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Some convex hull badness measures

🔗Gene Ward Smith <gwsmith@svpal.org>

2/4/2004 2:02:51 PM

I took all pairs n,m<=100 of integers and calculated the corresponding
wedgie; excluding [0,0,0,0,0,0] gave me 2280 wedgies. The convex hull
of this in the log(comp)-log(error) plane has 16 verticies. If I
exclude the slopes greater than -2 (the logflat slope) and those going
between the "bad" verticies, I am left with two possibilities:

Line from the {9/8, 15/14} temperament to the {10/9, 16/15}
temperament, with slope -5.99, and the line from {10/9, 16/15} to
ennealimmal, with slope -2.69; the first something which would make
sense only if you were interested in high-error, low-complexity.

If I take only temperaments with error less than that of {9/8, 15/14}
and redo the proceedure, I get a line from {15/14, 25/24} to
ennealimmal with a slope of -2.79. If I now boot {15/14, 25/24}, I get
the beep-ennealimmal line, with slope -2.88. Booting beep gives me a
blackwood-ennealimmal line with slope -2.91. If I boot blackwood, I
get a more complicated situation again; I have the unusuably large
slope of
-42.23 from augmented to dominant seventh, and then a line of slope
-3.03 running from dominant seventh to ennealimmal.

I think this suggests it is better to treat ennealimmal as a friend
than an enemy; rather than trying to cook up a way to exclude it, it
seems to work to make its very strong goodness a basis for seeing what
kinds of badness functions make sense. One question is do we want to
keep beep, or exclude it? What about father?

🔗Dave Keenan <d.keenan@bigpond.net.au>

2/4/2004 3:51:14 PM

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

Hi Gene,

To be able to comment on any of this, I really need to see them
plotted in the (linear) error vs. complexity plane.

Could you just post something like that list of 114 TOP 7-limit linear
temps again, but with Paul's latest favourite complexity measure (the
spooky one) and with the log-flat badness cutoff raised somewhat so we
get a few previously-unnamed temps in the middle region.

One temp per line, tab delimited? We only need columns for complexity,
error, comma-pair, and name (if already in use). Thanks.

🔗Paul Erlich <perlich@aya.yale.edu>

2/5/2004 1:37:32 PM

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> Hi Gene,
>
> To be able to comment on any of this, I really need to see them
> plotted in the (linear) error vs. complexity plane.
>
> Could you just post something like that list of 114 TOP 7-limit
linear
> temps again, but with Paul's latest favourite complexity measure

It should be a new list based on that complexity measure. The list
should agree with the complexity measure. Otherwise things will be
missing.