 Archytas’ final contribution to music theory has to do with the structure of the scale (for a more detailed account than what follows see Huffman 2005: 402–25 and Barker 2007: 292–302). The Greeks used a number of different scales, which were distinguished by the way in which the fourth, or tetrachord, was constructed. These scales were grouped into three main types or genera. One genus was called the diatonic; one example of this is the Pythagorean diatonic described above, which is built on the tetrachord with the intervals 9 : 8, 9 : 8 and 256 : 243 and was used by Philolaus and Plato. There is no doubt that Archytas knew of this diatonic scale, but his own diatonic tetrachord was somewhat different, being composed of the intervals 9 : 8, 8 : 7 and 28 : 27. Archytas also defined scales in the two other major genera, the enharmonic and chromatic. Archytas’ enharmonic tetrachord is composed of the intervals 5 : 4, 36 : 35 and 28 : 27 and his chromatic tetrachord of the intervals 32 : 27, 243 : 224, and 28 : 27. There are several puzzles about the tetrachords which Archytas adopts in each of the genera. First, why does Archytas reject the Pythagorean diatonic used by Philolaus and Plato? Second, Ptolemy, who is our major source for Archytas’ tetrachords (A16), argues that Archytas adopted as a principle that all concordant intervals should correspond to superparticular ratios. The ratios in Archytas’ diatonic and enharmonic tetrachords are indeed superparticular, but two of the ratios in his chromatic tetrachord are not superparticular (32 : 27 and 243 : 224). Why are these ratios not superparticular as well? Finally, Plato criticizes Pythagorean harmonics in the Republic for seeking numbers in heard harmonies rather than ascending to generalized problems (531c). Can any sense be made of this criticism in light of Archytas’ tetrachords? The basis for an answer to all of these questions is contained in the work of Winnington-Ingram (1932) and Barker (1989, 46-52). The crucial point is that Archytas’ account of the tetrachords in each of the three genera can be shown to correspond to the musical practice of his day; Ptolemy’s criticisms miss the mark because of his ignorance of musical practice in Archytas’ day, some 500 years before Ptolemy (Winnington-Ingram 1932, 207). Archytas is giving mathematical descriptions of scales actually in use; although mathematical considerations did play a role (Barker 2007: 295–302), he arrived at his numbers in part by observation of the way in which musicians tuned their instruments (Barker 1989, 50–51). He did not follow the Pythagorean diatonic scale because it did not correspond to any scale actually in use, although it does correspond to a method of tuning. The unusual numbers in Archytas’ chromatic tetrachord do correspond to a chromatic scale in use in Archytas’ day. Ptolemy was wrong to suggest that Archytas adhered to the principle that all concordant intervals should have superparticular ratios (Huffman 2005: 422–3), although Barker suggests that he may have been following a different but related principle (2007: 301). Archytas thus provides a brilliant analysis of the music of his day, but it is precisely his focus on actual musical practice that draws Plato’s ire. Plato does not want him to focus on the music he hears about him (“heard harmonies”) but rather to ascend to consider quite abstract questions about which numbers are harmonious with which. Plato might well have welcomed a principle of concordance based solely on mathematical considerations, such as the principle that only superparticular ratios are concordant, but Archytas wanted to explain the numbers of the music he actually heard played. There is an important metaphysical issue at stake here. Plato is calling for the study of number in itself, apart from the sensible world, while Archytas, like Pythagoreans before him, envisages no split between a sensible and an intelligible world and is looking for the numbers which govern sensible things. For discussions of Archytas as the target of Plato’s complaints in the  Republic see Huffman 2005: 423–5 and Barker 2014:192–3.