'In the size of the intervals and the pitch of the notes the elements of melody seem to be infinite; but in respect of the values (_i.e._ the relative places of the notes) and in respect of the forms (_i.e._ the succession of the intervals) and in respect of the positions they are limited and settled.' Aristoxenus goes on to illustrate this by supposing that we wish to continue a scale downwards from a [Greek: pyknon] or pair of small intervals (Chromatic or Enharmonic). In this case, as the [Greek: pyknon] forms the lower part of a tetrachord, there are two possibilities. If the next lower tetrachord is disjunct, the next interval is a tone; if it is conjunct, the next interval is the large interval of the genus ([Greek: hê men gar kata tonon eis diazeuxin agei to tou systêmatos eidos, hê de kata thateron diastêma ho ti dêpot' echei megethos eis synaphên]). Thus the succession of intervals is determined by the relative position of the two tetrachords, as to which there is a choice between two definite alternatives. This then is evidently what is meant by the words [Greek: kata tas theseis][1]. On the other hand the [Greek: thesis] of Ptolemy's nomenclature is the absolute pitch (_Harm._ ii. 5 [Greek: pote men par' autên tên thesin, to oxyteron haplôs ê baryteron, onomazomen]), and this is one of the elements which according to Aristoxenus are indefinite. [Footnote 1: So Bacch. p. 19 Meib. [Greek: theseis de tetrachordôn hois to melos horizetai eisin hepta? synaphê, diazeuxis, hypodiazeuxis, k.t.l.] (see the whole passage).] Westphal also finds the nomenclature by position implied in the passage of the Aristotelian _Problems_ (xix. 20) which deals with the peculiar relation of the Mesê to the rest of the musical scale. The passage has already been quoted and discussed (_supra_, p. 43), and it has been pointed out that if the Mesê of the Perfect System ([Greek: mesê kata dynamin]) is the key-note, the scale must have been an octave of the _a_-species. If octaves of other species were used, as Westphal maintains, it becomes necessary to take the Mesê of this passage to be the [Greek: mesê kata thesin], or Mesê by position. That is, Westphal is obliged by his theory of the Modes to take the term Mesê in a sense of which there is no other trace before the time of Ptolemy. But-- (1) It is highly improbable that the names of the notes--Mesê, Hypatê, Nêtê and the rest--should have had two distinct meanings. Such an ambiguity would have been intolerable, and only to be compared with the similar ambiguity which Westphal's theory implies in the use of the terms Dorian, &c.