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Message: 4725 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 15:14:07

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Regarding the problem of apotome complement symbols for rational 
> tunings, please see 
> Yahoo groups: /tuning-math/files/Dave/Compleme... * [with cont.] 
> It should be self-explanatory.


I just uploaded a new version of that file, which now contains not 
only a statement of the problem, but a solution, which turns out to be 
related to 453-ET. It requires the addition of one new symbol as the 
complement of the 25-comma symbol. The new symbol is a combination of 
left wavy and left concave flags, and at around 12.1 cents, it goes in 
the middle of the largest remaining gap.

So now everything that needs a complement has one. There are no simple 
complementation rules beyond 13 limit, but I can live with that. Those 
symbols that don't have complements should be avoided.

There are some alternatives for complements in some cases. The 17 
comma symbol (wavy left) has several other 3-flag options for its 
complement besides the 37' comma symbol (vL+xR+wR). These are 
xL+vR+vR, wL+sR+vR, sL+wL+wR, which don't yet exist. 

I haven't checked whether this system of complements lets us give each 
flag a constant value when it occurs on a double-shaft. An examination 
of this might cause one to choose some different alternative to those 
I have chosen.

I've also uploaded a new version of
Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.] 
showing the new symbol, with all the others, on the staff.

Message: 4726 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 16:13:02

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> At 01:26 22/04/02 -0000, George Secor wrote:


[DK, msg #4148:]

> I've now exhaustively searched all combinations of up to 3 of our 
flags.
> 
> Here's what I end up with.
> 
> Symbol  Left      Right       Schisma
> for     flags     flags       (cents)
> -------------------------------------
> 5   =       5                   0
> 7   =              7            0
> 11  =       5  +  (11-5)        0
> 11' =      29  +   7            0.34
> 13  =       5  +   7            0.42
> 13' =      29  +  (11-5)        0.08
> 17  =      17                   0
> 17' =      17  +  (17'-17)      0
> 19  =      19                   0
> 19' =      19  +  23            0.16
> 23  =             23            0
> 23' =      17  +  (11-5)        0.49
>    or      29  +  (17'-17)      0.52  *
> 29  =      29                   0
> 31  =      19  +  (11-5)        0.12
> 31' = 29 +  5                   0.03  *
>    or             7  +  7       0.44  *
>    or       5  +  (17'-17) + 7  0.19
>    or       5  +  23 + 23       0.37
> 37  =  5 +  5                   0.22  *
>    or 29 + 17                   0.57
> 37' =      19  +  23 + 7        0.25
>    or  5 + 17  +  23            0.65
> 41  =       5                   0.26  *
>     or  17  +  (17'-17) + (17'-17)  0.51
> 43  = 19 + 19  +  (17'-17)      0.72        [schisma too big]
> 47  =      17  +   7            0.45
>    or 19 + 29                   0.42
>    or      19  +  23 + 23       0.02
>    or  5 + 17  +  (17'-17)      0.21


Okay, I'm with you 100 percent on this now.  (I haven't checked all 
of these schismas, but trust that you have been thorough with this.)  
Something that I especially like is that everything through the 29 
limit works without requiring two flags on the same side.


> pythagorean
> comma = 17 + 17 + (17'-17)      0
> diaschisma = 19 + 23            0.37        [same symbol as 19']
> diesis = 17  +  (11-5)          0.56        [same symbol as 23']
> 
> * doesn't vanish in 1600-ET.

Very nice!

> So, in addition to c37 = c5 + c5, there are some other schismas 
available
> to us, that don't vanish in 1600-ET and are smaller than those that 
do.
> Namely:
> 
> 31' = 29 +  5                   0.03 cents
> 41  =       5                   0.26 cents
> 
> We should definitely stop at prime 41, since there is no way to get 
43 with
> sufficient accuracy using our 8 existing flags. We're under the 
half cent
> otherwise.


I see no point in going to 43, so I agree.


> In the application you (or Erv) found for 41, would a 0.26 cent 
error in
> the 41 have rendered it useless? Why not simply reuse the 5 comma 

as the 41

> comma?


My original regarding this was in message #3985:

<< While we are on the subject of higher primes, I have one more 
schisma, just for the record.  This is one that you probably won't be 
interested in, inasmuch as it is inconsistent in both 311 and 1600, 
but consistent and therefore usable in 217.  It is 6560:6561 
(2^5*5*41:3^8, ~0.264 cents), the difference between 80:81 and 81:82, 
the latter being the 41-comma, which can be represented by the sL 
flag.  I don't think I ever found a use for any ratios of 37, but Erv 
Wilson and I both found different practical applications for ratios 
involving the 41st harmonic back in the 1970's, so I find it rather 
nice to be able to notate this in 217. >>

Inasmuch as the sL flag *is* the 5-comma, what you now suggest is 
exactly what I originally proposed to do for 217-ET.  So, yes, we are 
in agreement on this.  (And I don't see how anybody could have a 
problem with an error of only 0.26 cents.)


> If we do that we eliminate one major reason for choosing (17'-17) 
as our
> final comma (over 17'-19 or simply 17'). No other comma symbols 
depend on
> it. But it is the only one that has good complementation rules in 
217-ET.


In addition to this, I would argue in favor of the 17'-17 comma in 
that it nicely fills the size gap between the 19 and 17 commas.  
(Although the 17'-19 comma does fill the size gap between the 17 and 
17' commas, the combination of 17+19 can also do this.)  Who knows 
what interval someone might want in the future (e.g., to notate 
2deg224 as vR or 2deg311 as vL+vR), and having the 17'-7 comma just 
might make their day.


> Actually, it might be better to stop at 31, since symbols with more 
than 2
> flags (e.g. 37') are getting too difficult, for my liking.


At least we could list these as possiblilities for applications in 
which precise higher-prime ratios are desired (e.g., for computer 
music in which ASCII versions of the comma-symbols might be used as 
input to achieve the appropriate frequencies) -- just to say we've 
covered as many of the bases as possible.


> I've uploaded a new version of
> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/SymbolsBySize.bmp
> based on the first option for each symbol, up to the prime 41, in 
the table
> above.
> 
> I realised recently that some of those alternate commas (the primed 
ones
> that are intended for a diatonic-based notation) should not really 
be
> defined as they currently are, but as their apotome complements, 
because
> that's how they will be used. They are 17', 19', 23' and 25. Let's 
call the
> apotome complements of these 17", 19", 23" and 25". For diatonic-
based
> purposes, these should be defined as 17:18, 18:19, 23:24 and 24:25
> respectively, and should be assigned appropriate double-shaft 
symbols.
> 
> The question is, can their symbols be sensibly based on the 
complementation
> rules which we derived in the context of 217-ET?


Before answering this question, let me present a rationale for 
selection of a standard set of 217-ET symbols.

In the standard (or preferred) set of symbols for 217-ET, we will 
want to follow the complementation rules strictly.  We will also want 
to use the same sequence of flags in the second half-apotome as 
occurs in the corresponding (i.e., 2-to-10-degree) portion of the 
first half-apotome.  There are two ways in which this can be 
accomplished (with the differences indicated by asterisks next to the 
degree number in the first column):

deg  Plan A   Plan B
--------------------
 1     |v       |v
 2    w|       w| 
 3*    |w      w|v
 4    s|       s| 
 5     |x       |x    (or  s|v)
 6     |s       |s
 7*   s|w      w|x
 8    w|s      w|s
 9    s|x      s|x
10    s|s      s|s
11    x|x      x|x
12    x|s      x|s
13    w||      w||
14*    ||w     w||v
15    s||      s||
16     ||x      ||x   (or  s||v)
17     ||s      ||s
18*   s||w     w||x
19    w||s     w||s
20    s||x     s||x
21    s||s     s||s

Note:  The symbols |x and s|v, which convert to complements of s||v 
and ||x, respectively, are virtual equivalents of one another, 
differing by the schisma 163840:163863, ~0.243 cents.  This enables 
||x to be used (in either plan) as both the 217-ET and the JI 
complement of |x.

Plan A is essentially different from plan B *only* in the symbol 
chosen for 3deg:  |w vs. w|v.  The other differences are derived from 
from this as follows:

1) The aptotome complements (or 20deg) for 3deg in plan A and plan B 
are s||w and w||x, respectively.

2) Keeping a uniform flag sequence between the half-apotomes, the 
flags for 14deg must match those for 3deg, i.e., ||w and w||v, 
respectively.

3) Keeping a uniform flag sequence between the half-apotomes, the 
flags for 7deg must match those for 18deg, i.e., s|w and w|x, 
respectively.

Plan A has four more pairs of laterally confusible symbols than does 
plan B:  between 2 and 3deg, 7 and 8deg, 13 and 14deg, and between 18 
and 19deg.  This would make plan A less desirable than plan B.

Although it might be considered more desirable to use a single-flag 
rather than a double-flag symbol for 3deg, the combination (as the 
sum of the 1deg and 2deg symbols) is easier to remember.

The sequence of symbols in plan B beginning with 5deg and continuing 
through 12deg (and likewise for 16 through 21deg) is rather simple to 
memorize, since the right flags alternate between convex and 
straight, while the left flags change every second degree.  The 
sequence in plan A appears more random.

It is also interesting to note that plan B uses the lowest possible 
prime symbols, avoiding altogether those that define the 19 and 23 
commas.

For this reason, I would consider plan B as the standard set of 217-
ET symbols.

Of course, the 23-comma (wR) flag would still follow the 
complementation rules that you gave earlier (in msg. #4071), with the 
flags being:

         |  Left    Right
---------+---------------
Convex   |  29        7
Straight |   5     (11-5)
Wavy     |  17       23
Concave  |  19     (17'-17)

and the complementation rules being:

               Complementary
Flag    Size   Size   Flag
comma   in steps of   comma
name      217-ET      name
----------------------------
Left
----
29       6   -2       none available with same side and direction
5        4    0       blank
17       2    2       17
19       1    3       none available with same side and direction

Right
-----
7        5    1       (17'-17)
(11-5)   6    0       blank
23       3    3       23
(17'-17) 1    5       7

By modifying the complementation rules slightly, the following 
additional pairs of JI and auxiliary 217-ET complements may be 
defined having the vL and xL flags:

apotome - v|  = x||w
apotome - v|w = x||
apotome - x|  = v||w
apotome - x|w = v||

Note that the right wavy (23-comma) flag involved here is not used in 
the standard set of symbols in plan B, so it would be a simple matter 
to remember that any complements involving this flag are not among 
the standard 217-ET set.

Now, to repeat your question:

<< I realised recently that some of those alternate commas (the 
primed ones that are intended for a diatonic-based notation) should 
not really be defined as they currently are, but as their apotome 
complements, because that's how they will be used. They are 17', 19', 
23' and 25. Let's call the apotome complements of these 17", 19", 23" 
and 25". For diatonic-based purposes, these should be defined as 
17:18, 18:19, 23:24 and 24:25 respectively, and should be assigned 
appropriate double-shaft symbols.

The question is, can their symbols be sensibly based on the 
complementation rules which we derived in the context of 217-ET? >>

Yes, three of the four will convert consistently, as follows:

apotome - 17' = 17:18, by converting w|v to w||x
apotome - 19' = 18:19, by converting v|w to x||
apotome – 23' = 23:24, by converting w|s to w||

And the fourth one, which is not a new prime, can still be 
represented as:

apotome - 25  = 24:25, by w|| (non-unique, but consistent)

which should be okay, since 217-ET is unique only through the 19 
limit anyway.

--George

Message: 4727 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 21:29:46

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4161]:

>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> The problem is that that middle line needs to be noticed as much 
as 
>> the other two, so that we can see that there are three of them, 
and 
>> making it shorter tends to de-emphasize it.
> 

> I think we need a third (and fourth and fifth ...) opinion on this 
one.
> From a performer who sight reads.
> 

>> Why don't we just make all three of the arrow shafts the same 
length, 
>> and I'll forget about making the symbols shorter than 17 pixels.
> 

> I have found our cooperation on this notation to be remarkable ego-
less,
> with both of us concerned only with what will be best for the end-
user, and
> not concerned with "getting our own way". But we've always given 
reasons
> for rejecting the other's proposal, so as to avoid any hurt 
feelings. I
> feel that any compromises we have made so far, e.g in lengths, 
widths,
> thicknesses or curvatures, have been made because we believe the 
best
> option most likely lies in between our two extremes.
> 
> Now I may be reading it wrong, but the above seems to be suggesting 
a
> trade-off of two completely unrelated things, purely on the 

basis, "you let

> me have my way on this and I'll let you have your way on that". If 
we can't
> agree on something, I'd prefer to seek other opinions, rather than 
engage
> in such a tradeoff.


If you look at my reason for suggesting that the symbols be shortened 
to less than 17 pixels, you will then see that the two things are 
closely related (from my message #4133):

<< I believe that shortening the middle line makes it more difficult 
to see it, thereby making it *more* difficult to distinguish three 
from two.  This is particularly true when the symbol modifies a note 
on a line and the middle line terminates at a staff line (so you see 
only two lines sticking out).  In fact, after looking at this again, 
I think I would be in favor of shorting all of the symbols from 17 to 
16 pixels so that no vertical line would terminate at a staff line.  
(This would also keep symbols modifying notes a fifth apart from 
colliding.  But you made a comment below regarding how the length of 
a new symbol looks when placed beside a conventional flat, so I need 
to evaluate this further.) >>

In other words, if vertical lines terminate at a staff line, they 
might not appear to "stick out" as much as they would if they fell 
one pixel short of the staff line.  However, while a (shorter) middle 
line *would not* terminate at a staff line in instances where the 
outer lines *do*, it *would* in instances where the outer ones 
*don't*, making it doubly obscure by the point of its termination 
*and* by its shorter length.  What I was advocating in the above 
paragraph was both *shorter* and *equal-length* lines, and if I had 
to concede one of the two, then it would be the shortness, but not 
the equal length.

For the life of me, I just can't understand how you are so insistent 
that something can be made more noticeable by making it *smaller* or 
*shorter*, especially when you *don't even want* symbols with triple 
shafts or X's.  Would Ted Mook have been able to read a Tartini 
sesquisharp more easily by making its center vertical line shorter?  
I would think that the change would make it more confusible with a 
conventional sharp.  I have done quite a bit of sight-reading in my 
time, both on keyboard and wind instruments, and I think that I'm 
arguing in the best interest of the end-user.

Quote for the day:  "Be reasonable -- do it my way."

Anyway, if we can't agree on this, and if you think I haven't given 
good enough reasons, then we should get some opinions from a few 
other people.


>> So would you then be satisfied with a difference in width alone to 
>> aid in making the lateral distinction?
> 
> Yes.


Okay, that's one more thing on which we can agree!


>> Why not just go with my version of the concave symbols:
>> 
>> Yahoo groups: /tuning- * [with cont.] 
>> math/files/secor/notation/SymAllSz.bmp
> 

> As I wrote in
> Yahoo groups: /tuning-math/message/4117 * [with cont.] 
> 
> I'm not averse to a slight recurve on the concaves, but I'm afraid 
I find
> your current proposals so extreme in this regard, that they are 
quite
> ambiguous in their direction. With a mental switch akin to the 
Necker cube
> illusion, I can see them as either a recurved concave pointing 

upwards or a

> kind of wavy pointing down. Apart from any nub, I don't think that 
they
> should go more than one pixel back in the "wrong" direction. Those 
at the
> extreme lower left of Symbols2.bmp look ok.


I don't understand this -- the symbols that you seem to be referring 
to each have the curve going upward 6 pixels from its lowest point, 
yet you think that they are okay?  Or perhaps you are referring to 
the "wrong" direction laterally?

In my subsequent file SymAllSz I made the left flag symbol one pixel 
narrower and the nub on the right flag symbol smaller (which I think 
we would both consider an improvement, even if that has nothing to do 
with the "wrong" direction).  My next comment refers to this file:


>> (see upper right, top staff)?  The left flag is 3 pixels wide, and 
>> the right flag is 4 pixels wide, yet they are clearly 
identifiable.  
>> (I also threw in a complement symbol.)
> 

> These are 4 and 5 pixels wide by my reckoning (including the part
> coincident with the shaft). One must define a flag as including a 
part
> coincident with the shaft so one knows what it will look like when 
it is
> sharing a || or X shaft with another flag. But I did overstate the 
case
> when I said that 4 pixels wide doesn't work. We now have agreement 
that the
> concaves on a single shaft should both be 4 pixels wide.


I don't think we do.  Referring again to SymAllSz, at the right end 
of the bottom staff my vL flag is 4 pixels and my vR flag is 5, and 
the combination of 4+5 is a total of 8 (the center pixel being 
occupied by both).  If you reduce the total width by 1 pixel by 
putting a mirror image of the left pixel on the right side, you will 
find that the result looks like a U-shaped curve rather than two 
separate curves, so I would not recommend this.  (However, see what I 
have to say below.)


> Maybe I went too far in reducing the height of the wavy and concave 
to 7
> pixels (including shortening the shaft at the pointy end). I see 
that your
> concaves are 9 pixels high, and your wavys are 10 pixels high, only 
one
> pixel shorter than the straight and concaves. In fact when it isn't 
used
> with the left wavy, your right wavy is the full 11 pixels in height 
and 6
> pixels wide. I think these lead to too many symbols whose apparent 
visual
> size is too far out of keeping with their size in cents.
> 
> I find that:
> 9.4 14.7 20.1 all look bigger than 21.5
> 27.5 and 30.6 look bigger than 31.8
> 
> I am proposing something between yours and mine. See
> 
> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/SymbolsBySize.bmp


Okay, those look good, including the vL+vR symbol.  Let's go with 
them!


> I don't think we actually need any lateral distinction between the 
two
> concaves because in rational tunings the (17'-17) flag will never 
occur on
> its own, and I don't think any ETs of interest below 217-ET will 
need to
> use both 19 and (17'-17). What do you think? 


I'd keep them distinct -- you'll never know what somebody is going to 
want to use this notation for (assuming that anyone is going to use 
it at all).


> But it wouldn't hurt if they were distinct. The biggest problem 

(for me) is

> trying to make the 19 comma (left concave) look as small as it 
really is
> without it disappearing. If its width was in proportion to the 

width of the

> 5 comma flag, you wouldn't see it for the shaft! If we look at 
areas and
> ignore the part coincident with the shaft, the 5 comma flag is 4 
pixels by
> 11 pixels. The 19 comma flag would have to fit in a rectangle 7 
pixels in
> area. In my 22-Apr proposal I've allowed those 7 pixels to blow out 
to 12,
> 3 wide by 4 high (excluding shaft). Yours is 18 pixels, 3 wide by 6 
high.
> 
> If we simply count black pixels (excluding shaft) we find that the 
5 comma
> flag has 15, which means the 19 comma flag should only have 2.4, 
which we
> might generously round up to 3 black pixels. Mine has 7, yours has 
9.
> 
> So my (17'-17) (right concave) flag is about the right size, but my 
19 flag
> is about double the size it should be. I can live with double, but 
I'n not
> sure I can handle triple.
> 
> Now you probably think I'm being too literal with this 
representative size
> stuff, but the problems occur when you have the 19 flag combined 
with
> another flag and the result looks much bigger than some single flag 
that it
> should be much smaller than. In particular 
> 9.4 looking bigger than 21.5, 
> 20.1 looking bigger than 27.3.
> 
> I suppose we can have a 19 comma flag that is lrger when used alone 
than
> when combined with others, but I'd prefer not.


I think that the way you have it is fine -- just so it's big enough 
to see.  After all, it's not going to be confused with anything else.


>> I wouldn't make the vertical arrow shaft shorter, though.
> 
> OK.
> 

>> To the right of our convex symbols are our latest versions of the 
>> wavy flags for comparison.  I made the left wavy flag 4 pixels 
wide, 
>> like the concave right flag, and the right wavy flag is 5 pixels 
>> wide.  Both of our wL+wR symbols have flags 4 pixels wide on each 
>> side.  
> 
>> As with the concave symbols, I also threw in a complement 
>> symbol.
> 

> Ah, but what exactly are they complements _of_?


I didn't have anything in mind at the time -- I just wanted to 
illustrate the curve with 2 shafts.  (However, my last posting *does* 
include this one as the complement of x|w.)


> I assume it was an oversight that left the wavy side of the 36.0 
symbol
> unmodified.


Yes.  I was a bit hasty.


>> I also experimented with taking the curves out of the wavy 
symbols, 
>> making them right-angle symbols, which I put at the far right.  
(The 
>> left vs. right line lengths are different in both the horizontal 
and 
>> vertical directions to aid in telling them apart.)  We already 
have 
>> two kinds of curved-line symbols, and substituting these for the 
wavy 
>> symbols would give us two kinds of straight-line symbols as well.  
>> It's not that I don't like the wavy symbols (I do like them), but 
I 
>> thought that this would make it easier -- both to remember and to 
>> distinguish them.  (This one's your call.)
> 

> You're right about them being more distinct, but the aesthetics are 
the
> killer. Given more resolution, I'd go for something in between the 
existing
> wavys and these right-angle ones, but not these totally sharp 
corners.


You're right about the aesthetics -- that's the reason why I also 
prefer wavy to right-angle symbols.  If you're happy with what you 
have now (they look like mine from the staff above), then we can go 
with them.


>> I copied your symbols (unaltered) into the second staff.  Below 
that 
>> I put my versions of the symbols for comparison.
>> 
>> I found that when I draw *convex* flags free-hand that I tend to 
>> curve the end of the flag inward slightly to make sure that it 
isn't 
>> mistaken for a straight flag, and I have been doing something on 
this 
>> order for some time with my bitmap symbols as well.  I have 
modified 
>> these also to reflect this, and you can let me know what you 
think. 
> 

> I think they look good, aesthetically speaking. The trouble is it 
makes the
> down versions look too much like flats and backward flats. Also, you
> decreased the size difference between the 7 flag and the 29 flag by 
adding
> curvature on the outside of the 7 flag and the inside of the 29 
flag.
> 
> I find the fact that the convex flags start off at right-angles to 
the
> shaft and end parallel to the shaft, sufficient to make them 
distinct from
> straight flags, without tending towards flats.


Okay, we can leave them as they were.  It was just a suggestion.


>> (I notice that the right flag of your 47.4-cent symbol has this 
sort 
>> of feature -- was that a mistake?)
>                   

> That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted 
from
> mindlessly overlaying wR and xR. Being 37', my heart wasn't in it. 
I've had
> a better go at it now, based on what you did for 25 and 31'.


Yes, now I can tell what it's supposed to be.


>> Or possibly only the left convex 
>> flag could be given this feature to further distinguish it from 
the 
>> convex right flag.
> 

> That would at least retain the full 2 pixel difference in width 
between XL
> and xR, but still has the problem of looking too much like a 
backwards flat.
> 
> There is a way to make the convex more distinct from straight 
without
> taking them closer to flats. We make them closer to being right-
angles,
> i.e. reduce the radius of the corner. I've shown comparisons with 
straight
> flags and flats at top right of my latest bitmap.


I like the original version (DK22) better.  I wasn't having any 
problem with the bitmap distinction -- it was just when I was drawing 
them freehand that sometimes they didn't look as different from 
straight flags as I would have liked them to.  But that's no reason 
to change the bitmap version; what you originally had looks better, 
so leave well enough alone!


>> Also, observe my 43-cent and 55-cent symbols -- the ones with two 
>> flags on the same side.
> 

> Yes. I wasn't very happy with mine. I like yours better, but I've 
modified
> them very slightly. Tell me what you think.


The 43-cent one is good (it looks like the one I did).  For the 55-
cent symbol, why don't you try removing the top straight flag from 
the 43-cent symbol and adding a reversed 27.3-cent symbol to it, so 
that the two flags cross.  (Also try the same thing with one of my 
27.3-cent symbols reversed and see if you that the effect is even 
better, since the two tend to cross more at right angles.)


> Notice that it's OK for 31' down to look like a backwards flat, 
because it
> _is_ a half-flat.

Sure!


>> Yes, good point, and one reason why I'm not reluctant to discard 
the 
>> idea of making the symbols any shorter than 17 pixels.  When I put 
a 
>> 5-comma-down symbol next to a flat the new symbol has a shorter 
stem 
>> than the flat.  I don't think that this is inappropriate, inasmuch 
as:
>> 
>> 1) the two symbols are in about the same proportion length-to-
width; 
>> and
>> 
>> 2) the difference in height is the same as that in the two 
vertical 
>> lines of a conventional sharp symbol.
> 

> Good points. OK. I'll forget the idea of giving down arrows longer 
shafts
> than up arrows. Are we agreed then that all sagittals should be 17 
pixels
> high?


Okay, then, even if we don't agree about the equal vs. unequal 3 
vertical lines.


>> And I can't imagine that anyone else has ever worked out a 
notation 
>> in this much detail.
> 
> Me neither.
> 

> It will of course be rejected out of hand by others, for reasons we 
haven't
> even considered. :-)


But of course.  We can't think of everything or please everybody, can 
we?  We just do the best we can.

--George

Message: 4728 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 18:32:01

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 22:15 24/04/02 -0000, George Secor wrote:

>Okay, I'm with you 100 percent on this now.  (I haven't checked all 
>of these schismas, but trust that you have been thorough with this.)  
>Something that I especially like is that everything through the 29 
>limit works without requiring two flags on the same side.


Yes. That is more than I would have expected if you'd asked me at the
start. It's certainly a nice vindication of your sagittal idea.


>> pythagorean
>> comma = 17 + 17 + (17'-17)      0
>> diaschisma = 19 + 23            0.37        [same symbol as 19']
>> diesis = 17  +  (11-5)          0.56        [same symbol as 23']
>> 
>> * doesn't vanish in 1600-ET.
>
>Very nice!


I gave up on trying to make an actual symbol for the pythagorean comma
based on the above identity. Maybe you want to have a go.


>Inasmuch as the sL flag *is* the 5-comma, what you now suggest is 
>exactly what I originally proposed to do for 217-ET.  So, yes, we are 
>in agreement on this.


Great! Sorry I forgot your original proposal re 41.


>  (And I don't see how anybody could have a 
>problem with an error of only 0.26 cents.)


Just don't say that too loudly around here. 


>> If we do that we eliminate one major reason for choosing (17'-17) 
>as our

>> final comma (over 17'-19 or simply 17'). No other comma symbols 
>depend on

>> it. But it is the only one that has good complementation rules in 
>217-ET.
>

>In addition to this, I would argue in favor of the 17'-17 comma in 
>that it nicely fills the size gap between the 19 and 17 commas. 

Yes.
 
>(Although the 17'-19 comma does fill the size gap between the 17 and 
>17' commas, the combination of 17+19 can also do this.)


Yes. As you have probably read by now, I am proposing precisely that; a
17+19 symbol (2 left flags) to serve as the rational complement of 25.


>Who knows 
>what interval someone might want in the future (e.g., to notate 
>2deg224 as vR or 2deg311 as vL+vR), and having the 17'-7 comma just 
>might make their day.


Yes. Or 2deg453 as vR and 4deg453 as vL+wL.

There is no doubt in my mind now, that (17'-17) is the best choice for the
last flag. We're in complete agreement now on what the 8 flags mean (when
on a single shaft).


>> Actually, it might be better to stop at 31, since symbols with more 
>than 2

>> flags (e.g. 37') are getting too difficult, for my liking.
>

>At least we could list these as possiblilities for applications in 
>which precise higher-prime ratios are desired (e.g., for computer 
>music in which ASCII versions of the comma-symbols might be used as 
>input to achieve the appropriate frequencies) -- just to say we've 
>covered as many of the bases as possible.


Yes we should list them, but beyond 31 we do not have unique symbols. 35 is
also 13, 37 is also 25, 41 is also 5, so the above application wouldn't work.

One minor point to note in connection with relegating primes above 31 to
second-class citizen status is that the 37' symbol _is_ unique, and I'm
currently using it as the rational complement of the 17 symbol. But there's
no need to call it 37' in that context, and anyway a different 3-flag
symbol may turn out to be better as the rational complement of the 17 symbol.


>In the standard (or preferred) set of symbols for 217-ET, we will 
>want to follow the complementation rules strictly.  We will also want 
>to use the same sequence of flags in the second half-apotome as 
>occurs in the corresponding (i.e., 2-to-10-degree) portion of the 
>first half-apotome.

Agreed.

>There are two ways in which this can be 
>accomplished (with the differences indicated by asterisks next to the 
>degree number in the first column):
>
>deg  Plan A   Plan B
>--------------------
> 1     |v       |v
> 2    w|       w| 
> 3*    |w      w|v
> 4    s|       s| 
> 5     |x       |x    (or  s|v)
> 6     |s       |s
> 7*   s|w      w|x
> 8    w|s      w|s
> 9    s|x      s|x
>10    s|s      s|s
>11    x|x      x|x
>12    x|s      x|s
>13    w||      w||
>14*    ||w     w||v
>15    s||      s||
>16     ||x      ||x   (or  s||v)
>17     ||s      ||s
>18*   s||w     w||x
>19    w||s     w||s
>20    s||x     s||x
>21    s||s     s||s
>
>Note:  The symbols |x and s|v, which convert to complements of s||v 
>and ||x, respectively, are virtual equivalents of one another, 
>differing by the schisma 163840:163863, ~0.243 cents.  This enables 
>||x to be used (in either plan) as both the 217-ET and the JI 
>complement of |x.

Agreed.

>Plan A is essentially different from plan B *only* in the symbol 
>chosen for 3deg:  |w vs. w|v.  The other differences are derived from 
>from this as follows:
>
>1) The aptotome complements (or 20deg) for 3deg in plan A and plan B 
>are s||w and w||x, respectively.


s||w works as a rational complement to |w, but w||x doesn't work as
rational complement to w|v. Instead I propose w||s (or possibly x||v) as
rational complement of w|v.

However this is fairly irrelevant since neither plan A nor plan B can agree
with the rational complement rules, since rational complementation must
deny that wavy left is its own flag-complement.


>2) Keeping a uniform flag sequence between the half-apotomes, the 
>flags for 14deg must match those for 3deg, i.e., ||w and w||v, 
>respectively.
>
>3) Keeping a uniform flag sequence between the half-apotomes, the 
>flags for 7deg must match those for 18deg, i.e., s|w and w|x, 
>respectively.
>
>Plan A has four more pairs of laterally confusible symbols than does 
>plan B:  between 2 and 3deg, 7 and 8deg, 13 and 14deg, and between 18 
>and 19deg.  This would make plan A less desirable than plan B.
>
>Although it might be considered more desirable to use a single-flag 
>rather than a double-flag symbol for 3deg, the combination (as the 
>sum of the 1deg and 2deg symbols) is easier to remember.
>
>The sequence of symbols in plan B beginning with 5deg and continuing 
>through 12deg (and likewise for 16 through 21deg) is rather simple to 
>memorize, since the right flags alternate between convex and 
>straight, while the left flags change every second degree.  The 
>sequence in plan A appears more random.
>
>It is also interesting to note that plan B uses the lowest possible 
>prime symbols, avoiding altogether those that define the 19 and 23 
>commas.
>
>For this reason, I would consider plan B as the standard set of 217-
>ET symbols.


I'm convinced. Plan B it is.


>Of course, the 23-comma (wR) flag would still follow the 
>complementation rules that you gave earlier (in msg. #4071), with the 
>flags being:
>
>         |  Left    Right
>---------+---------------
>Convex   |  29        7
>Straight |   5     (11-5)
>Wavy     |  17       23
>Concave  |  19     (17'-17)
>
>and the complementation rules being:
>
>               Complementary
>Flag    Size   Size   Flag
>comma   in steps of   comma
>name      217-ET      name
>----------------------------
>Left
>----
>29       6   -2       none available with same side and direction
>5        4    0       blank
>17       2    2       17
>19       1    3       none available with same side and direction
>
>Right
>-----
>7        5    1       (17'-17)
>(11-5)   6    0       blank
>23       3    3       23
>(17'-17) 1    5       7
>
>By modifying the complementation rules slightly, the following 
>additional pairs of JI and auxiliary 217-ET complements may be 
>defined having the vL and xL flags:
>
>apotome - v|  = x||w
>apotome - v|w = x||
>apotome - x|  = v||w
>apotome - x|w = v||


These all agree with my proposed rational complements.


>Note that the right wavy (23-comma) flag involved here is not used in 
>the standard set of symbols in plan B, so it would be a simple matter 
>to remember that any complements involving this flag are not among 
>the standard 217-ET set.

Right.

>Now, to repeat your question:
>
><< I realised recently that some of those alternate commas (the 
>primed ones that are intended for a diatonic-based notation) should 
>not really be defined as they currently are, but as their apotome 
>complements, because that's how they will be used. They are 17', 19', 
>23' and 25. Let's call the apotome complements of these 17", 19", 23" 
>and 25". For diatonic-based purposes, these should be defined as 
>17:18, 18:19, 23:24 and 24:25 respectively, and should be assigned 
>appropriate double-shaft symbols.
>
>The question is, can their symbols be sensibly based on the 
>complementation rules which we derived in the context of 217-ET? >>
>
>Yes, three of the four will convert consistently, as follows:
>
>apotome - 17' = 17:18, by converting w|v to w||x


I propose instead that 17:18 should be w||s.


>apotome - 19' = 18:19, by converting v|w to x||

Agreed.

>apotome – 23' = 23:24, by converting w|s to w||


I propose instead that 23:24 should be w||v. This is the inverse of the 17'
complement.


>And the fourth one, which is not a new prime, can still be 
>represented as:
>
>apotome - 25  = 24:25, by w|| (non-unique, but consistent)
>
>which should be okay, since 217-ET is unique only through the 19 
>limit anyway.


By now I guess you've read my rational complement proposal based on 453-ET.
I'd prefer 24:25 to have a unique symbol and have proposed a new symbol
wv|| for this.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4729 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 21:41:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 22:17 24/04/02 -0000, you wrote:

>If you look at my reason for suggesting that the symbols be shortened 
>to less than 17 pixels, you will then see that the two things are 
>closely related (from my message #4133):

My apologies.

>For the life of me, I just can't understand how you are so insistent 
>that something can be made more noticeable by making it *smaller* or 
>*shorter*, especially when you *don't even want* symbols with triple 
>shafts or X's.  Would Ted Mook have been able to read a Tartini 
>sesquisharp more easily by making its center vertical line shorter?  
>I would think that the change would make it more confusible with a 
>conventional sharp.  I have done quite a bit of sight-reading in my 
>time, both on keyboard and wind instruments, and I think that I'm 
>arguing in the best interest of the end-user.


You're certainly more qualified than me in that regard. It was the way the
middle stroke is always shortened in an uppercase E that got me thinking.
Also, I find that something with an apparent V notched out of its tail is
somewhat distinct from something with a square tail, no matter the number
of shafts.


>Quote for the day:  "Be reasonable -- do it my way."


Good one. ;-)


>Anyway, if we can't agree on this, and if you think I haven't given 
>good enough reasons, then we should get some opinions from a few 
>other people.


I've emailed Ted Mook for his opinion. You should have received a copy.


>I don't understand this -- the symbols that you seem to be referring 
>to each have the curve going upward 6 pixels from its lowest point, 
>yet you think that they are okay?  Or perhaps you are referring to 
>the "wrong" direction laterally?


Sorry. I must have screwed up here. I was probably looking at a version of
Symbols2.bmp I had already edited myself and forgotten. None of the
concaves in it look ok to me. The ""wrong" direction" was referring to
vertical direction only.


>In my subsequent file SymAllSz I made the left flag symbol one pixel 
>narrower and the nub on the right flag symbol smaller (which I think 
>we would both consider an improvement, even if that has nothing to do 
>with the "wrong" direction).


Yes I found it to be an improvement.


>> I am proposing something between yours and mine. See
>> 
>> Yahoo groups: /tuning- * [with cont.] 
>math/files/Dave/SymbolsBySize.bmp
>

>Okay, those look good, including the vL+vR symbol.  Let's go with 
>them!

Yikes! OK.


>> I don't think we actually need any lateral distinction between the 
>two

>> concaves because in rational tunings the (17'-17) flag will never 
>occur on

>> its own, and I don't think any ETs of interest below 217-ET will 
>need to

>> use both 19 and (17'-17). What do you think? 
>

>I'd keep them distinct -- you'll never know what somebody is going to 
>want to use this notation for (assuming that anyone is going to use 
>it at all).

OK. Yeah.


>> I suppose we can have a 19 comma flag that is lrger when used alone 
>than

>> when combined with others, but I'd prefer not.
>

>I think that the way you have it is fine -- just so it's big enough 
>to see.  After all, it's not going to be confused with anything else.

OK. Great.

>You're right about the aesthetics -- that's the reason why I also 
>prefer wavy to right-angle symbols.  If you're happy with what you 
>have now (they look like mine from the staff above), then we can go 
>with them.


Yes. They are yours. Except I think I took one pixel off the end of the
right hand ones so they are the same height as the left ones.


>> That's 37' = 19 + 23 + 7 = vL + wR + xR, so what you saw resulted 
>from

>> mindlessly overlaying wR and xR. Being 37', my heart wasn't in it. 
>I've had

>> a better go at it now, based on what you did for 25 and 31'.
>

>Yes, now I can tell what it's supposed to be.

Good.


>>> Or possibly only the left convex 
>>> flag could be given this feature to further distinguish it from 
>the 

>>> convex right flag.
>> 

>> That would at least retain the full 2 pixel difference in width 
>between XL

>> and xR, but still has the problem of looking too much like a 
>backwards flat.
>> 

>> There is a way to make the convex more distinct from straight 
>without

>> taking them closer to flats. We make them closer to being right-
>angles,

>> i.e. reduce the radius of the corner. I've shown comparisons with 
>straight

>> flags and flats at top right of my latest bitmap.
>

>I like the original version (DK22) better.  I wasn't having any 
>problem with the bitmap distinction -- it was just when I was drawing 
>them freehand that sometimes they didn't look as different from 
>straight flags as I would have liked them to.  But that's no reason 
>to change the bitmap version; what you originally had looks better, 
>so leave well enough alone!


I was getting to like the squarer DK23 ones, as more distinct from flats,
but OK. Does this mean we now have a full set of single-shaft symbols that
we both find acceptable? I think maybe we're still tinkering with some of
the two-flags-on-the-same-side ones. What do you think of the new vw|
symbol, the complement to the ss|| symbol?
 

>The 43-cent one is good (it looks like the one I did). 


Yes I only moved a few pixels so it looks smoother in all alignments.


>For the 55-
>cent symbol, why don't you try removing the top straight flag from 
>the 43-cent symbol and adding a reversed 27.3-cent symbol to it, so 
>that the two flags cross.  (Also try the same thing with one of my 
>27.3-cent symbols reversed and see if you that the effect is even 
>better, since the two tend to cross more at right angles.)


I prefer them meeting, rather than crossing. Why do you like the crossing?

What I imagine happening when two flags are combined on one side, is that
the two flags are scaled down to about two thirds of their height including
scaling the vertical line thickness. And they are scaled up (out) in the
horizontal direction slightly to compensate for the loss of area due to the
vertical scale-down. Then one of them is moved to the top of the available
space and the other to the bottom, and overlaid. Then a little bit of
license is used to make it look like something sensible and be sufficiently
distinct from everything else.

I thought I extracted that from what you were doing.


>But of course.  We can't think of everything or please everybody, can 
>we?  We just do the best we can.

Indeed.


-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4730 - Contents - Hide Contents

Date: Wed, 24 Apr 2002 06:59:25

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

Regarding the problem of apotome complement symbols for rational 
tunings, please see 
Yahoo groups: /tuning-math/files/Dave/Compleme... * [with cont.] 
It should be self-explanatory.

Message: 4731 - Contents - Hide Contents

Date: Thu, 25 Apr 2002 20:47:58

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> 
> ... I was more
> disappointed to find that the following 3 messages from you and 2 
from
> Paul, in this thread, were only trapezoid trivia. :-)


Well, sorry to disappoint you again today.  I'm having a time trying 
to keep up with answering all of this (besides being a week behind in 
reading the digests from 3 different tuning lists).  I need a day 
off, so I'm sending no big/serious message(s) today.


> Here's some more. The U.S. definitions of trapezoid and trapezium 
are
> exactly swapped relative to the British/Australian definitions.
> 
> In the OED and Macquarie dictionaries, a trapezium has only one 
pair of
> sides parallel, while a trapezoid has none. Websters has it the 
other way
> 'round. There's no requirement for any right-angles anywhere.


Interesting.  Well, at least we understand what we're both talking 
about, even if it doesn't come out quite right sometimes.  (By the 
way, do you know any more trapezoid jokes, or do you think I should 
just leave that subject and shut my trapezoid?)


> At least it's good to know Paul's reading the thread. I've been 
wondering
> whether no-one else was contributing because 
> (a) they think we're doing such a wonderful job without them, or
> (b) they have no interest whatsoever in the topic, and think we're 
a couple
> of looneys?


Speaking of Paul (at least I didn't put all seriousness aside and 
say "speaking off looneys", which would have been most unkind!), now 
that we have made such wonderful (and unexpected) progress agreeing 
on single-shaft standard 217-ET symbols, I realized that Paul's 
request to see an adaptive-JI progression (message #3950) can be 
filled.  Would you care to do the honors, or shall I?


> Hey, I've become so obsessed about this notation that I was lying 
in bed
> this morning thinking how my various sleeping postures could be 
read as
> various sagittal symbols. I was imagining children being taught them
> kinaesthetically. Sagittal aerobic workout videos by Jane Fonda! :-)


Now that's scary.  It sounds like you need a day off, too.


>>> I'm not averse to a slight recurve on the concaves, but I'm 

afraid I find

>>> some of those in symbols2.bmp, so extreme in this regard, that 
they are
>>> quite ambiguous in their direction. With a mental switch akin to 
the Necker
>>> cube illusion, I can see them as either a recurved concave 
pointing upwards
>>> or a kind of wavy pointing down. ...


Whoa! This sort of thing is too convoluted for me today.  Hopefully, 
I'll be back tomorrow for more -- more serious, that is.

--George

Message: 4732 - Contents - Hide Contents

Date: Thu, 25 Apr 2002 22:47:31

Subject: Re: A common notation for JI and ETs

From: David C Keenan

-----------------------------------------------------------------
The continuing search for the ideal rational complement symbols
-----------------------------------------------------------------

Hi George,

I was wrong about being able to notate 453-ET. It would need the addition
of a 22 step symbol |sx, which would be like the c31' symbol flipped
horizontally, and would look way too much like a conventional flat.
Fortunately notating 453-ET wasn't the point.

The point is that whatever rational complements we decide on, should also
be the true complements in some ET (I think). It doesn't matter whether we
have a symbol for every degree of that ET, in fact it's probably better if
we don't. I think the higher that ET is, the better, except that if we go
too high we find that too many symbols don't _have_ a complement. 

We know 217-ET doesn't work for rational complements because it is only
19-limit unique and so doesn't provide enough unique complements. 

I proposed 453-ET and found I needed an additional symbol vw|| to get a
complement for c25. But 453-ET isn't that great. It would be nice to use an
ET that was 31-limit unique, like our symbols. 

Thanks to Gene Smith's search for good 31-limit unique ETs, I tried 653-ET
and found that it works! It needed only the same additional symbol vw||,
but this time it is the complement of c23', not c25.

You can see the 653-ET complements in the latest version of
Yahoo groups: /tuning-math/files/Dave/Compleme... * [with cont.] 

4095:4096 doesn't actually vanish in 653-ET. It seems to be the only one of
our sub-symbol schismas that doesn't. So the complementarity of s|x (c13)
and ||v, (|v is c(17'-17)), is based on s|x being 653-ET's best 13-comma,
not on it being the sum of the 5 and 7 commas.

One thing to note is that in 653-ET |x and ||x are not complements.

All these rational complement schemes seem very unsatisfactory to me. And
the thing is, I don't think anyone will use them. Not even the relatively
simple ones in 217-ET. Just like I don't think anyone will use the ||| and
X shaft symbols. Expecting people to learn the single-shaft symbols is more
than enough. A sharp with a down symbol next to it is going to be way more
easily parsed than some double shaft symbol with a combination of flags
that they are used to associating with some other prime when on a single
shaft (or more likely have never seen before), and if taken as simply a
number of cents, must be added onto, not an actual half apotome, but an
11'-diesis.

So basically, I've now got what I wanted from this, and what I think the
microtonal world might want, but if you are still determined to persue
multishaft symbols I'm still willing to comment on your proposals.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4733 - Contents - Hide Contents

Date: Thu, 25 Apr 2002 06:22:59

Subject: Temperaments and notations

From: genewardsmith

Here is yet another twist on that useful device I've dubbed a "notation", this one finding a notation associated to a linear temperament.

If
we have a temperament defined in terms of an octave and period, where
the period is expressed in JI terms, then we may complete this to a
basis fora notation. The notation mapping then becomes an extended
mapping of generators to primes, which could be useful for planar
temperaments.

For example, meantone temperament with generators <2, 3/2> and
kernel <81/80> leads to the notation basis <2,3/2,81/80>, the inverse
of which is


[1, 0, 0]
[1, 1, 0]
[0, 4, -1]

The coumns of this matrix are vals giving mappings to primes, and the
rows show the primes in terms of <2,3/2,81/80>.

For 7-limit Orwell, we have generators <2, 7/6> and a reduced basis <225/224, 1728/1715>. Putting these together to form the 
notation basis <2,7/6,225/224,1728/1715> and taking the inverse gives
us

[1, 0, 0, 0]
[0, 7, 1, 2]
[3, -3, 0, -1]
[1, 8, 1, 2]

The rows of this show us that

3 = (7/6)^6 (225/224) (1728/1715)^2
5 = 2^3 (7/6)^(-3) (1728/1715)^(-1)
7 = 2 (7/6)^8 (225/224) (1728/1715)^2

We can now measure the rms steps to 7-limit consonances for 225/224
and 1728/1715, leading to .8165 and 2.1213 respectively. This shows
that keeping 1728/1715 as a comma leads to something closer to Orwell
than 225/224, even though 225/224 is a more significant comma in
general. 

This conclusion is strengthed by the fact that it 7/6 is a 7-limit
consonance which clearly is the generator; in the case of miracle, we
might choose either 15/14 or 16/15 for our generator, leading to
different maps. From <2,15/14,225/224,243/242,385/384> we get

[1, 0, 0, 0, 0]
[1, 6, -4, 1, 2]
[3, -7, 5, -1, -2]
[3, -2, 1, 0, 0]
[2, 15, -10, 2, 5]]


While from <2, 16/15, 225/224, 243/242, 385/384> we get

[1, 0, 0, 0, 0]
[1, 6, 2, 1, 2]
[3, -7, -2, -1, -2]
[3, -2, -1, 0, 0]
[2, 15, 5, 2, 5]

Similarly, from <2,15/14,225/224,1029/1024> we get

[1, 0, 0, 0]
[1, 6, -3, 1]
[3, -7, 4, -1]
[3, -2, 1, 0]

while from <2,16/15,225/224,1029/1024> we get

[1, 0, 0, 0]
[1, 6, 3, 1]
[3, -7, -3, -1]
[3, -2, -1, 0]

Even with the added uncertainty, it is clear 225/224 is the most
significant for miracle.

Finally, there is no requirement that octaves rather than fractions of
octaves be generators. For pajara, we have <2,7/5,50/49,64/63>,
leading to

[2, 0, 1, 0]
[2, 1, 1, 0]
[7, -2, 4, -1]
[8, -2, 4, -1]

Clearly 50/49 seems to be the more significant comma, so pajara would
be more closely related to the 50/49 planar temperament than to the
64/63 planartemperament.

Message: 4734 - Contents - Hide Contents

Date: Thu, 25 Apr 2002 06:37:35

Subject: Re: Temperaments and notations

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> If
we have a temperament defined in terms of an octave and period, where
the period is expressed in JI terms, then we may complete this to a
basis for a notation. The notation mapping then becomes an extended
mapping of generators to primes, which could be useful for planar
temperaments.


In these terms, what George and Dave are up to is related to a basis
for the 12-et version of meantone, for instance
<2,3/2,81/80,64/63,33/32>^(-1) gives us

[1, 0, 0, 0, 0]
[1, 1, 0, 0, 0]
[0, 4, -1, 0, 0]
[4, -2, 0, -1, 0]
[4, -1, 0, 0, 1]

Message: 4735 - Contents - Hide Contents

Date: Thu, 25 Apr 2002 06:37:35

Subject: Re: Temperaments and notations

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> If we have a temperament defined in terms of an octave and period, where =

the period is expressed in JI terms, then we may complete this to a basis f=
or a notation. The notation mapping then becomes an extended mapping of gen=
erators to primes, which could be useful for planar temperaments.

In these terms, what George and Dave are up to is related to a basis for th=
e 12-et version of meantone, for instance
<2,3/2,81/80,64/63,33/32>^(-1) gives us

[1, 0, 0, 0, 0]
[1, 1, 0, 0, 0]
[0, 4, -1, 0, 0]
[4, -2, 0, -1, 0]
[4, -1, 0, 0, 1]

Message: 4737 - Contents - Hide Contents

Date: Fri, 26 Apr 2002 01:15:18

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:

> Interesting.  Well, at least we understand what we're both talking 
> about, even if it doesn't come out quite right sometimes.  (By the 
> way, do you know any more trapezoid jokes, or do you think I should 
> just leave that subject and shut my trapezoid?)


Son, I think you oughta make like one a' them trapezoid monks I heered 
about.


>> At least it's good to know Paul's reading the thread. I've been 
> wondering

>> whether no-one else was contributing because 
>> (a) they think we're doing such a wonderful job without them, or
>> (b) they have no interest whatsoever in the topic, and think we're 
> a couple
>> of looneys?
> 

> Speaking of Paul (at least I didn't put all seriousness aside and 
> say "speaking off looneys", which would have been most unkind!), now 
> that we have made such wonderful (and unexpected) progress agreeing 
> on single-shaft standard 217-ET symbols, I realized that Paul's 
> request to see an adaptive-JI progression (message #3950) can be 
> filled.  Would you care to do the honors, or shall I?


Since you're taking the day off. I guess I'd better do it.


>> Hey, I've become so obsessed about this notation that I was lying 
> in bed

>> this morning thinking how my various sleeping postures could be 
> read as

>> various sagittal symbols. I was imagining children being taught 
them
>> kinaesthetically. Sagittal aerobic workout videos by Jane Fonda! 
:-)
> 

> Now that's scary.  It sounds like you need a day off, too.

Yeah.


>>>> I'm not averse to a slight recurve on the concaves, but I'm 

> afraid I find

>>>> some of those in symbols2.bmp, so extreme in this regard, that 
> they are

>>>> quite ambiguous in their direction. With a mental switch akin 
to 
> the Necker

>>>> cube illusion, I can see them as either a recurved concave 
> pointing upwards

>>>> or a kind of wavy pointing down. ...
> 

> Whoa! This sort of thing is too convoluted for me today.  Hopefully, 
> I'll be back tomorrow for more -- more serious, that is.


We've already dealt with that one. So you can relax.

Message: 4738 - Contents - Hide Contents

Date: Fri, 26 Apr 2002 19:02:32

Subject: Re: Adaptive JI notated on staff

From: jpehrson2

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

Yahoo groups: /tuning-math/message/4174 * [with cont.] 



> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

>> right, but i'd like to see this actually notated, on a staff.
> 

> Here it is.
> Yahoo groups: /tuning-math/files/Dave/Adaptive... * [with cont.] 


***What on earth is going on here?

Could we move some of this over to the "main" list for our 
appreciation??

jp

Message: 4739 - Contents - Hide Contents

Date: Fri, 26 Apr 2002 19:47:34

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> At 22:15 24/04/02 -0000, George Secor wrote:


This is just a quick comment on the single-shaft symbols.

I was just noticing how large a few of the symbols are in comparison 
to the conventional sharp and flat symbols.  I suggest making the 
convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and 
65.3-cent symbols.  (I tried it by replacing the left halves with the 
left half of the 60.4-cent symbol.  The 55.0 and 60.4-cent symbols 
can remain the way they are.)  I think that this alleviates the 
conventional-saggital size disparity somewhat, in addition to making 
a better size progression, but is not enough of a change to cause 
lateral confusibility.

And while I am on the subject of fine-tuning symbols:


> I gave up on trying to make an actual symbol for the pythagorean 
comma
> based on the above identity. Maybe you want to have a go.


Try this:  Make a copy of the 17' symbol (wL+vR).  Move all of the 
pixels in the 4 leftmost columns up 1 position (thus raising the wavy 
flag by 1).  Then copy these and paste them so that you have a second 
wavy flag 4 pixels lower than the top one.  The two wavy flags 
together are considerably smaller than a single 29 flag (even with my 
proposed reduction in size for the latter), and together they clearly 
indicate the staff position of the note being altered.


>>> Actually, it might be better to stop at 31, since symbols with 
more 
>> than 2

>>> flags (e.g. 37') are getting too difficult, for my liking.
>> 

>> At least we could list these as possiblilities for applications in 
>> which precise higher-prime ratios are desired (e.g., for computer 
>> music in which ASCII versions of the comma-symbols might be used 
as 
>> input to achieve the appropriate frequencies) -- just to say we've 
>> covered as many of the bases as possible.
> 

> Yes we should list them, but beyond 31 we do not have unique 

symbols. 35 is

> also 13, 37 is also 25, 41 is also 5, so the above application 
wouldn't work.


I don't see this as a problem -- the whole point of identifying the 
schismas is to minimize the number of symbols required for the 
notation, which by its very nature reduces and eventually eliminates 
uniqueness once the harmonic limit reaches a certain size.  With the 
schismas as small as they are, we are entitled to assert that, for 
all practical purposes, it is impossible to tell these intervals 
apart, whereby they no longer have separate and distinct identities 
(i.e., bridging at the point of inaudibility).  It is entirely 
appropriate for the notation to reflect this reality, and there 
should be no need to apologize for it.


> One minor point to note in connection with relegating primes above 
31 to
> second-class citizen status is that the 37' symbol _is_ unique, and 
I'm
> currently using it as the rational complement of the 17 symbol. But 
there's
> no need to call it 37' in that context, and anyway a different 3-
flag
> symbol may turn out to be better as the rational complement of the 
17 symbol.


I'll need to look at this further.


>> ... For this reason, I would consider plan B as the standard set 
of 217-
>> ET symbols.
> 

> I'm convinced. Plan B it is.


It's my turn to say yikes! OK.

And with that, I hope that Paul likes the adaptive JI figure that you 
made.  (I notice that the 17-comma wavy symbols possess the same 
slope directionality as the 5-comma symbols, which makes an effective 
emphasis of the direction of pitch alteration, particularly in the D-
minor triad.  (This same directionality cue also works with the wavy 
flags in the 3-degree symbols; and there would be little problem with 
misinterpreting the direction from the slope of the single-degree 
concave symbols, since a concave flag slopes both ways.)  These new 
symbols may take a little bit of time to get used to, but everything 
fits together so logically that it should be a relatively easy matter 
to learn the 12 symbols -- e.g., relative to learning the alphabet.

(Your reply continued with a discussion of rational complements, but 
I should leave off here until I have studied the rest of your 
messages.)

--George

Message: 4740 - Contents - Hide Contents

Date: Fri, 26 Apr 2002 08:14:46

Subject: Adaptive JI notated on staff

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> right, but i'd like to see this actually notated, on a staff.


Here it is.
Yahoo groups: /tuning-math/files/Dave/Adaptive... * [with cont.] 

Message: 4741 - Contents - Hide Contents

Date: Fri, 26 Apr 2002 20:00:15

Subject: Re: Adaptive JI notated on staff

From: gdsecor

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:

> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> Yahoo groups: /tuning-math/message/4174 * [with cont.] 
> 
> 

>> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

>>> right, but i'd like to see this actually notated, on a staff.
>> 

>> Here it is.
>> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/AdaptiveJI.bmp
> 

> ***What on earth is going on here?


Something really amazing -- In the course of our notational dicussion 
& debate, Dave Keenan and I have finally agreed over the past couple 
of days on quite a few things, and we now have something to show for 
it.


> Could we move some of this over to the "main" list for our 
> appreciation??
> 
> jp


If there's still any file space.  It's your move, Dave!

--George

Message: 4742 - Contents - Hide Contents

Date: Sun, 28 Apr 2002 04:20:21

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>> At 22:15 24/04/02 -0000, George Secor wrote:
>

>This is just a quick comment on the single-shaft symbols.
>
>I was just noticing how large a few of the symbols are in comparison 
>to the conventional sharp and flat symbols.  I suggest making the 
>convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and 
>65.3-cent symbols.  (I tried it by replacing the left halves with the 
>left half of the 60.4-cent symbol.  The 55.0 and 60.4-cent symbols 
>can remain the way they are.)  I think that this alleviates the 
>conventional-saggital size disparity somewhat, in addition to making 
>a better size progression, but is not enough of a change to cause 
>lateral confusibility.


I've done it. See 
Yahoo groups: /tuning-math/files/Dave/SymbolsB... * [with cont.] 

It _does_ increase lateral confusability somewhat.

Which is one thing that causes me to re-propose the convex with the
slightly squarer corners. Your only comment about them has been to "leave
well enough alone". Can you give a more detailed reason for rejecting them?


>And while I am on the subject of fine-tuning symbols:
>

>> I gave up on trying to make an actual symbol for the pythagorean 
>comma

>> based on the above identity. Maybe you want to have a go.
>

>Try this:  Make a copy of the 17' symbol (wL+vR).  Move all of the 
>pixels in the 4 leftmost columns up 1 position (thus raising the wavy 
>flag by 1).  Then copy these and paste them so that you have a second 
>wavy flag 4 pixels lower than the top one.  The two wavy flags 
>together are considerably smaller than a single 29 flag (even with my 
>proposed reduction in size for the latter), and together they clearly 
>indicate the staff position of the note being altered.


See the above file for my best attempt. Not precisely what you suggested,
but close.


I don't want to jump the gun and go to the main list just yet, and when I
do, I'll want a staff showing the odd harmonics of G up to 41, including
all optional spellings (using single shaft symbols with conventional sharps
and flats), as well as the 217-ET notation and a couple of other ETs.

You might want to check out
Shareware.com - truety * [with cont.]  (Wayb.)
pe+font+editor

and

http://www.sibelius.com

to get the free download which is fully functional except for save.
-- Dave Keenan
Brisbane, Australia
http://uq.net.au/~zzdkeena&tag=ex.sa.sr.srch.sa_all&q=truety
pe+font+editor

and

Welcome to Sibelius * [with cont.]  (Wayb.)

to get the free download which is fully functional except for save.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4743 - Contents - Hide Contents

Date: Mon, 29 Apr 2002 16:58:30

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4179]:

>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

>>> At 22:15 24/04/02 -0000, George Secor wrote:
>> 

>> I was just noticing how large a few of the symbols are in 
comparison 
>> to the conventional sharp and flat symbols.  I suggest making the 
>> convex left flag one pixel narrower for the 33.5, 39.5, 50.0, and 
>> 65.3-cent symbols.  (I tried it by replacing the left halves with 
the 
>> left half of the 60.4-cent symbol.  The 55.0 and 60.4-cent symbols 
>> can remain the way they are.)  I think that this alleviates the 
>> conventional-saggital size disparity somewhat, in addition to 
making 
>> a better size progression, but is not enough of a change to cause 
>> lateral confusibility.
> 

> I've done it. See 
> Yahoo groups: /tuning- * [with cont.] 
math/files/Dave/SymbolsBySize3.bmp
> 
> It _does_ increase lateral confusability somewhat.


But there's stil quite a difference, so I don't think that it would 
be a problem.  Note that the change does make the vertical shaft look 
more centered in that large 65.3-cent symbol.

By the way, it's been bugging me that we've yet to agree on the 
spelling of confusable vs. confusible.  I finally looked up the -able 
vs. -ible rules.  There were two that applied (source: _The Grammar 
Bible_, Strumpf & Douglas, Knowledgeopolis, 1999):

Rule 2: If the base itself is a complete English word, use the 
suffix -able.  Examples: changeable, flyable

Which would result in "confuseable".  However, see

Rule 4: If you can add the suffix -ion to the base to make a 
legitimate English word, then you should use the suffix -ible.  
Examples: corruptible (corruption), perfectible (perfection)

Which results in "confusible".

I hope that rule 4 will end the confusion, even if it doesn't 
eliminate all of the confusibility.


> Which is one thing that causes me to re-propose the convex with the
> slightly squarer corners. Your only comment about them has been 
to "leave
> well enough alone". Can you give a more detailed reason for 
rejecting them?


Just an aesthetic consideration: the flags with the squarer corners 
tend to look like right-angle flags with rounded corners, as opposed 
to flags that are curved along their entire length.  (I notice that 
you did make a difference for this in the 55-cent symbol also, which 
is good; I later realized that I was mistaken in suggesting that it 
didn't need to be changed to conform to the others.)


>> And while I am on the subject of fine-tuning symbols:
>> 

>>> I gave up on trying to make an actual symbol for the pythagorean 
>> comma

>>> based on the above identity. Maybe you want to have a go.
>> 

>> Try this:  Make a copy of the 17' symbol (wL+vR).  Move all of the 
>> pixels in the 4 leftmost columns up 1 position (thus raising the 
wavy 
>> flag by 1).  Then copy these and paste them so that you have a 
second 
>> wavy flag 4 pixels lower than the top one.  The two wavy flags 
>> together are considerably smaller than a single 29 flag (even with 
my 
>> proposed reduction in size for the latter), and together they 
clearly 
>> indicate the staff position of the note being altered.
> 

> See the above file for my best attempt. Not precisely what you 
suggested,
> but close.


It is appropriate that you limited the downward travel of the lower 
flag on the left side to conform to the rest of the symbols.  I was a 
little hesitant to put the top wavy flag any higher, lest it be 
confused with a convex flag, but I now realize that the two are so 
different in size that this wouldn't be a problem.  What you have 
looks good.


> I don't want to jump the gun and go to the main list just yet, and 
when I
> do, I'll want a staff showing the odd harmonics of G up to 41, 
including
> all optional spellings (using single shaft symbols with 
conventional sharps
> and flats), as well as the 217-ET notation and a couple of other 
ETs.


Then I think that we should decide on standard (or preferred) sets of 
symbols for as many ET's as we can before doing this.  I would also 
like to get the rest of the single symbols taken care of, too.  (The 
question about the length of the middle shaft of the sesqui-symbols 
shouldn't hold us back from designing the flags.  There also remains 
the question about the design of the X-symbols -- I don't recall that 
you replied to my diagonals-of-a-trapezoid answer; we were both 
getting a little punchy from overwork, and this is something that we 
need to get back to.)


> You might want to check out
> Shareware.com - * [with cont.]  (Wayb.)
cat=247&tag=ex.sa.sr.srch.sa_all&q=truety
> pe+font+editor


It looks like there are a few packages that could be used.  Do you 
have any suggestions or preferences?


> and
> 
> Welcome to Sibelius * [with cont.]  (Wayb.)
> 
> to get the free download which is fully functional except for save.
> -- Dave Keenan


This, I presume, would give us a chance to see how a new font would 
work with their product.


> Brisbane, Australia
> Dave Keenan's Home Page * [with cont.]  (Wayb.)

Message: 4747 - Contents - Hide Contents

Date: Mon, 29 Apr 2002 03:38:10

Subject: Re: what's up with the paper?

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> Shall we move on to a full consideration of {2,3,5,7}, 
> preferably with dave keenan and graham breed looking over 
> gene's shoulder? or am i just being a pain in the :-B ?


I'd prefer to do {2,3,5,7} next. I don't have a good feel for {2,3,7} 
(or {2,5,7} or {3,5,7}). In fact I'd prefer to do the full 9-limit and 
11-limit after that, and hopefully by then we'll have figured out how 
to interpolate the cutoffs for the less familiar subsets.

So Gene, how about hitting us with a wide-open list of 7-limit linear 
temperaments, so we can consider where the cutoffs might need to go.

Message: 4749 - Contents - Hide Contents

Date: Mon, 29 Apr 2002 05:27:17

Subject: Re: what's up with the paper?

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> Also, i'd like to propose that we report the 
> complexity of the simplest temperament that we left off the end of 
> each list (in addition to my proposal that we order by complexity).


You mean the simplest one that comes inside the badness cutoff, but 
outside the complexity cutoff?
This doesn't need to be _in_addition_ to the complexity cutoff. It can 
_be_ the complexity cutoff (as in "less than and not equal to").

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