HEWM

I've given this analysis before, but JoeM might want something like it
on his HEWM page.

HEWM is Pythagorean plus 81/80, 64/63 and 33/32. To convert an
11-limit interval to it we can first convert to the
[2,3/2,81/80,64/63,33/32] system and then rewrite the [2,3/2] part in
terms of nominals and sharp/flats based on 2187/2048. The first
conversion can be done as matrix math, or equivalently in terms of vals.

If we convert 2,3/2,81/80,64/63 and 33/32 to monzos and use these as
rows of a matrix, we get

[[1, 0, 0, 0, 0], [-1, 1, 0, 0, 0], [-4, 4, -1, 0, 0],
[6, -2, 0, -1, 0], [-5, 1, 0, 0, 1]]

This is a unimodular matrix (it has integer entries and a determinant
of one), and if we invert it we obtain a matrix whose columns are the
vals w2, w3, w5, w7, w11 where

w2 = [1, 1, 0, 4, 4]
w3 = [0, 1, 4, -2, -1]
w5 = [0, 0, -1, 0, 0]
w7 = [0, 0, 0, -1, 0]
w11 = [0, 0, 0, 0, -1]

Then for an 11-limit interval q, we have

q = 2^w2(q) (3/2)^w3(q) (81/80)^w5(q) (64/63)^w7(q) (33/32)^w11(q)

From this representation, conversion to HEWM is straightforward. The
vals w2, w3, etc. can be thought of as dot products with the monzo for
q. For instance, if q = 77/75, it has monzo qm = [0, -1, -2, 1, 1],
then (w2, qm) = w2 . qm = 7 and (w3, qm) = -12. The others, up sign,
are simply the corresponding 5, 7, and 11 monzo coordinates, or
"valuations" in math-speak. We end up with

77/75 = 2^7 (3/2)^(-12) (81/80)^2 (64/63)^(-1) 33/32

If C is 1, then 2^7 (3/2)^(-12) is a Pythagorean comma down from 1,
and a Pythagorean comma down is D (= 9/8) double flat. Hence 77/75 is
D double flat, up two 81/80, down a 64/63 and up a 33/32 again.

hi Gene,

thanks for this! i have a spreadsheet that does HEWM math,
in fact one that converts from HEWM to Johnston. but this
is good that you spelled it all out. on more thing below ...

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> I've given this analysis before, but JoeM might want
> something like it on his HEWM page.
>
> <snip>
>
> From this representation, conversion to HEWM is
> straightforward. The vals w2, w3, etc. can be thought
> of as dot products with the monzo for q.
>
> [editorial space added for better clarity]
>
> For instance,if q = 77/75, it has monzo qm = [0, -1, -2, 1, 1],
> then (w2, qm) = w2 . qm = 7 and (w3, qm) = -12. The others,
> up sign, are simply the corresponding 5, 7, and 11 monzo
> coordinates, or "valuations" in math-speak. We end up with
>
> 77/75 = 2^7 (3/2)^(-12) (81/80)^2 (64/63)^(-1) 33/32
>
> If C is 1, then 2^7 (3/2)^(-12) is a Pythagorean comma down
> from 1,and a Pythagorean comma down is D (= 9/8) double flat.
> Hence 77/75 is D double flat, up two 81/80, down a 64/63 and
> up a 33/32 again.

so then in actual JI-HEWM notation, that's Dbb^<++ ??

i get Dbb^<-- .

in 72edo-HEWM this seems to simplify to Dbb^ , or better, C^ .

can you check those, Gene?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> > If C is 1, then 2^7 (3/2)^(-12) is a Pythagorean comma down
> > from 1,and a Pythagorean comma down is D (= 9/8) double flat.
> > Hence 77/75 is D double flat, up two 81/80, down a 64/63 and
> > up a 33/32 again.
>
>
> so then in actual JI-HEWM notation, that's Dbb^<++ ??

Right. D is 9/8, and b is 2048/2187, so Dbb is (9/8)*(2048/2187)^2 =
524288/531411, a Pythag comma down. Then times ^<++ means

(524288/531411)*(33/32)*(63/63)*(81/80)^2 = 77/75/

> i get Dbb^<-- .

Better check your system.

> in 72edo-HEWM this seems to simplify to Dbb^ , or better, C^ .

It's C^ all right, since Dbb and C are the same in any system where
the fifth is the 12-equal fifth.