Although the language of Ptolemy's exposition is studiously impersonal, it may be gathered that his reduction of the number of keys from fifteen to seven was an innovation proposed by himself[2]. If this is so, the rest of the scheme,--the elimination of the element of pitch, and the 'nomenclature by position,'--must also be due to him. Here, however, we find ourselves at issue with Westphal and those who agree with him on the main question of the Modes. According to Westphal the nomenclature by position is mentioned by Aristoxenus, and is implied in at least one important passage of the Aristotelian _Problems_. We have now to examine the evidence which he adduces to support his contention. [Footnote 1: _Harm._ ii. 7 [Greek: pros tên toiautên diaphoran hê tôn organôn holôn epitasis ê palin anesis aparkei.]] [Footnote 2: This may be traced in the occasionally controversial tone; as _Harm._ ii. 7 [Greek: hoi men ep' elatton tou dia pasôn phthasantes, hoi d' ep' auto monon, hoi de epi to meizon toutou, prokopên tina schedon toiautên aei tôn neôterôn para tous palaioterous thêrômenôn, anoikeion tês peri to hêrmosmenon physeôs te kai apokatastaseôs; hê monê perainein anankaion esti tên tôn esomenôn akrôn tonôn diastasin]. We may compare c. 11.] § 30. _Nomenclature by Position._ 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 ([Greek: megethos]), and for each compass the differences in form and composition and position ([Greek: 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 [Greek: 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 ([Greek: 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: [Greek: 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.]