Ratios also play a critical role in Plato's Timeas (360 BCE) where he describes God as inserting in gaps "two mean forms in each interval, one exceeding one extreme and being exceeded by the other by the same fraction of the extremes and the other being exceeded by the same numerical amount".
Returning to music, these ratios are indeed a foundation of music as noted by Archytus of Tarentum (c. 350 BCE): There are three 'means' in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call 'harmonic'. The arithmetic mean is when there are three terms showing successively the same excess: the second exceeds the third by the same amount as the first exceeds the second. In this proportion, the ratio of the larger numbers is less, that of the smaller numbers greater. The geometric mean is when the second is to the third as the first is to the second; in this, the greater numbers have the same ratio as the smaller numbers. The subcontrary, which we call harmonic, is as follows: by whatever part of itself the first term exceeds the second, the middle term exceeds the third by the same part of the third. In this proportion, the ratio of the larger numbers is larger, and of the lower numbers less. [Ancilla to the Pre-Socratic Philosophers, by Kathleen Freeman, , at sacred-texts.com; http://www.sacred-texts.com/cla/app/app42.htm]
In order to understand these descriptions, it is necessary to realize that for the Greeks, a mean for two numbers B>A, was a third number C satisfying B>C>A and a further property. The above description of the arithmetic mean states that B-C = C-A (this is "the same numerical amount" referred to by Plato). The above description of the geometric mean states that C/A = B/C (this is "the same fraction of the extremes" referred to by Plato). The above description of the harmonic mean states that (B-C)/B = (C-A)/A
Algebraic manipulation recharacterizes the arithmetic mean in a more familiar form. B-C = C-A => B+A = 2C => C = (A+B)/2.
Algebraic manipulation recharacterizes the geometric mean in a more familiar form. C/A = B/C => C^2 = AB => C = (AB)^.5.
Algebraic manipulation recharacterizes the harmonic mean. (B-C)/B = (C-A)/A => (AB-AC)/AB = (BC-AB)/AB => AB-AC = BC-AB => 2AB = C(A+B) => C = 2AB/(A+B) = 1/(.5(1/B+1/A).
The arithmetic and harmonic means of 6 and 12 are readily verified to be equal to 9 and 8 as mentioned in the legend of Pythagoras. In general, the arithmetic mean frequency of an octave gives the perfect fifth ((1+2)/2=3/2), the harmonic mean frequency of an octave gives the perfect fourth (1/((1/1+1/2)/2) = 4/3), and the geometric mean frequency of an octave gives the tritone (Hungarian minor fourth with equal temperament) ((1×2)^.5). Other intervals in music can be characterized similarly.
These means have of course been generalized to more than two numbers. You are familiar with the arithmetic mean as the sum of n numbers divided by n (e.g., (2+6+4+7+3)/5 = 4.4). The geometric mean is the n-th root of the product of n numbers (e.g., (2×6×4×7×3)^(1/5) = 3.99. The harmonic mean of n numbers is the reciprocal of the arithmetic mean of their reciprocals (e.g., 1/((1/2+1/6+1/4+1/7+1/3)/5) = 3.59. Indeed, it can be shown that in general the arithmetic mean is greater than or equal to the geometric mean which is greater than or equal to the harmonic mean.
Different problems require different means for their solution. If a car drives 30 mph for one hour and 60 mph for one hour, its average speed (total distance divided by total time) is the arithmetic mean 45 mph. If a car drives 30mph for one mile and 60 mph for one mile, its average speed (total distance divided by total time) is the harmonic mean 40 mph. If you put your money in the bank for two years and it earns 5% interest the first year and 2% the second year, you calculate the average interest rate (interest rate which if constant for those two years would give the same return) by employing the geometric mean ((1.05×1.02)^.5 = 1.0349, hence 3.49%)
There is a larger family of generalized (also called power or Hölder) means to which the above belong. A Hölder mean with power p of a set of numbers is the p-th root of the arithmetic mean of the p-th power of the numbers. If p=1, this gives the arithmetic mean. If p=-1, this gives the harmonic mean. The limit as p goes to 0 yields the geometric mean. The limit as p approaches infinity produces the maximum. The limit as p approaches negative infinity produces the minimum.
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