 Archytas took harmonic theory to a whole new level of theoretical and mathematical sophistication. Ptolemy, writing in the second century AD, identifies Archytas as having “engaged in the study of music most of all the Pythagoreans” (A16). First, Archytas provided a general explanation of pitch, arguing that the pitch of a sound depends on the speed with which the sound is propagated and travels (B1). Thus, if a stick is waved back and forth rapidly, it will produce a sound that travels rapidly through the air, which will be perceived as of a higher pitch than the sound produced by a stick waved more slowly. Archytas is correct to associate pitch with speed, but he misunderstood the role of speed. The pitch does not depend on the speed with which a sound reaches us but rather on the frequency of impacts in a given period of time. A string that vibrates more rapidly produces a sound of a higher pitch, but all sounds, regardless of pitch, travel at an equal velocity, if the medium is the same. Although Archytas’ account of pitch was ultimately incorrect, it was very influential. It was taken over and adapted by both Plato and Aristotle and remained the dominant theory throughout antiquity (Barker 1989, 41 n. 47; Barker 2014: 187). Second, Archytas introduced new mathematical rigor into Pythagorean harmonics. One of the important results of the analysis of music in terms of whole number ratios is the recognition that it is not possible to divide the basic musical intervals in half. The octave is not divided into two equal halves but into a fourth and a fifth, the fourth is not divided into two equal halves but into two whole tones and a remainder. The whole tone cannot be divided into two equal half tones. On the other hand, it is possible to divide a double octave in half. Mathematically this can be seen by recognizing that it is possible to insert a mean proportional between the terms of the ratio corresponding to the double octave (4 : 1) so that 4 : 2 :: 2 : 1. The double octave can thus be divided into two equal parts each having a ratio of 2 : 1. The ratios which govern the basic musical intervals (2 : 1, 4 : 3, 3 : 2, 9 : 8), all belong to a type of ratio known as a superparticular ratio – roughly speaking, ratios of the form (n + 1) : n. Archytas made a crucial contribution by providing a rigorous proof that there is no mean proportional between numbers in superparticular ratio (A19) and hence that the basic musical intervals cannot be divided in half. Archytas’ proof was later taken over and modified slightly in the Sectio Canonis ascribed to Euclid (Prop. 3; see Barker 1989, 195). On Archytas’ proof see Huffman 2005: 451–70 and Barker 2007: 303–5.