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Some remarks on the basic intervals by the greek and eastern music
Musicology (Μουσικολογία) No. 2, May 1985 (text in greek)
During the entire history of the greek music theory two methods have been used for the calculation of intervals.
a) The multiplicative-geometrical, according to which the intervals are expressed as ratios of the strings' lengths, or -reversely- to their frequencies.
b) the additive-arithmetical, where the octave is divided in equal minimal units.
In parallel use since antiquity, those two methods have produced considerable confusion. It is observed that the intervals of the diatonic tetrachord -in the basis of the musical system in ancient Greek, Byzantine and Eastern, in general, civilization- are given, already since Pythagoras or Aristoxenos, different values, varying with the method of calculation. Similar discrepancies occur between the values given to the intervals by Chrysanthos from Madytos and the 1881 Committee of the Constantinople Patriarchate.
The division of the octave into 53 units (the tetrachord into 22), attributed to Mercator (17th s.) is the only one permitting a nearly perfect coincidence between the two methods. Adopted by the Turks, this method contributed to their coherent and lucid music theory. Besides, it indicates to the fact that the Pythagorean and the natural intervals have up to this date dominated the musical practice of the East.