Tuning-Math Digests messages 6275 - 6299

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Message: 6275

Date: Sun, 02 Feb 2003 08:52:33

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Another correction.

I wrote:
"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)."

That should have been:

... while mistaking a triple shaft for a double is serious (about 50 cents).
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6276

Date: Sun, 02 Feb 2003 14:22:14

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Some suggestions for a term that means a diesis larger than a half-apotome.

biesis    (contraction of "big diesis")
diesoma   (ending somewhat like "comma",
            also -oma = growth, unfortunately an unnatural one)
oediesis  (swollen diesis)
ediasis   (same as oediasis but with modern spelling)

Apparently the "di" in "diesis" doesn't mean two, but is "dia" meaning 
"through" or "across". And "esis" means something like "into". This is 
gleaned from the Shorter Oxford.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6277

Date: Mon, 03 Feb 2003 19:56:48

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Now that we have all those single ASCII characters representing the most 
common single-shaft sagittals, it suggests we might use that for the 
keyboard mapping of the font. Shift could supply the apotome complement of 
each symbol and Ctrl could add an apotome to those. The less common symbols 
would be mapped in some other way and be available when Alt was pressed.

What do you think?
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6281

Date: Tue, 4 Feb 2003 16:46:31

Subject: Re: A common notation for JI and ETs

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

George Secor wrote:
>I wanted a notation that would be the best one possible -- one that
>would, in effect, be good enough to *replace* other notations that
>also use 7 nominals, so there would be no need for competing systems.

I see your and Dave's notation as a complementary approach to the 
Scala notation. You are using lots of symbols and achieve a very 
accurate representation. Scala on the other hand is very economical
with symbols, less accurate, but much more quickly to learn in my
opinion. There's value in both, no problem with a bit of competition.

Manuel


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Message: 6282

Date: Tue, 04 Feb 2003 04:46:03

Subject: Re: heuristic and straightness

From: Carl Lumma

> carl, here's an old message where i explained the error
> heuristic:
> 
> Yahoo groups: /tuning-math/message/1437 *

Great, thanks!  I hadn't seen this, as "heuristic" doesn't
appear in it.
 
> and you can see that gene, in his reply, was the one who
> actually suggested the word "heuristic" in connection
> with this . . .

I do see that... you were already using the term for the
complexity heuristic at that time, right?


I understand everything but a few details...

>log(n) +  log(d) (hence approx. proportional to log(d)

Is it any more proportional to log(d) than log(n) in this
case?  Since n~=d?

>w=log(n/d)

Got that.

>w~=n/d-1

How do you get this from that?

>w~=(n-d)/d

Ditto.

-Carl


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Message: 6283

Date: Tue, 4 Feb 2003 18:13:37

Subject: Re: A common notation for JI and ETs

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

Gene wrote:
>Any name for 100/99 should be part of a pair with 99/98. 
>This is a problem with "small unidecimal comma" for 99/98; 
>if 99/98 is "small", what is 100/99--smaller?

Then we have a pair now, Ptolemy's comma and small undecimal comma.
100/99 could also be called "2nd small undecimal comma" but
George's idea is better.

Manuel


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Message: 6284

Date: Tue, 04 Feb 2003 09:33:03

Subject: Re: heuristic and straightness

From: Graham Breed

Carl Lumma  wrote:

>>log(n) +  log(d) (hence approx. proportional to log(d
> 
> 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.  And in that 
case it would have been easier to go straight to log(n*d).

>>w=log(n/d)
> 
> 
> Got that.
> 
> 
>>w~=n/d-1
> 
> 
> How do you get this from that?

It's the first order approximaton where n/d ~= 1.  See (8) in

Natural Logarithm -- from MathWorld *

or check on your calculator.

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

That's subtracting fractions.  Did you do fractions at school?

4/3 - 1 = 4/3 - 3/3 = (4-3)/3 = 1/3

5/4 - 1 = 5/4 - 4/4 = (5-4)/4 = 1/4

n/d - 1 = n/d - d/d = (n-d)/d


                          Graham


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Message: 6285

Date: Tue, 04 Feb 2003 19:10:35

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.

>>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?
>
>log2(x) = log(x)/log(2).  In this case we're only estimating
>the complexity, so the common factor if 1/log(2) isn't important.

Ok, but I still don't get how the "Mercator series" shown in (8)
dictates the rules for this approximation.

-Carl


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Message: 6286

Date: Tue, 04 Feb 2003 10:08:37

Subject: Re: heuristic and straightness

From: Carl Lumma

>>>log(n) +  log(d) (hence approx. proportional to log(d
>> 
>> 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))?

>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))?

>>>w=log(n/d)
// 
>>>w~=n/d-1
>>
>>
>>How do you get this from that?

Oh, (n/d)-1, not n/(d-1).

>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?

>>>w~=(n-d)/d
>> 
>>Ditto.
> 
>That's subtracting fractions.  Did you do fractions at school?

Yes; I was still seeing n/(d-1).

-Carl


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Message: 6287

Date: Tue, 04 Feb 2003 21:04:13

Subject: Re: heuristic and straightness

From: Graham Breed

Carl Lumma  wrote:

> 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.  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

If we really wanted the logarithm to base 2, that'd be

log2(n/d) = ln(n/d)/ln(2) =~ (n/d - 1)/ln(2)

or

log2(n/d) ~ n/d - 1

where ~ is "roughly proportional to" which is all we need to know.  And 
the same's true whatever base logarithm you use.



                     Graham


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Message: 6288

Date: Tue, 04 Feb 2003 10:26:03

Subject: Re: heuristic and straightness

From: Graham Breed

Carl Lumma  wrote:

> So it could just as well be (n-d)/(d*log(n))?

This is approximately log(n/d)/log(n)

The order will be reversed.  If you can calculate it, see if it holds 
the ordering.  With numerators it's easy as they're already in the database.

>>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)

The intervals start out in prime factor notation.  So log(n*d) can be 
calculated as log(2)*abs(x_1) + log(3)*abs(x_2) + log(5)*abs(x_3) + 
log(7)*abs(x_4) where x_i is the ith prime component of n/d.

> 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?

log2(x) = log(x)/log(2).  In this case we're only estimating the 
complexity, so the common factor if 1/log(2) isn't important.

For estimating comma sizes using mental arithmetic, remembering that 
1200/log(2) is approximately 1730 comes in handy.  1730/80 = 21.625, so 
a syntonic commma's around 22 cents.


                               Graham


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Message: 6296

Date: Wed, 05 Feb 2003 01:06:01

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:
>As I indicated in another message, this sounds like a good name,
>although I thought that we should be more specific about a couple of
>things.  After a more careful reading of your message, I see that
>you've done that by proposing a boundary, which sounds okay (unless
>some rational interval ~1.0 cent could be specified).  The only
>question that remains is whether others would be willing to exclude
>the term "schisma" as applying to intervals of less than ~1 cent, and
>if not, then we would have to make "schismina" a subclass
>of "schisma".

Since the term schismina is not required in the _use_ of sagittal, but only 
in describing the theory behind it, I don't think it matters much whether 
others use the term at all, or whether they accept a boundary at 
sqrt(32805/32768), or whether they consider schismina a subclass of 
schisma, or indeed whether the _only_ thing they consider as a schisma is 
32768:32805 itself, which is, I think, where things stood before we 
started. The term may never be used anywhere outside of the XH paper and 
yet I think we should leave the possibility open that it may be used 
elsewhere and therefore not define it purely as something that vanishes in 
sagittal.

>Yes, the word "prime" leaves something to be desired, since it is
>also liable to be confused with a "primary" (as opposed to secondary)
>comma role for a symbol.

Indeed.

> > 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.
>
>That sounds like it'll work.
>
> > 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.
>
>That boundary comes almost exactly between two interpretations of ~)
>|, -- the 17:19 comma (~11.352c) and the 17+19 comma (~12.108c)

Not almost exactly, but exactly. Good point. But only the smaller of these 
is of interest. The larger has too high a slope and too high a power of 3.

There are many combinations of primes whose useful "commas" come in pairs 
whose absolute values sum to a Pythagorean comma. Here's an even worse 
example. An 11:35 above C can be described as either a Pythagorean G# + 
11.82 cents or a Pythagorean Ab - 11.64 cents. These are so close it 
doesn't matter if you get the wrong one. So they could both be called the 
11:35-kleismas.

>or a
>possible alternate 5:17 comma (135:136, ~12.777c, although I see
>that .~|( would be much better for this last one), so we might want
>to adjust it somewhat (see below).

Yes .~|( would notate it exactly. And since the other one is 36.24 cents I 
agree it would be good to call them kleisma and comma.

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 the idea of a kleisma-comma
>boundary is good.  Recall that I had something to say not too long
>ago (msg. #5202, 16 Dec 2002) about boundaries.  I separated the
>eight flags into two groups, between which your proposed boundary
>falls:
>   small flags:  '|  )|  |(  ~|  are the schismas and kleismas, and
>   large flags:  |~  /|  |)  |\  (|  are the commas.

Right!

>A diesis would be the sum of two large flags, i.e., two commas, but a
>kleisma plus a comma would still be a comma.  (The exception would
>be /|~, ~38.051c, but we aren't using it in the notation.)  So the
>largest two-flag comma (i.e., comma + kleisma) would be ~|\,
>~40.496c, and the smallest remaining two-flag diesis would be //|,
>~43.013c.  This is consistent with your proposed upper boundary for a
>comma, < 125:128, ~41.058c, so I can agree to that.  This isn't
>actually the boundary that I suggested in that message, which was
>anything larger than the 5:11 comma (~38.906c), which would make ~|\
>a diesis, even though it is the sum of a kleisma and a comma (which I
>didn't notice).  But this would give us the convenience of
>distinguishing between a 23 comma (729:736, ~16.544c) and a 23 diesis
>(16384:16767, ~40.004c), even if we shifted the upper boundary for a
>comma to anything infinitesimally smaller than 16384:16767, i.e., for
>all practical purposes 40 cents.

Yes. I'll go with a boundary at 16384:16767 - 1/oo so that so we have a 
23-comma and a 23-diesis.

It's a pity the thing about only two large commas making a diesis doesn't 
quite work, but I don't think that's important.

>Can we justify anything this small as a diesis?  Well, yes:1deg31
>(~38.710c) has been called a diesis.  In fact it's below all of the
>comma-diesis boundaries that we've proposed, but it's a tempered
>interval, so I don't think that we should let that bother us, since
>the just intervals (or dieses) that it approximates are above the
>boundary.

I agree this is a red herring.

>  So the 40-cent boundary gets my vote.

Mine too. It does cause 11:19 to have two dieses 40.33 49.89 (and it has an 
ediesis 63.79) but 11:19 is much further down the popularity list and there 
are lots of other ratios that have two dieses and an ediasis or two.

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.

>Now for the kleisma-comma boundary.  Let me quote from that earlier
>message that I mentioned above (in which I refer to a kleisma as a
>small comma):
>
><< Another basis for establishing a boundary between large and small
>commas (which agrees with this) goes back to the original definition
>of comma: the difference in size between the two largest steps in a
>diatonic tetrachord.  About the smallest that these steps can get is
>in Ptolemy's diatonic hemiolon, where they are 9:10 and 10:11, with a
>comma of 99:100 (~17.399 cents).  The next smallest superparticular
>pair are 11:12 and 10:11, making a lesser comma of 120:121 (~14.367
>cents, which is not only significantly smaller than 1deg72 (~16.667
>cents), but also closer in size to 1deg94 (~12.766 cents), in which
>system both the 5 and 7 commas are 2deg (and 120:121 is only slightly
>more than one-half the size of a 7 comma.)  So I think this is
>getting a bit small to be considered a comma in the original sense.
>What we really need is a separate name for commas smaller than
>~1deg72, and I don't think "kleisma" fills the bill. >>

Some things in the kleisma size range have been called semicommas.

>I made this last remark about the term "kleisma", because I had the
>impression that the upper limit for a kleisma should probably be
>smaller, but perhaps I was mistaken.  (And by the way, I thought that
>99:100 might be a good interval to name "Ptolemy's comma", since
>Pythagoras, Didymus, and now Archytas each have one.  9:10 and 10:11
>is also the largest pair of superparticular ratios that are the same
>number of degrees in 41-ET -- hence Ptolemy's comma vanishes in 41
>just as Didymus' comma does in 19, 31, and meantone.  But I digress.)
>
>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? Why must they even be simple ratios?

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.

>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 I can probably live with that. Or would you rather 
have two 7:13 commas?

>If not, then this would set both the upper and lower boundaries for
>the term "comma" based on both historical considerations and prime-
>number-comma size groupings.

I think I missed something there. What upper limit for a comma do you get 
from historical considerations?

>It would also be good to have input from others regarding what the
>upper size limit for a kleisma should be.

Sure.

>Merely to state that the
>term has not previously been applied to anything as large as 14 cents
>would probably not be enough to disqualify its use --

No. I wouldn't try to argue that.

>  I believe that
>it would be necessary to demonstrate some specific reason to insist
>on that, just as I have given a reason for the lower limit for a more
>specific usage of the term "comma" such as we require.

I think it's fine for us to just define it "for our purposes" and let 
others worry about whether they want to also adopt it for their own purposes.

> > 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.
>
>I already addressed this (above).

There are many ratios with pairs of commas that add up to a limma. In most 
cases these are not very close to the half limma so the 40 cent boundary 
between comma and diesis serves to separate them. Here are the most popular 
ones that don't get separated:

       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. There's 
a similar problem with the apotome complements of these with the new 
boundary required near 68.57 cents. So we would end up needing four 
categories of what used to be just dieses, except that we can keep it down 
to 3 by refusing to notate anything bigger than 68.57 cents (we've not 
wanted to so far.

>But I wonder whether we should put the upper limit on a diesis at
>half an apotome (~56.843c) and use another term for anything larger.

Sure. e.g. diesis below, ediasis above. Again I think we can afford to be 
vague about whether edieses are really a subset of dieses, but "for our 
purposes" we should consider them disjoint.

>   My reason for this is that by the time you reach ~63 cents (27:28,
>1deg19), the interval has a melodic effect much more like a small
>semitone (and a very effective one at that) than a quartertone (or
>diesis).  By contrast, the single degree of 22-ET (~54.545c) can
>function either as a very small diatonic semitone *or* as a
>quartertone (i.e., 11 diesis), so I would consider an interval of
>this size to be at or near the borderline.  If we want a rational
>interval, then the upper limit for a diesis could be the 5:49 diesis
>(392:405, ~56.482c) that I proposed to notate the hemififth family of
>temperaments.  (We would still need to settle whether '(/| would be
>its rational symbol, as well as (/| for the 49 diesis.  I am inclined
>to go with it, if only because of its accuracy.)

OK. Lets drop |)) completely, in favour of (/|.

But I see no need for a rational boundary. sqrt(2187/2048) seems ideal.

> > The upper limit for a big diesis would be 70.17 cents for our
>purposes.
>
>I wouldn't put any sort of boundary there for whatever we might call
>this interval class, and we really don't need one there, since there
>will be no larger class of single-shaft symbols from which to
>distinguish this one.

Well no, it's just the boundary of what we are willing to notate with two 
flags and a single shaft. As I mentioned before, 70.17 cents is the most 
that '((| could possibly represent, but I'm happy to stay below an apotome 
minus a half limma, 68.57 cents so we never have more than one ediasis for 
a ratio.

>   I consider the ideal melodic and most
>harmonically dissonant "semitone" somewhere in the range of 63-78
>cents.  This is actually what would more accurately be called a third-
>tone (1 degree of 17, 18, or 19), the sort of interval that's
>melodically very effective in the enharmonic genus, which is what a
>label for this interval range might suggest.  ("Limma" won't do;
>that's for the chromatic genus).  Or perhaps a prefix or suffix to
>modify the word diesis, as was done to get schismina. Any ideas?

Done.

> > 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.
>
>I didn't agree not to use ( on account of reason 1, but only
>because ) was not a suitable opposite.  If ( were used, it would
>always be as the rightmost character of a symbol, in which position
>it would never be an opening parenthesis, whereas an opening
>parenthesis would always be leftmost (since it is always preceded by
>a space).  This is similar to why a period used as the 5' comma ascii
>symbol would never be confused with a period ending a sentence.

Alright. I can agree with that argument.

>As for reason 2, I don't think that using ( in someone else's ascii
>symbol system is a good enough reason to discard something that would
>work so well in ours.  (More about this below.)

Weeell, it's not just that it's used in someone elses system. It's that 
everyone, whenever they want to use one of the brackets (){}[]<> as an 
accidental just naturally takes ({[< as down symbols and )}]> as up 
symbols. It's similar to the reason why / is up and \ is down. Because we 
read from left to right, the left parentheses are taken as arrows pointing 
to the left which is conventionally the negative direction. This would be 
the case even if no one had actually used them before.

> > 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
> > `,~!|@#%^&()+-{}{}\/'.";?<>.
>
>Hey, watch your language!  ;-)

Sorry. :-)

> > ... 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.
> > ...
>I still think that ( and c are best.  It wasn't my intention to have
>a notation that should indefinitely *coexist* with other notations --
>I wanted a notation that would be the best one possible -- one that
>would, in effect, be good enough to *replace* other notations that
>also use 7 nominals, so there would be no need for competing systems.

Dream on.

>I see no particular reason why ( and ) should have been chosen to
>represent a diesis in Scala,

It's because they look most like the symbols in Rapoport's paper, but as I 
say it's not that they are used in Scala that is important but just that 
they are so obviously paired in people's minds and it's so obvious what 
directions they should mean.

>  but we have a very good reason to use (
>and c for the 5:7 comma rather than something else from the bottom of
>the barrel that everybody will have a much harder time remembering.

I'm sorry. I just think that they will have a hard time remembering 
that  "(" is up, not down, and that ")" is not its partner.

>Supposing that we're successful in getting a lot of others to adopt
>our notation, we'd later regret not making the best choice from the
>start (and having to justify a change -- over complaints and
>objections, such as, why didn't we do it right the first time?).  And
>supposing that hardly anybody uses our notation, then what does it
>matter what we chose?

If we can't agree on this, I'd prefer to roll back to where we used " and ; 
for the 5:7-kleisma and had no symbols for the 19-schisma.

But here's another attempt at the 5:7-kleisma without using " or ; or *. 
How about k for down and p for up?

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


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Message: 6298

Date: Wed, 05 Feb 2003 01:37:46

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 03:37 AM 4/02/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
><wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote:
>
> > way and we would still have the advantage, when talking about the
> > development of sagittal, that only schisminas vanish.
>
>hmm . . . i think we've been over this before, but for any schismina,
>no matter how tiny, there'll be some excellent temperament where it
>doesn't vanish. does this matter?

No. I believe it doesn't matter. Take 2400:2401 (the 5^2:7^4 schismina) 
which, when untempered is only 0.72 cents and so cannot itself be notated 
in sagittal. It may correspond to a step of some temperament which needs to 
be notated, however unless the step really is that small it is most likely 
that the step will also correspond to other commas which are larger when 
untempered and which therefore have saggital symbols.

But we'd appreciate it if you'd find us some tunings that you think may 
cause difficulties for sagittal.

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


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