Arrow equivalences

If we look at what are equivalent to the 5, 7, 11 and 13 commas under
arrow, we get the following:

81/80 ~ 78/77

(81/80)^2 ~ 40/39

(81/80)^3 ~ 27/26 ~ 80/77

(81/80)^4 ~ 81/77

This gives one sequence of related commas; another starts from 64/63:

64/63 ~ 65/64 ~ 66/65

(64/63)^2 ~ 33/32 ~ 65/63

(64/63)^3 ~ 22/21

Note in particular (81/80)^3 ~ 27/26 and (64/63)^2 ~ 33/32, which
express the 11 and 13 commas in terms of the 5 and 7 commas.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> Note in particular (81/80)^3 ~ 27/26 and (64/63)^2 ~ 33/32, which
> express the 11 and 13 commas in terms of the 5 and 7 commas.

In fact, any 13 limit interval can be expressed in terms of 81/80 and
64/63 if we assume arrow equivalence, since these are a pair of
generators. 99/70 and 44/39 are the traditional octave and period,
and these can be expressed as

(81/80)^3 (64/63)^20 ~ 99/70

(81/80) (64/63)^7 ~ 44/39.

The mapping to primes is

(81/80)^6 (64/63)^40 ~ 2
(81/80)^10 (64/63)^63 ~ 3
(81/80)^15 (64/63)^92 ~ 5
(81/80)^16 (64/63)^113 ~ 7
(81/80)^20 (64/63)^139 ~ 11
(81/80)^21 (64/63)^149 ~ 13

Also we might want

(81/80)^4 (64/63)^23 ~ 3/2
(81/80)^4 (64/63) ~ 2187/2048