“We need now to look at another series of intervals. These are linked to a mysterious side of music.  How they work in this way cannot always be fully explained with words. They arise from the JI series but in a different way. Once again I have found no documentation which clearly designates this type of interval in musical systems.  I have discovered that they are the proper Pythagorean way in which the intervals described in the oral tradition can be mathematically portrayed.”

“The causal intervals come from the fact that the Pythagoreans also recognised the ratio 3:1 as having a degree of consonance, because it represents an octave and a perfect fifth thus :- 2:1  + 3:2  = 3:1, or 2/1 x 3/2 = 3/1.  If the perfect fifth is consonant in relation to the tonic note (starting or base note) it is also, in a different, way consonant in relation to the octave below that note.  Hence, this interval is the first of the series which I have termed ‘causal’.”

“If we combined the next three pairs on the JI series in turn, we would get 4:3 + 5:4 = 5:3 then 6:5 + 7:6 = 7:5  and thirdly  8:7 + 9:8 = 9:7. All of the resulting new intervals are formed from the ratio of two odd numbers with a difference of two between them. When played on an instrument which has the capability of playing them accurately, such as the tanbur, they do not sound dissonant.”

(Dr. Prosser, The Music of Rumi - preview)
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