> 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~=(n-d)/d
Ditto.
-Carl
Message: 6283 - Contents - Hide Contents
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
Message: 6284 - Contents - Hide Contents
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 * [with cont.]
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
Message: 6285 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 19:10:35
Subject: Re: heuristic and straightness
From: Carl Lumma
>>> >nd 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
Message: 6286 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 10:08:37
Subject: Re: heuristic and straightness
From: Carl Lumma
>>> >og(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).
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
Message: 6287 - Contents - Hide Contents
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
Message: 6288 - Contents - Hide Contents
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
Message: 6289 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 21:18:35
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> 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?
no, gene introduced the word "heuristic".
> 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?
no.
> Since n~=d?
yes.
i prefer log(odd limit) over either log(n) or log(d).
>> w=log(n/d)
>
> Got that.
>
>> w~=n/d-1
>
> How do you get this from that?
standard taylor series approximation for log . . . if x is close to
1, then log(x) is close to x-1 (since the derivative of log(x) near
x=1 is 1/1 = 1).
>> w~=(n-d)/d
>
> Ditto.
arithmetic. n/d - 1 = n/d - d/d = (n-d)/d.
Message: 6290 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 10:31:51
Subject: Re: A common notation for JI and ETs
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:
(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.
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?
Message: 6291 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 21:23:11
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> 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.
>
> So it could just as well be (n-d)/(d*log(n))?
a very different sorting. that would be heuristic error, not
heuristic complexity. of course it's a very different sorting, since
knowing log(n) or log(d) tells you nothing about (n-d).
>> 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))?
meaning the sorting by log(n*d) would be virtually identical to the
sorting by log(n) or by log(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?
i meant ln. i always use matlab, in which "log" means ln.
Message: 6292 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 21:28:26
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>>> And in that case it would have been easier to go straight
>>>> to log(n*d).
>>>
>>> Straight to where (do you see log(n*d))?
>>
>> log(n*d) = log(n) + log(d)
>
> Of course... so we're coming from log(n*d), not going to it.
what do you mean, we're coming from log(n*d)??
Message: 6293 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 22:37:59
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
>
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
>> > --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>> wrote:
> 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.
Okay, I just wanted you to clarify this so that we would both be
using the term the same way. The 1/2-of-32768:32805 upper limit for
a schismina should be okay (unless we find something else).
> ...
> There are two slightly more popular pairs that would benefit from a
higher
> kleisma-comma boundary.
>
> N kleisma comma
> 245 14.19 37.65
> 7:13 14.61 38.07
>
> I note that neither of us is willing to bring the comma diesis
boundary
> down below 38.07 or 37.65 cents.
But I'm willing to consider it if there's a good reason for it.
> ...
Otherwise:
>> ... 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.
Okay. (Somebody, *please* suggest something better; ediasis sounds
too much like a disease.)
>> Now for the kleisma-comma boundary. ...
>
> Some things in the kleisma size range have been called semicommas.
I just think that "kleisma" sounds better.
>> ...
>> The point here is that I thought that the comma (120:121, ~14.367c)
>> between the next smaller pair of superparticular ratios (10:11 and
>> 11:12) should be smaller than the lower size limit for a comma. If
>> they were used as the two ("whole") tones in a tetrachord, their
sum
>> would be 5:6, which would leave 9:10 as the remaining interval
>> (or "semitone") of the tetrachord. But to have a "semitone" in a
>> tetrachord that is larger than either of the "whole" tones is
absurd,
>> hence a practical basis for a boundary.
>
> I find this argument interesting but not convincing. Why must the
whole
> tones be superparticular?
They don't have to, but since there are actual examples of ancient
Greek tetrachords with diatonic steps of 7:8 with 8:9 (Archytas), of
8:9 with 9:10 (Didymus), and of 9:10 with 10:11 (Ptolemy's hemiolon),
these being all of the possible cases, I found that it was possible
to draw a conclusion from them.
Interestingly, the next larger pair, 6:7 with 7:8 -- difference of
48:49, ~35.697c -- adds up to an exact 3:4, leaving a semitone that
vanishes. Their difference is still a bit smaller than we were
considering for the comma-diesis boundary, which makes me wonder if
40 cents is still too large.
> Why must they even be simple ratios?
Because a comma, by definition, is the difference between two
rational intervals similar in size.
But let's get back to the kleisma-comma boundary discussion.
> Anyway, some very small intervals have been called commas for a
long time.
> e.g. We have Mercator's comma at about 3.6 cents and Wuerschmidt's
comma at
> about 11.4 cents. These are from Scala's intnam.par.
These are just examples that the term can have a broad or generic
usage in addition to the more specific definition that we're seeking.
>> You want the boundary to be somewhere between what we have been
>> calling the 17 comma (~8.7c) and 17' comma (~14.730c). To
>> accommodate both of these requirements, we could put the lower
>> boundary for a comma at infinitesimally above 120:121, >14.37c.
>> Would this be too large an upper limit for a "kleisma?" If so,
why?
>
> No I can't really argue that, although it is getting close to
double the
> size of _the_ kleisma. I now want to put the boundary even a little
higher
> than you suggest, at just above 28431:28672 (or 14.614 c) so we
have a 7:13
> kleisma and a 7:13 comma (38.07 c) as mentioned above.
>
> This does mean we have the 17-comma and the 7:13-kleisma being
notated with
> the same symbol ~|(
But it doesn't make much sense, though.
> but I can probably live with that. Or would you rather
> have two 7:13 commas?
This might be a good reason to make the comma-diesis boundary
somewhere around 37 to 38 cents. This would then put 1deg31 in the
diesis range (at the lower end) -- 1deg31 is also functions as a 7-
comma, but I think its dual use demonstrates that it's appropriate to
have the boundary somewhere around this size.
If 1664:1701 (~38.073c) is the 7:13 diesis, then 1377:1408 (~38.543c)
and 44:45 (~38.906c) would become the 11:17 and 5:11 dieses. Do you
know of any potential problems with these designations?
>> 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?
I didn't get it directly -- it was inferred as being coincident with
the lower limit for a diesis.
>> It would also be good to have input from others regarding what the
>> upper size limit for a kleisma should be.
>
> Sure.
IF ANYONE OUT THERE HAS ANY INPUT ABOUT THIS (upper limit of ~14.5
cents for a kleisma), PLEASE SAY SOMETHING SOON!
>> 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.
If nobody else says anything, then that's what's going to happen. I
can't see making another boundary between semicomma and kleisma
without a good reason.
>>> 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.
I'll have to look these over to see if there's really any musical
need some of these. For example, the 46.39-cent 5:13 diesis would
notate 10:13 as an interval of a third, but I can't imagine that
anyone would want a third this large in a diatonic or heptatonic
scale very often.
> ...
>> 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.
Okay.
>> ... (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 (/|.
Fine!
> But I see no need for a rational boundary. sqrt(2187/2048) seems
ideal.
Since an apotome is so important in the scheme of things, I agree.
>>> 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.
Okay.
> ...
>>> But I really don't like using ( for 5:7-kleisma up.
>>> ... Folks are already used to thinking of ([<{ as meaning down
and )]
>>> ] as
>>> up. Scala uses ( for diesis down.
>>>
>> ... 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. ...
>
> 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.
Okay, I get your point. This would also allow user-defined symbols
using any of the laterally mirrored pairs in conjunction with our
sagittal shorthand ones, should that be desired.
> 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.
No, I think we should include the 19 comma.
> But here's another attempt at the 5:7-kleisma without using " or ;
or *.
> How about k for down and p for up?
Maybe. Let me think about it.
--George
Message: 6294 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 23:17:37
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> IF ANYONE OUT THERE HAS ANY INPUT ABOUT THIS (upper limit of ~14.5
> cents for a kleisma), PLEASE SAY SOMETHING SOON!
these cutoffs are totally arbitrary and need not even be based
strictly on JI cents considerations. but you guys may be interested
in studying the following links (which, however, concern 5-limit
only):
Yahoo groups: /tuning/database? * [with cont.]
method=reportRows&tbl=10&sortBy=5&sortDir=up
(be sure through scroll through all the pages)
Onelist Tuning Digest # 483 message 26, (c)200... * [with cont.] (Wayb.)
Message: 6295 - Contents - Hide Contents
Date: Tue, 04 Feb 2003 23:30:11
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> (be sure through scroll through all the pages)
sorry -- that should be *click* through all the pages!
Message: 6296 - Contents - Hide Contents
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 * [with cont.] (Wayb.)
Message: 6297 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 12:58:43
Subject: Schisminas and rational notation
From: Gene Ward Smith
If the four "schisminas" 2401/2400, 3025/3024, 4375/4374 and 9801/9800
vanish,
we are in the Hemiennealimmal temperament or something compatible with
it, such as the 612 et. I proposed some time back a notation for
11-limit JI based on nine nominals for which these are ignored, and
Graham has a similar 10-nominal system based on 494. Is there some
kind of convergence going on here?
Message: 6298 - Contents - Hide Contents
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 * [with cont.] (Wayb.)
Message: 6299 - Contents - Hide Contents
Date: Wed, 05 Feb 2003 13:21:50
Subject: Mercator
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
> On the mathworld page, it says "the Mercator series gives a
> Taylor series for the natural logarithm", and in fact makes
> it look like the Taylor series is the Mercator series.
It is in this case--it's a special name for this series alone, sort of like "Gregory/Leibniz" for
the arctangent (or the special value when x=1, namely pi/4.)
This is "our" Mercator, of the Mercator comma, incidentally, and not
the map Mercator.
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