The Harmonic Mean
The Harmonic Mean was developed by Palladio, and it involves different intervals between notes. It states that the mean of three numbers will exceed one extreme and be exceeded by another extreme by the same fraction of the two extremes. The formula contains a, b, and c, in which b is the mean between the extremes a and c. The formula is: (b-a)/a = (c-b)/c. It can also be expressed as: b= 2ac/(a+c). For example, pretend you are given three interval numbers: 6, 8, and 12. Using the first equation we would get: (8-6)/6 = (12-8)/12 or 1/3 = 1/3. Since the two sets on either side of equal sign are equal, the three numbers work in the Harmonic Mean. The Arithmetic Mean states that a<b = b<c and is different from the Harmonic Mean, but it is often associated with the Harmonic Mean. Another mean that is commonly seen with the Harmonic Mean is the Geometric Mean, which states that a:b = b:c. |
