User:Frostburn/Collatz scales: Difference between revisions
Formulate the musical equivalent of the Collatz conjecture. |
Add notes about the 3/1 iteration being special and give a counter example for 5/3 iteration. |
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| Line 66: | Line 66: | ||
2/1 | 2/1 | ||
</pre> | </pre> | ||
== Additional results == | |||
Variations of the recursion do not seem work for all seeds e.g. | |||
<math> | |||
s_{n+1} = \frac{p_{n+1}}{q_{n+1}} = \frac{1 + 5 p_n}{3 q_n}\ \mathrm{red}\ 2 | |||
</math> | |||
Works for <math>\frac{14}{11}</math> | |||
<pre> | |||
! varcollatz14_11.scl | |||
! | |||
Collatz scale 5/3 variation of 14/11 | |||
7 | |||
! | |||
71/66 | |||
7/6 | |||
14/11 | |||
4/3 | |||
3/2 | |||
178/99 | |||
2/1 | |||
</pre> | |||
But produces an infinite scale for <math>\frac{19}{17}</math>. | |||