White keys and black keys

I thought I'd mention again my approach to this, which does not
involve either matricies or wedge products, and does not get into any
unfortunate confusion of scales with linear temperaments. Linear
temperaments do arise, but in terms of a generator for them in a
particular equal temperament.

Let's take 5 and 19. The convergents of 5/19 are 1/3, 1/4, 5/19; so we
take 1/4 and consider 1/5 and 4/19. Here 1/5 < 4/19 are adjacent in
the Farey sequence, and the mediant is 5/24. Hence, *in terms of
scales*, we get a 5/24 generator for the 19 white key, 5 black key
scale partition of 24 equal. The generator 5/24 is not associated to
any particular prime limit, and hence not with any particular linear
temperament. We can, however, associate it to various linear
temperaments, but we *should not* confuse it with a linear temperament
as it stands. In the 5-limit, we may regard it as a contorted
meantone. In the 7-limit, more reasonably, as hemifourths, with TM
basis 49/48 and 81/80. In the 11-limit, we get 11-limit hemifourths,
with the comma 56/55 added to the list, and in the 13-limit we can add
91/90 to that. The choice of temperament is not unique; it's simply a
matter of picking the most reasonable alternative.

Here's the mapping for 13-limit hemifourths:

[<1 2 4 3 2 6|, <0 -2 -8 -1 7 -11|]

The complexity is low enough that the 19 note DE is getting the benefit.

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

> Let's take 5 and 19. The convergents of 5/19 are 1/3, 1/4, 5/19; so we
> take 1/4 and consider 1/5 and 4/19.

Incidentally, though I didn't do this before we *could* allow
semiconvergents into the mix, and I think this shows 5/24 is not
really so unique. The semiconvergents from 1/4 to 5/19 are
2/7, 3/11 and 4/15, and these lead to generators of 9/24, 14/24 and
19/24 respectively. If we derive temperaments from these generators,
from 9/24=3/8 one gets augmented in the 5-limit, with a cheesy
extension adding 1323/1250 as a comma in the 7-limit. 14/24=7/12 gives
us 648/625 and 1323/1280 as commas and [<8 13 19 22|, <0 -1 -1 1|] as
a mapping, and 19/24 = 1-5/24, so this is the same as 5/24.