Hi Susan, Your message leaves the impression that non-Western tuning systems are unrelated to the Western system and do not follow small-integer ratio principles. But that's not quite true. First, our modern Western tuning does not follow Pythagorean intervals, either. It uses equal temperament, as I'm sure you know. Second, the tuning systems of the rest of the world are related in many ways to the Western tuning system and tend to use small integer ratios (with some exceptions, such as gamelan). From something I'm currently working on: The oldest Western theory of musical consonance is due to Pythagoras: Musical consonance is determined by the ratios of small whole numbers. The principle was that intervals of small integer ratios produced harmonies that were pleasing and mathematically pure. The Pythagorean tuning system is the oldest extant Western system devised explicitly according to this principle, and is thought to have been devised by Pythagoras himself. Interestingly, the available evidence suggests that a scale similar to this was already in use in the West before any mathematical principle was advanced to describe its structure. According to Iambiclus’s Life of Pythagoras (Guthrie, 1987), Pythagoras did not invent the Pythagorean scale, he discovered a principle to explain a scale that was already in use. A related tuning system, Just Intonation (JI), originally proposed by Ptolemy (Hunt, 1992), derives all interval ratios in relation to one single tonic, and chooses the smallest possible integer ratios that divide the octave (approximately) equally. Just intonation satisfies the principle of small integer ratios more nearly than the Pythagorean, and it is sometimes referred to as the natural scale. The three largest non-Western tuning systems are Indian, Chinese and Arab-Persian. Each of these has inclusive 12-tone scales whose frequency relationships are similar to the Western chromatic scales. Two of these systems, the Indian and the Arab-Persian, use more than 12 intervals per octave (Burns, 1999). The musical systems of India are theoretically based on 22 intervals per octave. However, the basic scale consists of 12 tones tuned according to a form of just intonation. The remaining 10 tones are slight variations of certain intervals, the exact frequencies of which depend upon the individual melodic framework (raga) being played. The Arab-Persian system theoretically employs intervals that bisect the distance between Western chromatic intervals. However, there is some controversy as to the exact number of possible intervals and the actual intervals produced in performance. Most sources list the small integer ratio tuning relationships. Ed On Feb 28, 2007, at 1:41 AM, Susan Allen wrote:
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