Two passages of Aristoxenus are quoted by Westphal in support of his contention. The first (p. 6 Meib.) is one in which Aristoxenus announces his intention to treat of Systems, their number and nature: 'setting out their differences in respect of compass (megethos), and for each compass the differences in form and composition and position (tas te kata schÊma kai kata synthesin kai kata thesin), so that no element of melody,—either compass or form or composition or position,—may be unexplained.' But the word thesis, when applied to Systems, does not mean the 'position' of single notes, but of groups of notes. Elsewhere (p. 54 Meib.) he speaks of the position of tetrachords towards each other (tas tÔn tetrachordÔn pros allÊla theseis), laying it down that any two tetrachords in the same System must be consonant either with each other or with some third tetrachord. The other passage quoted by Westphal (p. 69 Meib.) is also in the discussion of Systems. Aristoxenus is pointing out the necessity of recognising that some elements of melodious succession are fixed and limited, others are unlimited:

kata men oun ta megethÊ tÔn diastÊmatÔn kai tas tÔn phthongÔn taseis apeira pÔs phainetai einai ta peri melos, kata de tas dynameis kai kata ta eidÊ kai kata tas theseis peperasmena te kai tetagmena.

'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 pyknon or pair of small intervals (Chromatic or Enharmonic). In this case, as the 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 (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 kata tas theseis [33]. On the other hand the thesis of Ptolemy's nomenclature is the absolute pitch (Harm. ii. 5 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.

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 (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 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.

(2) If the different species of the octave were the practically important scales, as Westphal maintains, the position of the notes in these scales must have been correspondingly important. Hence the nomenclature by position must have been the more usual and familiar one. Yet, as we have shown, it is not found in Aristotle, Aristoxenus or Euclid—to say nothing of later writers.

(3) The nomenclature by position is an essential part of the scheme of Keys proposed by Ptolemy. It bears the same relation to Ptolemy's octaves as the nomenclature by 'value' bears to the old standard octave and the Perfect System. It was probably therefore devised about the time of Ptolemy, if not actually by him.