Our next source of information is the technical writer Aristoxenus, a contemporary and pupil of Aristotle. Of his many works on the subject of music three books only have survived, bearing the title harmonika otoicheia [2]. In the treatment adopted by Aristoxenus the chapter on keys follows the chapter on 'systems' (systÊmata). By a systÊma he means a scale consisting of a certain succession of intervals: in other words, a series of notes whose relative pitch is determined. Such a system may vary in absolute pitch, and the tonoi or keys are simply the different degrees of pitch at which a particular system is taken (tous tonous eph' Ôn tithemena ta systÊmata melÔdeitai). When the system and the key are both given it is evident that the whole series of notes is determined.

Aristoxenus is the chief authority on the keys of Greek music. In this department he is considered to have done for Greece what Bach's Wohltemperirtes Clavier did for modern Europe. It is true that the scheme of keys which later writers ascribe to him.

'No one,' says Aristoxenus (p. 37 Meib.), 'has told us a word about the keys, either how they are to be arrived at (tina tropon lÊpteon), or from what point of view their number is to be determined. Musicians assign the place of the keys very much as the different cities regulate the days of the month. The Corinthians, for example, will be found counting a day as the tenth of the month, while with the Athenians it is the fifth, and in some other place the eighth. Some authorities on music (harmonikoi) say that the Hypo-dorian is the lowest key, the Mixo-lydian a semitone higher, the Dorian again a semitone higher, the Phrygian a tone above the Dorian, and similarly the Lydian a tone above the Phrygian. Others add the Hypo-phrygian flute [i. e. the scale of the flute so called] at the lower end of the list. Others, again, looking to the holes of the flute (pros tÊn tÔn aulÔn trupÊsin blepontes), separate the three lowest keys, viz. the Hypo-phrygian, Hypo-dorian, and Dorian, by the interval of three-quarters of a tone (trisi diesesin), but the Phrygian from the Dorian by a tone, the Lydian from the Phrygian again by three-quarters of a tone, and the Mixo-lydian from the Lydian by a like interval. But as to what determines the interval between one key and another they have told us nothing.'

It will be seen that (with one marked exception) there was agreement about the order of the keys in respect of pitch, and that some at least had reduced the intervals to the succession of tones and semitones which characterises the diatonic scale. The exception is the Mixo-lydian, which some ranked immediately below the Dorian, others above the Lydian. Westphal attributes this strange discrepancy to the accidental displacing of some words in the MSS. of Aristoxenus [3]. However this may be, it is plain that in the time of Aristoxenus considerable progress had been made towards the scheme of keys which was afterwards connected with his name. This may be represented by the following table, in which for the sake of comparison the later Hypo-lydian and Hypo-dorian are added in brackets:

Mixo-lydian
semitone - {
Lydian
tone -{
Phrygian
tone -{
Dorian
semitone - {
Hypo-dorian[Hypo-lydian]
tone -{
Hypo-phrygian
tone -{
[Hypo-dorian]