A quick look at Wikipedia tells me that the Byzantine Empire was 'was the continuation of the Roman Empire in the East during Late Antiquity and the Middle Ages, when its capital city was Constantinople (modern-day Istanbul, which had been founded as Byzantium).' It existed from 330 to 1453 AD.
So there you have it, a history lesson as well as a guitar lesson! This scale's eastern roots are immediately apparent the moment you start playing with it. It sounds even more exotic than the similar Phrygian dominant scale.
Try this scale over: a major chord with the same root; a power chord / 5th chord with the same root; a minor chord or progression with a root a perfect fourth higher (e.g. use an A Byzantine scale over a Dm chord or progression; or an 'eastern'-sounding ostinato / riff / unchanging drone accompaniment.
The scale spelling for a Byzantine scale is: 1, b2, 3, 4, 5, b6, 7.
Byzantine Scale TAB & Scale Diagrams
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The Byzantine scale doesn't exactly fall under the fingers, so take a bit of time practicing it and finding a way of playing it that works for you.
Because it contains the major seventh, rather than the minor seventh of the Phrygian dominant scale, the Byzantine scale sounds even more exotic than the Phrygian dominant.
It's a great scale to use in order to spice up a standard rock guitar solo, and will really make your audience's ears prick up!
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