The statement that the ancient diagrams gave a series of twenty-eight successive dieses or quarter-tones has not been explained. The number of quarter-tones in an octave is only twenty-four. Possibly it is a mere error of transcription ([Greek: [=kê]] for [Greek: [=kd]]). If not, we may perhaps connect it with the seven intervals of the ordinary octave scale, and the simple method by which the enharmonic intervals were expressed in the instrumental notation. It has been explained that raising a note a quarter of a tone was shown by turning it through a quarter of a circle. Thus, our _c_ being denoted by [Symbols: E], _c_* was [Symbols: w], and _c_[Symbols: c] was [Symbols: 3]. Now the ancient diagrams, which divided every tone into four parts, must have had a character for _c_[Symbols: S]*, or the note three-quarters of a tone above _c_. Naturally this would be the remaining position of [Symbols: E], namely [Symbols: m]. Again, we have seen that when the interval between two notes on the diatonic scale is only a semitone, the result of the notation is to produce a certain number of duplicates, so to speak. Thus: [Symbols: K] stands for _b_, and therefore [Symbols:)1] for _c_: but _c_ is a note of the original scale, and as such is written [Symbols: q]. It may be that the diagrams to which Aristoxenus refers made use of these duplicates: that is to say, they may have made use of all four positions of a character (such as [Symbols: K 7g]) whether the interval to be filled was a tone or a semitone. If so, the seven intervals would give twenty-eight characters (besides the upper octave-note), and apparently therefore twenty-eight dieses. Some traces of this use of characters in four positions have been noticed by Bellermann (_Tonleitern_, p. 65).] This representation of the musical diagrams is borne out by the passage in the _Republic_ in which Plato derides the experimental study of music: _Rep._ p. 531 a [Greek: tas gar akouomenas au symphônias kai phthongous allêlois anametrountes anênyta, hôsper hoi astronomoi, ponousin. Nê tous theous, ephê, kai geloiôs ge, pyknômat' atta onomazontes kai paraballontes ta ôta, hoion ek geitonôn phônên thêreuomenoi, hoi men phasin eti katakouein en mesô tina êchên kai smikrotaton einai touto diastêma, hô metrêteon, hoi de k.t.l.]