In another place (_Probl._ xix. 32) the question is asked, why the interval of the octave is called [Greek: dia pasôn], not [Greek: di' oktô],--as the Fourth is [Greek: dia tessarôn], the Fifth [Greek: dia pente]. The answer suggested is that there were anciently seven strings, and that Terpander left out the Tritê and added the Nêtê. That is to say, Terpander increased the compass of the scale from the ancient two tetrachords to a full Octave; but he did not increase the number of strings to eight. Thus he produced a scale like the standard octave, but with one note wanting; so that the term [Greek: di oktô] was inappropriate. Among later writers who confirm this account we may notice Nicomachus, p. 7 Meib. [Greek: mesê dia tessarôn pros amphotera en tê heptachordô kata to palaion diestôsa]: and p. 20 [Greek: tê toinyn archaiotropô lyra toutesti tê heptachordô, kata synaphên ek duo tetrachordôn synestôsê k.t.l.] It appears then that two kinds of seven-stringed scales were known, at least by tradition: viz. (1) a scale composed of two conjunct tetrachords, and therefore of a compass less than an octave by one tone; and (2) a scale of the compass of an octave, but wanting a note, viz. the note above Mesê. The existence of this incomplete scale is interesting as a testimony to the force of the tradition which limited the number of strings to seven. § 18. _The Perfect System._ The term 'Perfect System' ([Greek: systêma teleion]) is applied by the technical writers to a scale which is evidently formed by successive additions to the heptachord and octachord scales explained in the preceding chapter. It may be described as a combination of two scales, called the Greater and Lesser Perfect System. The Greater Perfect System ([Greek: systêma teleion meizon]) consists of two octaves formed from the primitive octachord System by adding a tetrachord at each end of the scale. The new notes are named like those of the adjoining tetrachord of the original octave, but with the name of the tetrachord added by way of distinction. Thus below the original Hypatê we have a new tetrachord Hypatôn ([Greek: tetrachordon hypatôn]), the notes of which are accordingly called Hypatê Hypatôn, Parhypatê Hypatôn, and Lichanos Hypatôn: and similarly above Nêtê we have a tetrachord Hyperbolaiôn. Finally the octave downwards from Mesê is completed by the addition of a note appropriately called Proslambanomenos. The Lesser Perfect System ([Greek: systêma teleion elasson]) is apparently based upon the ancient heptachord which consisted of two 'conjunct' tetrachords meeting in the Mesê. This scale was extended downwards in the same way as the Greater System, and thus became a scale of three tetrachords and a tone. These two Systems together constitute the Perfect and 'unmodulating' System ([Greek: systêma teleion ametabolon]), which may be represented in modern notation[1] as follows: