Defects of ETs

Using diadic harmonic entropy, odd-limit 223, s=0.06%, we get the following
defects relative to 1200-tET:

ET 3-limit 5-limit 7-limit 9-limit 11-limit

4.0000 0.3875 0.6249 0.7432 NaN NaN
5.0000 0.3811 0.6144 0.7162 1.0919 NaN
6.0000 0.3663 0.5760 0.7019 NaN NaN
9.0000 0.4223 0.6093 0.6891 NaN NaN
10.0000 0.3811 0.6583 0.7666 NaN NaN
12.0000 0.0082 0.2317 0.3538 0.3594 NaN
15.0000 0.3811 0.4997 0.6056 NaN NaN
16.0000 0.4293 0.6289 0.7277 NaN NaN
18.0000 0.4266 0.6574 0.7429 NaN NaN
19.0000 0.1071 0.1475 0.2744 0.3959 NaN
22.0000 0.1111 0.2105 0.3041 0.4245 0.3550
26.0000 0.1740 0.3383 0.4165 0.6191 0.6043
27.0000 0.1680 0.2900 0.3164 0.4811 NaN
29.0000 0.0065 0.2234 0.3261 0.3707 0.3734
31.0000 0.0583 0.0892 0.0957 0.1734 0.1805
35.0000 0.3386 0.4377 0.4792 NaN NaN
36.0000 0.0082 0.2317 0.2771 NaN NaN
37.0000 0.2371 0.2984 0.3242 NaN NaN
41.0000 0.0013 0.0586 0.0720 0.0764 0.0747
42.0000 0.2584 0.3640 0.4252 NaN NaN
43.0000 0.0406 0.1143 0.1721 NaN NaN
45.0000 0.1451 0.2551 0.2884 NaN NaN
46.0000 0.0150 0.0431 0.0824 0.1070 0.1196
49.0000 0.1419 0.1737 0.2134 NaN NaN
50.0000 0.0754 0.0917 0.1326 0.2199 NaN
53.0000 0.0001 0.0026 0.0319 0.0351 NaN
56.0000 0.0629 0.0893 0.1322 NaN NaN
57.0000 0.1071 0.1475 0.1806 NaN NaN
58.0000 0.0065 0.0683 0.0756 0.0852 0.0878
59.0000 0.1899 0.2548 0.2931 NaN NaN
60.0000 0.0082 0.0540 0.0980 0.1098 NaN
62.0000 0.0583 0.0892 0.0957 NaN NaN
63.0000 0.0201 0.0913 0.1166 NaN NaN
68.0000 0.0376 0.0442 0.0472 0.0936 NaN
70.0000 0.0025 0.0921 0.1263 0.1381 NaN
72.0000 0.0082 0.0161 0.0191 0.0296 0.0307

where at the n-limit we're evaluating the complete n-limit otonal chord
1:3:...:n, or equivalently, the utonal chord 1/n:...:1/3:1. NaN means the ET
is inconsistent at that limit.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
http://www.egroups.com/message/tuning/12940

> Using diadic harmonic entropy, odd-limit 223, s=0.06%, we get the
following defects relative to 1200-tET:

This is a very interesting chart... I made a copy of it...
_________ ___ __ __ _
Joseph Pehrson

This is very interesting Paul. I'd love to see the columns as barcharts (4
decimal places seems like overkill).

I'd be interested in how it compares with using non-octave-equivalent
harmonic entropy while taking the chord to be in "standard" inversion as
4:5:6:7:9:11.

Did you mean s=0.6%. If not then I don't know what s is.

I know you were using it in a mathematical sense, but the term "defect"
seems inadvisable given the different perspectives on this list. As Margo
has recently shown me, one persons defect is another's beauty spot. :-)

It's good to be back, old friend. But it probably won't be for long.

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>This is very interesting Paul. I'd love to see the columns as barcharts (4
>decimal places seems like overkill).

Why don't you make some and add them to your webpage?

>Did you mean s=0.6%.

Oh yes, sorry.

Paul Erlich wrote:

> Using diadic harmonic entropy, odd-limit 223, s=0.06%, we get the following

David Keenan responded:

> This is very interesting Paul. I'd love to see the columns as barcharts.

I thought so, too. It's really pretty easy to do with Excel. Here's a quick and dirty version
of it:

http://personal.bna.bellsouth.net/bna/d/f/dfin/DefectsETs.htm

I'd forgotten how "good" 41 equal is!

--
David J. Finnamore
Nashville, TN, USA
http://members.xoom.com/dfinn.1
--

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:
> http://www.egroups.com/message/tuning/13078
>
>
> Paul Erlich wrote:
>
> > Using diadic harmonic entropy, odd-limit 223, s=0.06%, we get
> > the following
>
> David Keenan responded:
>
> > This is very interesting Paul. I'd love to see the columns as
> > barcharts.
>
> I thought so, too. It's really pretty easy to do with Excel.
> Here's a quick and dirty version of it:
>
> http://personal.bna.bellsouth.net/bna/d/f/dfin/DefectsETs.htm

Amazing!... I made precisely the same kind of graphs for ETs
almost 10 years ago, but because I 'invented' my first lattice
diagrams shortly afterward, just sort of put them on the back
burner...

I'm personally really glad to have this forum so that so much
'reinventing of the wheel' no longer has to happen.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

David wrote,

>I thought so, too. It's really pretty easy to do with Excel. Here's a
quick and dirty >version
>of it:

>http://personal.bna.bellsouth.net/bna/d/f/dfin/DefectsETs.htm

>I'd forgotten how "good" 41 equal is!

Thanks so much, David!!! If you like, I'll e-mail you the data for the
in-between ETs -- I only included those consistent in the 7-limit, leaving
out ones like 34-tET.