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The tritone is one of those fascinating problems that music theory has dealt with from the very beginning. It is an interval, the distance between two notes, and it has been a problem because it pops up in any scale or mode. The problem is that it has a very harsh sound--is dissonant. The tritone is just three whole tones, hence the name. Like this:
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The problem comes up when you try to do a primitive kind of polyphony with two voices in parallel motion.
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Several solutions emerged. One is to lower the B natural to a B flat, which is why in the earliest written music the only accidental is a flat on B. This was also the earliest key signature. Another is to avoid parallels and from that came the rule about parallel fifths. Instead, have the voices move in oblique or contrary motion. Another solution which came later was to only move in parallel thirds or sixths, but that had to wait until these intervals, dubbed 'imperfect', became acceptable as consonant. At first the only intervals called consonant were the 'perfect' ones: the unison, fourth, fifth and octave.
Much, much later the tritone became prized as a special kind of intense dissonance and is an essential part of the most important progression that defines tonality: the V7 - I cadence.
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It is the very intensity of the dissonance built into the V7 chord with the tritone from F to B, that makes the cadence powerful because this tension resolves into the consonant tonic harmony.
The intensity of the tritone interval can be used in a melody. Leonard Bernstein used it in the song "Maria" from West Side Story. You can hear it in this excerpt right at the 35 - 36 second mark. The tritone is between "Ma" and "ri".
Sometimes the tritone is used just for shock value as in this song by George Harrison, "I Want to Tell You". Listen for the piano part between the 26 and 32 second mark and similar places. Yes, it's that same pesky F to B natural.
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