Superparticular

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. ❋ Huffman, Carl (2007)

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). ❋ Huffman, Carl (2007)

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. ❋ Huffman, Carl (2007)

Barker tries to save Archytas 'adherence to the principle that all concordant intervals should have superparticular ratios, but there is no direct evidence that he was using such a principle, and Ptolemy may be mistaken to apply it to him. ❋ Huffman, Carl (2007)

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. ❋ Huffman, Carl (2007)

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. ❋ Huffman, Carl (2007)

Page 262, Volume 3 that it was the perfect tuning, because both perfect and imperfect consonances were in simple ratios of the class n + 1/n, known as superparticular. ❋ CLAUDE V. PALISCA (1968)

For example, I've written pieces that generate harmonies from combination tones, and I've recently been inspired by Greg Scheimer's work to focus on epimores (i.e., superparticular ratios). ❋ Unknown (2008)