--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> My (incomplete) understanding is that flatness is flatness. It's
what
> you acheive when you hit the critical exponent. The "logarithmic"
> character that we see is simply a by-product of the criticality.
>
> I look forward to a fuller and more accurate reply from Gene.
When you measure the size of an et n by log(n), and are at the
critical exponent, the ets less than a certain fixed badness are
evenly distributed on average; if you plotted numbers of ets less
than the limit up to n versus log(n), it should be a rough line. If
you go over the critical exponent, you should get a finite list. If
you go under, it is weighted in favor of large ets, in terms of the
log of the size.
... the ... gonna ... a ... interest ... My (incomplete) understanding is that flatness is flatness. It's what you acheive when you hit the critical exponent....
... what ... When you measure the size of an et n by log(n), and are at the critical exponent, the ets less than a certain fixed badness are evenly distributed...
... This is only true if you choose a very low value for your "certain fixed badness", right? ... What if you used n instead of log(n)? Would there still be...
... Or start a bit away from 0. ... same ... This is what I was talking about in a previous posting; if we look at ... condition that |h(q)-n*log2(q)|^3 <...
... is ... It is among functions n^e, for some fixed e; the value e=-1 is the critical exponent where n^(e+1)/(e+1) no longer works as an antiderivative, and...
... like ... more ... It should be noted that these work only "almost always", whereas 1/n works without exception, giving us an infinite set. It is highly ...
In-Reply-To: <9v741u+62fl@eGroups.com> ... A sharp cutoff won't be what most people want. For example, in looking for an 11-limit temperament I might have...
... It ... will ... looking ... want more ... keyboard ... that scale ... pleased if ... 22, ... That's why I suggested that we place our sharp cutoffs where...
... Wow- how did that happen? One heckuva switch from the last post I can find in this thread. Not that I understand what any of this is about. Badness?? ...
... The "badness" of a linear temperament is a function of two components -- how many generators it takes to get the consonant intervals, and how large the...
... Hey. I had a coupla days to cool off. :-) I'm only saying its a starting point and I probably should have written "I now understand that Gene's flat...
... more ... Hey Paul, that's what I had originally but see what Gene wrote in http://groups.yahoo.com/group/tuning-math/message/1833 But as far as I can tell,...
... He was talking about linear temperaments there, not ETs (right, Gene?). ... That's "flat" for all ETs overall (though the wiggles aren't), but what we...
... Well the size of wiggles and the best in each range look pretty damn flat to me for steps * cents (and not for steps^(4/3)*cents or steps^2*cents). Take a...
... Paul, I assume goodness = 1/badness? How could any reasonable transformation from badness to goodness change whether it looks flat or not? I've added...
... flat ... You'll be looking at the opposite extremes of the graph. ... easily. ... the ... Not really. At 612, you can't really see the difference yet. Go...
I wrote, ... much ... Well I extended the graph out to 32768, and 4/3 starts to make more sense as an exponent. But I noticed something else -- something...
I wrote, ... about ... within ... Take a look at the two pictures in http://groups.yahoo.com/group/tuning-math/files/Paul/ (I didn't enforce consistency, but...
In-Reply-To: <9vaje4+gncf@eGroups.com> ... I don't know either, but I'll register an interest in finding out. I've thought for a while that the set of...
... I've ... properties ... Well, this pattern I found shows up regardless of whether you look at consistent ETs only, or fail to enforce consistency at all. ...
Furthermore, noting a striking symmetry centered just above 50,000, I surmised that there must be an especially exceptional ET just above 100,000. And in fact...
... 1664 is 128*13. So 103169 is 13*256*31. Interesting, don't know if it's meaningful, that it's lots of 2s and two prime numbers. The obvious reason for...
... suggest ... survey? ... No, but 103168 is. 103169 is 11*83*113. ... obvious ... from ... equal is ... Confused . . . you mean 2*103168-equal? That's not...
... about ... within ... This partly makes sense to me and partly doesn't; it should have wave frequencies corresponding to the good 7-limit ets, but why 1680?...
... wave ... Matlab has fft. The FFT of the set of results up to 2^17 has a few extremely sharp peaks. With what formula should I interpret the results?...
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