2. The notation does not represent only the _species_ of a scale, that is to say, the relative pitch of the notes which compose it, but it represents also the absolute pitch of each note. Thus the octaves which are defined by the successive pairs of letters, [Greek:b-g, d-e], and the rest, are octaves of definite notes. If they were framed with a view to the ancient modes, as Westphal thinks, they must be the actual scales employed in these modes. If so, the modes followed each other, in respect of pitch, in an order exactly the reverse of the order observed in the keys. It need hardly be said that this is quite impossible. § 29. _Ptolemy's Scheme of Modes._ The first writer who takes the Species of the Octave as the basis of the musical scales is the mathematician Claudius Ptolemaeus (fl. 140-160 A.D.). In his _Harmonics_ he virtually sets aside the scheme of keys elaborated by Aristoxenus and his school, and adopts in their place a system of scales answering in their main features to the mediaeval Tones or Modes. The object of difference of key, he says, is not that the music as a whole may be of a higher or lower pitch, but that a melody may be brought within a certain compass. For this purpose it is necessary to vary the succession of intervals (as a modern musician does by changing the signature of the clef). If, for example, we take the Perfect System ([Greek: systêma ametabolon]) in the key of _a_ minor--which is its natural key,--and transpose it to the key of _d_ minor, we do so, according to Ptolemy, not in order to raise the general pitch of our music by a Fourth, but because we wish to have a scale with _b_ flat instead of _b_ natural. The flattening of this note, however, means that the two octaves change their species. They are now of the species _e - e_. Thus, instead of transposing the Perfect System into different keys, we arrive more directly at the desired result by changing the species of its octaves. And as there are seven possible species of the Octave, we obtain seven different Systems or scales. From these assumptions it follows, as Ptolemy shows in some detail, that any greater number of keys is useless. If a key is an octave higher than another, it is superfluous because it gives us a mere repetition of the same intervals[1]. [Footnote 1: _Harm._ ii. 8 [Greek: hoi de hyperekpiptontes tou dia pasôn tous ap' autou tou dia pasôn apôterô parelkontôs hypotithentai, tous autous aei ginomenous tois proeilêmmenois.]] If we interpose a key between (_e.g._) the Hypo-dorian and the Hypo-phrygian, it must give us over again either the Hypo-dorian or the Hypo-phrygian scale[1]. Thus the fifteen keys of the Aristoxeneans are reduced to seven, and these seven are not transpositions of a single scale, but are all of the same pitch. See the table at the end of the book.