>>> "Poptimal" is short for "p-optimal". The p here is a real variable
>>> p>=2, which is what analysts normally use when discussing these
>> Holder
>>> type normed linear spaces.
>>>
>>> A pair of generators [1/n, x] for a linear temperament is
>>> *poptimal* if there is some p, 2 <= p <= infinity,
>>
>> why not go all the way to 1? MAD, or p=1, error certainly seems
> most
>> appropriate for dissonance curves such as vos's or secor's -- which
>> are in fact even pointier at the local minima (resembling exp
>> (|error|)) . . .
>
> Possibly because no one in the history of this endeavour has ever
> before now suggested that mean-absolute error corresponds in any way
> to the human perception of these things.
no one in history? you've gotta be kidding me. sum or mean of absolute
errors is a quite common error criterion.
>> "A "poptimal" generator can lay claim to being absolutely and ideally
>> perfect as a generator for a given temperament ..."
>
> When we're talking about human perception, as we are, it should be
> obvious that nothing can be absolutely and ideally perfect for
> everyone. Even a single person might prefer slightly different
> generators for different purposes. To validate such a claim
> of "perfection" you would at least need to produce statistics on the
> opinions of many listeners.
clearly dave missed the clever mockery hidden in gene's statement.
Message: 5880 - Contents - Hide Contents
Date: Tue, 07 Jan 2003 06:02:33
Subject: Nonoctave scales and linear temperaments
From: Gene Ward Smith
I pointed out over on PostTonality that the 88CET is closely related to
the Octafifths temperament; you might say it is Octafifths without the
octave reduction. If Octafifths (with a generator around 8/109) has a
generator small enough for this, so does Quartaminorthirds, with a
generator arounf 7/108. Moreover, if the nonoctave people have not yet
tried the secor, they are missing a bet. Slender, with a generator of
about 4/125 gives something perversely related to 31-et, and we might
even try Porcupine, Tertiathirds or Hemikleismic.
The Wendy Carlos alpha is related to an 11-limit temperament which
appeared on my best-20 list; it is what you get by wedging 121/120,
126/125 and 176/175, [9,5,-3,7,-13,-30,-20,-21,-1,30], and we might
call it Wendy or Alpha. Which is best?
I also looked at beta and gamma, but I didn't find much. Beta can be
related to a cheeseball system obtainable as h75&h94, and it is
possible to regard gamma as 5/171. Maybe Graham or Dave can point out
something I am missing here.
Message: 5881 - Contents - Hide Contents
Date: Tue, 07 Jan 2003 23:57:22
Subject: Re: thanks manuel
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:
> Sure, I've enjoyed it too.
>
>> so one might either be interested in the *average*
>> complexity of the intervals formed by the note in question from all
>> the other notes in the scale, or, in special cases, the complexity of
>> the interval formed by the note from the tonic (1/1).
>
> There's an idea, it could be added to the output of
> "show/attribute intervals" which I didn't show you.
>
> Manuel
i think it would be good to have a graphical scale analysis tool. the
dyadic one would look like the blackjack interval matrix (.jpg file to
be referenced in my next messaage), and you could move the scale tones
around by dragging the lines (or indicators on a separate slider, which
would then move the lines accordingly). that would be cool.
Message: 5882 - Contents - Hide Contents
Date: Tue, 7 Jan 2003 10:24:49
Subject: Re: thanks manuel
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
Sure, I've enjoyed it too.
> so one might either be interested in the *average*
>complexity of the intervals formed by the note in question from all
>the other notes in the scale, or, in special cases, the complexity of
>the interval formed by the note from the tonic (1/1).
There's an idea, it could be added to the output of
"show/attribute intervals" which I didn't show you.
Manuel
Message: 5883 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 09:55:44
Subject: Notating Kleismic
From: Gene Ward Smith
I've been pondering this, and I think there is a strong argument in
favor of using 53. The top of the poptimal range for septimal kleismic
and the bottom of the poptimal range for 5-limit kleismic coincide at
the minimax generator of 3^(1/6), which is the same for both. This is
the only generator which is poptimal in both limits, but of course 53,
which has a much better fifth than 72, comes a lot closer. Moreover,
kleismic is more important as a 5-limit system (where it is very
strong) than as a 7-limit system, and I think we should try to use one
et for all of the versions of a temperament when we can. I vote (if
that is how this is done) for 53. Anyone else care to chime in?
Message: 5885 - Contents - Hide Contents
Date: Wed, 8 Jan 2003 12:28:29
Subject: Re: thanks manuel
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
Paul wrote:
>i think it would be good to have a graphical scale analysis tool.
Do you mean with a colour representation of attribute values,
like in your gif picture?
There are some other graphical analysis things in the to-do list
already, don't think I'll get to it soon.
Manuel
Message: 5886 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 21:15:20
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
> In case anyone has already looked at the .bmp for my latest
suggestion
> regarding symbolising the 5'-comma (5-schisma) up and down:
>
> It was riddled with vertical alignment errors so I've had another
go at it.
I've looked at it and thought about it for a couple of days.
Good points:
1) The +-5' symbols can easily be used in conjunction with any
existing symbol.
2) They clearly indicate (by vertical position) the line or space of
the note being modified.
3) There is also a helpful indication (a difference in vertical
position in addition to slope) of whether the 5' is plus or minus.
Comment on point 3: The slope is most meaningful if the new symbols
are placed to the left, as in the upper staff (which you also favor).
Problems:
1) The +-5' symbols are detached from the others, so are too easy to
overlook (particularly if this is the only thing modifying a natural
note).
2) Since they are detached from the others, we technically have two
new modifying symbols used together, so the double-symbol version of
the notation might now become a triple-symbol version -- something to
think about.
Since looking at this I also tried something else, which I have added
to this file (on the third staff):
Yahoo groups: /tuning- * [with cont.]
math/files/secor/notation/Schisma.gif
Note: If you don't see 4 staves in the figure, then click on the
refresh button on your browser to ensure that you're looking at the
latest version of the file.
I tried small arrowheads to indicate the 5' down and up symbols. In
the 3rd staff I attached them to the point of an existing sagittal
symbol; for the up-arrow I removed the pixel at the end of the shaft
to clarify the symbol. The big advantage here is that we would avoid
having detached symbol elements.
In the 4th staff (up to the first double bar) I placed the arrows to
the left of existing sagittal symbols, but they could just as easily
be placed to the right, or on either side, depending on where they
would look or fit best.
Wherever you put them, I think that these small arrowheads are easier
to see than those tiny slanted lines, and they give a better
indication of direction of alteration.
While I was writing this I got a couple of other ideas that use 5'
flags, so I quickly added them on the fourth staff. I lowered the
short straight -5' flag to the same vertical position that we seem to
be agreeing on to see how that would look and made 3 symbols that way.
Then, after the next double bar, I used the small arrowheads as right
flags and tried some symbols that way. (The 5:7 comma is also there
for comparison.) After I looked at them for a little while, I
decided to move the 5' flags one pixel to the right, so that they are
almost, but not quite touching the rest of the sagittal symbol (to
avoid confusion with a concave right flag). I think that this last
group is my preference in that:
1) The 5' flags are clear and logical;
2) The 5' symbol elements aren't off by themselves, therefore don't
get overlooked;
3) Their vertical positions are well placed;
4) They aren't larger than concave flags.
These are just a bunch of ideas that I'm tossing out there. Let me
know what you think.
--George
Message: 5887 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 00:02:14
Subject: blackjack interval matrix as promised
From: wallyesterpaulrus
Yahoo groups: /tuning/files/perlich/secor2.gif * [with cont.]
Message: 5888 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 00:08:32
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> The problem with this is that it makes the question of what the
>> consonances of the temperament are murky--we can't simply use
>> everything in the odd limit for some odd n. Of course, we could
>> simply create a set of intervals and then temper, or use n-limit
>> including evens.
>
> Perhaps I'm not seeing it, but I don't think we need to change
> our concept of limit.
we certainly would, and could use "integer limit" as gene suggests, or
use product limit (tenney).
Message: 5889 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 00:31:52
Subject: Re: Poptimal generators
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
>>
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
>>>
>>>> "Poptimal" is short for "p-optimal". The p here is a real variable
>>>> p>=2, which is what analysts normally use when discussing these
>>> Holder
>>>> type normed linear spaces.
>>>>
>>>> A pair of generators [1/n, x] for a linear temperament is
>>>> *poptimal* if there is some p, 2 <= p <= infinity,
>>>
>>> why not go all the way to 1? MAD, or p=1, error certainly seems
>> most
>>> appropriate for dissonance curves such as vos's or secor's -- which
>>> are in fact even pointier at the local minima (resembling exp
>>> (|error|)) . . .
>>
>> Possibly because no one in the history of this endeavour has ever
>> before now suggested that mean-absolute error corresponds in any way
>> to the human perception of these things.
>
> no one in history? you've gotta be kidding me. sum or mean of absolute
> errors is a quite common error criterion.
By "this endeavour" I meant specifically the mathematical modelling of
perceptual optimality of generators for musical temperaments, not
mathematic or statistics in general. I also only said "possibly". I'm
happy to be corrected.
>>> "A "poptimal" generator can lay claim to being absolutely and ideally
>>> perfect as a generator for a given temperament ..."
>>
>> When we're talking about human perception, as we are, it should be
>> obvious that nothing can be absolutely and ideally perfect for
>> everyone. Even a single person might prefer slightly different
>> generators for different purposes. To validate such a claim
>> of "perfection" you would at least need to produce statistics on the
>> opinions of many listeners.
>
> clearly dave missed the clever mockery hidden in gene's statement.
Sorry. I must have missed some previous discussion that would have
made it clear that irony was intended. A smiley or winky after it
wouldn't have gone astray. I just read it and thought, hey this is the
sort of talk that gets the non-math folk pissed at us math folk.
Thanks Gene, for being kind enough to ignore my patronising pedantry.
Message: 5890 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:15:21
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >erhaps I'm not seeing it, but I don't think we need to change
>> our concept of limit.
>
>we certainly would, and could use "integer limit" as gene
>suggests, or use product limit (tenney).
Maybe so, but I don't see why. I'm suggesting we think only of
the map, and let it do the walking. We get to pick what goes
in the map. Picking 2, 3, 5, 7 and calling it "7-limit" seems
fine to me.
>> If you weight the error right, you shouldn't have to weight
>> the complexity.
>
>I'll return to this after I've had my breakfast coffee. :)
Sorry, all I should have said is, * is communitive. So it's
really...
Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) )
I'll wager a coke this eliminates the need for an averaging
function over the intervals of the limit. If so, it would
approximate traditional badness, and this could be checked.
-Carl
Message: 5891 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:17:59
Subject: 8-limit meantone
From: Gene Ward Smith
If we look at all-number optimization, we can consider septimal meantone inthe 7, 8, 9 and 10 limits, for all the different p values.
Since
anything other than p=2 or p=infinity is asking for a headache, Idid
those for the 8-limit. The minimax calculation simply gave us our old
friend, 1/4-comma meantone. For rms I got stretched octaves:
2~1200.416 cents 3~1897.321 cents. If we take the ratio, we get
.580551, which still doesn't quite do it for us.
Message: 5892 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:22:48
Subject: Re: Poptimal generators
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:
> Sorry. I must have missed some previous discussion that would have
> made it clear that irony was intended. A smiley or winky after it
> wouldn't have gone astray.
"Irony" is too strong, but a knowing grin would do excellently.
Message: 5893 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:25:53
Subject: Re: Nonoctave scales and linear temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
> Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) )
>
> I'll wager a coke this eliminates the need for an averaging
> function over the intervals of the limit. If so, it would
> approximate traditional badness, and this could be checked.
I've
had my coffee, but still can't see the advantage of this system. We
want to have a measure of absolute error independent of complexity in
any case, do we not?
Message: 5894 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:41:08
Subject: Re: 8-limit meantone
From: Carl Lumma
>If we take the ratio, we get .580551, which still doesn't
>quite do it for us.
What ratio is that? Blackwood's r?
-Carl
Message: 5895 - Contents - Hide Contents
Date: Wed, 08 Jan 2003 01:58:10
Subject: Re: 8-limit meantone
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
>> If we take the ratio, we get .580551, which still doesn't
>> quite do it for us.
>
> What ratio is that? Blackwood's r?
It's
the ratio between the approximation for 3 and the approximation for
2,minus one; or in other words the log base the approximation for two
of theapproximation for 3/2. This is relevant since in the end we want
to relateit all back to a division of the octave.
Message: 5896 - Contents - Hide Contents
Date: Thu, 9 Jan 2003 13:29:40
Subject: Re: thanks manuel
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
>to start with, it would be good enough to simply have all the
>consonant intervals (say within a given odd limit) show up as
>diagonal lines -- the rest of the chart can be all white or all black
>for now . . . the point is you could visually tweak the scale with an
>eye toward approximating this consonance here and that consonance
>there . . . donīt know of a better way to achieve this goal than an
>applet like this!
Ok I understand. It probably won't be much work to expand the triad
player to do this.
Manuel
Message: 5898 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 00:04:23
Subject: Re: Poptimal generators
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
>>
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
>>>
>>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
>>>>
>>>>> "Poptimal" is short for "p-optimal". The p here is a real
variable
>>>>> p>=2, which is what analysts normally use when discussing
these
>>>> Holder
>>>>> type normed linear spaces.
>>>>>
>>>>> A pair of generators [1/n, x] for a linear temperament is
>>>>> *poptimal* if there is some p, 2 <= p <= infinity,
>>>>
>>>> why not go all the way to 1? MAD, or p=1, error certainly
seems
>>> most
>>>> appropriate for dissonance curves such as vos's or secor's --
which
>>>> are in fact even pointier at the local minima (resembling exp
>>>> (|error|)) . . .
>>>
>>> Possibly because no one in the history of this endeavour has
ever
>>> before now suggested that mean-absolute error corresponds in
any way
>>> to the human perception of these things.
>>
>> no one in history? you've gotta be kidding me. sum or mean of
absolute
>> errors is a quite common error criterion.
>
> By "this endeavour" I meant specifically the mathematical modelling
of
> perceptual optimality of generators for musical temperaments, not
> mathematic or statistics in general. I also only said "possibly".
I'm
> happy to be corrected.
consider yourself corrected :)
>>>> "A "poptimal" generator can lay claim to being absolutely and
ideally
>>>> perfect as a generator for a given temperament ..."
>>>
>>> When we're talking about human perception, as we are, it should
be
>>> obvious that nothing can be absolutely and ideally perfect for
>>> everyone. Even a single person might prefer slightly different
>>> generators for different purposes. To validate such a claim
>>> of "perfection" you would at least need to produce statistics
on the
>>> opinions of many listeners.
>>
>> clearly dave missed the clever mockery hidden in gene's statement.
>
> Sorry. I must have missed some previous discussion that would have
> made it clear that irony was intended.
nope. itīs just that an infinite number of tunings, a continuous
range of generators, can be considered poptimal for a given
temperament -- thus "absolutely and ideally perfect" cannot be taken
seriously, and was obviously poking fun of similar claims by people
like lucy.
paul (stuck in mannheim right now due to trains not running on time)
Message: 5899 - Contents - Hide Contents
Date: Thu, 09 Jan 2003 17:54:46
Subject: Re: Minimax generator
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:
>
> Thanks. How would one calculate the rms generator for say, Kleismic?
Kleismic tells us that
3 ~ k^6
5 ~ 2 k^5
6/5 ~ k
where k is the generator. If we take log base 2 of both sides and
call x=log2(k), we get
u1 = 6*x-log2(3) ~ 0
u2 = 5*x+1-log2(5) ~ 0
u3 = x-log2(6/5) ~ 0.
The least squares solution to this is found by finding the minimum of
the polynomial function
F(x) = u1^2 + u2^2 + u3^2
which we may easily do by taking the derivative and setting F'(x)=0.
If you have a program which solves linear programming problems, you can setup the standard minimization problem with objective function
y and constraint set
{y>=u1, y>=u2, y>=u3, y>=-u1, y>=-u2, y>=-u3, x>=0, y>=0}
and solve for the minimax generator x.
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