There is here no explicit statement that the melody ended on the Hypatê, or even that it began with the Mesê. In what sense, then, was the Mesê a 'beginning' ([Greek: archê]), and the Hypatê an 'end'? In Aristotelian language the word [Greek: archê] has various senses. It might be used to express the relation of the Mesê to the other notes as the basis or ground-work of the scale. Other passages, however, point to a simpler explanation, viz. that the order in question was merely conventional. In _Probl._ xix. 44 it is said that the Mesê is the beginning ([Greek: archê]) of one of the two tetrachords which form the ordinary octave scale (viz. the tetrachord Mesôn); and again in _Probl._ xix. 47 that in the old heptachord which consisted of two conjunct tetrachords (_e-a-d_) the Mesê (_a_) was the end of the upper tetrachord and the beginning of the lower one ([Greek: hoti ên tou men anô tetrachordou teleutê, tou de katô archê]). In this last passage it is evident that there is no reference to the beginning or end of the melody. [Footnote 1: The term [Greek: hêgemôn] or 'leading note' of the tetrachord Mesôn, here applied to the Mesê, is found in the same sense in Plutarch, _De Mus._ c. 11, where [Greek: ho peri ton hêgemona keimenos tonos] means the disjunctive tone. Similarly Ptolemy (_Harm._ i. 16) speaks of the tones in a diatonic scale as being [Greek: en tois hêgoumenois topois], the semitones [Greek: en tois hepomenois] (sc. of the tetrachord): and again of the ratio 5:4 (the major Third) as the 'leading' one of an Enharmonic tetrachord ([Greek: ton epitetarton hos estin hêgoumenos tou enarmoniou genous]).] Another instance of the use of [Greek: archê] in connexion with the musical scale is to be found in the _Metaphysics_ (iv. 11, p. 1018 _b_ 26), where Aristotle is speaking of the different senses in which things may be prior and posterior: [Greek: Ta de kata taxin; tauta d' estin hosa pros ti hen hôrismenon diestêke kata ton logon, hoion parastatês tritostatou proteron, kai paranêtê nêtês; entha men gar ho koryphaios, entha de hê mesê archê.] 'Other things [are prior and posterior] in _order_: viz. those which are at a varying interval from some one definite thing; as the second man in the rank is prior to the third man, and the Paranêtê to the Nêtê: for in the one case the coryphaeus is the starting-point, in the other the Mesê.' Here the Mesê is again the [Greek: archê] or beginning, but the order is the ascending one, and consequently the Nêtê is the end. The passage confirms what we have learned of the relative importance of the Mesê: but it certainly negatives any inference regarding the note on which the melody ended.