In illustration of his theory Ptolemy gives tables showing in numbers the intervals of the octaves used in the different keys and genera. He shows two octaves in each key, viz. that from Hypatê Mesôn (kata thesin) to Nêtê Diezeugmenôn (called the octave apo nêtês), and that from Proslambanomenos to Mesê (the octave apo mesês). As he also gives the divisions of five different 'colours' or varieties of genus, the whole number of octaves is no less than seventy.
Ptolemy does not exclude difference of pitch altogether. The whole instrument, he says, may be tuned higher or lower at pleasure [31]. Thus the pitch is treated by him as modern notation treats the tempo, viz. as something which is not absolutely given, but has to be supplied by the individual performer.
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 [32]. 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.
§ 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 (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—