Tuning-Math Digests messages 4725 - 4749

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Message: 4725

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/Complements.bmp *
> 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/SymbolsBySize.bmp *
showing the new symbol, with all the others, on the staff.


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Message: 4728

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 *


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Message: 4729

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- *
>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 *


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Message: 4730

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/Complements.bmp *
It should be self-explanatory.


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Message: 4732

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/Complements.bmp *

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 *


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Message: 4733

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.


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Message: 4734

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]


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Message: 4735

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]


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Message: 4737

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.


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Message: 4738

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 *


> --- 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/AdaptiveJI.bmp *

***What on earth is going on here?

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

jp


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Message: 4740

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/AdaptiveJI.bmp *


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Message: 4742

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/SymbolsBySize3.bmp *

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 *
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 *

to get the free download which is fully functional except for save.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4747

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.


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Message: 4749

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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