recurrent sequence of ETs from george secor

george secor wrote me off-list, and i replied:

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>By the way, what would you call a series where the nth member is the
>sum of the (n-3) plus the (n-2) members?

it's a type of "plastic" sequence, because all sequences obeying this recurrence, such as

http://mathworld.wolfram.com/PadovanSequence.html

and

http://mathworld.wolfram.com/PerrinSequence.html

have the limiting ratio of successive terms equal to the plastic constant:

http://mathworld.wolfram.com/PlasticConstant.html

>I found that such a series
>gave me most of my "top-rated" divisions under 100 if I started with
>12, 19, 22:

>12 19 22 31 41 53 72 94

cool! if you like 3, 9, and 10, you could start off with them . . .

[i just realized you could start with 1, 2, 7 -- that might be easier to remember]

we've discussed a lot of patterns like these on tuning and tuning-math. gene has brought a lot of insight into them: why they work, etc.

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--- In tuning-math@y..., wally paulrus <wallyesterpaulrus@y...> wrote:

> http://mathworld.wolfram.com/PlasticConstant.html

The constant here is particularly interesting as the smallest PV number:

http://mathworld.wolfram.com/Pisot-VijayaraghavanConstant.html

Also of possible musical interest are Salem numbers:

http://mathworld.wolfram.com/SalemConstants.html

I've mentioned some of this stuff in connection with "metallic" tunings (gold, platinum, etc.) and if I gave the name "Osmium" to a sequence coming from x^3-x-1, since Osmium is (probably) the densest metal, and the "plastic constant" is the smallest PV number.

> >I found that such a series
> >gave me most of my "top-rated" divisions under 100 if I started with
> >12, 19, 22:
>
> >12 19 22 31 41 53 72 94
>
> cool! if you like 3, 9, and 10, you could start off with them . . .

I mentioned this sequence in

/tuning-math/message/1297

as being reasonable, but not leading to an Osmium generator.

George might also like

/tuning-math/message/1375

if only because of the name.