3HE values for saturated MI triads

Recently in Facebook XA-II, Dave Keenan said:
"The list of saturated MI (merciful/metastable intonation) triads at the end of this post may be of interest:
/tuning/topicId_76975.html#77421
It would be interesting to see where these are on a (non-octave-equivalent?) triadic harmonic entropy plot."

I haven't plotted them, but I calculated 3HE values as follows (with s=1% and N=1M):

MI triads:
Lower,Upper,3HE,maxprob,MLtriad
338,606,5.921244,0.019944,14:17:24
606,338,5.916083,0.020325,7:10:12
833,833,5.850374,0.031012,8:13:21
560,943,5.84254,0.032931,8:11:19
943,560,5.928962,0.01466,11:19:26
422,1002,5.848999,0.033061,7:9:16
1002,422,5.922365,0.018142,14:25:32
339,1039,5.839292,0.031614,9:11:20
1039,339,5.910623,0.025756,5:9:11
607,771,5.904161,0.022316,5:7:11
771,607,5.913622,0.027172,9:14:20

For comparison, here are the values for the local maxima and minima of 3HE found by starting from each of those MI points:

Local Maxima:
Lower,Upper,3HE,maxprob,MLtriad,Distance from original MI triad
341,618,5.93077,0.015289,19:23:33,15.87
602,353,5.929396,0.018365,12:17 21,15.53
837,859,5.943835,0.012535,21:34:56,32.58
540,924,5.951034,0.012231,22:30:51,39.00
931,569,5.935836,0.013173,21:36:50,12.49
451,1017,5.949965,0.012256,20:26:47,44.74
1013,400,5.932531,0.017933,15:27:34,22.00
353,1061,5.94236,0.013075,22:27:50,36.30
1059,316,5.932023,0.014636,13:24:29,25.01
624,777,5.929506,0.018239,16:23:36,23.86
832,597,5.935751,0.015054,18:29:41,65.43

Local Minima:
Lower,Upper,3HE,maxprob,MLtriad,Distance from original MI triad
315,604,5.902438,0.035238,10:12:17,27.78
615,317,5.883994,0.046649,7:10:12,21.07
813,841,5.828929,0.055123,5:8:13,20.13
583,933,5.816001,0.059728,5:7:12,23.07
967,529,5.903622,0.034586,8:14:19,32.52
386,1018,5.78197,0.078729,4:5:9,36.07
969,435,5.834579,0.070356,4:7:9,33.25
317,1049,5.809662,0.064305,5:6:11,22.03
1017,348,5.868072,0.056079,5:9:11,22.12
583,782,5.852314,0.06158,5:7:11,24.03
764,617,5.910082,0.032728,9:14:20,10.26

Also for comparison, here are the values for 1:1:1 and 2:3:4 and the local maxima of 3HE found by starting from each of those two points:

Comparison triads:
Lower,Upper,3HE,maxprob,MLtriad
0,0,4.258453,0.422631,1:1:1
702,498,5.554099,0.153356,2:3:4
28,28,6.014333,0.00764,56:57:58
688,463,5.937923,0.011284,18:27:35

I hope this is of some interest to somebody!

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> Recently in Facebook XA-II, Dave Keenan said:
> "The list of saturated MI (merciful/metastable intonation) triads at the end of this post may be of interest:
> /tuning/topicId_76975.html#77421
> It would be interesting to see where these are on a (non-octave-equivalent?) triadic harmonic entropy plot."
>
> I haven't plotted them, but I calculated 3HE values ...

I have now plotted them:
/tuning/files/SteveMartin/dkeenan.png

yellow labels are A:B:C ratios, blue labels are L,U cents pairs

Steve M.

That's awesome Steve! Thank you so much for going to so much trouble.

It confirms that they are very near high local maxima, although not the highest ones.

The nearby simple rational triads are interesting too. But a metastable triad, being based on simple noble numbers, should find itself poised within about equal probability of two or more simple rational triads.

Regards,
-- Dave Keenan
http://dkeenan.com

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>> > Recently in Facebook XA-II, Dave Keenan said:
> > "The list of saturated MI (merciful/metastable intonation) triads at the end of this post may be of interest:
> > /tuning/topicId_76975.html#77421
> > It would be interesting to see where these are on a (non-octave-equivalent?) triadic harmonic entropy plot."
>
> I have now plotted them:
> /tuning/files/SteveMartin/dkeenan.png
>
> yellow labels are A:B:C ratios, blue labels are L,U cents pairs
>
> Steve M.