Tuning-Math Digests messages 5925 - 5949

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Message: 5925

Date: Fri, 10 Jan 2003 12:36:13

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Working down the ratio popularity list, of those we don't yet have a symbol 
for:

There are two 125 commas of interest
125-diesis     125:128    41.06 c  .//|    exact, no symbol without 5'
125'-diesis    243:250    49.17 c  /|) or (|~

two 49 commas
49-diesis       48:49    35.69 c  ~|)
49'-diesis    3963:4096  54.53 c  (/| or |))

one 7:25 comma
7:25-comma    224:225     7.71 c  '|(      exact, no symbol without 5'

two 5:49 commas
5:49-comma 321489:327680 33.02 c  (|
5:49'-diesis  392:405    56.48 c  '(/| or '|)) exact, no symbol w/o 5'

Perhaps we should ditch the (/| symbol entirely and use |)) for the 31' 
comma since |)) is the more obvious symbol for the 49'-diesis.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5926

Date: Fri, 10 Jan 2003 04:40:05

Subject: Re: Notating Kleismic

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> 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?
<

The way this standardisation stuff has been done in the past, is not
by voting as such, but by consensus. That is, we keep presenting
arguments for and against various options, with as little ego
investment and as much praise of the other people's ideas as possible,
until everyone who has ever expressed an opinion on it, either agrees
or says they no longer care.

So here goes me.

While I liked the pattern that kleismic makes in 72-ET, as pointed out
by George, I'm swayed by Gene's arguments above. By most people's
reckoning kleismic is one of the top four 5-limit temperaments. At the
7-limit there are two extensions of kleismic that might be considered.
Both are down past number ten on anyone's list.

However, if you _were_ using kleismic for 7-limit with the least
complex 7's, you would probably be dissatisfied with a 53-ET based
notation.

Lets try approaching it in an ET-independent manner, considering only
the 5-limit map
2:3 is 6 gens
4:5 is 5 gens
5:6 is 1 gen

BTW the two 7-limit extensions have
4:7 is 3 gens
or 
4:7 is 22 gens

I'll use / and \ as 5-comma (80:81) up and down symbols and FCGDAEB
and #b have their usual Pythagorean relationships. Then the following
is clearly a correct notation for a chain of 19 notes of kleismic.
Alternative names are given underneath some note names.

 A#\\\  C#\\  E\   G   Bb/  Db//  E#\\\  G#\\  B\   
Bbb///                            Fb///        


D   F/   Ab//  Cb///  D#\\  F#\  A   C/   Eb//  Gb///
               B#\\\                            Fx\\\

Which ET notation best preserves this?
53: /|  /|\  (|)
or
72: /|   |)  /|\
or should we use something else such as
    /|  //| .//|

Since we've got symbols for 11 commas in both of those ET notations,
we really should check whether they are valid in any sensible 11-limit
extensions of kleismic.

Gene or Graham, have any 11-limit kleismics turned up in your
searches. If so, what maps?


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Message: 5927

Date: Fri, 10 Jan 2003 05:49:44

Subject: Re: Notating Kleismic

From: Dave Keenan

I wrote:

> Which ET notation best preserves this?
> 53: /|  /|\  (|)
> or
> 72: /|   |)  /|\
> or should we use something else like
>     /|  //| .//|
> 
> Since we've got symbols for 11 commas in both of those ET notations,
> we really should check whether they are valid in any sensible 11-limit
> extensions of kleismic.
> 
> Gene or Graham, have any 11-limit kleismics turned up in your
> searches. If so, what maps?

In the case of the 53-ET based notation it could also be a 1,3,5,11
temperament (no 7s).


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Message: 5930

Date: Fri, 10 Jan 2003 10:23:44

Subject: Re: thanks manuel

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

>the triad player? . . . what's weird is that the idea above deals
>with dyads, while i had a similar idea actually dealing with triads --

Yes, that doesn't matter. The triad player also plays dyads by the way.
It can only play triads with one fixed tone, 1/1 or the octave.

> plotting all the triads in the scale on top of the snowflake, and
>seeing how the points (scale's triads) move around as you fiddle with
>the scale (possibly with an eye towards approximating a number of
>otonal triads) . . .

So this is quite another story.

Manuel


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Message: 5931

Date: Fri, 10 Jan 2003 19:31:54

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
> > At 01:16 AM 9/01/2003 +0000, Dave Keenan <d.keenan@u...> wrote:
> > >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.
> >
> > Yes. But unfortunately they make it look like you're modifying a
>note
> > aligned with the place between the 5' arrowhead and the rest of the
>flags.
>
>I just gave them the same vertical placement that you used for
>your "accent" marks.

No. I was referring here to where you attached them to the point of the 
existing symbol. But I think we've both rejected these by now.

> > Based on making symbols proportional to their size in cents
>relative to
> > strict Pythagorean, the 5' symbol should only have about 6 pixels
>because
> > the 19 comma flag has 10 and corresponds to 3.4 cents. The small
>arrowheads
> > (or circumflex and caron) contain 8 pixels.
>
>At least it's fewer pixels.

Yes. The 8 pixels wasn't a big deal.

> > Full arrowheads already have a sagittal association with the prime
>11
> > whereas the slanted lines preserve the association with 5.
>
>Also true.  Another problem that I see with these full arrowheads is
>how to represent them in ascii -- ^ and v would need to be used, and
>although this doesn't pose any conflict with sagittal ascii, it would
>pose a problem for those who want to use these as shorthand for the
>11 diesis.

Good point. But I don't think we should allow the limitations of ASCII to 
exert much influence, if any.

> > Well, I'd go along with kerning the acute nearer to (the left of)
>the
> > symbol being modified, when that symbol has a left flag (as in the
> > pythagorean comma symbol), but I'd still prefer that the 5' symbols
>were
> > defined as separate symbols in the font, for what are, I hope,
>obvious
> > reasons,
>
>Yes.
>
> > and I'd still prefer that the unkerned distance was two pixels
> > (such as in the diaschisma symbol).
> >
> > Pythag comma   '/|
> > Diaschisma     `/|
>
>To evaluate all of these issues, I added a fifth staff to my figure:
>
>Yahoo groups: /tuning- *
>math/files/secor/notation/Schisma.gif
>
>Note:  If you don't see 5 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 put on the fifth staff 5 different versions of symbols for each of
>five commas, along with the 19 and 5:7 comma symbols for comparison.
>The five versions are (left to right):
>
>1) Your 5' "accent marks" with the largest separation from the rest
>of the symbol that I believe would be acceptable.
>
>The separation for some of these is still more than I would like, so
>the next one is:
>
>2) Same as 1), but with 1 pixel less separation.
>
>One problem I have with your accent marks is that part of the mark is
>lost because it coincides with a staff line when the note is on a
>line, since the accent is 4 pixels high.  This doesn't occur with my
>arrowheads, which are 3 pixels high.  Therefore in the next one:
>
>3) The accent mark is redrawn 3 pixels high by 4 wide

Good. I prefer this to mine.

>  and given an
>amount of separation that I judged to be best, which is never greater
>than in 2), and sometimes less.
>
>Observe that with equal separation with 2) the 5' symbols (except for
>the pythagorean comma) appear to have an amount of separation
>intermediate between 1) and 2).  For the pythagorean comma symbol the
>separation is one pixel less than 2), such that the rightmost pixel
>of the accent mark is aligned with the leftmost pixel of the 5
>comma.  This would require a separate symbol in a font, much as some
>fonts have the letter combination "fi" as a single character.

You won't need to do that if the score software (e.g. Sibelius) recognises 
the font's kerning table. When designing the font you list specific pairs 
and their required negative offset relative to standard spacing. It doesn't 
matter if this causes them to overlap. But we could also do them as 
ligatures (single glyphs combining two characters) in some obscure location 
in the font, just in case.

>The next two use my small arrowhead marks:
>
>4) To the left, using an amount of separation that I judged to be
>best, and
>
>5) As in 4), but to the right.
>
>After studying these, I reached the conclusion that I like 3) best.
>If you agree in principle, we would need to finalize what should be
>the amount of separation between the accent mark and the rest of the
>symbol.

I definitely go for something very much like 3). I might prefer an extra 
half pixel of separation when it comes to the outline font, but otherwise 
great!

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


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Message: 5935

Date: Sat, 11 Jan 2003 11:53:44

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 11:13 PM 10/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > Working down the ratio popularity list, of those we don't yet have
>a symbol
> > for:
> >
> > There are two 125 commas of interest
> > 125-diesis     125:128    41.06 c  .//|    exact, no symbol without
>5'
> > 125'-diesis    243:250    49.17 c  /|) or (|~
>
>Now that we've agreed on the 5' comma symbols, may I suggest that the
>ascii symbols for -5' and +5' be ' and ` respectively,

I assume you meant to write ` and ' respectively?

>regardless of
>the direction of alteration of the main symbol (particularly since
>the actual accents don't appear aligned with the point of the arrow
>in the actual symbols)?  I think that the period and comma are too
>difficult to remember, especially the way you've done the 125-diesis
>above (which is different than before), and I think `//| and '\\!
>should be clear enough for a 125-diesis up and down, respectively.

Yes it's different than before. I find that `//| and '\\! don't look like 
inverses of each other. My thinking is that, with these tiny ASCII symbols, 
the vertical position is a much stronger cue than the slope, particularly 
since neither ' nor . have any slope. I find that `//| and ,\\! look like 
inverses, but unfortunately position and slope cues conflict with each 
other in these two symbols. That only leaves .//| and '\\!

So I'm proposing that the ascii symbols for -5' and +5' be . and ' 
respectively, regardless of the direction of alteration of the main symbol.

Consider distinguishing the Pythagorean comma from the diaschisma. Which 
pair makes it clearer which is which.
'/|  `/|
or
'/|  ./|
and in the other direction
`\!  '/!
or
.\!  '\!

I have to say both options are pretty unsatisfactory.

>For the 125' diesis, many divisions (including 171, 217, 224, 270,
>282, 342, 388, and 612) would allow either /|) or (|~, but 53, 99,
>and 494 all require /|), while 311 allows neither.  So I believe
>that /|) is the clear choice.

That's fine by me, for a completely different reason. Namely that it would 
be bizarre to introduce a wavy flag at the 5-prime-limit when these 
generally correspond to primes 17, 19, 23 and only appear in very large ETs.

> > two 49 commas
> > 49-diesis       48:49    35.69 c  ~|)
> > 49'-diesis    3963:4096  54.53 c  (/| or |))
>
>For the 49 comma ~|) is obviously the right size.
>
>The 49' diesis should be 3969:4096.  More on this one below.

Yes. My typo. Sorry.

> > one 7:25 comma
> > 7:25-comma    224:225     7.71 c  '|(      exact, no symbol without
>5'
> >
> > two 5:49 commas
> > 5:49-comma 321489:327680 33.02 c  (|
> > 5:49'-diesis  392:405    56.48 c  '(/| or '|)) exact, no symbol w/o
>5'
> >
> > Perhaps we should ditch the (/| symbol entirely and use |)) for the
>31'
> > comma since |)) is the more obvious symbol for the 49'-diesis.
>
>For the 31' comma only the divisions that have any semblance of
>consistency up to the 31 limit would have any practical bearing on
>this decision.  For 270 and 311 |)) is required, while for 217, 388,
>and 653 either one is valid; 494 requires (/|, but is not 1,7,31,49-
>consistent.  It looks like |)) takes it.  But this would require
>other symbols for 23 and 24deg494; any ideas?

I don't think that notating 494-ET is a high enough priority to delay the 
adoption of |)) as both the 49' and 31' diesis symbol. I can only think 
that we might be forced to use some symbols involving 5' for 494-ET. We 
could wait and see if suitable symbols come up as we work our way down the 
ratio popularity list.

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


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Message: 5936

Date: Sat, 11 Jan 2003 11:55:32

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 11:14 PM 10/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > > > Are there any ETs in which we should now prefer )|( over some
>other
> > >symbol
> > > > given that it now has such a low prime-limit or low product
> > >complexity?
> > > >
> >
> > I'll just note that neither of us have answered the above yet, in
>case the
> > way I edited things might have made it look like the following was
> > answering it, which of course it is not.
>
>There are none that I see for this as a 7':11' comma (or whatever
>we're going to call it).  It has a dual role with the 7+5+19 comma in
>212, 224, 311, 342, 612, and 624, where )|( has already been agreed
>on or is the obvious choice.  And it is not valid as the 7':11' comma
>in either 217 or 494.

OK. Good. Thanks for checking that.

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


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Message: 5937

Date: Sun, 12 Jan 2003 09:22:47

Subject: Notating linear temperaments

From: David C Keenan

Here are some more sagittal notations that are independent of any ET (as 
far as they go). They use only 5-comma and 5^2-comma symbols /| and //| in 
addition to sharps and flats.

Since they only use 5-prime symbols they are valid for any 7-limit or 
higher extension of these 5-limit temperaments.

Diaschismic (including Pajara)
3s   5s
0    1   periods
1   -2   gens

... Eb  Bb  F   C   G   D   A   E   B   F#  C# ...
... A\  E\  B\  F#\ C#\ G#\ Eb/ Bb/ F/  C/  G/ ...
                         Ab/

Compatible with 22-ET notation.


Augmented (including Tripletone)
3s  5s
0   1  periods
1   0  gens

                 G#\\
... C#\\G#\\C#\\Ab/ Eb/ Bb/ F/  C/  G/  D/  A/  ...
... Eb  Bb  F   C   G   D   A   E   B   F#  C#  ...
... G\  D\  A\  E\  B\  F#\ C#\ G#\ Eb//Bb//F// ...
                                 Ab//

Not compatible with any existing sagittal notation for an ET, but 27-ET 
notation is closest. We don't have a notation for 33-ET that uses its 
native fifth. We notate it as a subset of 99-ET.

As a 7-limit temperament, I think Tripletone (the 7-limit version of 
Augmented that exists in 33-ET) is garbage anyway. However, the above 
scheme will still notate it.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5938

Date: Sun, 12 Jan 2003 11:04:17

Subject: Notating linear temperaments

From: David C Keenan

Here are some more sagittal notations that are independent of any ET (as 
far as they go). They use only 5-comma and 5^2-comma symbols /| and //| 
(abbreviated below) in addition to sharps and flats.

Since they only use 5-prime symbols they are valid for any 7-limit or 
higher extension of these 5-limit temperaments.

Schismic
3s   5s
0    0   periods
1   -8   gens
...
Eb\\Bb\\F\\ C\\ G\\ D\\ A\\ E\\ B\\ F#\\C#\\G#\\
Eb\ Bb\ F\  C\  G\  D\  A\  E\  B\  F#\ C#\ G#\
Eb  Bb  F   C   G   D   A   E   B   F#  C#  G#
Eb/ Bb/ F/  C/  G/  D/  A/  E/  B/  F#/ C#/ G#/
Eb//Bb//F// C// G// D// A// E// B// F#//C#//G#//
...

Major thirds = Magic (including Narrow major thirds (= Muggles or Wizard?))
3s  5s
0   0  periods
5   1  gens

... C   E\  G#\\Cb//Eb/ G   B\  D#\\Gb//Bb/ D   F#\ A#\\Db//F/  A   C#\ 
E#\\Ab//C/  E ...


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Message: 5946

Date: Mon, 13 Jan 2003 20:57:34

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Map for 5-limit meantone...
>>
>>      2  3  5
>>gen1  1  1 -2
>>gen2  0  1  4
>
>hmm . . . gen1 is an octave, gen2 is a fifth . . . right?

Right.

>>Complexity for each identity...
>>
>>2= 1
>>3= 2
>>5= 6
>
>defined how?

Those should be 2/1, 3/2, and 5/4.  It's the taxicab distance
on the rectangular lattice of generators.  Which I cooked up
as a generalization of Graham complexity for temperaments that
don't necessarily have octaves.  How have you been calculating
Graham complexity for temperaments with more than one period
to an octave?

>>Let's weight by 1/base2log(i)...
>>
>>2= 1.00
>>3= 1.26
>>5= 2.58
>
>>Now gen1 and gen2 are variables, and minimize...
>> 
>>error(2) + 1.26(error(3)) + 2.58(error(5))
>> 
>>I don't know how to do such a calculation, or even
>>if it's guarenteed to have a minimum.  It would
>>give us minimum-badness generators, not minimum
>>error gens.
>
>i'm not sure what you're getting at. given the mapping,
>there's no way to change the complexity, so we'd be
>holding complexity constant.  so isn't minimizing
>badness then the same thing as minimizing error?

Doesn't the presence of the weighting factors change the
result?


How would you calculate complexity, error, badness, and
optimum generators for 5-limit meantone?

-Carl


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Message: 5948

Date: Mon, 13 Jan 2003 21:52:21

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Which I cooked up as a generalization of Graham complexity for
>>temperaments that don't necessarily have octaves.
> 
>there seems to be a problem, in that by defining the generators
>as an octave and a fifth, you get different numbers than by
>defining them as an octave and a twelfth, say.

I thought of that, but I thought also that as long as one always
uses the same set of targets across temperaments, one is ok.
Whaddya think?

>plus, graham complexity doesn't operate on a per-identity basis.

Indeed.  That's part of my inquiry into the order of operations.

>>How have you been calculating Graham complexity for temperaments
>>with more than one period to an octave?
>
>multiply the generator span of the otonal (or utonal) n-ad by the
>number of periods per octave.

That's what I thought.  How does this compare to the taxicab
approach?  Say, for Pajara.

>>>>error(2) + 1.26(error(3)) + 2.58(error(5))
//
>>>i'm not sure what you're getting at. given the mapping,
>>>there's no way to change the complexity, so we'd be
>>>holding complexity constant.  so isn't minimizing
>>>badness then the same thing as minimizing error?
>> 
>>Doesn't the presence of the weighting factors change the
>>result?
> 
>well, if you mean minimizing the expression above (assuming you
>meant to the errors to be absolute values or squares),

yep.

>basically you've just come up with a different weighting scheme
>for the error.

Ok.

>not one which i like, by the way, since you don't penalize the
>error in 3 and the error in 5 for being in opposite directions;
>that is, you don't take into account the error in 5:3.

That's what I said at the beginning of the thread.  So how do
you do weighted error?  Do you weight the error for an entire
limit by the limit, for intervals individually?

>>How would you calculate complexity, error, badness, and
>>optimum generators for 5-limit meantone?
>
>well, certainly woolhouse's derivation is known by everyone
>by now, isn't it? that gives you the error and optimum generator,
>in the equal-weighted RMS case. the complexity is a function of
>the mapping, and can be defined in various ways (the graham
>complexity is 4), but does not depend on the precise choice of
>generator. badness also has several definitions -- log-flat
>badness is pretty much gene's territory -- but is typically error
>times complexity to some power.

Okay, that's what I thought.

-Carl


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