72 et 7-limit linear temperaments

I computed what we affectionately call the "wedgie" for each
generator from 1 to 35 of the 72-et, used that to go to the
corresponding linear temperament, and calculated the gens^2 cents
badness measure of the temperament, to get the following list:

1 1767.388627
2 1114.213323
3 33.46017221
4 651.6853668
5 686.8635997
6 --
7 94.69660055
8 2833.372653
9 650.8689382
10 2833.372653
11 2469.631953
12 --
13 94.69660055
14 651.6853668
15 650.8689382
16 1114.213560
17 1661.269624
18 --
19 292.8526656
20 1114.213560
21 33.46020340
22 651.6853668
23 1059.495335
24 --
25 2996.901727
26 2833.372653
27 33.46017221
28 2833.372658
29 548.0249935
30 --
31 1172.245300
32 651.6853567
33 650.8689382
34 1114.213560
35 442.8818121

In first place, at 3/72, 21/72 and 27/72 we have Ennealimmal, in
second place at 7/72 and 13/72, Miracle. This already requires a
little explanation. 13/72?? What happens is that 13/72 has generator
mappings which are exactly five times further out than 7/72, so my
sytem decides 13/72 is a sort of occult version of Miracle, and
classifies it there.

I previously discussed 46+26, with generator 22/72; the associated
linear temperament scores 652. 50+22 with generator 26/72 is well in
the back at 2833, and 53+19 with generator 19/72 (kleismic) is
looking good at 293.

However the above list suggests we look at 29/72, which scores 548,
and 35/72, which scores 443, and 9/72, 15/72 or 33/72, with a score
of 651.

We have 2/5 < 29/72 < 27/67, and so 29/72 is the 67+5 system. The
mapping to primes is

[ 0 1]
[-6 4]
[-29 14]
[ 2 2]

It favors 3 and 7 at the expense of 5.

For 35 we have 17/35 < 35/72 < 18/37, and we have the 37+35 system.
The mapping to primes is

[ 0 1]
[ 30 -13]
[ 13 -4]
[ 14 -4]

It abounds in intervals of 7/5, which is the genrator, and is not too
bad for 5/4, 7/4, 5/3 or 7/6; it is the fifth which is missing in
action.

Finally, the mystery system which scores 651 turns out to be quite a
find, a cross between Ennealimmal and Miracle--Enneamiracle, maybe?
It has two generators, the limma 1/9 and a version of the secor. Its
map to primes is

[ 0 9]
[ -2 16]
[ 1 20]
[ -2 27]

It has commas of 225/224 and 40353607/40310784, the latter telling us
that 2^11 ~ (7/3)^9 in this system.

I think this demonstrates the value of using ets such as 72 to give
us wedgies.

genewardsmith wrote:

> I computed what we affectionately call the "wedgie" for each
> generator from 1 to 35 of the 72-et, used that to go to the
> corresponding linear temperament, and calculated the gens^2 cents
> badness measure of the temperament, to get the following list:

Sorry to lower the tone of the conversation, but I used to mix a lot in swimming and other
athletic circles where a "wedgie" is when your swimming costume gets wedged or stuck up your rear
end, causing the swimmer great discomfort or offering an incentive to get the race over quickly.

Not being facetious here, but hoping to avoid sniggers and other covert laughter should the name
reach a wider audience.

Kind Regards

Alison,

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> Not being facetious here, but hoping to avoid sniggers and other
> covert laughter should the name reach a wider audience.

"Goodness", "Badness", "wedgie" -- these people have a reason for
their terms (apparantly). I actually asked Paul about the first two
and got chided, so if they want to wear mathematical thongs, who are
we to stop them? :)

All kidding aside, if the end purpose is to publish a paper, I'm sure
they'll think about their terms carefully. Won't they?

Cheers,
Jon

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

/tuning/topicId_31224.html#31253

> Alison,
>
> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> > Not being facetious here, but hoping to avoid sniggers and other
> > covert laughter should the name reach a wider audience.
>
> "Goodness", "Badness", "wedgie" -- these people have a reason for
> their terms (apparantly). I actually asked Paul about the first two
> and got chided, so if they want to wear mathematical thongs, who
are we to stop them? :)
>
> All kidding aside, if the end purpose is to publish a paper, I'm
sure they'll think about their terms carefully. Won't they?
>
> Cheers,
> Jon

How are we defining "end purpose..." ?

JP

Joe,

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> How are we defining "end purpose..." ?

Excellent point. I won't put words into Paul/Gene/Dave/Graham/s mouth
(s), but my impression is at least some of them are interested in
finding more resources for making music. But the development, at
least currently and the way it is being done, seems to point to
publishing the results in a paper/article. For some, that may be the
end purpose; beyond the paper, how many composers would refer to a
wedge product? I don't have the answer to that one...

Be sure to share this with jpehrson1 as well... :)

Cheers,
Jon (who has listened to the progression, while consulting
the 'written' progression, and will write later tonight...)

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> 13/72?? What happens is that 13/72 has generator
> mappings which are exactly five times further out than 7/72, so my
> sytem decides 13/72 is a sort of occult version of Miracle, and
> classifies it there.

Gene, wouldn't the "gens" measure for 13/72 be five times as large as
the "gens" measure for 7/72 ????

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

/tuning/topicId_31224.html#31256

For some, that may be the
> end purpose; beyond the paper, how many composers would refer to a
> wedge product? I don't have the answer to that one...
>
> Be sure to share this with jpehrson1 as well... :)
>

I would... but he's not available...

JP

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
>
> genewardsmith wrote:
>
> > I computed what we affectionately call the "wedgie" for each
> > generator from 1 to 35 of the 72-et, used that to go to the
> > corresponding linear temperament, and calculated the gens^2 cents
> > badness measure of the temperament, to get the following list:
>
> Sorry to lower the tone of the conversation, but I used to mix a
lot in swimming and other
> athletic circles where a "wedgie" is when your swimming costume
gets wedged or stuck up your rear
> end, causing the swimmer great discomfort or offering an incentive
to get the race over quickly.
>
> Not being facetious here, but hoping to avoid sniggers and other
covert laughter should the name
> reach a wider audience.
>
> Kind Regards

I was very well aware of this meaning of the term when I introduced
it as a shorthand for "wedge invariant", and I'm hoping for as many
sniggers, and as much laughter, as possible.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Gene, wouldn't the "gens" measure for 13/72 be five times as large
as
> the "gens" measure for 7/72 ????

It would, but the system I use, which standardizes the wedgie and
removes torsion, in efect decides this is not a legitimate system,
and reduces it to Miracle. And who can blame it?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Gene, wouldn't the "gens" measure for 13/72 be five times as
large
> as
> > the "gens" measure for 7/72 ????
>
> It would, but the system I use, which standardizes the wedgie and
> removes torsion, in efect decides this is not a legitimate system,
> and reduces it to Miracle. And who can blame it?

Oh, so rather than a score next to 13/72, perhaps a note informing
the reader that this was "collapsed" to 7/72 would have been less
misleading?

In-Reply-To: <9v36j6+pmnu@eGroups.com>
jon szanto wrote:

> Excellent point. I won't put words into Paul/Gene/Dave/Graham/s mouth
> (s), but my impression is at least some of them are interested in
> finding more resources for making music. But the development, at
> least currently and the way it is being done, seems to point to
> publishing the results in a paper/article. For some, that may be the
> end purpose; beyond the paper, how many composers would refer to a
> wedge product? I don't have the answer to that one...

No indeed, wedge products are only used for an intermediate step of the
calculation. Composers may refer to one if they also happen to be
mathematicians. Like they may refer to the inevitable collapse of the
bourgeoisie in the face of global, communistic revolution if they also
happen to subscribe to the basic tenets of dialectical materialism.

And we can always say "wedge product" or "wedge invariant" in a journal
that doesn't appreciate frivolity. But in that case, "Miracle", "Magic"
and "ASS" will get us into trouble as well.

It may be significant that Paul Erlich has a background in physics. After
"quarks" and "naked bottoms", you get used to important ideas having silly
names. Even biology's coming round to that idea, with "sonic hedgehog".

The end purpose is for people to use these ideas to make music. For that,
the ideas have to be made available to them. I'm always expanding my
website, but dead-tree publication may reach a different audience, which
would be a good thing. We're thinking of Xenharmonikon, which has run
some profoundly silly articles in the past. Brink McGoogy springs to
mind.

Graham

Graham,

--- In tuning@y..., graham@m... wrote:
> And we can always say "wedge product" or "wedge invariant" in a
> journal that doesn't appreciate frivolity. But in that
> case, "Miracle", "Magic" and "ASS" will get us into trouble as well.

!!!

> It may be significant that Paul Erlich has a background in
> physics. After "quarks" and "naked bottoms"...

Excellent point.

> but dead-tree publication may reach a different audience, which
> would be a good thing

Yup.

> We're thinking of Xenharmonikon, which has run
> some profoundly silly articles in the past. Brink McGoogy springs
> to mind.

ROFLOL!

Cheers,
Jon