one step to the close; and the dominant