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Message: 6225 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 21:00:42
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
> Hi George,
>
> I agree with most of your suggestions re single ASCII characters
for
> sagittal. Previously I figured there were so few approximate up-
down pairs
> that the left-right pairs <> [] {} had to get used somewhere. Sure
they're
> laterally confusable, but so are / and \.
True, but / and \ are also vertical mirrors, and they exhibit a
vertical directionality that is related to the sagittal symbols,
whereas the other pairs are strictly lateral opposites.
> However I think you've shown that
> we can get by without using them.
>
> How's this? (in order of size relative to strict Pythagorean)
>
> '| ' 5'-comma sharp 32768:32805
> .! . 5'-comma flat
> |( ` 5:7-comma sharp 5103:5120
> !( , 5:7-comma flat
There may or may not be trouble in keeping a clear distinction
between ` and ' and between . and , -- but we're scraping the bottom
of the barrel looking for characters. I think that " for up and ;
for down might be somewhat better (and certainly no worse than ' and
`) for the 5:7 comma.
> ~| ~ 17-comma sharp 2176:2187
> ~! $ or z 17-comma flat
Of course, ~ couldn't be any better. But how would s work as the
down symbol? It does combine the best features of both $ and z.
> ~|( h 17'-comma sharp 4096:4131
> ~!( y 17'-comma flat
If you make y up and h down, then the characters will more closely
resemble the symbols, according to which direction the shaft sticks
out. I realize that the y has a lower vertical placement relative to
the h, but consider how the actual symbols would be placed relative
to one another for a notehead in a given position. (See also my
comments for the 5:11 comma below.)
> /| / 5-comma sharp 80:81
> \! \ 5-comma flat
> |) f 7-comma sharp 63:64
> !) t 7-comma flat
> |\ & 55-comma sharp 54:55
> !/ % 55-comma flat
> (| ? 7:11-comma sharp 45056:45927
> (! j 7:11-comma flat
> (|( d 5:11-comma sharp 44:45
> (!( q 5:11-comma flat
I would make q the up symbol and d the down character (according to
which direction the arrow shaft protrudes). Then compare the
resemblance between the 7:11 and 5:11 comma characters, specifically
the part of each character where the convex curve is located:
up: ? q
down: j d
(It would also help to remember that "d" could stand for "down.")
> //| // 25-diesis sharp 6400:6561
> \\! \\ 25-diesis flat
This is a combination of two characters, but it's an exception that
is easily justified.
> /|) n 13-diesis sharp 1024:1053
> \!) u 13-diesis flat
> /|\ ^ 11-diesis sharp 32:33
> \!/ v 11-diesis flat
> (|) @ 11'-diesis sharp 704:729
> (!) U or o 11'-diesis flat
@ is very good! I would use o rather than U, since
1) All of the other down symbols that are letters are lower case; and
2) There is already a lower-case u being used, so 'o' would be less
confusing.
> (|\ m 13'-diesis sharp 26:27
> (!/ w 13'-diesis flat
> /||\ # apotome sharp 2048:2187
> \!!/ b apotome flat
>
> Which do you prefer out of $ or z and U or o?
>
> I note that some folk have in the past used t for the tartini half-
sharp
> and d for the backwards-flat (meaning half-flat), but I don't think
that
> should stop us using them in other ways here.
Not at all.
>> For 217-ET that's 12 pairs of characters. I don't think I would
want
>> to see ) paired with (, for example.
Not counting the 5' and apotome pairs, it's actually 13 pairs. I
snuck both the 55 and 7:11 comma symbols in there for 6deg217. As a
consequence, we also have all of the symbols needed for a 15-limit
tonality diamond.
This then covers all of the ETs in Table 3 and around half of those
in Table 4 (in general, the ones that don't use the 19-comma symbol).
> ...
>> Maybe a 17 comma
>> up could be S and down s, or would that be better for the 23 comma?
>
> I don't like using uppercase-lowercase pairs. Too confusable. I
know we're
> using x and X in the multi-ASCII, but these should be very rare and
will
> often have additional direction cues provided by straight flags,
e.g. \x/ /X\
Yes, and besides, this character-based shorthand simulates only the
double-symbol version of the sagittal notation.
> The 23-comma is so rare it doesn't need a single ASCII character.
>
>> After that it gets more difficult.
>
> Yes. No need to go beyond 17.
Agreed.
>> Do we really need shorthand ascii notation for anything more than
>> straight and convex-flag symbols?
>
> It would be nice to have the full 217-ET set provided they're
reasonably
> memorable, or figure-out-able.
>
>> (Or are you intending to combine
>> those single-character ascii symbols in any way?)
>
> No.
Okay. That way we keep the shorthand simple.
>> I think that it
>> would get pretty complicated (hence difficult to remember), and the
>> result would have very resemblance to what sagittal symbols look
>> like.
>
> Well how do you think we're doing above?
It looks like it'll fly.
--George
Message: 6226 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 05:29:58
Subject: Re: Graham's top 20, with standard vals
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> Say what? I thought a val was the complement of a vector.
>>
>> It's the dual of a vector, if by a vector you mean an interval
>> in Monzo notation. Due to the magic of Poincare duality, you
>> can wedge with either a vector or a val.
>
> I may be dangerously close to understanding vals. I've read
> the definition in monz's dictionary. Anybody care to give
> an example?
the 5-limit standard val for 12 is [12, 19, 28] since that's how many
semitones are in a 2:1, a 3:1, and a 5:1. a non-standard val for 12
would be [12, 19, 27] since you'd be using 27 semitones, instead of
28 semitones, to approximate 5:1.
> Maybe Paul could shed some light on a layman's
> definition.
>
> -Carl
Message: 6227 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 22:35:47
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>>> >>>
>>>>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS,
>>>>> HERE'S A QUESTION: What other ETs above 494 besides
>>>>> 612 and 624 would you want to notate -- ones in which
>>>>> the 5' comma (a.k.a, historical 5-schisma, 32768:32805)
>>>>> is either a single degree of the ET or vanishes?
>>>>
>>>> 665, 684, 730, 742 and 836.
>>>
>>> Thanks, Gene. I'll also add 653 to that list.
>>>
>>> But we won't be able to notate 684, because the 5' comma
>>> vanishes and no other symbol in the sagittal notation
>>> would represent a single degree, either.
>>>
>>> --George
>>
>> how about 768? ... because it's the tuning resolution for a
>> number of popular electronic instruments.
>>
>> -monz
>
> Sorry, that one can't be done with the commas that we have. The 5'
> comma (32768:32805) is actually -1 degrees, so it would be too
> confusing to use it. And while the 19 comma (512:513) could be
used
> as 1 degree, there's nothing for 2, 3, and 4 degrees.
>
> Anyway, 768 isn't even 1,3,9-consistent, hence not very desirable
> musically.
>
> But after taking a quick look at 653, 665, 730, 742 and 836, I
think
> those five are all do-able.
>
> --George
based on the graphs and stuff i've been posting, 600 and 1000 are
both quite interesting in the 5-limit.
Message: 6228 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 06:01:52
Subject: Re: Graham's top 20, with standard vals
From: Carl Lumma
>the 5-limit standard val for 12 is [12, 19, 28] since that's how
>many semitones are in a 2:1, a 3:1, and a 5:1.
Ah, ok. THANKS PAUL.
>a non-standard val for 12 would be [12, 19, 27] since you'd be
>using 27 semitones, instead of 28 semitones, to approximate 5:1.
What's a non-standard val? Why?
-Carl
Message: 6229 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 22:36:53
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
> <gdsecor@y...> wrote:
>> Anyway, 768 isn't even 1,3,9-consistent, hence not very desirable
>> musically.
>>
>> But after taking a quick look at 653, 665, 730, 742 and 836, I
> think
>> those five are all do-able.
>>
>> --George
>
> based on the graphs and stuff i've been posting, 600 and 1000 are
> both quite interesting in the 5-limit.
and they've been used historically -- 600 is called "centitones", and
1000 is called "millioctaves".
Message: 6230 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 06:02:36
Subject: Re: Graham's Top 20 13-limit temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> wallyesterpaulrus wrote:
>
>> calculate the numerators and denominators here which came out in
>> scientific notation, making it impossible for yahoo to sort by
>> denominator:
>>
>>
Yahoo groups: /tuning/database? * [with cont.]
>> method=reportRows&tbl=10&sortBy=4
>
>
" large limma", "0 3 -2", ... * [with cont.] (Wayb.)
>
> Graham
thanks dude!
try now:
Yahoo groups: /tuning/database? * [with cont.]
method=reportRows&tbl=10&sortBy=4
Message: 6231 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 06:03:47
Subject: Re: Graham's top 20, with standard vals
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> the 5-limit standard val for 12 is [12, 19, 28] since that's how
>> many semitones are in a 2:1, a 3:1, and a 5:1.
>
> Ah, ok. THANKS PAUL.
>
>> a non-standard val for 12 would be [12, 19, 27] since you'd be
>> using 27 semitones, instead of 28 semitones, to approximate 5:1.
>
> What's a non-standard val? Why?
>
> -Carl
gene defined "standard val" (not something i'd be particularly
interested in) as where each prime is mapped to its best
approximation in N-equal -- in this case, 12-equal.
Message: 6233 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:20:03
Subject: Re: Graham's top 20, with standard vals
From: Carl Lumma
> gene defined "standard val" (not something i'd be particularly
> interested in) as where each prime is mapped to its best
> approximation in N-equal -- in this case, 12-equal.
??? Could you explain your reasoning here?
-C.
Message: 6234 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:21:59
Subject: Re: Graham's top 20, with standard vals
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> gene defined "standard val" (not something i'd be particularly
>> interested in) as where each prime is mapped to its best
>> approximation in N-equal -- in this case, 12-equal.
>
> ??? Could you explain your reasoning here?
>
> -C.
my reasoning in not being particularly interested in this? it's that,
like graham, i don't think the standard val is necessarily the best
val for any ET. try 64-equal in the 5-limit.
Message: 6235 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:26:24
Subject: Re: New file uploaded to tuning-math
From: Carl Lumma
>
Yahoo groups: /tuning-math/files/Paul/zooms.gif * [with cont.]
Awesome. -C.
Message: 6236 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:28:46
Subject: Re: Graham's Top 20 13-limit temperaments
From: Carl Lumma
**Now we're talking**. Sorting by the denominator
of the comma really works better than anything I
tried on Dave's spreadsheet.
Aside from the order, what bounds were used to
select temperaments for this list?
-Carl
Message: 6237 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:30:08
Subject: Re: Graham's top 20, with standard vals
From: Carl Lumma
>my reasoning in not being particularly interested in this?
>it's that, like graham, i don't think the standard val is
>necessarily the best val for any ET. try 64-equal in the
>5-limit.
This wouldn't happen in a linear temperament, though, right?
Message: 6238 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:44:39
Subject: Re: Graham's Top 20 13-limit temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> **Now we're talking**. Sorting by the denominator
> of the comma really works better than anything I
> tried on Dave's spreadsheet.
better than any other complexity measure! cool, so you must *really*
like the heuristic for complexity . . .
> Aside from the order, what bounds were used to
> select temperaments for this list?
my best recollection, off the top of my head:
log-flat badness < 3500, rms error < 50 cents, geometric complexity <
104-151 (doesn't matter where you draw the line in this range).
Message: 6239 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:46:33
Subject: Re: Graham's top 20, with standard vals
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> my reasoning in not being particularly interested in this?
>> it's that, like graham, i don't think the standard val is
>> necessarily the best val for any ET. try 64-equal in the
>> 5-limit.
>
> This wouldn't happen in a linear temperament, though, right?
i don't know what you mean. you need two vals to define a linear
temperament, just like you need two commas to define a temperament
with "codimension 2" (number of primes minus dimensionality of tuning
equals two).
Message: 6240 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:53:55
Subject: Re: Graham's top 20, with standard vals
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:
>> gene defined "standard val" (not something i'd be particularly
>> interested in) as where each prime is mapped to its best
>> approximation in N-equal -- in this case, 12-equal.
>
> ??? Could you explain your reasoning here?
My reasoning is that the defintion is straightforward, and in effect other people use it.
Message: 6241 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 07:55:05
Subject: Re: Graham's top 20, with standard vals
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> my reasoning in not being particularly interested in this? it's that,
> like graham, i don't think the standard val is necessarily the best
> val for any ET. try 64-equal in the 5-limit.
Do you have an alternative definition?
Message: 6242 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 12:04:50
Subject: Re: Calculating geometric complexity II
From: Graham Breed
Gene Ward Smith wrote:
>> Oh, that's good. It should be the same as my invariant. But are
>> 7-limit wedge products taken from vectors or vals?
>
> Either, but I follow the val ordering.
Okay, so that's the one that give the mapping correctly
>> I get 7-limit meantone as 21.97, 11-limit meantone as 31.72 and
>> h12^h19^h22 in the 11-limit as 29.52. The planar temperament with
>> 441:440 and 225:224 is 34.44.
>
> I'm afraid I don't know what these numbers mean. I have
They're geometric complexity, calculated from your algorithm! Look at
the subject line!!!!
Why are you using natural logarithms in the definition? I have my
standard arrays as logarithms to base 2, and that's the metric that
gives interval sizes in octaves. It'd be much easier if geometric
complexity stayed in base 2.
> 7-limit meantone: h50^h31 = 126/125^81/80 = [1, 4, 10, 12, -13, 4]
Okay, we're already in trouble. I make this val
(0, 1): 1
(0, 2): 4
(0, 3): 10
(1, 2): 4
(1, 3): 13
(2, 3): 12
That'd be an invariant of
[1, 4, 10, 4, 13, 12]
Which isn't what you give! You must be using (3,1) instead of (1,3) for
the sign to match. And the ordering isn't the same, and I don't see how
that can be numerical order of the bases, whatever the bases are. So
what other surprises do you have up your sleeve?
I'm fully in agreement with the calculations for wedge product and
complement given in
Grassmann Algebra Book * [with cont.] (Wayb.)
where my bases are the coefficients of his e's (except that I start with
0 instead of 1). So can you please state your algorithms in terms of these?
> h12^h19^h22 = 100/99^225/224 = [-1,2,-2,2,2,-8,-5,-2,14,-6]
(0, 1, 2): 1 *
(0, 1, 3): 2
(0, 1, 4): 2 *
(0, 2, 3): -2 *
(0, 2, 4): 2
(0, 3, 4): 8 *
(1, 2, 3): -5
(1, 2, 4): 2 *
(1, 3, 4): 14
(2, 3, 4): 6 *
Here, it's the right order but the signs are wrong. It's also different
to the invariant I defined, which always converts to the smaller bases.
I can change that to always have the dual flag set and I don't think
there'll be any repercussions. And the signs don't matter for the
complexity calculation, so this one's okay.
> 225/224^441/440 = h41^h31^h12 = [1,-2,3,-2,6,-6,5,-13,11,-4]
> I
said I was using duality to identify compliments. I started out trying
to do things the right way, as you seem to be doing, but it gave me
trouble, so I settled for a fast, simple-minded approach, which means
I have a separate program for each kind of wedge product I want to
take.
No, you've repeatedly said things like the the wedge product of 2 vals
is the same as that of n-1 commas, which only works if you use
duality.
I'd rather keep the complements in there, explicit is better than
implicit and all that. But now I've implemented duality to fit in
with
you, you say you've been taking complements all the time!
Oh, and "compliment" and "complement" are different words.
>> reversing the list will do the trick? Some of the coefficients
should
>> be negated if you aren't using a special ordering.
>
> Sometimes I do. I simply make the wedge product of "vectors", or
what I would call intervals, correspond to the wedge product for vals,
which I take as the basis.
So can you give the general algorithm for geometric complexity? I
think
I've got an interior product worked out, but (as with the complement)
only for the Euclidian metric. Which Browne says is the identity
matrix, although you have some other metric that you also say is
Euclidian.
Graham
Message: 6243 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 13:36:16
Subject: Re: Calculating geometric complexity II
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
> I'm fully in agreement with the calculations for wedge product and
> complement given in
>
>
Grassmann Algebra Book * [with cont.] (Wayb.)
>
> where my bases are the coefficients of his e's (except that I start with
> 0 instead of 1). So can you please state your algorithms in terms of these?
> No, you've repeatedly said things like the the wedge product of 2 vals
> is the same as that of n-1 commas, which only works if you use duality.
People
define things in different ways; you could introduce interior
products, for instance. The way Browne does things is a little unusual
and I haven't been trying to follow him. For our musical purposes, as
far as I can see we only need to multiply by either vals or intervals,
and we can, if we like, consider that the kind of product is
determined by whether we are taking it with a val or an interval.
> I'd rather keep the complements in there, explicit is better than
> implicit and all that. But now I've implemented duality to fit in
with
> you, you say you've been taking complements all the time!
It's duality which allows us to identify a compliment with a dual. In
other words, an interval, which is the dual to a val, can be
identified with an n-1 grade element in the exterior power of the
vals, and vice-versa.
> So can you give the general algorithm for geometric complexity? I
think
> I've got an interior product worked out, but (as with the
complement)
> only for the Euclidian metric. Which Browne says is the identity
> matrix, although you have some other metric that you also say is
Euclidian.
If you consider both vals and intervals to be 1-vectors, then the
mapping of an interval by the val is an inner product. Mostly,
mathematicians would simply leave the vals and intervals to be dual
spaces, and identify the double dual with the original space. That's
one possible kind of metric, but we don't actually need to make it a
metric if we don't do things in Browne's way but in a more usual way.
The metric I am using for geometric complexity is a metric not on
intervals, but on octave classes. The classes are 1-vectors in a space
of one less dimension, and the wedge products are being taken in this
different space. This now is the situation as Browne envisions it,
except that we are not using an orthonormal basis. However, a
coordinate transformation will take us to such a basis, which means
that elements of any grade can be measured--we are in a Euclidean
setting, which from my point of view we are *not* in when we start
with vals and intervals, where we are in a vector space V and its dual
space V^* setting.
Message: 6244 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 14:35:19
Subject: Re: Calculating geometric complexity II
From: Graham Breed
Gene Ward Smith wrote:
> People define things in different ways;
you could introduce interior products, for instance. The way Browne
does things is a little unusual and I haven't been trying to follow
him. For our musical purposes, as far as I can see we only need to
multiply by either vals or intervals, and we can, if we like, consider
that the kind of product is determined by whether we are taking it
with a val or an interval.
So far, all we need are wedge products and complements. But you only
defined wedge products -- I had to work out the complement for myself.
If the wedge product works on general multivectors, there's no need to
distinguish vals from intervals. Even if the distinction is
preserved,
you need to do complements, they're just hidden away.
The only text I have is Browne's, so if you're using different
definitions, how are any of us supposed to know what you mean?
>> So can you give the general algorithm for geometric complexity? I
think
>> I've got an interior product worked out, but (as with the
complement)
>> only for the Euclidian metric. Which Browne says is the identity
>> matrix, although you have some other metric that you also say is
Euclidian.
>
>
> If you consider both vals and intervals to be 1-vectors, then the
mapping of an interval by the val is an inner product. Mostly,
mathematicians would simply leave the vals and intervals to be dual
spaces, and identify the double dual with the original space. That's
one possible kind of metric, but we don't actually need to make it a
metric if we don't do things in Browne's way but in a more usual way.
The inner product can be calculated from wedge products and
complements.
There's no need to bring dual spaces into it. It's the only way I
know to get the size of a wedgie, and for it to work as a complexity
measure a metric needs to be applied to the complement operation. It
does need to work beyond 1-vectors because we need to get the
complexity
of other wedgies.
> The metric I am using for geometric complexity is a metric not on
intervals, but on octave classes.
What's an octave class?
> The classes are 1-vectors in a space of one less dimension, and the
wedge products are being taken in this different space.
You mean it's octave-equivalent space? Well, we've always been able
to
rate the complexity of the mapping. We need to get the complexity of
intermediate wedgies, like spatial temperaments in 13-equal.
> This now is the situation as Browne envisions it, except that we
are
not using an orthonormal basis.
We aren't?
> However, a coordinate transformation will take us to such a basis,
which means that elements of any grade can be measured--we are in a
Euclidean setting, which from my point of view we are *not* in when we
start with vals and intervals, where we are in a vector space V and
its dual space V^* setting.
So what's the transformation? And how do you transform coordinates of
wedgies anyway?
Graham
Message: 6245 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 16:24:33
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
>>
>>> Anyway, 768 isn't even 1,3,9-consistent, hence not very
desirable
>>> musically.
>>>
>>> But after taking a quick look at 653, 665, 730, 742 and 836, I
>> think
>>> those five are all do-able.
>>>
>>> --George
>>
>> based on the graphs and stuff i've been posting, 600 and 1000 are
>> both quite interesting in the 5-limit.
>
> and they've been used historically -- 600 is called "centitones",
and
> 1000 is called "millioctaves".
It appears that 600 can be done with the symbols we have, but
definitely not 1000 (since the 5' comma would be 2 degrees).
--George
Message: 6246 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 20:40:00
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote [#5684]:
> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...>
> wrote:
>> There may or may not be trouble in keeping a clear distinction
>> between ` and ' and between . and , -- but we're scraping the
bottom
>> of the barrel looking for characters.
>
> There would be trouble. We can either use obviously different
characters as
> you've suggested, or we could attempt to outlaw the use of the 5'-
comma
> symbols in the shorthand. But I don't think the latter will work.
It's too
> obvious a thing to want to do.
Yes, and very convenient for notating some intervals that have been
frequently mentioned by theorists for many, many years.
> 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 think that " for up and ;
>> for down might be somewhat better (and certainly no worse than '
and
>> `) for the 5:7 comma.
>
> I don't know where you got ' and `) from. Typos?
> Oh. Now I see that ) was a closing parenthesis, another good reason
not to
> use ( or ) in the shorthand.
Yes.
> Similarly, we shouldn't use a comma , for anything since it would
make
> punctuating sentences rather fraught. But why " and ;? Why not "
and :? I
> suppose one reason is that we sometimes want to write C:G just as
we write 2:3.
The comma has a little bit of curvature in it, which might suggest a
curved flag. All I was interested in was a symbol that had two
things in it, to have some commonality with the two strokes in the "
mark. But I suppose that a colon would do just as well.
> Semicolon is rarely used for punctuation or anything else. Except
we do
> have Paul Erlich's usage, which I quite like, which is instead of :
in
> commas. For example we write 80;81 to make it clear that we mean
the
> interval in some tuning that functions as the syntonic comma but is
not
> necessarily 21.5 cents. The semicolon there only appears between
numbers,
> not letters, so there's no problem.
This would tend to favor the semicolon, then.
> Since we need small symbols for the 5:7 comma and I can't think of
anything
> better, I reluctantly agree with " and ; although they bear no
resemblance
> to the graphicals.
My thoughts exactly.
>> If you make y up and h down, then the characters will more
closely
>> resemble the symbols, according to which direction the shaft
sticks
>> out. I realize that the y has a lower vertical placement
relative to
>> the h, but consider how the actual symbols would be placed
relative
>> to one another for a notehead in a given position. (See also my
>> comments for the 5:11 comma below.)
>
> Agreed.
>
>>> ~| ~ 17-comma sharp 2176:2187
>>> ~! $ or z 17-comma flat
>>
>> Of course, ~ couldn't be any better. But how would s work as the
>> down symbol? It does combine the best features of both $ and z.
>
> The problems I have with s are
> 1. It can be confused with plurals, e.g. one C two Cs. Writing one
C two
> C's doesn't help either since the apostrophe is the 5'-up symbol.
Yes, a potential comedy of errors. Attention to context would be
important in avoiding this.
> 2. Just as we have d for down, we have s for sharp (the wrong
direction).
> And for this reason I must now reject $ which even _looks_ like a
kind of
> half-sharp symbol.
>
> "z" doesn't carry any of this baggage and while it doesn't look as
much
> like the sagittal I can accept it because we already have some
other
> angular characters paired with rounded ones. h and y, w and m.
>
> But if, after considering the above, you still prefer "s", and no-
one else
> objects ...
I haven't seen a reason compelling enough to make me want to
reject "s".
> Hello everyone else, you're welcome to give your opinions on these.
>
> ... then I'll go with the "s".
Pending any other opinions.
>> I would make q the up symbol and d the down character (according
to
>> which direction the arrow shaft protrudes). Then compare the
>> resemblance between the 7:11 and 5:11 comma characters,
specifically
>> the part of each character where the convex curve is located:
>>
>> up: ? q
>> down: j d
>>
>> (It would also help to remember that "d" could stand for "down.")
>
> Agreed.
>
>>> //| // 25-diesis sharp 6400:6561
>>> \\! \\ 25-diesis flat
>>
>> This is a combination of two characters, but it's an exception
that
>> is easily justified.
>
> Yes. Not to mention that we can't find a single character that
looks
> anything like them! "F" looks a bit like //| but can't be used for
obvious
> reasons.
>
>>> (|) @ 11'-diesis sharp 704:729
>>> (!) U or o 11'-diesis flat
>>
>> @ is very good! I would use o rather than U, since
>> 1) All of the other down symbols that are letters are lower
case; and
>> 2) There is already a lower-case u being used, so 'o' would be
less
>> confusing.
>
> Yes. It is good not to use any uppercase. This also allows the
sagittals to
> be used for linear temperament notations that might use more that 7
> nominals. Other uppercase letters can then be used for the nominals.
An excellent point!
>> Not counting the 5' and apotome pairs, it's actually 13 pairs. I
>> snuck both the 55 and 7:11 comma symbols in there for 6deg217.
As a
>> consequence, we also have all of the symbols needed for a 15-
limit
>> tonality diamond.
>>
>> This then covers all of the ETs in Table 3 and around half of
those
>> in Table 4 (in general, the ones that don't use the 19-comma
symbol).
>
> Wonderful.
>
>>>> (Or are you intending to combine
>>>> those single-character ascii symbols in any way?)
>>>
>>> No.
>>
>> Okay. That way we keep the shorthand simple.
>
> Of course I intend that # or b (apotome) and ' or . (5'-comma) may
be
> combined with any of the others, and // may occur,
Yes.
> 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. Since the two versions differ
according to whether or not a symbol contains either |, !, X, or x
(or, rather, whatever is replacing x), then there's no problem in
determining which ascii version of the notation is being used.
> It's unfortunate that we can't allow the traditional use of x
instead of ##
> in this shorthand notation without creating ambiguous symbols. Is
there any
> chance we could find some other ASCII character for the down x-
shaft? How
> about k?
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).
> 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. 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.
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.
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);
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;
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., \!#).
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.
> Here's what we've got now (in order of size relative to strict
Pythagorean)
>
> '| ' 5'-comma sharp 32768:32805
> .! . 5'-comma flat
> |( " 5:7-comma sharp 5103:5120
> !( ; 5:7-comma flat
> ~| ~ 17-comma sharp 2176:2187
> ~! z or s 17-comma flat
> ~|( y 17'-comma sharp 4096:4131
> ~!( h 17'-comma flat
> /| / 5-comma sharp 80:81
> \! \ 5-comma flat
> |) f 7-comma sharp 63:64
> !) t 7-comma flat
> |\ & 55-comma sharp 54:55
> !/ % 55-comma flat
> (| ? 7:11-comma sharp 45056:45927
> (! j 7:11-comma flat
> (|( q 5:11-comma sharp 44:45
> (!( d 5:11-comma flat
> //| // 25-diesis sharp 6400:6561
> \\! \\ 25-diesis flat
> /|) n 13-diesis sharp 1024:1053
> \!) u 13-diesis flat
> /|\ ^ 11-diesis sharp 32:33
> \!/ v 11-diesis flat
> (|) @ 11'-diesis sharp 704:729
> (!) o 11'-diesis flat
> (|\ m 13'-diesis sharp 26:27
> (!/ w 13'-diesis flat
> /||\ # apotome sharp 2048:2187
> \!!/ b apotome flat
> /X\ ## or x apotome sharp 2048:2187
> \x/ bb apotome flat
>
> George, whatever you decide on for the 17-comma flat and the
apotome sharp,
> could you please add these symbols to your quick reference.
Okay. Perhaps that's the best way to elicit comments about these.
> And when you
> have time, could you add the sequence of these single-character
ASCII
> symbols for some of the most common ETs. In particular the ones
that have
> come up in linear temperament notation discussions.
I'll do all the ETs that we have that can be notated with these
characters (apart from 5' characters, since we haven't agreed on any
yet for ET notation).
--George
Message: 6247 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 22:27:27
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
> I've added one more pair below. Since, according to Manuel's
statistics,
> the ratios it notates (49 with various powers of 2 and 3) are more
common
> (1.6%) than many others on this list, and it's a no-brainer to
notate. This
> list has 86% of the ratio ocurrences covered.
> ...
Okay, I've updated the following file:
Yahoo groups: /tuning- * [with cont.]
math/files/secor/notation/quickref.txt
with all of the latest abbreviations, including ET symbol sequences.
(And following those are single-symbol sequences for the same ETs.)
--George
Message: 6248 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 12:10:58
Subject: Re: A common notation for JI and ETs
From: David C Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>" <gdsecor@y...>
wrote:
> There may or may not be trouble in keeping a clear distinction
> between ` and ' and between . and , -- but we're scraping the bottom
> of the barrel looking for characters.
There would be trouble. We can either use obviously different characters as
you've suggested, or we could attempt to outlaw the use of the 5'-comma
symbols in the shorthand. But I don't think the latter will work. It's too
obvious a thing to want to do.
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.
> I think that " for up and ;
> for down might be somewhat better (and certainly no worse than ' and
> `) for the 5:7 comma.
I don't know where you got ' and `) from. Typos?
Oh. Now I see that ) was a closing parenthesis, another good reason not to
use ( or ) in the shorthand.
Similarly, we shouldn't use a comma , for anything since it would make
punctuating sentences rather fraught. But why " and ;? Why not " and :? I
suppose one reason is that we sometimes want to write C:G just as we write 2:3.
Semicolon is rarely used for punctuation or anything else. Except we do
have Paul Erlich's usage, which I quite like, which is instead of : in
commas. For example we write 80;81 to make it clear that we mean the
interval in some tuning that functions as the syntonic comma but is not
necessarily 21.5 cents. The semicolon there only appears between numbers,
not letters, so there's no problem.
Since we need small symbols for the 5:7 comma and I can't think of anything
better, I reluctantly agree with " and ; although they bear no resemblance
to the graphicals.
> If you make y up and h down, then the characters will more closely
> resemble the symbols, according to which direction the shaft sticks
> out. I realize that the y has a lower vertical placement relative to
> the h, but consider how the actual symbols would be placed relative
> to one another for a notehead in a given position. (See also my
> comments for the 5:11 comma below.)
Agreed.
>> ~| ~ 17-comma sharp 2176:2187
>> ~! $ or z 17-comma flat
>
> Of course, ~ couldn't be any better. But how would s work as the
> down symbol? It does combine the best features of both $ and z.
The problems I have with s are
1. It can be confused with plurals, e.g. one C two Cs. Writing one C two
C's doesn't help either since the apostrophe is the 5'-up symbol.
2. Just as we have d for down, we have s for sharp (the wrong direction).
And for this reason I must now reject $ which even _looks_ like a kind of
half-sharp symbol.
"z" doesn't carry any of this baggage and while it doesn't look as much
like the sagittal I can accept it because we already have some other
angular characters paired with rounded ones. h and y, w and m.
But if, after considering the above, you still prefer "s", and no-one else
objects ...
Hello everyone else, you're welcome to give your opinions on these.
... then I'll go with the "s".
> I would make q the up symbol and d the down character (according to
> which direction the arrow shaft protrudes). Then compare the
> resemblance between the 7:11 and 5:11 comma characters, specifically
> the part of each character where the convex curve is located:
>
> up: ? q
> down: j d
>
> (It would also help to remember that "d" could stand for "down.")
Agreed.
>> //| // 25-diesis sharp 6400:6561
>> \\! \\ 25-diesis flat
>
> This is a combination of two characters, but it's an exception that
> is easily justified.
Yes. Not to mention that we can't find a single character that looks
anything like them! "F" looks a bit like //| but can't be used for obvious
reasons.
>> (|) @ 11'-diesis sharp 704:729
>> (!) U or o 11'-diesis flat
>
> @ is very good! I would use o rather than U, since
> 1) All of the other down symbols that are letters are lower case; and
> 2) There is already a lower-case u being used, so 'o' would be less
> confusing.
Yes. It is good not to use any uppercase. This also allows the sagittals to
be used for linear temperament notations that might use more that 7
nominals. Other uppercase letters can then be used for the nominals.
> Not counting the 5' and apotome pairs, it's actually 13 pairs. I
> snuck both the 55 and 7:11 comma symbols in there for 6deg217. As a
> consequence, we also have all of the symbols needed for a 15-limit
> tonality diamond.
>
> This then covers all of the ETs in Table 3 and around half of those
> in Table 4 (in general, the ones that don't use the 19-comma symbol).
Wonderful.
>>> (Or are you intending to combine
>>> those single-character ascii symbols in any way?)
>>
>> No.
>
> Okay. That way we keep the shorthand simple.
Of course I intend that # or b (apotome) and ' or . (5'-comma) may be
combined with any of the others, and // may occur, 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).
It's unfortunate that we can't allow the traditional use of x instead of ##
in this shorthand notation without creating ambiguous symbols. Is there any
chance we could find some other ASCII character for the down x-shaft? How
about k?
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.
Here's what we've got now (in order of size relative to strict Pythagorean)
'| ' 5'-comma sharp 32768:32805
.! . 5'-comma flat
|( " 5:7-comma sharp 5103:5120
!( ; 5:7-comma flat
~| ~ 17-comma sharp 2176:2187
~! z or s 17-comma flat
~|( y 17'-comma sharp 4096:4131
~!( h 17'-comma flat
/| / 5-comma sharp 80:81
\! \ 5-comma flat
|) f 7-comma sharp 63:64
!) t 7-comma flat
|\ & 55-comma sharp 54:55
!/ % 55-comma flat
(| ? 7:11-comma sharp 45056:45927
(! j 7:11-comma flat
(|( q 5:11-comma sharp 44:45
(!( d 5:11-comma flat
//| // 25-diesis sharp 6400:6561
\\! \\ 25-diesis flat
/|) n 13-diesis sharp 1024:1053
\!) u 13-diesis flat
/|\ ^ 11-diesis sharp 32:33
\!/ v 11-diesis flat
(|) @ 11'-diesis sharp 704:729
(!) o 11'-diesis flat
(|\ m 13'-diesis sharp 26:27
(!/ w 13'-diesis flat
/||\ # apotome sharp 2048:2187
\!!/ b apotome flat
/X\ ## or x apotome sharp 2048:2187
\x/ bb apotome flat
George, whatever you decide on for the 17-comma flat and the apotome sharp,
could you please add these symbols to your quick reference. And when you
have time, could you add the sequence of these single-character ASCII
symbols for some of the most common ETs. In particular the ones that have
come up in linear temperament notation discussions.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6249 - Contents - Hide Contents
Date: Tue, 28 Jan 2003 12:47:23
Subject: Re: A common notation for JI and ETs
From: David C Keenan
I've added one more pair below. Since, according to Manuel's statistics,
the ratios it notates (49 with various powers of 2 and 3) are more common
(1.6%) than many others on this list, and it's a no-brainer to notate. This
list has 86% of the ratio ocurrences covered.
'| ' 5'-comma sharp 32768:32805
.! . 5'-comma flat
|( " 5:7-comma sharp 5103:5120
!( ; 5:7-comma flat
~| ~ 17-comma sharp 2176:2187
~! z or s 17-comma flat
~|( y 17'-comma sharp 4096:4131
~!( h 17'-comma flat
/| / 5-comma sharp 80:81
\! \ 5-comma flat
|) f 7-comma sharp 63:64
!) t 7-comma flat
|\ & 55-comma sharp 54:55
!/ % 55-comma flat
(| ? 7:11-comma sharp 45056:45927
(! j 7:11-comma flat
(|( q 5:11-comma sharp 44:45
(!( d 5:11-comma flat
//| // 25-diesis sharp 6400:6561
\\! \\ 25-diesis flat
/|) n 13-diesis sharp 1024:1053
\!) u 13-diesis flat
/|\ ^ 11-diesis sharp 32:33
\!/ v 11-diesis flat
|)) ff 49'-diesis sharp 3969:4096 might be (/|
!)) tt 49'-diesis flat might be (\!
(|) @ 11'-diesis sharp 704:729
(!) o 11'-diesis flat
(|\ m 13'-diesis sharp 26:27
(!/ w 13'-diesis flat
/||\ # apotome sharp 2048:2187
\!!/ b apotome flat
/X\ ## or x apotome sharp 2048:2187
\x/ bb apotome flat
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