### Testing Algorithm Effect on Several Notes in an Overtone Series

Visualization of Beat Frequency Reduction on an A overtone Series.  The left diagram shows the piano approximation of the overtone series, and the right diagram shows the output of these notes from our algorithm.  The green band is the sum of the frequencies.  When all of the pitches in the chord are a multiple of a common frequency, the sum of the pitch signals has a period equal to the fundamental.  This lets the pitches stay in phase with each other, causing more predictable and stable interaction (right) then an un-optimized chord (left).

### Testing Algorithm on A Major Chord

In the A dominant 7th chord, we can see the fundamental hit at around 0.11 seconds.  In the uncorrected chord, the individual notes do not cross 0 at the same point, but in the corrected chord, they do.

### Algorithm Timing and Effectiveness Tests

#### Timing And Convergence Rates

This graph shows the centering improvements provided by running algorithm 2 on the randomly generated chords.  As the chords get larger, on average we can expect less improvement, as a note is more likely to start closer to the 49-cent threshold, and therefore the shift terminates before the chord is completely centered.  In all cases, there were many examples where the chord was able to restore its shift value to 0, but these examples were relatively rare.

## Run Time of Algorithm

(n=1000, Randomized Trial)

 Number of Notes Average Runtime (milliseconds) Convergence Rate 2 0.1575 100% 3 0.1925 97% 4 0.1711 92% 5 0.1896 82%