harmonic entropy by geometric mean (JI)

Paul just today e-mailed a listing which uses the geometric mean to
determine degree of concordance for simple JI sonorities represented
in the Tuning Lab experiment:

http://artists.mp3s.com/artists/140/tuning_lab.html

Here's the "new" ordering:

bass tenor alto soprano otonal_rep g.m.

0 388 702 970 4:5:6:7 5.3836
0 318 816 1020 5:6:8:9 6.8173
0 498 702 886 6:8:9:10 8.1072
0 202 702 974 8:9:12:14 10.4872
0 204 702 1088 8:9:12:15 10.6697
0 386 702 1088 8:10:12:15 10.9545
0 184 498 886 9:10:12:15 11.2818
0 316 702 1018 10:12:15:18 13.4164
0 498 886 1384 9:12:15:20 13.4164
0 502 1002 1390 9:12:16:20 13.6346
0 268 702 970 12:14:18:21 15.8745
0 388 702 886 12:15:18:20 15.9549
0 388 886 1274 12:15:20:25 17.3205
0 498 888 1282 12:16:20:25 17.6022
0 318 818 1320 15:18:24:32 21.3394
0 500 816 1316 15:20:24:32 21.9089
0 388 776 1090 16:20:25:30 22.1336
0 186 576 888 18:20:25:30 22.7951
0 498 702 1086 24:32:36:45 33.3979
0 386 884 1088 24:30:40:45 33.7405
0 312 702 888 30:36:45:50 39.4822
0 184 388 886 36:40:45:50 42.4264
0 384 588 1086 32:40:45:60 43.1165
0 272 772 974 36:42:56:63 48.0585
0 388 888 1390 36:45:60:80 52.8067
0 204 702 1020 40:45:60:72 52.8067
0 314 702 1090 40:48:60:75 54.2161
0 502 1002 1320 45:60:80:96 67.4810
0 394 784 1282 48:60:75:100 68.1732
0 268 582 970 60:70:84:105 78.0152
0 302 502 1004 not JI
0 502 702 1004 not JI
0 492 980 1472 not JI
0 500 886 1320 not JI
0 434 820 1320 not JI
0 442 884 1326 not JI

After going through and listening to the tetrads with the above
ordering, I have to conclude that the "progression" of discordance
seems a lot more sensible than the ordering method by the previous
logarithmic method of diadic harmonic entropy.

At least, there are no simple JI sonorities that "stick out" as being
"suddenly" more consonant in the listings as in the other ordering.

This seems pretty "simple," yes?? Or am I misunderstanding some
complexity in what seems like the simplicity of the above method ?????
______________ ____ __ _ _ _ _
Joseph Pehrson

Joseph wrote,

>After going through and listening to the tetrads with the above
>ordering, I have to conclude that the "progression" of discordance
>seems a lot more sensible than the ordering method by the previous
>logarithmic method of diadic harmonic entropy.

Logarithmic?

>At least, there are no simple JI sonorities that "stick out" as being
>"suddenly" more consonant in the listings as in the other ordering.

I think you'll find some if you look. For example, 9:11:13:15, a chord that
Monz just posted, is not more consonant than tha minor seventh chord
10:12:15:18, is it?

But your cursory impression does mean that tetradic harmonic entropy is
likely one of the more important factors in determining the overall
concordance of a tetrad.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13584

> Joseph wrote,
>
> >After going through and listening to the tetrads with the above
> >ordering, I have to conclude that the "progression" of discordance
> >seems a lot more sensible than the ordering method by the previous
> >logarithmic method of diadic harmonic entropy.
>
> Logarithmic?

Gee... I thought we said that the ordering we have now for the
"Tuning Lab" tetrads resulted from the Tenney method, which I thought
you said resulted from multiplication and was logarithmic...

I must be getting confused... but ask that little guy in the corner
with his hand up instead. Why... it's Dr. Pepper!

>
> >At least, there are no simple JI sonorities that "stick out" as
being "suddenly" more consonant in the listings as in the other
ordering.
>
> I think you'll find some if you look. For example, 9:11:13:15, a
chord that Monz just posted, is not more consonant than tha minor
seventh chord 10:12:15:18, is it?
>

Ummm. Yes, you mentioned that to me in your e-mail. Hmmm. Well,
nothing's perfect. Apparently NONE of the simpler methods is totally
reliable by itself... We're lucky that one wasn't in the exercise we
were doing! :)

> But your cursory impression does mean that tetradic harmonic
entropy is likely one of the more important factors in determining
the overall concordance of a tetrad.

It sure does sound like your otonal ranking by the geometric mean has
the most AUDIBLE logic so far...!
__________ ___ __ _ _
Joseph Pehrson

Joseph wrote,

>Gee... I thought we said that the ordering we have now for the
>"Tuning Lab" tetrads resulted from the Tenney method, which I thought
>you said resulted from multiplication and was logarithmic...

Yeah, a guess a lot of things are logarithmic in a lot of different ways --
but in this case the only way is through the definition of entropy. So I
would leave out the "logarithmic" and mention entropy instead.

Actually the ordering was based on the exponential of entropy so it's not
logarithmic at all, since exponential is the opposite of logarithmic -- in
this case the exponential turns the entropy function from

-sum(p*log(p))

to

prod(p^(-p)).

>It sure does sound like your otonal ranking by the geometric mean has
>the most AUDIBLE logic so far...!

Well, tetradic harmonic entropy is a strong component of concordance. But it
should break down toward the bottom of the list -- no one really perceives
1/7:1/6:1/5:1/4 as 60:70:84:105 -- those numbers are just too high.

>Well, tetradic harmonic entropy is a strong component of concordance. But it
>should break down toward the bottom of the list -- no one really perceives
>1/7:1/6:1/5:1/4 as 60:70:84:105 -- those numbers are just too high.

Asking for a term for chords which sound consonant, by (at least) lacking
roughness, but whose harmonic representation is too high to be tonally
significant, and which do not sound like nearby lower-numbered chords on the
voronoi plot. Perhaps my earlier suggestion of "minor" is not good.
Somebody mentioned Sethares' album, _Xentonality_. A lot of it sounds sort
of "minor", to my ear...

-Carl

I wrote,

>>Well, tetradic harmonic entropy is a strong component of concordance. But
it
>>should break down toward the bottom of the list -- no one really perceives
>>1/7:1/6:1/5:1/4 as 60:70:84:105 -- those numbers are just too high.

Carl wrote,

>Asking for a term for chords which sound consonant, by (at least) lacking
>roughness, but whose harmonic representation is too high to be tonally
>significant, and which do not sound like nearby lower-numbered chords on
the
>voronoi plot.

60:70:84:105 is close to 5:6:7:9, and even closer to other chords with
intermediate-sized numbers.