Another passage of the Problems has been thought to show that in Greek music the melody ended on the Hypatê. The words are these (Probl. xix. 33):
Dia ti euarmostoteron apo tou oxeos epi to bary ê apo tou
bareos epi to oxy; poteron hoti to apo tês archês ginetai archesthai? hê gar mesê kai hêgemôn oxytatê tou tetrachordou; to de ouk ap' archês all' apo teleutês.
'Why is a descending scale more musical than an ascending one? Is it that in this order we begin with the beginning,—since the Mesê or leading note [19] is the highest of the tetrachord,—but with the reverse order we begin with the end?'
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' (archê), and the Hypatê an 'end'? In Aristotelian language the word 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 (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 (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.
Another instance of the use of 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:
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 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.