Tuning-Math Digests messages 6310 - 6334

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Message: 6310

Date: Wed, 05 Feb 2003 03:02:49

Subject: Re: heuristic and straightness

From: Carl Lumma

>>>> Is it any more proportional to log(d) than log(n) in this
>>>> case?  Since n~=d?
>>>
>>>No, and the spreadsheet sorted by d is also sorted by n.
>> 
>>So it could just as well be (n-d)/(d*log(n))?
>
>a very different sorting. that would be heuristic error, not
>heuristic complexity. of course it's a very different sorting,
>since knowing log(n) or log(d) tells you nothing about (n-d).

?  I was asking if the *error* heuristic could become
(n-d)/(d*log(n)) if we substituted log(n) for log(d).

>>>It's the first order approximaton where n/d ~= 1.  See (8) in
>>>Natural Logarithm -- from MathWorld *
>> 
>>The Mercator series??  And all the stuff on this page applies
>>only to ln, not log in general (which is what I assume Paul
>>meant), right?
>
>i meant ln. i always use matlab, in which "log" means ln.

Oh.  How odd.

-Carl


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Message: 6311

Date: Wed, 05 Feb 2003 03:07:34

Subject: Re: heuristic and straightness

From: Carl Lumma

>>>>>And in that case it would have been easier to go straight
>>>>>to log(n*d).
>>>> 
>>>>Straight to where (do you see log(n*d))?
>>>
>>>log(n*d) = log(n) + log(d)
>> 
>>Of course... so we're coming from log(n*d), not going to it.
>
>what do you mean, we're coming from log(n*d)??

In your original message, you *start from* "is proportional to
log(n) + log(d)" and *arrive at* "Hence the amount of tempering
implied by the unison vector is approx. proportional to
(n-d)/(d*log(d))".

-Carl


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Message: 6312

Date: Wed, 05 Feb 2003 03:10:55

Subject: Re: heuristic and straightness

From: Carl Lumma

> > I do see that... you were already using the term for the
> > complexity heuristic at that time, right?
> 
> no, gene introduced the word "heuristic".

K.

> > >w=log(n/d)
> > 
> > Got that.
> > 
> > >w~=n/d-1
> > 
> > How do you get this from that?
> 
> standard taylor series approximation for log . . . if x is
> close to 1, then log(x) is close to x-1 (since the derivative
> of log(x) near x=1 is 1/1 = 1).

Cool.

> > >w~=(n-d)/d
> > 
> > Ditto.
> 
> arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.

Yeah, I thought the 1 was in the denominator.

-C.


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Message: 6313

Date: Wed, 05 Feb 2003 03:29:37

Subject: Re: heuristic and straightness

From: Carl Lumma

>>Ok, but I still don't get how the "Mercator series" shown in (8)
>>dictates the rules for this approximation.
>
>Oh, I thought you had followed that.
>
>It's usually called the Taylor series.  I don't know what
>Mercator's got to do with it.

On the mathworld page, it says "the Mercator series gives a
Taylor series for the natural logarithm", and in fact makes
it look like the Taylor series is the Mercator series.

>But anyway it's
> 
> ln(1+x) = x - x**2/2 + x**3/3 + ...
> 
> where x is small.  Because x is small, x**2 must be even
> smaller, so you can use the first order approximation
> 
> ln(1+x) =~ x
> 
> In this case, 1+x is n/d, so x = n/d - 1
> 
> ln(n/d) =~ n/d - 1

Thanks!

-Carl


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Message: 6314

Date: Wed, 05 Feb 2003 04:02:23

Subject: Re: heuristic and straightness

From: Carl Lumma

>>It's usually called the Taylor series.  I don't know what
>>Mercator's got to do with it.
>
>On the mathworld page, it says "the Mercator series gives a
>Taylor series for the natural logarithm", and in fact makes
>it look like the Taylor series is the Mercator series.

...by the way the page is formatted.

Turns out that the Mercator series is what we're talking
about; a special case of the Taylor series.

Taylor Series -- from MathWorld *

"A Taylor series is a series expansion of a function about
a point.  ... "

-Carl


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Message: 6315

Date: Wed, 05 Feb 2003 18:01:03

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Paul Erlich wrote:
>these cutoffs are totally arbitrary and need not even be based
>strictly on JI cents considerations. but you guys may be interested
>in studying the following links (which, however, concern 5-limit
>only):
>
>Yahoo groups: /tuning/database? *
>method=reportRows&tbl=10&sortBy=5&sortDir=up
>
>(be sure [to click 'Next'] through all the pages)
>
>Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *

Thanks. Those were very useful. In case anyone is wondering, the usage I 
gave, that is referred to on Monz's page above, was that of Scala's 
intnam.par which is that the difference between 3 major thirds and an 
octave is the minor diesis (41.06 c untempered) and the difference between 
4 minor thirds and an octave is the major diesis (62.57 c untempered).

Since for our purposes, we _are_ basing it strictly on "JI cents 
considerations" (i.e. the untempered size), I think we can take it that the 
"minimal diesis" at 27.66 c and the "small diesis" at 29.61 cents can 
afford to be considered as some kind of comma, since there's a big gap 
between them and all the other 5-prime-limit dieses. Given that the prime 
exponent vectors for these are [5 -9 4] and [-10 -1 5] respectively, there 
is unlikely to be much call for using them to notate rational pitches, and 
certainly not for notating temperaments.

The next largest is the minor diesis of 41.06 c. By the way, calling this a 
great or major diesis just seems silly to me since there are two larger 
than it, and they are not particularly obscure or complex. I think 
Mandelbaum and/or Würschmidt and/or Helmholtz and/or Ellis screwed up.

Monz's chart showing his size categories for 5-prime-limit "anomalies" is 
very interesting and has the following agreement with ours so far.

The skhisma-kleisma boundary is between 2 and 6 cents.
The kleisma-comma boundary is between 13 and 18 cents.
There is a boundary between 53 and 59 cents.

Differences are:

We have found no need of a boundary between 23 and 28 cents.
We don't have a boundary between 34 and 39 cents, although we should 
probably move our 40 cent boundary down to here.
We could use a boundary between 43 and 47 cents .
We have one between 65 and 69 cents that Monz doesn't.

The places where we want different or additional boundaries, _do_ appear as 
gaps on Monz's chart, even when he has not seen fit to make them nominal 
boundaries.

Here's how the names correspond so far:

Monz             George and Dave
--------------------------------
skhisma          schisma
kleisma          kleisma
comma            comma
small diesis     comma
great diesis     diesis
small semitone   ediesis (and other larger anomalies)

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > There are two slightly more popular pairs that would benefit from a
>higher
> > kleisma-comma boundary.
> >
> > N     kleisma  comma
> > 245   14.19    37.65
> > 7:13  14.61    38.07
> >
> > I note that neither of us is willing to bring the comma diesis
>boundary
> > down below 38.07 or 37.65 cents.
>
>But I'm willing to consider it if there's a good reason for it.
> > ...

Monz's chart has helped me to see that lowering the comma-diesis boundary 
to just below 37.65 cents is a better solution than raising the 
kleisma-comma boundary above 14.61.

Then we have
N     comma    diesis
245   14.19    37.65
7:13  14.61    38.07


> > I'll use the term ediasis (pron. ed-I-as-is, not ee-DI-as-is) for a
>diasis
> > larger than a half apotome, until someone tells me they like
>something else
> > better.
>
>Okay.  (Somebody, *please* suggest something better; ediasis sounds
>too much like a disease.)

You saw my other suggestions didn't you? "biesis" and "diesoma". I think 
they sound even more like diseases. There's not a lot you can do about that 
when "diesis" itself sounds like a disease?

The prefix "edi" or "oedi" meaning swollen has the same etymology as 
"Edipus" of the ancient Greek story. Edipus is literally "swollen feet" 
(from walking so much).

> > Some things in the kleisma size range have been called semicommas.
>
>I just think that "kleisma" sounds better.

Me too. Good ole Shohe' Tanaka.

> > >...
> > >The point here is that I thought that the comma (120:121, ~14.367c)
> > >between the next smaller pair of superparticular ratios (10:11 and
> > >11:12) should be smaller than the lower size limit for a comma.  If
> > >they were used as the two ("whole") tones in a tetrachord, their
>sum
> > >would be 5:6, which would leave 9:10 as the remaining interval
> > >(or "semitone") of the tetrachord.  But to have a "semitone" in a
> > >tetrachord that is larger than either of the "whole" tones is
>absurd,
> > >hence a practical basis for a boundary.
> >
> > I find this argument interesting but not convincing. Why must the
>whole
> > tones be superparticular?
>
>They don't have to, but since there are actual examples of ancient
>Greek tetrachords with diatonic steps of 7:8 with 8:9 (Archytas), of
>8:9 with 9:10 (Didymus), and of 9:10 with 10:11 (Ptolemy's hemiolon),
>these being all of the possible cases, I found that it was possible
>to draw a conclusion from them.
>
>Interestingly, the next larger pair, 6:7 with 7:8 -- difference of
>48:49, ~35.697c -- adds up to an exact 3:4, leaving a semitone that
>vanishes.

A tetrachord with only 3 notes isn't exactly kosher is it? And we don't 
have any other reason to go quite that low.

>   Their difference is still a bit smaller than we were
>considering for the comma-diesis boundary, which makes me wonder if
>40 cents is still too large.

Yes. I'm down to 37.65 c now. Just under the 245-diesis.

> > Why must they even be simple ratios?
>
>Because a comma, by definition, is the difference between two
>rational intervals similar in size.

However they don't have to be _simple_. But never mind.

> > Anyway, some very small intervals have been called commas for a
>long time.
> > e.g. We have Mercator's comma at about 3.6 cents and Wuerschmidt's
>comma at
> > about 11.4 cents. These are from Scala's intnam.par.
>
>These are just examples that the term can have a broad or generic
>usage in addition to the more specific definition that we're seeking.

I'll buy that.

> > >You want the boundary to be somewhere between what we have been
> > >calling the 17 comma (~8.7c) and 17' comma (~14.730c).  To
> > >accommodate both of these requirements, we could put the lower
> > >boundary for a comma at infinitesimally above 120:121, >14.37c.
> > >Would this be too large an upper limit for a "kleisma?"  If so,
>why?
> >
> > No I can't really argue that, although it is getting close to
>double the
> > size of _the_ kleisma. I now want to put the boundary even a little
>higher
> > than you suggest, at just above 28431:28672 (or 14.614 c) so we
>have a 7:13
> > kleisma and a 7:13 comma (38.07 c) as mentioned above.
> >
> > This does mean we have the 17-comma and the 7:13-kleisma being
>notated with
> > the same symbol ~|(
>
>But it doesn't make much sense, though.

You're right.

> > but I can probably live with that. Or would you rather
> > have two 7:13 commas?
>
>This might be a good reason to make the comma-diesis boundary
>somewhere around 37 to 38 cents.

Yes.

>   This would then put 1deg31 in the
>diesis range (at the lower end) -- 1deg31 is also functions as a 7-
>comma, but I think its dual use demonstrates that it's appropriate to
>have the boundary somewhere around this size.

But the size in cents of 1deg31 is irrelevant because it's tempered. I 
expect it is called a diesis because it corresponds to some 5-prime-limit 
comma whose untempered size would have made it a diesis even with our 
earlier 40 cent cutoff. In the case of sagittal it remains a diesis because 
it's a tempered 11-diesis.

>If 1664:1701 (~38.073c) is the 7:13 diesis, then 1377:1408 (~38.543c)
>and 44:45 (~38.906c) would become the 11:17 and 5:11 dieses.  Do you
>know of any potential problems with these designations?

No problem.

>If nobody else says anything, then that's what's going to happen.  I
>can't see making another boundary between semicomma and kleisma
>without a good reason.

I agree there is no need for a separate semicomma category.

> >       diesis 1  diesis 2
> > N     cents     cents
> > ------------------------------
> > 5:13  43.83     46.39
> > 37    42.79     47.43
> > 11:19 40.33     49.89
> > 25:77 44.66     45.56
> >
> > It doesn't seem like a good idea to have a category that only
>covers the
> > range 40.00 to 45.11 cents, but that's what we need for the above.
>
>I'll have to look these over to see if there's really any musical
>need some of these.  For example, the 46.39-cent 5:13 diesis would
>notate 10:13 as an interval of a third, but I can't imagine that
>anyone would want a third this large in a diatonic or heptatonic
>scale very often.

It _does_ seem like a good idea to me now to have a category of small 
dieses between 37.65 c and 45.11 c. For now I'll call these "carcinomas". 
That should make folks think _real_hard_ to come up with a better term. :-)

With this category, we don't have anything with two useful "anomalies" in 
the same category until we get down to 32:49, which is waaaay down the 
popularity list at number 145 with a 0.02% ocurrence. It has two kleismas, 
from C, Cb - 10.81 c and B + 12.65 c.

Although, when I say that, I am not counting 11:35 which is considerably 
more popular and has been mentioned before and has, from C, Cb - 11.64 c 
and B + 11.82 c. But we can get away with this because these are 
essentially a single kleisma.

To summarise:
0
schismina
0.98
schisma
4.50
kleisma
13.58
comma
37.65
carcinoma
45.11
diesis
56.84
ediasis
68.57

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6316

Date: Wed, 05 Feb 2003 18:08:32

Subject: Re: A common notation for JI and ETs

From: David C Keenan

On second thoughts, 13.47 cents might be a better choice for the 
kleisma-comma boundary.


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Message: 6317

Date: Wed, 05 Feb 2003 18:08:32

Subject: Re: A common notation for JI and ETs

From: David C Keenan

On second thoughts, 13.47 cents might be a better choice for the=20
kleisma-comma boundary.


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Message: 6320

Date: Thu, 06 Feb 2003 17:04:18

Subject: Re: A common notation for JI and ETs

From: Graham Breed

Me:
> Ratios are one of the insufficiently 
>>general representations -- they don't work for inharmonic timbres.

Gene:
> Isn't this a complete red herring? We are now talking about lists or sets of partial tones, not vectors.

Intervals are defined as vectors in terms of a minimal subset of the 
partials relative to the fundamental (which, for inharmonic timbres, 
will probably be the whole set).

The only part of the definition here:

Vector -- from MathWorld *

they don't comply with is that you can't construct unit vectors.  But 
that can be fixed if we have to as there are cases where fractions do 
creep in.  They can be added, subtracted and multiplied by scalars.  A 
list, it appears, is merely an ordered set, and so doesn't support these 
operations.

If all vectors are also lists, so they must be lists.  But they look 
like vectors as well.

As for sets, from

Set -- from MathWorld *

"A set is a finite or infinite collection of objects in which order has 
no significance, and multiplicity is generally also ignored (unlike a 
list or multiset)"

Well, we certainly can't ignore multiplicity.  [1, 2, 0] is very 
different to [1, 1, 0].  Even if we don't, are partials allowed to be 
present a negative number of times?  Even if that is a set, we're 
writing it as a list or vector (except when we write it as a ratio, but 
that won't work in general).


I'm sorry the CGI doesn't support this yet.  For now, here's an old list 
of tubulong temperaments:

2 4 6 10 14 16 18 20 22 26 30 34 36 38 44 46 50 52 54 62 *

The unison vectors are not frequency ratios, so there's no other way to 
represent them.  The partials are:

2.82843 5.42326 8.77058 12.86626 17.70875 23.29741

The first unison vector is [-3, 2, 0, 0]  That's -3*log2(2) + 
2*log2(2.82843) = 2.9 microoctaves.  I know it isn't quite a unison, but 
if it isn't a vector, what is it?


                           Graham


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Message: 6321

Date: Thu, 06 Feb 2003 17:12:06

Subject: Re: A common notation for JI and ETs

From: Graham Breed

I wrote:

> 2.82843 5.42326 8.77058 12.86626 17.70875 23.29741
> 
> The first unison vector is [-3, 2, 0, 0]  That's -3*log2(2) + 
> 2*log2(2.82843) = 2.9 microoctaves.  I know it isn't quite a unison, but 
> if it isn't a vector, what is it?

Bwahahaha!!!!

This particular unison vector, as an emergent property of the system, 
really is a unison.


                      Graham


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