The statement that the 'relaxed Lydian' was the opposite of the Mixo-lydian, and similar to the Ionian, has given rise to much speculation. In what sense, we naturally ask, can a key or a mode be said to be 'opposite' or 'similar' to another? I venture to think that it is evidently a mere paraphrase of Plato's language. The relaxed Lydian is opposed to the Mixo-lydian because it is at the other end of the scale in pitch; and it is similar to the Ionian because the two are classed together (as [Greek: chalarai]) by Plato. The Mixo-lydian, according to Aristoxenus, was employed by the tragic poets in close union with the Dorian mode ([Greek: labontas syzeuxai tê Dôristi]). The fact that the Mixo-lydian was just a Fourth higher than the Dorian must have made the transition from the one to the other a natural and melodious one. As Aristoxenus suggested, it would be especially used to mark the passage from grandeur and dignity to pathos which is the chief characteristic of tragedy ([Greek: hê men to megaloprepes kai axiômatikon apodidôsin, hê de to pathêtikon, memiktai de dia toutôn tragôdia]). It is worth noticing that this relation obtained in the scheme of the musicians who did not arrange the keys according to the diatonic scale, but in some way suggested by the form of the flute ([Greek: hoi pros tên tôn aulôn trypêsin blepontes]). It may therefore be supposed to have been established before the relative pitch of other keys had been settled. [Footnote 1: It seems not impossible that this difficulty with regard to the 'slack Lydian' and Hypo-lydian may be connected with the contradiction in the statement of Aristoxenus about the schemes of keys in his time (p. 18). According to that account, if the text is sound, some musicians placed the Mixo-lydian a semitone below the Dorian--the Hypo-dorian being again a semitone lower. In this scheme, then, the Mixo-lydian held the place of the later Hypo-lydian. The conjecture may perhaps be hazarded, that this lower Mixo-lydian somehow represents Plato's 'slack Lydian,' and eventually passed into the Hypo-lydian.] So far the passage of Plutarch goes to confirm the view of the Platonic modes according to which they were distinguished chiefly, if not wholly, by difference of pitch. We come now, however, to a statement which apparently tends in the opposite direction, viz. that a certain Lamprocles of Athens noticed that in the Mixo-lydian mode the Disjunctive Tone ([Greek: diazeuxis]) was at the upper end of the scale ([Greek: epi to oxy]), and reformed the scale accordingly. This must refer to an octave scale of the form _b c d e f g a b_, consisting of the two tetrachords _b-e_ and _e-a_, and the tone _a-b_. Such an octave may or may not be in the Mixo-lydian key: it is certainly of the Mixo-lydian species (p. 57).