More on Ratios
Music is nothing but ratios and harmonic math, anyways.” ―Andrew Sega
The truest way to understand an interval or chord is with ratios. The numerator divided by the denominator generally ends up being between 1 and 2 (unison 1/1 to octave 2/1), to keep everything within the range of an octave. Each perfect fifth up is x3 for numerator (3, 9, 27, 81, etc. for each 5th) and the denominator is a multiple of 2 to bring the ratio into the octave (3/2, 9/8, 27/16, 81/64 etc.). Each perfect fifth down is 3x in the denominator (3, 9, 27, 81 etc.) and a multiple of 2 in the numerator, again to keep the ratio within the octave (4/3, 16/9, 32/27, 128/81). Each just third up is x5 in the numerator and a multiple of 2 in the denominator (5/4, 25/16, 125/64, etc.). Each just third down is x5 in the denominator and a multiple of 2 in the numerator (8/5, 32/25, 128/125, etc.). Each harmonic seventh up is x7 in the numerator and a multiple of 2 in the denominator (7/4, 49/48, etc.) and each seventh down is x7 in the denominator and a multiple of 2 in the numerator (8/7, 64/49, etc.).
To add two intervals, multiply their ratios and divide them down if necessary to bring them to within a single octave of each other (a fifth and a second is 3/2 x 9/8 = 27/16 or G + D = A; a fifth and a fourth is 3/2 x 4/3 = 12/6 or 2/1, G + F = C). To subtract one interval from another, invert the numerator and denominator of the interval being subtracted and multiply (a fifth minus a third is 3/2 – 5/4, or 3/2 x 4/5 = 12/10 or 6/5 E1b) If necessary, multiply or divide the numerator or denominator by 2 to get a ratio between 1 and 2, and divide both numerator and denominator together by whatever numbers possible to get the simplest number ratio (128/64 down to 2/1, for example).
A few of the most commonly used ratios appear in the Worldwide Musical Web, clustering around the starting note C, but there was no point in calculating ratios with numbers even bigger than the 2048/2025 of D2bb, for example.
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