harmonic series

If I add all the fractions

1+1/2+1/3+1/4+1/5+...

all the way to infinity, how far do I get?

The series is called the harmonic series because all the different harmonics of a string have these different wavelengths. So this series is almost like hearing all the harmonics of the vibrating string.

The question of how big this series is can be expressed physically too. If I have infinitely many books and I start stacking them on top of each other, how far can I get them to overhang? In the picture above I've used CD cases (Dorothy will be more interested in which CDs are being stacked probably - there was some Schubert, Shostokovich, Puccini, Stanley Turrentine, Elvis Costello, Dvorak and My Bloody Valentine.) In the picture, the top CD is completely clear of the CD at the base of the tower. As one descends the tower, the distance that each CD can be pushed so that the thing still balances is given by the harmonic series. So the reason the first CD is clear of the base is because the overhang of 4 CDs is 1/2x(1+1/2+1/3+1/4) which is bigger than 1. But how far can you go if you keep stacking the CDs or books?

Well, it turns out that I can actually get as far as I want. This is because

1+1/2+1/3+1/4+1/5+... gets infinitely big.

The first proof of this was given by the fourteenth century philosopher Nicole d'Oresme.

The proof is not too difficult but clever. I might give it in another blog entry if there is popular demand!

How far can we get with 19 steps: the top step will be 1.77387... units away from the step at the bottom. Although the series goes off to infinity, it takes a very long time to get there. How many steps do we need for the top step to be 2 units away? That takes 31 steps. And to go 5 units...that requires 12367 steps.

This paper explains some of the maths behind stacking books.

Some thoughts about Borges, Tango......

Been reading Borges A History of Tango which is in The Total Library Non Fiction 1922-1986 came across these quotes and thoughts not sure what they have to do with 19step yet but here they are anyway

"Schopenhaur (Welt als Wille und Vorstelling I, 52) has written that music is as near to us as the world itself; without the world, without a common stock of memories summoned by language, there would be no literature, but music does not need, could exist, without the world. Music is will and passion; the old tango, as music, immediately transmits that joy of combat which Greek and German poets, long ago tried to express in words."

“We read in one of Oscar Wilde’s dialogues that music reveals a personal past which, until then, each of us was unaware of, moving us to lament misadventures we have never suffered and wrongs we did not commit.”

Was discussing with Dorothy idea of Tango as an example of passion and humanness being controlled and tamed through the structure of language, learning steps and routines and nostalgia rather than its original passion Borges describes it originally as The Fighting Tango, which now can only lament for what is lost…………..

Suppose been thinking about the differences between learning ballroom tango and argentine tango, sequences and patterns as opposed to intuition and improvisation and the relationship between mathematics, music and art……..

Dorothy keeps mentioning a primordial state…….do we make sense of this through patterns and geometry?

I keep thinking of the drawing exercise Carol led were we linked the hexagon to parts of the body with letters, which creates a dance sequence, the letters and geometry/numbers take over the body movements.... does it in Dorothy’s score?

Is Borges hexagonal library built to control all our physical and emotional needs?

“ To the left and right of the entrance way are two miniature rooms. One allows standing room for sleeping; the other, the satisfaction of faecal necessities”