Music.Theory.Math.Nichomachus

Nichomachus of Gerasa (Νικόμαχος) c.60-c.120

https://pdfs.semanticscholar.org/5dac/8842ad857c822ab854ede3decadfe0464f15.pdf

Documentation

arithmetic_mean :: Fractional a => a -> a -> a

a-b = b-c ; b = a+c / 2

arithmetic_mean 2 6 == 4
arithmetic_mean 1 2 == (1+2)/2 -- 3/2

geometric_mean :: Floating a => a -> a -> a

ab = bc ; b = sqrt ac

geometric_mean 1 4 == 2
geometric_mean 1 2 == sqrt (1*2) -- sqrt 2

harmonic_mean :: Fractional a => a -> a -> a

a-b a = b-c c ; 2ac / a+c

harmonic_mean 2 6 == 3
harmonic_mean 1 2 == (2*1*2)/(1+2) -- 4/3

cont_harmonic_mean :: Fractional a => a -> a -> a

a-b c = b-c a ; a-b b-c = ca ; aa+cc / a+c

cont_harmonic_mean 3 6 == 5
cont_harmonic_mean 1 2 == (1*1+2*2)/(1+2) -- 5/3

cont_geometric_mean :: Floating a => a -> a -> a

a-b c = b-c b ; a-b b-c = cb ; c - a + (sqrt (5aa - 2ac + cc)) / 2

cont_geometric_mean 2 5 == 4
cont_geometric_mean 1 2 == (2-1+sqrt(5*1*1-2*1*2+2*2))/2 -- (1+sqrt 5)/2 -- GOLDEN RATIO -- 1.6180

subcont_geometric_mean :: Floating a => a -> a -> a

a-b c = b-c b ; a-b b-c = cb ; a - c + (sqrt (aa - 2ac + 5cc)) / 2

subcont_geometric_mean 1 6 == 4
subcont_geometric_mean 1 2 == (-1 + sqrt 17) / 2 -- 1.5616