Tuning-Math Digests messages 6252 - 6276

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Message: 6252

Date: Tue, 28 Jan 2003 01:10:43

Subject: dummy math question

From: monz

hi all,


here's what i think is probably a dummy math question,
meaning that i don't know how to do it 'cause i'm
a dummy ...


if x = y^z, and i know x and y, how do i find z?

thanks.



-monz


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Message: 6253

Date: Tue, 28 Jan 2003 10:18:44

Subject: Re: dummy math question

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

Assuming there is a solution:

    log x
z = -----
    log y

Manuel


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Message: 6254

Date: Tue, 28 Jan 2003 01:28:36

Subject: Re: dummy math question

From: monz

thanks, Manuel!  it was just what i needed.


-monz


> From: <manuel.op.de.coul@xxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, January 28, 2003 1:18 AM
> Subject: Re: [tuning-math] dummy math question
>
>
> Assuming there is a solution:
> 
>     log x
> z = -----
>     log y


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Message: 6255

Date: Wed, 29 Jan 2003 11:45:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> 
wrote:
Hi George,

Thanks for updating the quick reference. It's great.

 > > Incidentally, I think we should point out that the 5'-comma symbol
 > should
 > > stay to the left of any arrow symbol, even in text, so they're
 > always
 > > treated as a single compound symbol. e.g.
 > > Score:  ./| # C-notehead
 > > Text:   C#./
 > >
 > > This will also reduce the problem of . being taken as punctuation.
 >
 > Yes, absolutely!

I note that in the quick reference where you've given a combination of a 
shorthand ASCII sagittal with a conventional sharp, you've put the sagittal 
leftmost. You need a note to say that this is how it would appear on a 
score (but of course the ASCII characters should never appear on a score), 
but in text the sagittal should be rightmost.

On second thoughts, since the ASCII characters never appear on a score, 
wouldn't it be better to show them in text order and add a note to say that 
on a score the sagittal should be leftmost (i.e. furthest from the letter 
name or note head in both cases).

This is in contrast to the 5' accent marks which remain the leftmost 
component of a compound sagittal whether in text or on a score.

 > > but any other
 > > combinations of these shorthand symbols would represent multiple
 > sagittal
 > > symbols in the obvious way (we may yet find a use for this).
 >
 > For anything else, I would suggest going to the sagittal ascii that
 > we've previously been using, which has characters to indicate each
 > component of the actual symbol.

You may have misunderstood me. What I'm saying is that, for example, /f can 
be used as equivalent to the pair of symbols /|  |) (5-comma up, 7-comma 
up) but never the single symbol /|) for which n is available. The only 
exceptions to this are the pairs // \\ ff tt which are taken as equivalent 
to the single symbols //|  \\!  |))  !)) [or possibly (/|  (\! ] unless the 
author explicitly states otherwise.

There are two possible usages I have in mind for multiple sagittals against 
a single note.
1. A one-symbol-per-prime notation. Some may yet prefer it.
2. Linear-temperament-specific notations where more than 7 pseudo-nominals 
are produced by using one lot of sagittals (with the usual 7 nominals) to 
notate a small proper MOS. And then using a second lot of sagittals to 
represent the chroma (chromatic unison vector) and its multiples, that 
offset the other notes from that MOS.

For example, in miracle temperament the 10 pseudo-nominals could be (as a 
chain of secors):

F#\   G     Ab/   Af    Bv    Ct    C#\   D     Eb/   Ef

The above combinations of sagittals with sharps and flats would be better 
notated by double-shaft sagittals.

The chroma is t (the 7-comma down) and so the next row could be notated
F#\t  Gt    Ab/t  Aft   Bvt   Ctt   C#\t  Dt    Eb/t  Eft
where tt is to be taken as two separate sagittals.

Whether or not this is a good idea, I see no reason to forbid using the 
shorthand symbols in this way.

 > Since the two versions differ
 > according to whether or not a symbol contains either |, !, X, or Y,
 > then there's no problem in
 > determining which ascii version of the notation is being used.

Yes. A good feature.

 > By all means let's use x *only* for the double-sharp.  I initially
 > suggested Y for this purpose (I like its lateral symmetry,
 > particularly for the upward-pointing legs), which we can compare with
 > k for appearance:
 >
 > up       down
 > /|\      \!/
 > ||\      !!/
 > |||)     !!!)
 >  X\     Y/    k/
 >  X)     Y)    k)
 > /X\    \Y/   \k/
 >
 > I don't see any conflict between Y and y, because they won't ever
 > occur together, or even in two different symbols in the same ascii
 > version (as with X and x).

The conflict still exists in figuring out which of X and Y are up and down 
when no straight flags are present. Nothing about the X suggests any 
direction and if the user happens to have learnt the shorthand ASCII, she 
may well assume that the Y points up in the same way that lowercase y does.

However, the asymmetry of the k is ugly and I can't think of anything 
better so I'll go with the Y.

 > > By the way George, I hope you realise I still think there are
 > serious
 > > problems with the triple shafts and X shafts. It's only the
 > availability of
 > > the dual-symbol version of the notation that allows me to ignore
 > them.
 >
 > The problem that you had with single-symbol notation way back when
 > included double-shaft symbols, but you didn't mention those in the
 > above statement, so you need to explain what you mean by that.

This problem is not with the existence of single sagittal symbols between 
sharp and double-sharp or flat and double-flat, but specifically with the 
appearance of their shafts. There is a Ted Mook type objection 
(sight-reading in bad light) to the triple-shafts, which I refer to as 
2-3-confusability. We addressed that partially by agreeing not to pack the 
triple-shafts into the same width as the double-shafts, but I believe the 
problem still remains.

There is also the objection that the X-shaft up symbols look like they are 
_adding_ something to a conventional double sharp whereas they are intended 
to represent something smaller than a double sharp (or equal in the final 
case). e.g. /X looks like it ought to mean a double sharp _plus_ a 5-comma, 
whereas you have it meaning a double sharp _minus_ a 55-comma. Or looking 
at it another way, you want the X-shaft itself to mean sesqui-sharp so that 
/X means sesqui-sharp plus a 5-comma, but this is a very problematic 
redefinition of the X from double-sharp to sesqui-sharp.

There is no similar problem with the double-shaft || representing 
semi-sharp because it does not look like a complete sharp symbol, in fact 
it contains exactly half the strokes of a conventional sharp. Only when it 
attains the two straight flags as /||\ does it resemble (and have the same 
number of strokes as) a conventional sharp symbol. This is very good.

Ideally the sesqui-sharp shaft would only look like an X with the addition 
of the two straight flags. But this seems impossible to me.

 > I continue to have serious problems with the double-symbol notation
 > (especially when it results in an occasional _de facto_ triple symbol
 > whenever a double-flat is modified) which only the availability of
 > the single-symbol version allows me to ignore.

OK. I believe you suggested that the font should contain a single glyph for 
conventional double-flat which looks like two conventional flats touching 
each other (and maybe even squashed sideways a bit). I'll go along with 
that now.

 > The only problems that I see with the single-symbol version are that:
 > 1) The performance notation has a steeper learning curve;
 > 2) The ascii simulation is rather cumbersome, particularly for three-
 > shaft symbols;
 > 3) An ascii shorthand does not seem to be feasible;
 > 4) More symbols are required in a font.

Yeah. But it's still worth having. I just think the other problems I 
mentioned above might yet be ameliorated.

 > And the only problems that I have with the double-symbol notation
 > involve the performance version is actually a single problem that has
 > dual consequences:
 > 1) Lower efficiency (in contrast with the the single-symbol version,
 > in which every line segment conveys information);

Redundancy in communication is often an asset in preventing errors.

 >   a)  Less legibility, i.e., a more cluttered appearance on the
 > printed page;
 >   b)  Less clarity, i.e., in a polyphonic part or score, it is not
 > always obvious which symbols modify which notes;

I agree with those two.

 >   c)  Less intuitive, i.e., more symbols preceding a note-head often
 > symbolize a smaller amount of alteration (e.g., \!# is a smaller
 > alteration than #), and down-arrow symbols frequently appear when the
 > pitch is actually being altered upward (e.g., \!#).

I suspect that many musicians/composers have more of a 12-pseudo-nominals 
orientation, rather than 7-nominals. For example Joseph Pehrson. Such a 
person is likely to find the double-symbol notation _more_ intuitive.

 > But I would not want to abandon either version, because having both
 > available immediately puts off criticism from anyone else who might
 > have problems accepting one version or the other.

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


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Message: 6256

Date: Wed, 29 Jan 2003 14:38:49

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Of course this last-ditch effort to improve on the triple and X-shafts is 
motivated by the fact that I've agreed to draw the outline versions of 
these symbols for the font.

Another problem with the X shafts (these problems have all been mentioned 
before) is that it is unclear when looking at it, whether the note being 
modified is one aligned with the arrowhead or one aligned with the point 
where the shafts cross. Attention is unavoidably drawn to that crossing 
point, no matter how much one might know that we must look at the head, to 
be consistent with the other saggitals. This is partly intrinsic to the 
crossing itself and partly a matter of long habit from reading conventional 
double-sharp x symbols.

This kind of distraction caused by shaft features was the main reason you 
gave for rejecting a shortened middle shaft for the triple shaft symbols.

Any suggestions for better sharp and sesqui-sharp valued shafts. By shaft 
value I mean currently we have shaft values:

|   natural          1:1
||  semi-sharp     704:729
||| sharp         2048:2187
X   sesqui-sharp  2048:2187 * 704:729


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Message: 6263

Date: Thu, 30 Jan 2003 19:39:53

Subject: Re: Seven limit temperament graphs

From: Carl Lumma

>These show relationships between temperaments, with a line
>drawn between them if the bilinear form on 7-limit linear
>temperaments is zero. This means the temperaments have a
>common comma.

Cool.  These could come in handy for "transferring" (I think
that was Ivor's term) tunings mid-piece.  Have you considered:

() Labeling the lines by their commas?
(Wait, I suppose the Paul's dualzoom thing already shows this...)

() Making the lines longer in temp3da?

-C.


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Message: 6267

Date: Fri, 31 Jan 2003 05:56:58

Subject: Re: Seven limit temperament graphs

From: Carl Lumma

> > () Making the lines longer in temp3da?
> 
> Did you see temp3db?

I missed it; it needs longer lines too.  :)

-C.


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Message: 6271

Date: Fri, 31 Jan 2003 14:31:42

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > Thanks for updating the quick reference. It's great.
>
>I've updated it further, per your comments, plus a couple of
>corrections and extensive additions.  Have another look.

Good work!

To the end of the first paragraph you might add the words "They are all 
smaller than a cent."

There's typo in the paragraph immediately under
"ASCII SHORTHAND FOR SINGLE-SHAFT SYMBOLS"

"These single characters may also be used in combination
a 5'-comma accent mark"

should be:

"These single characters may also be used in combination
with a 5'-comma"
^^^^

The paragraph:
The 5' comma has traditionally been known as the "schisma", but in
the development of the sagittal notation we have used that term to
indicate the difference between two intervals that are represented
by the same symbol, i.e., one that vanishes in this notation.

I suggest:
The 5' comma has traditionally been known as the "schisma", but in
the development of the sagittal notation we have used that term to
indicate the the tiny difference between two commas that are represented
                  ^^^^                        ^^^^^^
by the same symbol, i.e., one that vanishes in this notation.

Or you could say "two commas or dieses" if you prefer.

This brings up the point that what applies "in the development" need not 
apply in the dissemination. I think there is an advantage in calling it the 
5-schisma. We could invent another term for the sub-cent ones like 
"schismina" (pron. skizmeena) literally "a small schisma".

The advantage relates to a wider consideration - how the symbols should be 
pronounced when reading them out loud. You wisely replaced all my "sharp" 
and "flat" with "up" and "down" and it is natural to want the whole name to 
be as short as possible to say. This leads me to prefer "55-comma" to "11-5 
comma". And those "prime"s (as in "five prime comma") sound silly and don't 
really tell you anything about the size.

It would be good if, when there are two notational commas available for a 
given prime number N (or combination of primes), the smaller is called the 
N-comma and the larger the N-diesis. This already occurs in many cases, so 
one can drop the "prime" for them. But in the case of N = 5 we can't do 
that so it would be good to call them 5-comma and 5-schisma.

This will also work for N = 17 and 19 although in the 17 case it would be 
better to call the small one the 17-kleisma. Incidentally the traditional 
kleisma is the 5^6-kleisma and the "septimal kleisma" is the 7:25-kleisma. 
Both of these are notated '|( so commas notated as |( (5:7) should probably 
be called kleismas too.

If we set the cutoff between a kleisma and comma at exactly half of a 
Pythagorean comma or 11.73 cents, this will work in the maximum number of 
cases.

The cutoff between schisma and kleisma doesn't matter too much for this 
purpose since no combination of primes ever has two useful commas in this 
range. So I go to Manuel's collected interval names in the file intnam.par 
that comes with Scala. Having hauled one into a spreadsheet some time ago 
and sorted it by size, I find that a 3.80 cent interval is the largest 
referred to as a schisma (33554432:33480783, septimal) and the smallest 
called a kleisma is 4.50 cents (384:385). Halfway between the 19-schisma )| 
and the 5:7-kleisma |( would be 4.57 cents, so I propose that the cutoff 
should be infinitesimally below 384:385 or at 4.50 cents.

The best cutoff between comma and diesis for this purpose would be exactly 
half a pythagorean limma or 45.11 cents. However this would omit the 
25-diesis and THE diesis (125:128) so I propose placing the cutoff 
infinitesimally below 125:128 or at 41.05 cents.

We then need an easy-to-say way to distinguish large dieses from small for 
things like the 11 and 13 dieses. This includes the 35, 5:49, 625 and 13:19 
dieses. The cutoff between these is obviously at exactly a half apotome or 
56.84 cents. Any suggestions? Is there a common suffix meaning "big" that 
we can tack on to "diesis"? I suppose we could just use "diesis" and "big 
diesis".

The upper limit for a big diesis would be 70.17 cents for our purposes.

The boundary between schisma and schismina (or whatever) would be half a 
5-schisma or 0.98 cents.

So here's the summary:
0 c
schismina
0.98 c
schisma
4.50 c
kleisma
11.73 c
comma
41.05 c
diesis
56.84 c
big diesis
70.17 c
semitones, limmas, apotome
to about 135 c

>There's now a section on order of symbol components.
>
> > I note that in the quick reference where you've given a combination
>of a
> > shorthand ASCII sagittal with a conventional sharp, you've put the
>sagittal
> > leftmost. You need a note to say that this is how it would appear
>on a
> > score (but of course the ASCII characters should never appear on a
>score),
> > but in text the sagittal should be rightmost.
> >
> > On second thoughts, since the ASCII characters never appear on a
>score,
> > wouldn't it be better to show them in text order and add a note to
>say that
> > on a score the sagittal should be leftmost (i.e. furthest from the
>letter
> > name or note head in both cases).
>
>I've addressed both of these -- see what you think.

I'm afraid I find it too complicated. I figure folks eyes will just glaze 
over. I think that when they are writing text they shouldn't have to worry 
about whether they are using it "abstractly" or not. They should just type 
the sharp or flat before the other accidental. That's how it's always been 
done on this list. This gets us down from three categories to two. Text is 
text and staff is staff ...

> > 1. A one-symbol-per-prime notation. Some may yet prefer it.
>
>They might then want shorthand characters to represent the 19 and 23
>commas, which we haven't covered.  At the very least I think that the
>19 comma should be available, since I recall that, when we started on
>this project, you thought that our notation would need to be 19-limit.
>
>Why don't we do the smallest commas this way:
>
>'|    '       5'-comma sharp 32768:32805
>.!    .       5'-comma flat
>)|    "       19-comma sharp  512:513
>)!    ;       19-comma flat
>  |(   (       5:7-comma sharp  5103:5120
>  !(   c       5:7-comma flat
>
>Since the 19-comma is about twice as large as the 5' comma, it would
>be appropriate if it were to use characters indicating an approximate
>double.  (If you think that the colon would be better than the
>semicolon for 19-comma-down, that would be okay with me.)  And the
>5:7 comma gets a better deal in the process.  This would also allow
>us to notate three more ETs with single characters:  80, 104, and 152
>(which I'm sure Paul would appreciate).

I'd like to have the 19-schisma in the single-ASCII, and if so ; and " 
would be the obvious choice (semicolon, not colon for reasons I gave 
earlier). But I really don't like using ( for 5:7-kleisma up.
1. It will get missed in text (i.e. parsed as an opening parenthesis).
2. Folks are already used to thinking of ([<{ as meaning down and )]>] as 
up. Scala uses ( for diesis down.

I thought we already agreed not to use () purely for reason 1.

As you say, we're scraping the bottom of the barrel. It can't be an 
uppercase character. In approximate keyboard order: It can't be 
`,~!|@#%^&()+-{}{}\/'.";?<>. It can't be qwtyuosdfhjxcvbnm. Already used or 
rejected for any use.

That only leaves $*_=:eripagklz.

A lowercase character shouldn't be used unless it has a descender, or 
no-ascender and is open at the bottom. That eliminates eaklz leaving ripg. 
p and g are too big to represent something that small. Cant use $ because 
it is wavy not concave. I want to reserve colon for placing between notes 
to form chords. _ is obviously down, not up. = suggests no direction and is 
utterly unlike an arrow.

That leaves *ri.

I note that k isn't a bad looking down symbol and might be paired with p 
for some use, for lack of anything else to pair it with and because p's 
obvious partner b is already taken. I also note that e and a or g and a 
might make a pair, and possibly $ and z. But none of these suit a small 
right concave flag.

r is more like |) or )|).  i looks like an inverted ! which should at least 
make it an up symbol, but I'm inclined to go with * because of its 
smallness and upwardness and because it seems better to use special 
characters rather than letters when possible.

Do we want to consider something other than c for its partner? k bears a 
vague resemblance to *, but it seems a bit too big. What do you think?

>I tried playing around with something like this a few weeks ago (when
>linear temperament notation was being discussed), but I came to the
>conclusion that it's much simpler just to use 72-ET notation for
>Miracle.  This is all so very specialized that I don't think very
>many are going to want to bother with it.

Yeah. It's pretty ugly. I guess if folks use combinations of these 
single-ASCIIs they just have to spell out what they mean by it.

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


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Message: 6272

Date: Fri, 31 Jan 2003 18:22:05

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi George,

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > This problem is not with the existence of single sagittal symbols
>between
> > sharp and double-sharp or flat and double-flat, but specifically
>with the
> > appearance of their shafts. There is a Ted Mook type objection
> > (sight-reading in bad light) to the triple-shafts, which I refer to
>as
> > 2-3-confusability. We addressed that partially by agreeing not to
>pack the
> > triple-shafts into the same width as the double-shafts, but I
>believe the
> > problem still remains.
>
>When I did some legibility testing with a few subjects last year, I
>found that the distance at which the number of shafts could be easily
>distinguished was greater than that at which wavy flags could easily
>be distinguished from concave ones.  Given that, I don't believe that
>there is a problem such as you describe.

I understood that these subjects were only involved in deciding whether 
they preferred the middle shaft shortened or not, and that the distance 
recognition tests were only conducted with yourself as subject. Is this 
correct?

Can you tell me something about your subjects and the test. How many? What 
were their musical backgrounds? Were they likely to be impartial or were 
they friends or relatives who might pick up subliminal cues as to which you 
preferred? What instructions were they given? How was the test conducted?

We both know the answers to these questions in the case of the subject Ted 
Mook. He has been involved in trying out various microtonal notations as a 
performer. He only knows me from one previous email exchange in which I 
asked him why he didn't like the tartini symbols. I expect you still have 
the email exchange relating to the test. The test was conducted by email, 
so subliminal cues would be difficult. Ted preferred the shortened middle 
shaft. But I acknowledge that a result from a single subject means almost 
nothing.

Even if we assume your results are valid above, your argument from them is 
a non-sequitur. It might only mean that we have a bigger problem with 
distinguishing wavy flags than we do with distinguishing triple shafts. But 
even this is not the case, because the consequences of mistaking a wavy 
flag for a straight one are not very serious musically (about 15 cents), 
while mistaking a triple shaft for a double is very serious (about 100 cents).

A useful test would provide random sagittal symbols, in dim light at a 
large enough distance for a significant error rate and ask the subject 
whether she thinks they are 1, 2, 3 or X shaft and whether they are up or 
down (having first allowed her to examine them closely to learn these 
categories). Then the same test would be repeated with the shortened middle 
shaft (again being allowed to examine them closely). Subjects would 
alternate according to which set they did first.

At best I only expect the shortened middle shaft to provide a minor 
improvement. And I have another suggestion (later) so I'll drop it.

> > There is also the objection that the X-shaft up symbols look like
>they are
> > _adding_ something to a conventional double sharp whereas they are
>intended
> > to represent something smaller than a double sharp (or equal in the
>final
> > case). e.g. /X looks like it ought to mean a double sharp _plus_ a
>5-comma,
> > whereas you have it meaning a double sharp _minus_ a 55-comma. Or
>looking
> > at it another way, you want the X-shaft itself to mean sesqui-sharp
>so that
> > /X means sesqui-sharp plus a 5-comma, but this is a very
>problematic
> > redefinition of the X from double-sharp to sesqui-sharp.
>
>Then we need to campaign for the elimination of these antiquated
>sharp and flat symbols (and especially their doubles) with all
>deliberate speed so as to eliminate the possibility of any confusion
>as soon as possible!  ;-)

Hee hee. And pigs might fly. :-)

> > There is no similar problem with the double-shaft || representing
> > semi-sharp because it does not look like a complete sharp symbol,
>in fact
> > it contains exactly half the strokes of a conventional sharp. Only
>when it
> > attains the two straight flags as /||\ does it resemble (and have
>the same
> > number of strokes as) a conventional sharp symbol. This is very
>good.
>
>And the conventional double-sharp symbol has half as many lines as
>(and is a lot smaller than) a sharp symbol.  This is not very good.
>More justification for our campaign!  (:You are with me on this,
>aren't you?:)

Yes. I agree it would be nice if the sagittals replaced the existing sharp 
and flat symbols but it isn't going to happen overnight. So you can't just 
ignore the problems that will occur during the (possibly decades-long) 
transition. In fact if you set up the sagittal notation for a head-on clash 
like the one I've described above, it is much less likely that a transition 
will ever be made. This is as much a political decision as it is an 
aesthetic or logical one (rather like much ancient-greek thought ;-).

> > Ideally the sesqui-sharp shaft would only look like an X with the
>addition
> > of the two straight flags. But this seems impossible to me.
>
>Hey, you're just nitpicking, now.

I don't think so. I get the feeling, from this and earlier exchanges on 
this topic, that the triple and X shafts are somewhat sacred cows to you. 
But that's the wrong pantheon, Hermes. :-)

>Anyone can see that it's logical
>enough once they are told that the shafts by themselves count for n-1
>semi-apotomes.

Huh? To me, your three-semi-apotome symbol has always had two shafts, not four.

The general n-1 idea is fine because conceptually a line has zero width and 
we're really looking at the overall width of the tail to roughly correspond 
to its value. I just thought the reason for the X was simply that four 
shafts is just too wide (and 3-4 confusability is much worse than 2-3) so 
we just find something completely different. Never mind how many strokes 
it's got.

>   It's the old symbol that makes little sense.

Whether the old symbols make sense or not is all but irrelevant given their 
ubiquitousness. But one way in which they do "make sense" is that they are 
very very different from each other and therefore very difficult to confuse.

> > ... I just think the other problems I
> > mentioned above might yet be ameliorated.
>
>I would say fix it only if we find that it doesn't work.

I find that X shafts don't work.

>   We're going
>to need to make the staves a little larger than usual in order to be
>able to read the flags.

Sure.

>   Oh, speaking of flags, I forgot to mention
>that single symbol notation results in larger (and thus more-
>readable) flags whenever a symbol has more than one shaft.

Sure. That helps with the 2-3 confusability but has no bearing on the two 
problems I've cited with the X shaft.
1. Clash of meanings with existing double-shaft.
2. Distraction caused by the crossing point.

>(Do we
>have to go through all this again?  I think we are still agreeing to
>disagree.  But now, after having written everything else in this
>message, I've come back to this point, because I think I've figured
>out why you've brought all of this up.  You're testing me to see if I
>still feel the same way about all of this, because you don't want to
>do a lot of work on the font, only to have me change my mind.)

It's the font work, yes - as I already said in a subsequent post. But I'm 
not testing you to see if you still feel the same way. I'm trying to 
brutally force you to have the rational discusion that I gave up on 
previously when I got the "sacred cow" feeling. Or maybe its not so much a 
sacred cow, but it's just that you have been using them yourself for a long 
time and would find in very inconvenient to change. But (and I'm always 
saying this to someone in these standardisation efforts) you are only one 
person. What is your inconvenience compared to that of all those who may 
come after you?

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
> > Of course this last-ditch effort to improve on the triple and X-
>shafts is
> > motivated by the fact that I've agreed to draw the outline versions
>of
> > these symbols for the font.
> >
> > Another problem with the X shafts (these problems have all been
>mentioned
> > before) is that it is unclear when looking at it, whether the note
>being
> > modified is one aligned with the arrowhead or one aligned with the
>point
> > where the shafts cross. Attention is unavoidably drawn to that
>crossing
> > point, no matter how much one might know that we must look at the
>head, to
> > be consistent with the other saggitals. This is partly intrinsic to
>the
> > crossing itself and partly a matter of long habit from reading
>conventional
> > double-sharp x symbols.
>
>This is alleviated by the fact that X symbols will not occur very
>often, so that:
>1) The alleged problem should not occur very often; and
>2) The habit of getting the alignment from the flags should be well
>established from the much more frequent occurrence of the other
>symbols.

I hope so. But if you could eliminate this distraction without significant 
bad side-effects, why wouldn't you?

I earlier wrote that it would be good if the sesqui-apotome-valued shaft 
only came to look like an X when topped by two straight shafts, i.e when it 
becomes a double-apotome symbol.

It seems from your draft XH article, that you are in favour of resemblances 
to existing symbols. You describe the correspondences between sagittal and 
four other systems that are otherwise unrelated to each other: the 
quartertone arrows, the Bosanquet slashes, the Tartini symbols and the 
Couper symbols (and hence the conventional symbols).

You specifically claim that a resemblance to the conventional double-sharp 
symbol is an advantage, which indeed it might be, if there was only the 
double-apotome symbol itself, and not all the other X-shaft symbols 
_smaller_ than it. A resemblance where the same feature or sub-symbol means 
different things is clearly not an advantage.

So I now have a proposal to replace the X shaft. It solves _both_ of the 
problems I mentioned. It has no features to distract from the arrowhead and 
it doesn't look like an X until it gets to be a double-apotome (but even 
then the resemblance is a bit strained).

Take a conventional double-sharp X symbol. Lose the square blobs on the 
ends of the strokes. Grab hold of the two top ends and bend them apart. 
Keep bending them down past horizontal until they become a pair of upward 
pointing straight flags on top of a V shaft (inverted V in this case).

> > This kind of distraction caused by shaft features was the main
>reason you
> > gave for rejecting a shortened middle shaft for the triple shaft
>symbols.
>
>This was something that I found much more distracting than an X.

It seems very odd to me that anyone would find the convergence of a pair of 
lines that don't actually appear, to be more distracting than ones that do.

>If
>anything, this made it more difficult to distinguish the ||| from the
>X, since the symbols have the same width and both have two lines
>sticking out one end.

Yes I expect it would increase confusability between 3 and X, but I figure 
there's a lot of room to play with there, and not much between 2 and 3. 
Anyway, we can probably forget about short middle shafts.

> > Any suggestions for better sharp and sesqui-sharp valued shafts? By
>shaft
> > value I mean currently we have shaft values:
> >
> > |   natural          1:1
> > ||  semi-sharp     704:729
> > ||| sharp         2048:2187
> > X   sesqui-sharp  2048:2187 * 704:729
>
>None of the symbols as you show them above will ever occur in any
>piece of music, ever.

No. That's why I refer to "shaft values", not symbol values, in the same 
way we talk about "flag values". Of course the bare single shaft may occur 
with a 5-schisma accent but that isn't relevant to this discussion.

>These were originally conceived as:
>  /|\   semi-sharp
>  /||\  sharp
>  /|||\ sesqui-sharp
>  /X\   double-sharp

Yes I understand that.

>inasmuch as you never see the shaft(s) without any flags, and the
>symbols are grouped (in one's mind) by rounding upward to the half-
>apotome, approximately.

In your mind perhaps, but not mine. In my mind, and I suspect many others, 
the flag values are _added_to_ the shaft value, approximately. I believe 
you have talked in those terms extensively yourself, during our long 
cooperative effort.

>The values you give above apply strictly only to the | and |||
>cases.  The || and X symbols are defined as apotome-complements or
>(double-apotome complements) of single-shaft symbols.  For example,
>||) is defined as 2048:2187 / 63:64, not 704:729 * 63:64.  So it is
>not very meaningful to give exact values for || and X, because their
>symbols are not defined that way.

Yes. Good point. Consider them deleted. Although we could argue whether the 
approximation (or offset) is in the flags or in the shafts, or partly in 
both, this is a meaningless distinction and is not relevant to the current 
discussion.

Here are some more alternative shaft ideas.

I can see that one objection to the V shaft might be that it is too narrow 
to have a value as large as a sesqui-apotome. So I propose moving the two 
shafts apart until they are as far apart at the head, as are the two 
parallel shafts of the existing double-shaft symbols.

This opens up the possibility of using the actual V shaft in place of the 
triple shaft.

There is a certain order to that. The tail area increases steadily with
shaft value.
|    0/2 apotome
||   1/2 apotome
\/   2/2 apotome  (note these are shafts not straight flags)
\ /  3/2 apotome

But note that the > 2/2 apotome symbols can have the same size flags as 
the > 0/2 apotome symbols, and the > 3/2 the same size as the > 1/2. This 
would make my job easier with the outline font, but of course I won't use 
that as an argument for accepting it.

These shafts could be represented in ASCII as V A W M.

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


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Message: 6273

Date: Fri, 31 Jan 2003 09:36:07

Subject: Re: A common notation for JI and ETs

From: Graham Breed

David C Keenan wrote:

> This brings up the point that what applies "in the development" need not 
> apply in the dissemination. I think there is an advantage in calling it the 
> 5-schisma. We could invent another term for the sub-cent ones like 
> "schismina" (pron. skizmeena) literally "a small schisma".

I'd certainly expect the 5-comma to be 81:80.  So if you're talking 
about schismas, it'd be much more straightforward if you called them 
schismas.


                   Graham


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Message: 6274

Date: Sat, 01 Feb 2003 09:16:23

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Some corrections.

I wrote:
"Sure. That helps with the 2-3 confusability but has no bearing on the two 
problems I've cited with the X shaft.
1. Clash of meanings with existing double-shaft.
2. Distraction caused by the crossing point."

That should have been

1. Clash of meanings with existing double-sharp.
                                           ^^^^^
I wrote:
"There is a certain order to that. The tail area increases steadily with
shaft value.
|    0/2 apotome
||   1/2 apotome
\/   2/2 apotome  (note these are shafts not straight flags)
\ /  3/2 apotome"

Since it would be natural to assume that I was giving the tails for up 
symbols, that should have been

|    0/2 apotome
||   1/2 apotome
/\   2/2 apotome  (note these are shafts not straight flags)
/ \  3/2 apotome

Hi Graham,

Good to hear from you in this thread. I assume you are saying that you 
greatly prefer the terms 5-comma and 5-schisma to 5-comma and 5'-comma for 
80:81 and 32768:32805 respectively. If so, good. It seems we're all agreed 
on that now.

I also assume that you were not commenting on the suggested distinction 
between schismas and schisminas. Is that correct?

I think that these should be distinguished purely on the basis of size, not 
whether they vanish in sagittal. They would be more generally useful that 
way and we would still have the advantage, when talking about the 
development of sagittal, that only schisminas vanish.


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