Tuning-Math Digests messages 6150 - 6174

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Message: 6150

Date: Tue, 21 Jan 2003 23:51:35

Subject: Re: Superparticular temperaments

From: Carl Lumma

> > I thought something had cast doubt on the poptimal stuff.
> > And didn't Paul say the entire p range tends to minimax?
> 
> didn't i say what?

That as p goes to infinity you get minimax.

-Carl


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Message: 6151

Date: Tue, 21 Jan 2003 23:54:52

Subject: Re: Superparticular temperaments

From: Carl Lumma

> > Look ye above.
> 
> yes, it's the exact opposite of what you thought -- if there's
> only one comma, you're not missing or ignoring anything, because 
> straightness doesn't even exist (let alone come into play).

OK.

>>>>Adaptive JI.  It should go without saying with me.  I
>>>>didn't even know about strict JI until joining this list,
>>>>and it took 2 years of confusion before we figured it out!!
>>>
>>you're kidding me, right?
>>
>>No, why would you say that?
> 
>you didn't refer to pitches as ratios?

Nope.  Well, I had seed scales in ratios, but I never even
thought of the idea of a global pitch set.

>i'm also not interested in the more accurate temperaments.
>they're mathematical curiosities. what interests me are
>temperaments where a "small-numbered" MOS already contains
>a good deal of the desired harmonies, and you can develop
>a grammar around the harmonic meaning of scalar alterations,
>etc.

Exactly!

-Carl


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Message: 6152

Date: Wed, 22 Jan 2003 00:01:47

Subject: Re: heuristic and straightness

From: Carl Lumma

> > So is this right:
> > 
> > Straightness...LengthUVs...Length+/-UV...Badness
> > Down...........Up..........Down..........Same
> > Up.............Down........Up............Same
> > 
> > ?
> 
> if you replace "badness" with "error", it's right.

I should have noted that this was for a given temperament,
not for all temperaments, though I take it you took it
that way.

So what should the badness column be?

-Carl


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Message: 6160

Date: Wed, 22 Jan 2003 18:48:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I suggest it's time to settle the issue of what single ASCII characters we 
recommend to stand for the more common single-shaft saggital symbols.

I describe certain sagittal symbols below by giving the 
multi-ASCII-character approximation we've been using. But if you want to 
get involved in this, you should check out the real symbols in
Yahoo groups: /tuning-math/files/Dave/SymbolsBySize3.bmp *
or
Yahoo groups: /tuning-math/files/secor/notation/AdaptJI.gif *
or
Yahoo groups: /tuning-math/files/secor/notation/Symbols3.bmp *
some of which are a little out of date in regard to what symbols are in and 
what are out, but are fine for seeing the symbols mentioned in this message.

I suggest we start with the ASCII sets used by Manuel in Scala, as far as 
they go, except that I think we should use < and > as an alternative to L 
and 7 for the 7-comma, to avoid confusion with 7 as an octave number or 7th 
chord, and to agree with the ASCII-Sims notation. And we should not use L 
or 7 for anything so at least we don't conflict with Scala in this regard.

Note that Scala uses < and > as single step symbols for some ETs in 
Rapoport notations, where they are considered to represent a 1/n fraction 
of the 5-comma 80:81 for some small n. We have no need to symbolise such 
things. Some time ago I suggested that Manuel use -/ and \- for these in 
the same way he has used -) and (- for diesis fractions, but he 
understandably doesn't like to change established usages in Scala and likes 
< and > for this purpose because of their resemblance to the filled 
triangles that Rapoport proposed (See Xenharmonikon 16, 1995). However I'm 
prepared to ride roughshod over Rapoport's symbols since they appear to 
have been chosen almost at random, as opposed to the systematic symbol 
structure of sagittal.

Unfortunately Scala has multiple uses for v and ^. When notating ETs in 
Rapoport's notation Scala uses them for the diaschisma 2025:2048. When 
notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET 
and 144-ET in Richter Herf notation it uses them for the quartertone. These 
latter two usages are compatible with each other but not with the former. 
We must choose the JI/Richter-Herf usage due to the strong graphical 
resemblance to the relevant sagittal symbol /|\, sans shaft.

Manuel uses v and ^ for the diaschisma due to their resemblance to those 
proposed by Rapoport. I pointed out that they better resemble Rapoport's 
Pythagorean comma symbols (which are however not used for notating ETs) and 
that u and n look more like Rapoport's diaschisma symbols. But I now want 
to use u and n for something else since the diaschisma symbol is rarely (if 
ever) used in sagittal notation.

We should avoid conflict with the ASCII Sims notation too, except in the 
one area where it is unavoidable. The characters v and ^ must stand for the 
11-diesis 32:33 in sagittal, as in Scala and in Monz's notation, not the 
5-comma 80:81 or 1/12 tone as in the ASCII Sims. However, we might use [ 
and ] for the 11'-diesis 704:729 without too much of a problem since this 
also corresponds to 3deg72, although it is not used in the _standard_ 
sagittal notation for 72-ET.

We should also avoid conflict with Monz's and Johnston's notations by not 
using + or - for anything.

So here's what this gives us so far.

/||\  #  apotome sharp 2048:2187
\!!/  b  apotome flat
/|    /  5-comma sharp 80:81
\!    \  5-comma flat
  |)   >  7-comma sharp 63:64
  !)   <  7-comma flat
/|\   ^  11-diesis sharp 32:33
\!/   v  11-diesis flat
(|)   ]  11'-diesis sharp 704:729
(!)   [  11'-diesis flat

Here's what Scala has for some higher prime commas. Note that each symbol 
can represent two different commas depending whether one is using the 
system JI1 or JI2. Note that half of these are two-characters and the other 
half do not really look like up and down pairs. I'm inclined to suggest 
some different symbol pairs, and simply to avoid clashing with these by not 
using any of those symbols for anything.

| 27/26 1053/1024
; 26/27 1024/1053
#' 17/16 2187/2176
b' 16/17 2176/2187
% 19/18 513/512
d 18/19 512/513
@ 24/23 736/729
* 23/24 729/736
#! 29/27 261/256
b! 27/29 256/261
|' 32/31 248/243
;' 31/32 243/248

I suggest the following for our 13-diesis symbols

/|)   n  13-diesis sharp 1024:1053
\!)   u  13-diesis flat
(|\   }  13'-diesis sharp 26:27
(!/   {  13'-diesis flat

Notice the correspondence with the 11' dieses. In both cases the smaller 
diesis is given by an up-down pair of (mostly) lowercase letters and the 
larger by a left-right pair of brackets. In the 11 case the up down pair v^ 
have two straight sides like the sagittal symbol (but no shaft) and in the 
13 case the up down pair have a straight side and a curved side un (but not 
on the correct sides in the case of the u, and no shaft). With the brackets 
[] and curly braces {}, the resemblance is more to the number 11 
(underlined and turned sideways) and the 3 of 13, than it is to the actual 
sagittal symbols.

So we have a bit of a system here with the 5, 11 and 13 commas resembling 
their sagittal symbols with the omission of the shaft. We cant do that for 
the 7-comma using ( and ) because once the shaft is gone these could 
equally well be the 7:11, 19 or 5:7 comma symbols (see below) in various 
states of up or down. Best not to use ( or ) at all in the single-ASCII 
version of sagittal, unless they're for the same use that Manuel makes of them.

) 125-diesis sharp 125:128
( 125-diesis flat

The 7-comma symbols < and > are like the smaller 11 and 13 symbols in being 
a kind of bracket that resembles the appropriate prime number.

Any comments/objections to the above?

There are no more bracket pairs available. The only ASCII character pairs I 
can suggest for use as further common sagittals are the following down-up 
pairs, in my order of preference:

jf
yh
wm
dq
o*
&%
J?

Can anyone think of any others that don't clash with existing uses?

I suggest we not have a single character for the 25'-diesis 6400:6561 //| 
but simply use \\ and //. Likewise << and >> for the |)) 49'-diesis 3969:4096.

The next most common sagittals, and therefore those most deserving of a 
single character, are, in order of popularity:

   |(   5:7 comma           5103:5120    ~5.758c
        11:13 comma          351:352     ~4.925c
        7:25 comma

  (|(   5:11 comma            44:45     ~38.906c
        7:13 comma          1664:1701   ~38.073c
        11:17 comma         1377:1408   ~38.543c

  (|    7:11 comma         45056:45927  ~33.148c
        13:17 comma           51:52     ~33.617c
        29 comma             256:261    ~33.487c

  ~|    17 comma            2176:2187,   ~8.730c

  ~|(   17' comma           4096:4131   ~14.730c

Can anyone suggest a convincing way to map single ASCII character pairs to 
these? I'd be happy to get this far, since it would see 217-ET notated 
along with the 15-limit diamond and the first 17 odd harmonics. But if you 
were keen, you might push on to single ascii characters for:

  )|    19 comma             512:513     ~3.378c

  )|~   19' comma          19456:19683  ~20.082c
        19+23 comma*         432:437    ~19.922c

   |~   23 comma             729:736    ~16.544c


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Message: 6161

Date: Wed, 22 Jan 2003 23:50:53

Subject: Re: Superparticular temperaments

From: Carl Lumma

> "as p goes to infinity you get minimax"
> 
> sure sounds very different to me than
> 
> "the entire p range tends to minimax"
> 
> !

Yeah, sorry 'bout that.  Just in a hurry,
or half asleep.

-Carl


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Message: 6162

Date: Wed, 22 Jan 2003 12:11:35

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Dave gave an accurate summary except for this:

>Unfortunately Scala has multiple uses for v and ^. When notating ETs in
>Rapoport's notation Scala uses them for the diaschisma 2025:2048. When
>notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET
>and 144-ET in Richter Herf notation it uses them for the quartertone. 
These
>latter two usages are compatible with each other but not with the former.

The symbols for 32:33 in JI were changed some time ago to [ and ], which 
are also used in ET notations.
I kept v and ^ for Richter Herf since this notation system stands on its 
own 
and they resemble the up and down arrows more.

>We must choose the JI/Richter-Herf usage due to the strong graphical
>resemblance to the relevant sagittal symbol /|\, sans shaft.

No reason to change this as far as I'm concerned but of course you 
can if you want to avoid confusion over [ and ].
The graphical translation in Scala looks about the same. Press Ctl+F7 
to see the graphical notation, it's a recently added dialog window.

Manuel


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Message: 6163

Date: Wed, 22 Jan 2003 23:55:41

Subject: Re: heuristic and straightness

From: Carl Lumma

> > > > Straightness...LengthUVs...Length+/-UV...Badness
> > > > Down...........Up..........Down..........Same
> > > > Up.............Down........Up............Same
> > > > 
> > > > ?
> > > 
> > > if you replace "badness" with "error", it's right.
> > 
> > I should have noted that this was for a given temperament,
> > not for all temperaments, though I take it you took it
> > that way.
> 
> yes, because of the last column being all "same".
> 
> > So what should the badness column be?
> 
> well, i guess that's all "same" too!

So if error is the same and badness is the same, then
complexity is the same (which I suppose makes since,
if the volume of the block is not to change).  So is it
safe to conclude that straightness is important for
heuristically searching temperaments, but not for
choosing a commatic basis for a given temperament?

-Carl


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Message: 6164

Date: Wed, 22 Jan 2003 10:00:07

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Here's a spreadsheet that calculates all the reasonable notational commas 
for the ratios in the popularity list.

Yahoo groups: /tuning-math/files/Dave/NotationalCommas.xls.zip *
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6169

Date: Fri, 24 Jan 2003 15:24:58

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Dave wrote:

>I apologise for not looking at the latest help file.
>Scala help *

Oh, this file is out of date. I don't update it with each
release, the latest version only comes with the program.
The webfile is only meant to give an impression what the
program can do. But maybe it's time for an update now.

>I'm hoping now that we've settled on the sagittal symbols that you will
>remove JI3 and replace it with whatever subset of the sagittal notation 
we
>manage to come up with single ASCII characters for.

Yes that might be done, sagittal ASCII notation for JI is probably 
surveyable. For more I don't have much hope since the current
Scala notations were already a huge amount of work.

>How does Scala use [] 32:33 for ET notations?

Like the others, it's based on the best approximations to 3 and 11.
Do you have a recent Scala? Do View->Staff->Select... and you can
browse the notations. E74 is the lowest one with [ and ].

Manuel


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Message: 6170

Date: Fri, 24 Jan 2003 09:20:42

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:
>Dave gave an accurate summary except for this:
>
> >Unfortunately Scala has multiple uses for v and ^. When notating ETs in
> >Rapoport's notation Scala uses them for the diaschisma 2025:2048. When
> >notating JI it uses them for the 11-diesis 32:33 and when notating 72-ET
> >and 144-ET in Richter Herf notation it uses them for the quartertone.
> >These latter two usages are compatible with each other but not with the
> >former.
>
>The symbols for 32:33 in JI were changed some time ago to [ and ],

Hi Manuel,

I apologise for not looking at the latest help file.
Scala help *

But when I do, I see that notations JI and JI2 are still the same, and show 
that v and ^ _are_ used for 32:33. But you have added:

QUOTE

The symbols which differ in JI3 are:
 > septimal comma sharp, 64/63
< septimal comma flat, 63/64
] 33/32
[ 32/33
} 1053/1024
{ 1024/1053
f 2187/2176
j 2176/2187
h 513/512
y 512/513

UNQUOTE

I believe these were only ever used by Gene and I in February 2002 in the 
"Notating ETs with one comma per prime" thread. At that time we were more 
concerned with the semantics than the symbols, but were assuming they would 
be an extension of the Sims symbols. Then George Secor came along and 
convinced us otherwise.

No one actually asked for JI3. I merely suggested some changes to JI and 
JI2. However I must acknowledge that I did not object when you added JI3.

I'm hoping now that we've settled on the sagittal symbols that you will 
remove JI3 and replace it with whatever subset of the sagittal notation we 
manage to come up with single ASCII characters for. But of course this is 
not only for JI, and has symbols for more than one comma per prime in some 
cases.

The full set of sagittal symbols is too large for single ASCII characters 
(although most symbols are rarely used). But we do have a system for this 
in which up to 5 ASCII characters are used for a single sagittal, and of 
course Scala can do proper graphical characters now.

>  which are also used in ET notations.

How does Scala use [] 32:33 for ET notations?

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


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