The Byzantine music system, called the Octoechos, is centered around 8 modes (from the prefix -octo which is Greek for 8). These modes are scales built around a unique melodic formula, and these modes serve as the foundation of all Byzantine hymns. Arab music also has a similar system of modes called the Maqam, although this system uses more scale varieties than Byzantine music. Both music systems share many similarities: they both have drone notes that serve as the sustained root notes of a song. No harmonization or chords exist in these styles of music. They also utilize vocal ornamentations and slides from one note to the next. They also both use quarter tones which are seldom used in Western music.
As far as history is concerned, we must remember that Christianity is older than Islam (1st century CE and 7th century CE, respectively), so it can be said with confidence that Byzantine music influenced the music of Arabic culture.
Turkish music draws from elements of Arab music and while this music system has its primary roots in the Ottoman Empire, experts believe that Turkish music began when the Seljuk Turks migrated from Central Asia and brought with them new instruments and music traditions. Byzantine and Ottoman influences played a role in the development of Turkish music.
I have not studied Persian music history in depth, but this system of music draws from various pre-Iranian civilizations and Indian musical influences. Since the Persians came before the Arabs, it can be assumed that the Persian culture had some degree of influence on the development of Arabic music, especially the development of the Maqam system. Since the 19th century, Dastgah has been the mode system used in Persian music.
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