[Footnote 1: It is foreign to our purpose to discuss the critical problems presented by the text of Aristoxenus. Of the three extant books the first is obviously a distinct treatise, and should probably be entitled [Greek: peri archôn]. The other two books will then bear the old title [Greek: harmonika stoicheia]. They deal with the same subjects, for the most part, as the first book, and in the same order,--a species of repetition of which there are well-known instances in the Aristotelian writings. The conclusion is abrupt, and some important topics are omitted. It seems an exaggeration, however, to describe the _Harmonics_ of Aristoxenus as a mere collection of excerpts, which is the view taken by Marquard (_Die harmonischen Fragmente des Aristoxenus_, pp. 359-393). See Westphal's _Harmonik und Melopöie der Griechen_ (p. 41, ed. 1863), and the reply to Marquard in his _Aristoxenus von Tarent_ (pp. 165-170).] 'No one,' says Aristoxenus (p. 37 Meib.), 'has told us a word about the keys, either how they are to be arrived at ([Greek: 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 ([Greek: 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 ([Greek: 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 ([Greek: 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.'