Yahoo Tuning Groups Ultimate Backup tuning-math Is there a limit for log-flat badness of temperaments with a given ET?

For a while I have known that for a for a given unison interval, only a finite quantity of temperaments which have that unison interval have a log-flat badness below a specific cutoff.

The above theorem can be proven rather easily, partly by noting that a given unison interval of a temperament sets a lower limit for the error of the temperament. However, temperaments can also be defined by a set of equal temperament mappings, and specifying an equal temperament sets an upper limit of error. While the error of temperaments compatible with a given equal tempermant has no lower limit, the existence or lack thereof of a lower limit of badness may be more difficult to prove for certain.

Here are some linear temperaments compatible with 5-equal ~~

Fourth-thirds "Father" 5&8
{16:15 \ 28:27}
[1 . -1 . 3]
Kees-max error: 28.598519 Â¢/oct
Kees-max complexity: 1.499298 s/oct
Log-flat badness: 64.286464 Â¢*s^2/oct^3

Bug "Beep" 5&9
{21:20 \ 27:25}
[2 . 3 . 1]
Kees-max error: 28.021183 Â¢/oct
Kees-max complexity: 1.292030 s/oct
Log-flat badness: 46.776901 Â¢*s^2/oct^3

Pentagram 5&10
{28:27 \ 49:48}
[0 . 5 . 0]
Kees-max error: 14.528978 Â¢/oct
Kees-max complexity: 2.153383 s/oct
Log-flat badness: 67.371706 Â¢*s^2/oct^3

Dominant sevenths 5&12
{36:35 \ 64:63}
[1 . 4 . -2]
Kees-max error: 9.580187 Â¢/oct
Kees-max complexity: 2.435121 s/oct
Log-flat badness: 56.808710 Â¢*s^2/oct^3

Semaphore "Godzilla" 5&19
{49:48 \ 81:80}
[2 . 8 . 1]
Kees-max error: 7.315317 Â¢/oct
Kees-max complexity: 3.445412 s/oct
Log-flat badness: 86.839160 Â¢*s^2/oct^3

Superpyth 5&22
{64:63 \ 245:243}
[1 . 9 . -2]
Kees-max error: 4.817408 Â¢/oct
Kees-max complexity: 4.588503 s/oct
Log-flat badness: 101.427469 Â¢*s^2/oct^3

Cynder (AKA Mothra) 5&31
{81:80 \ 1029:1024}
[3 . 12 . -1]
Kees-max error: 3.392239 Â¢/oct
Kees-max complexity: 5.524326 s/oct
Log-flat badness: 103.524964 Â¢*s^2/oct^3

Rodan 5&41
{245:243 \ 1029:1024}
[3 . 17 . -1]
Kees-max error: 1.788964 Â¢/oct
Kees-max complexity: 7.677709 s/oct
Log-flat badness: 105.454456 Â¢*s^2/oct^3

Quartal 5&53
{1728:1715 \ 5120:5103}
[4 . 21 . -3]
Kees-max error: 1.451979 Â¢/oct
Kees-max complexity: 10.112829 s/oct
Log-flat badness: 148.492933 Â¢*s^2/oct^3

5&72
{1029:1024 \ 19683:19600}
[6 . 29 . -2]
Kees-max error: 0.844418 Â¢/oct
Kees-max complexity: 13.202035 s/oct
Log-flat badness: 147.176702 Â¢*s^2/oct^3

5&99
{5120:5103 \ 420175:419904}
[7 . 38 . -4]
Kees-max error: 0.496541 Â¢/oct
Kees-max complexity: 17.790538 s/oct
Log-flat badness: 157.156874 Â¢*s^2/oct^3

5&171
{420175:419904 \ ???:???}
[13 . 67 . -6]
Kees-max error: 0.144626 Â¢/oct
Kees-max complexity: 30.992573 s/oct
Log-flat badness: 138.919199 Â¢*s^2/oct^3

5&3125
[235 . 1220 . -115]
Kees-max error: 0.001941 Â¢/oct
Kees-max complexity: 566.389227 s/oct
Log-flat badness: 622.657951 Â¢*s^2/oct^3

5&103169
[7757 . 40273 . -3794]
Kees-max error of 103169-equal: 0.000027616 Â¢/oct
Kees-max complexity: 18696.087 s/oct
Log-flat badness: 9652.942133 Â¢*s^2/oct^3

Note~~ The error of 103169-equal is used because attempting to find the actual error of the linear temperament leads to overflow problems; besides, the difference is negligible because 5-equal and 103169-equal are far, far, far apart

As we can see from above, beyond 171-equal the log-flat badness continues to increase as higher equal temperaments are paired with 5-equal. 103169-equal has by far the lowest log-flat badness from all ETs shown above, yet 5&103169 has the highest badness from the entire list.

The high-complexity temperaments have a complexity roughly proportional to their second equal temperament. This can be seen from the following
171 / 31 ~= 5.517
3125 / 566.39 ~= 5.517
103169 / 18696 ~= 5.518

Because log-flat badness of a 7-limit linear temperament is proportional to the complexity squared, whereas the log-flat badness of a 7-limit equal temperament is proportional to the 4/3 power of the number of steps per octave, and x^2 rises more quickly than x^(4/3), the log-flat badness of A&B should continue increasing, given that A is fixed and B increases while keeping the same log-flat badness. (The error of A&B differs less from the error of B as B increases with A staying fixed.)

Furthermore, the sequence of decreasing log-flat badness of equal temperaments is hyperexponential -- after 4, 31 and 171-equal, the next in line is 103169. What could be the smallest 7-limit equal temperament whose log-flat badness is better than that of 103169-equal? I have no idea which one that could be~~

The above shows some support for my hypothesis that for a given ET, there is a finite quantity of temperaments below a given log-flat badness cutoff. However, I have no conclusive proof of the hypothesis.

What additional information can support or invalidate this hypothesis?

Corrections to any errors the information above will be appreciated.

~~ Kariisa-tan as Sunny Milk

🔗Kees van Prooijen <keesvp@gmail.com>

🔗Carl Lumma <carl@lumma.org>

Hi Kariisa-tan,

Sorry your msg was delayed - caught in the spam filter I'm afraid.

>Here are some linear temperaments compatible with 5-equal ~~
>
>Fourth-thirds "Father" 5&8
>{16:15 \ 28:27}
>[1 . -1 . 3]
>Kees-max error: 28.598519 Â¢/oct
>Kees-max complexity: 1.499298 s/oct
>Log-flat badness: 64.286464 Â¢*s^2/oct^3

What are you using to calculate these? I ask because, if you wrote
it yourself, I'm very impressed you were able to gather all the
temperament names and so forth from various sources. Have you been
lurking here long?

I gather your question is: While there are an infinite number of
linear temperaments below any logflat badness cutoff, is this still
true if one of the vals (equal temperament mappings) is fixed?

My hunch is yes, simply since the unrestricted distribution is
logflat, any restriction is likely to make it finite. I don't know
how to go about proving it however.

>Furthermore, the sequence of decreasing log-flat badness of equal
>temperaments is hyperexponential -- after 4, 31 and 171-equal, the
>next in line is 103169. What could be the smallest 7-limit equal
>temperament whose log-flat badness is better than that of
>103169-equal?

If I use TOP damage for error and notes/octave for complexity, 31
doesn't show up on my list of improving 7-limit ETs, since 12 is
better. The next after 12 is 99.

-Carl

🔗sunny_red_faery <aerissunnymilk@yahoo.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Hi Kariisa-tan,
>
> Sorry your msg was delayed - caught in the spam filter I'm afraid.
>
> >Here are some linear temperaments compatible with 5-equal ~~
> >
> >Fourth-thirds "Father" 5&8
> >{16:15 \ 28:27}
> >[1 . -1 . 3]
> >Kees-max error: 28.598519 Â¢/oct
> >Kees-max complexity: 1.499298 s/oct
> >Log-flat badness: 64.286464 Â¢*s^2/oct^3
>
> What are you using to calculate these? I ask because, if you wrote
> it yourself, I'm very impressed you were able to gather all the
> temperament names and so forth from various sources. Have you been
> lurking here long?

I used a graphing application to calculate errors and to calculate complexity requires only simple arithmetic and logarithmic functions. I used Kees-max values because calculation is straightforward -- simply use maximum minus minimum deviation of values of prime number harmonics for error and complexity.

And yes~~ I have lurked on this list for so long, yet I had little motivation to post messages despite having looked at many hundreds of messages.

> I gather your question is: While there are an infinite number of
> linear temperaments below any logflat badness cutoff, is this still
> true if one of the vals (equal temperament mappings) is fixed?
>
> My hunch is yes, simply since the unrestricted distribution is
> logflat, any restriction is likely to make it finite. I don't know
> how to go about proving it however.

Yes~~ Setting an upper bound on complexity will make any set of octave-equivalent temperaments finite, and by ignoring temperaments more complex than 5&171 we can see that the "Beep" temperament, 5&9 has the lowest logflat badness from the list.

> >Furthermore, the sequence of decreasing log-flat badness of equal
> >temperaments is hyperexponential -- after 4, 31 and 171-equal, the
> >next in line is 103169. What could be the smallest 7-limit equal
> >temperament whose log-flat badness is better than that of
> >103169-equal?
>
> If I use TOP damage for error and notes/octave for complexity, 31
> doesn't show up on my list of improving 7-limit ETs, since 12 is
> better. The next after 12 is 99.

Hmm.. I never thought about that~ Perhaps can you show your calculations there?

~~ Kariisa-tan

🔗Carl Lumma <carl@lumma.org>

🔗Kari-sunny <aerissunnymilk@yahoo.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> >Furthermore, the sequence of decreasing log-flat badness of equal
> >> >temperaments is hyperexponential -- after 4, 31 and 171-equal, the
> >> >next in line is 103169. What could be the smallest 7-limit equal
> >> >temperament whose log-flat badness is better than that of
> >> >103169-equal?
> >>
> >> If I use TOP damage for error and notes/octave for complexity, 31
> >> doesn't show up on my list of improving 7-limit ETs, since 12 is
> >> better. The next after 12 is 99.
> >
> >Hmm.. I never thought about that~ Perhaps can you show your calculations there?
>
> It's a bunch of scheme code I've posted before. Not terribly readable
> but let me know if you're interested and I'll send it offlist.
> Here are the numbers
>
> <99 157 230 278| 155
> <12 19 28 34| 169
> <31 49 72 87| 176

Okay, I would like to see how these were calculated, and how that relates to the Kees-max method that I usually use =)

> Are using unweighted error? Notes/octave complexity?

I suppose so~ I simply selected the numbers 4, 31, 171 and 103169 from a list posted a while ago, which apparently may have used unweighted error.

~~ Kariisa-tan

🔗Graham Breed <gbreed@gmail.com>

"sunny_red_faery" <aerissunnymilk@yahoo.com> wrote:

> The above theorem can be proven rather easily, partly by
> noting that a given unison interval of a temperament sets
> a lower limit for the error of the temperament. However,
> temperaments can also be defined by a set of equal
> temperament mappings, and specifying an equal temperament
> sets an upper limit of error. While the error of
> temperaments compatible with a given equal tempermant has
> no lower limit, the existence or lack thereof of a lower
> limit of badness may be more difficult to prove for
> certain.

We can assume there will be a limit. Equal temperament
mappings are the algebraic dual of unison intervals. It
should be possible to take a theorem using one and convert
it to the other. Unfortunately I don't know how to do that.

> The above shows some support for my hypothesis that for a
> given ET, there is a finite quantity of temperaments
> below a given log-flat badness cutoff. However, I have no
> conclusive proof of the hypothesis.
>
> What additional information can support or invalidate
> this hypothesis?

This shows how you can prove you have all rank 2
temperaments within a given badness cutoff, working from
equal temperaments:

http://x31eq.com/complete.pdf

It only works for particular definitions of error,
complexity, and badness, but it's a starting point.

Note equation 31 on p.3. This converts from a given
complexity and error to a number of steps to the octave.
If you know what badness you're looking for, you can plot
a line of error against complexity consistent with it. So
you can use this relation of sqrt(complexity/error) to say
how many notes you expect to find in an equal temperament
for any point on the line. At some point, therefore, any
number of steps will be optimal. An equal temperament
mapping with that many steps will be less likely to give a
higher rank temperament with lower badness the further you
get from its optimal point.

This assumes the equal temperaments also have low badness.
You can choose a stupid mapping that gives you any result
you want. I think there's a rule that a consistent equal
temperament gives rank 2 temperaments with complexity lower
than the number of notes to the octave. You can see this
because if an interval maps to more than n generators for a
scale with n notes to the octave, you could have mapped it
more simply. But I don't think I ever proved this.

Another article here:

http://badness.pdf

may be relevant, but it's all about a different kind of
badness. You can think of it as a way of windowing logflat
badness if you like.

Graham

🔗Carl Lumma <carl@lumma.org>

--- In tuning-math@yahoogroups.com, "Kari-sunny" <aerissunnymilk@...> wrote:

> > Here are the numbers
> >
> > <99 157 230 278| 155
> > <12 19 28 34| 169
> > <31 49 72 87| 176
>
> Okay, I would like to see how these were calculated, and how
> that relates to the Kees-max method that I usually use =)

Maybe you can tell me how to calculate Kees-max. IIRC it's
what we're supposed to use when we want untempered octaves.

This Scheme procedure computes the TOP damage of a val,
when that val is TOP-tuned

(define top-damage
(lambda (val basis)
(let*
((weights (map log2 basis))
(ji (map (lambda (x) (* x 1200)) weights))
(errors (map abs (map - ji (top-val val basis)))))
(apply max (map / errors weights)))))

if that makes any sense to you. Oh, it uses this too

(define top-val
(lambda (val basis)
(let*
((log2-primes (map log2 basis))
(exact-val (map / val log2-primes))
(big (apply max exact-val))
(small (apply min exact-val))
(mid (/ (+ big small) 2))
(tuned-val (map (lambda (x)
(/ x mid))
val)))
(map (lambda (x)
(* x 1200))
tuned-val))))

which is Paul Erlich's algorithm.

To find the vals I listed before, I look at all vals up to
1000 notes/octave and sort them by notes/octave * top damage,
with the appropriate logflat exponents. "All" here means I
generate patent vals for ETs in 0.1-cent increments (12.1-ET,
12.2-ET and so on). The patent vals are true vals which means
they contain only integers. I then remove torsional ones by
excluding those whose integers have GCD > 1. I managed to
convince myself this catches everything interesting but I make
no claim it's the best method.

> > Are using unweighted error? Notes/octave complexity?
>
> I suppose so~ I simply selected the numbers 4, 31, 171 and
> 103169 from a list posted a while ago, which apparently may
> have used unweighted error.

Ah. Happen to have that msg number?

-C.