In this scheme the important feature--that which marks it as an advance on the others referred to by Aristoxenus--is the conformity which it exhibits with the diatonic scale. The result of this conformity is that the keys stand in a certain relation to each other. Taking any two, we find that certain notes are common to them. So long as the intervals of pitch were quite arbitrary, or were practically irrational quantities, such as three-quarters of a tone, no such relation could exist. It now became possible to pass from one key to another, _i. e._ to employ _modulation_ ([Greek: metabolê]) as a source of musical effect. This new system had evidently made some progress when Aristoxenus wrote, though it was not perfected, and had not passed into general use. § 11. _Names of Keys_ ([Greek: hypo-]). A point that deserves special notice at this place is the use of the prefix _Hypo-_ ([Greek: hypo-]) in the names of keys. In the final Aristoxenean system _Hypo-_ implies that a key is lower by the interval of a Fourth than the key to whose name it is prefixed. This convention served to bring out the special relation between the two keys, viz. to show that they are related (to use modern language) as the keys of a tonic and dominant. In the scheme of keys now in question there is only one instance of this use of _Hypo-_, namely in the Hypo-phrygian, the most recently introduced. It must have been on the analogy of this name that the term Hypo-dorian was shifted from the key immediately below the Dorian to the new key a Fourth below it, and that the new term Hypo-lydian was given to the old Hypo-dorian in accordance with its similar relation to the Lydian. In the time of Aristoxenus, then, this technical sense of _Hypo-_ had not yet been established, but was coming into use. It led naturally to the employment of _Hyper-_ in the inverse sense, viz. to denote a key a Fourth higher (the key of the sub-dominant). By further steps, of which there is no record, the Greek musicians arrived at the idea of a key for every semitone in the octave; and thus was formed the system of thirteen keys, ascribed to Aristoxenus by later writers. (See the scheme at the end of this book, Table II.) Whether in fact it was entirely his work may be doubted. In any case he had formed a clear conception--the want of which he noted in his predecessors--of the principles on which a theoretically complete scheme of keys should be constructed.