 Most scholars agree that Philolaus was not interested in harmonic theory for its own sake as were Archytas, Aristoxenus and much later Ptolemy (Barker 2007: 278–286; Creese 2010: 104). There is also no evidence that he used a monochord in order to determine or demonstrate harmonic relationships (Creese 2010: 104–117). However, even if Philolaus’ main interest was the application of harmonics to cosmology, the approach to harmonics found in Fragment 6a and in the disputed 6b and A26 has been of considerable interest to historians of harmonic theory, and their conclusions are highly relevant to the issue of the authenticity of 6b and A26. By the end of the fourth century there was a marked division between Pythagorean harmonic theorists who regarded musical intervals as corresponding to whole number ratios and followers of Aristoxenus who regarded them as linear magnitudes. For Aristoxenus the octave consisted of six equal tones added together and could be divided in half. For the Pythagoreans the octave was made up of ratios which were combined not by addition but by multiplication (e.g. the octave [2 : 1] = the product of the fifth [3 : 2] and the fourth [4 : 3], not the sum). The octave was a little less than six tones (9 : 8), and neither it nor the tone could be divided in halves. Fragment 6a of Philolaus mentions the ratios that correspond to the intervals (e.g., the tone is 9 : 8) whereas Fragment 6b and A26 present intervals as linear magnitudes (e.g., the tone is 27). Since 6a treats intervals as corresponding to ratios in a properly Pythagorean way, it has usually been accepted as authentic and the difference in approach found in 6b and A26 has been seen as incompatible with 6a and hence as indicative of their spuriousness. Recently some scholars, on the other hand, have suggested that in Philolaus’ time this split between regarding intervals as ratios and regarding them as linear magnitudes had not been fully developed and that Philolaus may have regarded the two approaches as compatible (Hagel 2009: 144; Creese 2010: 116, n. 119). Barker supports this point by arguing that Philolaus is employing both approaches already in Fragment 6a. Thus, while the fragment equates certain ratios with certain intervals, Philolaus does nothing mathematically with those ratios and instead emphasizes that intervals have size and can be combined to create the octave, which suggests that he thinks of them as linear magnitudes (Barker 2007: 264–271). This same combination of intervals as ratios and intervals as linear magnitudes can be seen in Fragment 6b and A26, where there is the additional emphasis on halving intervals. The halving of intervals seems to be directed at finding the mid-point of the octave and this might have been important in Philolaus’ application of the octave to the cosmos, where the center is important for the symmetry of the whole system (Barker 2007: 277; cf. Hagel 2009: 151). Moreover, Aristotle’s description of fifth-century Pythagoreanism seems to display the same odd mixture of ratios and individual numbers (Barker 2007: 286). However, there are some difficulties for this approach. While Fragment 6a does emphasize the size of intervals and their combination to form the octave, size need not mean linear magnitude and combination need not mean addition. In fact, no specific linear magnitudes (e.g. 27, which is equated with the tone in A26) are mentioned. The only sizes of intervals mentioned are precisely the ratios. Nor is there any evidence in Fragment 6a of any interest on Philolaus’ part in halving intervals. The situation is quite the reverse. He uniformly presents the octave and, indeed, all intervals as composed of unequal parts (e.g., the octave is composed of the fourth and the fifth). Thus, while it is impossible to deny that Philolaus might have tried to embrace both concepts of the nature of a musical interval, Fragment 6a still differs from 6b and A26 in being totally intelligible on the assumption that intervals correspond to ratios and in making no overt use of linear magnitudes or halves of such magnitudes (Huffman 2012: 235; cf. McKirahan 2012).