A Familiar Series of (Accurate) Temperaments

How many of you are familiar with this series of temperaments?

(4/3)^5 776; 3 (5-EDO, Blackwood)
(5/4)^5 b  3 (Magic)
(6/5)^6 b  3 (Hanson)
(7/6)^7 b  3 (Orwell)
(8/7)^8 b  3 (5&86, Blacksmith)
(9/8)^9 b  3 (17-EDO)
....

It is basically a series of temperaments where the generator is a super particular ratio, and where the numerator is the number of generators to reach 3/1. These temperaments tend to be pretty accurate. Here is why:

We can generalize the comma of each of these series to fit the following expression:

C = [(n+1)/n]^(n+1)/3

If we take the limit as n goes to infinity, we get e/3, which is about 171 cents.

So no matter how complex this series of temperaments becomes, the comma always stays approximately the same size. This allows the error to be distributed more evenly among the steps for the higher temperaments, which yields a low error for the majority of these temperaments.

Ryan

This would be a good technique to turn convergent series with integer
coefficients into series of temperaments in general. Anyone know any
good power series converging at 3?

-Mike

On Sat, Jun 2, 2012 at 8:51 PM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
>
>
> How many of you are familiar with this series of temperaments?
>
> (4/3)^5 ≈ 3 (5-EDO, Blackwood)
> (5/4)^5 ≈ 3 (Magic)
> (6/5)^6 ≈ 3 (Hanson)
> (7/6)^7 ≈ 3 (Orwell)
> (8/7)^8 ≈ 3 (5&86, Blacksmith)
> (9/8)^9 ≈ 3 (17-EDO)
> ....
>
> It is basically a series of temperaments where the generator is a super
> particular ratio, and where the numerator is the number of generators to
> reach 3/1. These temperaments tend to be pretty accurate. Here is why:
>
> We can generalize the comma of each of these series to fit the following
> expression:
>
> C = [(n+1)/n]^(n+1)/3
>
> If we take the limit as n goes to infinity, we get e/3, which is about 171
> cents.
>
> So no matter how complex this series of temperaments becomes, the comma
> always stays approximately the same size. This allows the error to be
> distributed more evenly among the steps for the higher temperaments, which
> yields a low error for the majority of these temperaments.
>
> Ryan