as a scatterplot in which one or both of the coordinate axes are binary logarithms of intensity ratios, or in visualizations such as the MA plot and RA plot that rotate and scale these log ratio scatterplots. Music theory In music theory, the interval or perceptual difference between two tones is determined by the ratio of their frequencies. Intervals coming from rational number ratios with small numerators and denominators are perceived as particularly euphonious. The simplest and most important of these intervals is the octave, a frequency ratio of . The number of octaves by which two tones differ is the binary logarithm of their frequency ratio. To study tuning systems and other aspects of music theory that require finer distinctions between tones, it is helpful to have a measure of the size of an interval that is finer than an octave and is additive (as logarithms are) rather than multiplicative (as frequency ratios are). That is, if tones , , and form a rising sequence of tones, then the measure of the interval from to plus the measure of the interval from to should equal the measure of the interval from to . Such a measure is given by the cent, which divides the octave into equal intervals ( semitones of cents each). Mathematically, given tones with frequencies and , the number of cents in the interval from to is The millioctave is defined in the same way, but with a multiplier of instead of . Sports scheduling In competitive games and sports involving two players or teams in each game or match, the binary logarithm indicates the number of rounds necessary in a single-elimination tournament required to determine a winner. For example, a tournament of players requires rounds to determine the winner, a tournament of teams requires rounds, etc. In this case, for players/teams where is not a power of 2, is rounded up since it is necessary to have at least one round in which not all remaining competitors play. For example, is approximately , which rounds up to , indicating that a tournament of teams requires rounds (either two teams sit out the first round, or one team sits out the second round). The same number of rounds is also necessary to determine a clear winner in a Swiss-system tournament. Photography In photography, exposure values are measured in terms of the binary logarithm of the amount