Calculating phi points

Post Reply New Topic

I need to divide 34 bars into golden sections. The ultimate aim is to subdivide those sections to create botha macro and micro structural process.
So first I need to divide 34 bars into the 2 sections and then those sections need to be divided and so on and so forth. Fractal like I suppose.

Can someone explain the method? I wont be satisfied unless I understand the proof inside out and can adapt it myself. I already feel tacky for asking

I joined student forum just to ask this question then realised that i'd much prefer to toss the idea about here. Keep in mind this is something I plan to integrate into my music coursework. Not sure whether I should show the resultant discussion to the lecturer to be totally transparent. I feel like a scholastic criminal heh

meh worked it out for myself - realised its just simple algebra ahaha I was thinking that i'd need complex postgrad maths.

x / 34 = phi
x = 34 * phi
x = 21
(phi = 0.6180339985218034)

Question: Why doesnt it work with 1.618? I arrive at a golden ratio but its in reverse. Can someone explain the mathematical principle behind that?


Are there any other similar irrational numbers and ratios that I could use to create mathematically interesting ratios

0.61803... is equal to 1/phi (I'm unclear whether you already knew that, if that's what you meant by "in reverse", but I will explain just in case). This can be derived thus:



Basically, you're dividing by phi, instead of multiplying (because you're multiplying by the reciprocal).
If you had actually multiplied by phi, your ratio would have been roughly 34/55, which works for 55-unit bars, but not 34 unit ones.
Did that explanation work? I'm sort of bad at understanding forum posts sometimes... Also, I'm not a professional mathemtician or anything, so everything I've said is subject to be wrong. There's no money-back guarantee

P.S. Sorry if my reply was late, my internet and/or WP's servers stopped working for a bit.

multiplying by a number greater than 1 increase the output.

ruveyn

heh cheers for the explanations I worked that out too in the end XD. PHI is a mysterious number at any rate. The amount of times I created processes and musical systems based on its ratios and found that said systems have incredible properties was astounding. For instance I managed to generate whole tone scales chromatic scales and even tables that had their own internal logic and patterning

34=f_9=f_8+f_7, where f_n is the nth fibonacci number
f_8=21

The ratio f_{n+1}/f_n tends to phi as n tends to infinity. Proof: (f_{n+1}/f_n) / (f_n/f_{n-1}) = (f_{n+1}f_{n-1}) / f_n^2 which tends to zero by Cassini's formula since the numerator and denominator are increasing therefore it converges to some real x. x<--f_{n+1}/f_n=1+f_{n-1}/f_n-->1+1/x

x=1+1/x gives x^2-x-1=0, the characteristic equation for which phi is defined as the positive root.