There is a reason why Barber Shop / Doowop singers shy away from instrumental accompaniment:
A little thought will show why no sequence of Perfect Fifths can ever exactly equal a sequence of Octaves, no matter how far you carry out both sequences: when you go up by Octaves, you’re multiplying the starting pitch by a 2 raised to a power equal to the number of Octaves. Going up one octave means multiplying by 2¹ = 2. Two octaves = starting pitch × 2² (= 4×), three octaves = starting pitch × 2³ (= 8×), and so on. But going up by Perfect Fifths (or, rather, Octave + Perfect Fifths) means raising powers of three. × 3¹ = 3× and takes you to the Octave + Perfect Fifth. × 3² = 9× and takes you to two octaves + Pythagorean Major Second (or one octave + Pythagorean Major Ninth, if you prefer). × 3³ = 27× and takes you up to three octaves + Pythagorean Major Sixth, and so on.
Notice something? The multipliers for Octaves are always even numbers (2×, 4×, 8×, 16×, …), while those for Perfect Fifths are always odd numbers (3×, 9×, 27×m etc.). No matter how far you carry out the sequences, you’ll never have an odd number equal an even number!
This means that the Circle of Fifths, the very basis for all of Western Civilization music, just plain doesn’t work!
For more, be sure to read this MeFi thread.