4. The Hindu-Greek diatonic scale

The relationship between number and tone common to the Vedic Aryans, Sumerians, Hebrews, and Greeks involves a reference scale of integers used for defining ratios between the numbers 1 and 2. Usually, these numbers are products of the prime numbers 2, 3, and 5. Also, the reference numbers corresponding to tones of a musical scale should be closed under reciprocation. Details are found in many musical texts, but McClain is particularly clear on this theory. For example, some reference scales found in the literature use the whole number intervals: [30, 60], [72, 144], [360, 720], and so on. Bypassing many complications, we choose one of these for our purposes, [72, 144], which is the smallest reference scale which may be interpreted as a scale of string lengths on a monochord for the diatonic scale. McClain calls this range of integers the Christian set. The heptatonic scale found by the usual Pythagorean method -- in rising order (decreasing string lengths, increasing tones) -- is thus:

This is the Hindu-Greek Diatonic Scale, also known as the Dorian Mode of ancient Greece and as Ptolemy's Diatonic Syntonon, and is shown in Fig. 1. (See McClain, 1976, Chart 1, p. xxi, and Chart 3, p. 13. See also Levin, 1994, p. 77.)