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Message: 4550 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 21:22:21

Subject: Re: A common notation for JI and ETs

From: David C Keenan

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

> Okay, that sounds like a good description of what we are are very 
> close to achieving.  I might prefer to call the 11-comma a diesis 
> (although it is plain that you are using the term "comma" in a 
> broader sense here), which would further justify the introduction of 
> the 11-5 comma that is used in achieving it, just as the 13-diesis 

is 
> also the (approximate) sum of two commas.

Yes. I like that idea.

> In the preceding sentence it should be obvious to you that I meant 

to 
> say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so no 
> one else misunderstands, I am correcting this here.

To be honest I had just read them as vL and vR without noticing the typos.
That's a worry. :-)

>> I have no objection to using multiple flags on the same side, to 
> notate

>> primes beyond 29. However I consider 999:1024 to be the standard 

37 
> comma

>> because it is smaller than 36:37, also because it only requires 2
>> lower-prime flags instead of 3. Can you explain why you want 36:37 
> to be

>> the standard 37 comma?
> 

> Using primes this high has more legitimacy, in my opinion, in otonal 
> chords than in utonal chords.

Certainly.

> If C is 1/1, then 37/32 would be D 
> (9/8) raised by 37:36.  With 1024:999 the 37 factor is in the 

smaller 
> number of the ratio, which is not where I need it.

But you would simply notate it as Eb lowered by 999:1024. The comma is in
the same direction as the flat. I don't see that this has anything to do
with otonal vs. utonal.

> For a similar reason I regard 26:27 as the principal 13-diesis.  
> Taking C as 1/1, to get 13/8 I want to lower A (27/16) by a semiflat 
> (26:27) instead of raising A-flat by a semisharp (1053:1024), even 

if 
> 1053:1024 is the smaller diesis.  

I find this more understandable since the comma is in the opposite
direction to the flat.

> But considering that 26:27 is more 
> than half an apotome (and that we are adequately representing both 

of 
> these in the notation anyway), I have no problem that you prefer to 
> state it the other way.

Right.

I see that the basic difference in our approaches is that you want the
simplest/smallest comma relative to a diatonic scale (Pythagorean-7), which
you then subtract an apotome from if it is bigger than an apotome, while
I'm happy with the simplest/smallest comma relative to a chromatic scale
(Pythagorean-12).

> 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.

You should definitely mention it wrt 217-ET, but make it clear it is not
universal. Whatever application did you or Erv find for the 41st harmonic?
Sounds crazy to me.

I think I lost some schismas for alternate 17 and 19 commas you found. Can
you remind me of those? 

> Why are you requiring that the new type of flag (whether for 19 or 
> 23) be smaller in size?  I would have the new flag represent 23 on 
> the basis that it is a *higher prime* than 19.  

When you added the (then) new type of flag, convex, to the existing
straight type, you didn't require that it represent 11 (or 11-5) on the
basis that it is a higher prime than 7. You used the new type for 7 so as
to eliminate lateral confusability from the 7-limit and from 72-ET.

I'm saying we should do the same thing with 19 and 23. i.e. use the new
type for 19 so as to eliminate (or at least greatly reduce) lateral
confusability from the 19-limit and from 217-ET.

The problem is of course the lateral confusability of the symbols for 1
step and 2 steps of 217-ET. There isn't even a consistent rule about which
is bigger, left or right. We already have xL > xR and sL < sR.

It is possible to choose valid alternatives for notating 217-ET so as to
completely eliminate lateral confusability, but only if 17 and 19 are
different types, which for convenience I'll call "v" and "n" respectively.

1   |n
2  v|
3  v|n
4  s|
5  s|n         (confusable alternative is |x)
6  x|          (confusable alternative is |s)
7  v|x or x|n
8  v|s or ss|
9  s|x
10 s|s
11 x|x
12  ||n        (confusable alternative is x|s)

The worst thing about this is not using the 7-comma |x for 5 steps. The
schisma that says 7 comma = 5 comma + 19 comma, only works in 217-ET, not
1600-ET.

However, if we do this (changing only 5 and 6)

1   |n
2  v|
3  v|n
4  s|
5   |x
6   |s
7  v|x or x|n
8  v|s or ss|
9  s|x
10 s|s
11 x|x
12  ||n

then at least we only have to deal with s| and |s. These will arguably be
the easiest to learn because they will occur most often, being of the
lowest prime limit.

If 17 and 19 commas are of the same type we have

1   |v
2  v|
3  v|v
4  s|
5   |x
6   |s
7  v|x or x|v
8  v|s or ss|
9  s|x
10 s|s
11 x|x
12  ||v

where we have two confusable pairs, and we have v| > |v while s| < |s. Or
we could use

1   |v
2  v|
3  v|v
4  s|
5   |x
6  x|
7  v|x or x|v
8  v|s or ss|
9  s|x
10 s|s
11 x|x
12  ||v

where at least we have v| > |v and x| > |x. They even differ by the same
amount.

But I still prefer zero or one confusable to 2 confusables.

On another matter: Can you tell me why the apotome symbol should not be
x||x instead of s||s?

> Then with 217-ET 
> (which is unique only through 19 and completely consistent only 
> through 21) we need only the three types of flags that are used for 
> the 19-limit notation, with a *newL* (different-looking *left*) flag 
> for the 23 comma being foreign to all three: the 19-limit, 217-ET, 
> and the single-symbol notation.

Well yes, minimising the number of flag-types is an advantage, but does it
sufficiently compensate us for the lateral confusability it allows? And if
so, why did we not consider it so in the 7-limit when we introduced a new
convex type of flag rather than use the other straight flag?

> Otherwise, I would need to have a way to incorporate the new flag 
> into the single-symbol notation, which will be discussed next.

I understand you've solved that problem now and we can just take it that
the second half apotome will follow the same sequence of flags as the
first, no matter what those flags may be. Is that correct?

> I would want xL in 217 anyway, since it does handle ratios of 29.  
> After all, this is supposed to allow 35-limit (nonunique) notation, 
> and it would be better not to have a new flag appearing out of the 
> blue, just for 29.

When dealing with 217-ET (or limits lower than 29) I think it's ok to
describe x| as the 13'-7 flag rather than the 29 flag. I expect you'd
prefer this.

> As it turns out, the complements that you propose are, as a whole, 
> much more intuitive than what I had, with the flag arithmetic being 
> completely consistent for the symbols that we prefer for each degree 
> of 217 (which we will hopefully agree upon within the next few days 
> as the "standard 217-ET set").  This standard set of 217-ET symbols 
> could then be used for determining the notation for any sharp/flat 
> (or other) equivalents that may be required for any JI interval (for 
> which the composer should be strongly encouraged to indicate the 
> symbol-ratio association in the score).
> 
> There are rules for arriving at the new complement symbols 

(including 
> the nonstandard ones), and they are a bit more convoluted that what 

I 
> had. But inasmuch as the overall result is much better than what I 
> previously had, I think that this is something that we should adopt 
> without any further hesitation.

I'm sure glad you found them by another route. I think I proposed them
mainly out of ignorance!

> The xL flag is already used for both the 11 and 12 degree symbols, 

so 
> it is definitely not foreign to 217, but I think you mean that there 
> is no need to use the apotome-complement symbol of one that has an 

xL 
> or vR flag.

Yes. I was forgetting about the 11 step symbol x|x. The 12 step one can be
replaced with a double-shaft symbol if need be, but you're right, there's
no way to do 217-ET without the 29 flag although we should probably call it
the 13'-7 flag in this context, except when we describe the non-unique
35-limit possibilities of 217-ET.

>  (These are among the "non-standard" symbols whose 
> complements have flags that are inconsistent with their order.)  I 
> wouldn't prohibit this altogether, should one want to indicate a 
> ratio of 29, for example -- apotome minus xL| equals xL|| is easy 
> enough to understand, but we will need to be careful to make others 
> aware that this is a "non-standard" complement symbol that doesn't 
> occur in the expected order.

Would you please describe the symbols you think should be used for 1 thru
21 steps of 217-ET? s||s type notation will do fine.

> I have tried the new complement symbols out with a few ET's (58, 94, 
> and 96), and I am delighted with the result, especially 94.

That's great news.

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



Message: 4551 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 04:31:40

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

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

>> I still need to prepare a diagram that illustrates the sequence of 
>> symbols in various ET's, and I'd like to do a full-octave diagram 
> for 

>> 217 as well, just so we have a better idea of how everything comes 
>> out.
>> 
>> --George
> 

> i think it would be cool if someone notated the adaptive-ji version 
> of the chord progression
> 
> Cmajor -> A minor -> D minor -> G major -> C major
> 
> in 217-equal. then we could all look at it and see if we have any 
> major problems with it.


You mean like

C:E\:G
AJ:C*:EJ
Dj:Ff:Aj
Go:BL:Do
C:E\:G

Where L \ J j o * f stand for the following arrows (George and I 
haven't agreed on all of these yet):

   |
   |
   |    L  5/217 oct down   (7 comma down)
   | |
   |_/

   |
   |
   |    \  4/217 oct down    (5 comma down)
 \ |
  \|

   |
   |
 _ |    J  3/217 oct down    (17 comma down + 19 comma down)
  \|_
   |

   |
   |
 _ |    j  2/217 oct down    (17 comma down)
  \|
   |

   |
   |
   |    o  1/217 oct down    (19 comma down)
   |_
   |
 
   |_
   |
   |    *  1/217 oct up      (19 comma up)
   |
   |
 
   |
 _/|
   |    f  2/217 oct up      (17 comma up)
   |
   |



Message: 4552 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 17:49 +0

Subject: Re: Decatonics

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <B8C71E03.38EA%mark.gould@xxxxxxx.xx.xx>
Mark Gould wrote (27th March):


> This is inconsistent with my rule ii: it contains segments of 3 and more
> adjacent PCs
> 
> 0 1 2, 4 5 6 7, 9 10 11 12, etc
> 
> So it will show up as intervallically inchoerent (as defined by 
> Balzano).

>From the discussion that followed, it emerged that Balzano's intervallic 

coherence is the same as Rothenberg's propriety.  The scale in question is

0 1 2 4 5 6 7 9 10 11 12 14 15 16 17 19 20 21 22 24 25 26

It's a bit tedious to work out the propriety by hand, but I've written a 
script to do it now.  The magic table is

 1  1  2  1  1  1  2  1  1  1  2  1  1  1  2  1  1  1  2  1  1
 2  3  3  2  2  3  3  2  2  3  3  2  2  3  3  2  2  3  3  2  2
 4  4  4  3  4  4  4  3  4  4  4  3  4  4  4  3  4  4  4  3  3
 5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  5  4  5
 6  6  7  6  6  6  7  6  6  6  7  6  6  6  7  6  6  6  6  6  6
 7  8  8  7  7  8  8  7  7  8  8  7  7  8  8  7  7  7  8  7  7
 9  9  9  8  9  9  9  8  9  9  9  8  9  9  9  8  8  9  9  8  8
10 10 10 10 10 10 10 10 10 10 10 10 10 10 10  9 10 10 10  9 10
11 11 12 11 11 11 12 11 11 11 12 11 11 11 11 11 11 11 11 11 11
12 13 13 12 12 13 13 12 12 13 13 12 12 12 13 12 12 12 13 12 12
14 14 14 13 14 14 14 13 14 14 14 13 13 14 14 13 13 14 14 13 13
15 15 15 15 15 15 15 15 15 15 15 14 15 15 15 14 15 15 15 14 15
16 16 17 16 16 16 17 16 16 16 16 16 16 16 16 16 16 16 16 16 16
17 18 18 17 17 18 18 17 17 17 18 17 17 17 18 17 17 17 18 17 17
19 19 19 18 19 19 19 18 18 19 19 18 18 19 19 18 18 19 19 18 18
20 20 20 20 20 20 20 19 20 20 20 19 20 20 20 19 20 20 20 19 20
21 21 22 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21
22 23 23 22 22 22 23 22 22 22 23 22 22 22 23 22 22 22 23 22 22
24 24 24 23 23 24 24 23 23 24 24 23 23 24 24 23 23 24 24 23 23
25 25 25 24 25 25 25 24 25 25 25 24 25 25 25 24 25 25 25 24 25

Which looks proper to me.  So this rule ii remains a spectacularly bad way 
of predicting propriety.


                    Graham

Message: 4553 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 17:26:56

Subject: Re: A common notation for JI and ETs

From: gdsecor

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

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

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

>>> ... you have proceeded ... to suggest changes in the symbols 
that: ...
>>> 2) Introduce new symbols that I have no idea how to incorporate 
into 
>>> a single-symbol notation (this makes it difficult to do 
something 
>>> that I was previously able to do with 217-ET); and
>> 
[dk:]

>> I think this is the big one, but I have a proposed solution. 
Later.
> 
[gs:]

> It doesn't work (see my reply below).

I was too hasty in jumping to a conclusion about this.  It does 
work!  (See below.)

[dk:]

>> Why can't we simply give a fixed comma value to the second shaft 

(and so on

>> for subsequent shafts), so the ordering of flag combinations 

learnt for the

>> first half-apotome is simply repeated in the second half-apotome 
(and all
>> other half-apotomes). To do this, the second shaft need only be 
declared
>> equal in value to xL+xR.
> 
[gs:]

> That's the way I did it way back (about 3 months ago) when life was 
> much simpler: I was using only straight flags and 72-ET was the 
most 
> complicated system I had to deal with.  The problem in doing that 
now 
> is that the ratios that we're trying to represent don't ascend in 
the 
> same order from a half-apotome (now what ratio is that anyway?) as 
> they do from a unison; instead they occur in reverse order from the 
> apotome downward.  So scratch that idea.

I spent some time messing around with the symbols last night, trying 
to see if I could find a way to make the apotome complements more 
intuitive and to eliminate the inconsistency from the sequence, and, 
lo and behold, I ended up with with exactly what you suggested, with 
the ordering of flag combinations in the first half-apotome repeated 
in the second half-apotome.

My profuse apologies -- this works beautifully!!!  When I read your 
proposal yesterday, it sounded so wacky that I didn't even bother to 
check it out.

I arrived at my somewhat arbitrary method for determining the apotome-
complement symbols when I was still tentatively working them out in 
183-ET, and at that time the vL and vR symbols were both being used 
for 17-commas.  The idea of making the sequence match in the two half-
apotomes never occurred to me, because I hadn't settled on mapping 
the symbols into any preferred division yet, and I (correctly) 
assumed that any such sequence would be rather arbitrary.

As it turns out, the complements that you propose are, as a whole, 
much more intuitive than what I had, with the flag arithmetic being 
completely consistent for the symbols that we prefer for each degree 
of 217 (which we will hopefully agree upon within the next few days 
as the "standard 217-ET set").  This standard set of 217-ET symbols 
could then be used for determining the notation for any sharp/flat 
(or other) equivalents that may be required for any JI interval (for 
which the composer should be strongly encouraged to indicate the 
symbol-ratio association in the score).

There are rules for arriving at the new complement symbols (including 
the nonstandard ones), and they are a bit more convoluted that what I 
had. But inasmuch as the overall result is much better than what I 
previously had, I think that this is something that we should adopt 
without any further hesitation.

[dk:]

>> Another advantage of this is that one does not need to use flags 
that
>> properly belong to higher limits in the second and subsequent 
half-apotomes
>> of lower limit rational notations, or of ET notations based on 
lower
>> limits. e.g. There will be no concave flags (or small flag) in 72-
ET. And
>> there will be no need for xL or vR in 217-ET.
> 
[gs:]

> I would want xL in 217 anyway, since it does handle ratios of 29.  
> After all, this is supposed to allow 35-limit (nonunique) notation, 
> and it would be better not to have a new flag appearing out of the 
> blue, just for 29.

The xL flag is already used for both the 11 and 12 degree symbols, so 
it is definitely not foreign to 217, but I think you mean that there 
is no need to use the apotome-complement symbol of one that has an xL 
or vR flag.  (These are among the "non-standard" symbols whose 
complements have flags that are inconsistent with their order.)  I 
wouldn't prohibit this altogether, should one want to indicate a 
ratio of 29, for example -- apotome minus xL| equals xL|| is easy 
enough to understand, but we will need to be careful to make others 
aware that this is a "non-standard" complement symbol that doesn't 
occur in the expected order.

[gs:]

> Now regarding 72-ET, you will recall that I said this earlier:
> 
> << Using curved flags in the 72-ET native notation to alleviate 
> lateral confusibility complicates this a little when we wish to 
> notate the apotome's complement (4deg72) of 64/63 (2deg72), a 
single 
> *convex right* flag.  I was doing it with two stems plus a *convex 
> left* flag, but the above rules dictate two stems with *straight 
> left* and *concave right* flags.  As it turns out, the symbol 
having 
> a single stem with *concave left* and *straight right* flags is 
also 
> 2deg72, and its apotome complement is two stems plus a *convex 
left* 
> flag (4deg72), which gives me what I was using before for 4 
degrees.  
> So with a little bit of creativity I can still get what I had (and 
> really want) in 72; the same thing can be done in 43-ET.  This is 
the 
> only bit of trickery that I have found any need for in divisions 
> below 100. >>
> 
> By using a "faux complement," I can avoid using any concave flags 
for 
> both 72-ET and 43-ET.  In fact, the only ET's under 100 that need 
> concave flags (that I have tried so far) are 50, 58, 94, and 96, 
and 
> none of the more important ones do.

This faux complement (apotome minus |xR equals xL|| ) could still be 
used in 72-ET (but not 43-ET), but I now feel very strongly that I 
would want to avoid this practice, inasmuch as the symbol for the 
true complement (apotome minus |xR equals ||xR ) is now highly 
intuitive.  Changing a convention for this one special case would be 
counter-productive if the true complement were used everywhere else.  
But in any case (as you note), there would be no concave or small 
flag in 72-ET.

I have tried the new complement symbols out with a few ET's (58, 94, 
and 96), and I am delighted with the result, especially 94.

--George



Message: 4554 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 17:36:44

Subject: Re: A common notation for JI and ETs

From: gdsecor

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

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

>> I still need to prepare a diagram that illustrates the sequence 
of 
>> symbols in various ET's, and I'd like to do a full-octave diagram 
> for 

>> 217 as well, just so we have a better idea of how everything 
comes 
>> out.
>> 
>> --George
> 

> i think it would be cool if someone notated the adaptive-ji version 
> of the chord progression
> 
> Cmajor -> A minor -> D minor -> G major -> C major
> 
> in 217-equal. then we could all look at it and see if we have any 
> major problems with it.


Once Dave and I agree on how to symbolize the 19-comma (512:513, the 
single degree of 217), then I would be happy to make a figure 
illustrating this on a real staff.

--George

Message: 4555 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 18:09:25

Subject: Re: A common notation for JI and ETs

From: gdsecor

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

> Hi George,
> 
> -----------
> The 19 flag
> -----------
> I don't require that the new type of flag be small irrespective of 
what it
> is used for. I only want the flag for the 3.3 cent 19 comma to be 
smaller
> than all the others, because it is less than half the size of any 
other
> flag comma and less than 1/6th of the size of all but the 17 comma. 
If this
> is allowed, then it follows that it must be a new kind of comma, not
> convex, striaght or concave.

Why don't you look at this way:  The four straight and convex flags 
are all greater than 20 cents.  The 17 and 19 flags are both less 
than 10 cents.  The 23 flag, 729:736, is ~16.5 cents, which is closer 
in size to the four larger flags than the two smaller ones.  So, as I 
see it, *both* the 17 and 19 flags should be smaller than the others, 
which is why I have been making concave flags laterally narrower than 
the others in my bitmap diagrams for the past several weeks.  You 
haven't seen them this way yet, but I think they look great.  In 
particular, the symbol with two concave flags is noticeably smaller 
than ones with two straight/convex flags -- more similar in size to 
one with a single straight or convex flag.

So I don't think there is any real problem here.  We could make the 
new 23 flag intermediate in size between a (smaller) concave flag and 
a (larger) straight flag.  All we need to do is to decide on a shape.

> It seems, from an RT point of view, that the 19 comma flag could 
equally
> well be a left flag or a right flag, I have no great attachment to 
either.
> However in notating 217-ET you need to use 19 flag + 17 flag to 
notate 3
> steps and so the 19 flag would be best on the opposite side from 
the 17
> flag. And if we want the large 23 comma not to have flags on the 
same side,
> then the 17 comma must be on the opposite side from the 11-5 flag, 
which
> means that the 17 flag must be a left flag and the 19 flag a right 
flag (I
> mistakenly had 19 as a left flag in my previous message).
> 
> ----------
> Priorities
> ----------
> It seems that there is a significant difference of priorities 
between an
> approach 
> (a) that seeks to notate a particular large ET, which is
> 19-odd-limit-unique, using single symbols spanning from double-flat 
to
> double-sharp (or even just from flat to sharp), and use subsets of 
it to
> notate lower ETs, and extend it to uniquely notate 19-or-higher-
prime-limit
> RTs (rational tunings),
> and an approach
> (b) that seeks to notate 19-or-higher-prime-limit RTs and use 

subsets of it

> to notate low enough ETs, and extend it to allow those ETs to be 
notated
> using single symbols spanning from double-flat to double-sharp (or 
even
> just from flat to sharp).
> 
> I believe I've understood your points, but I don't have any 
suggestions yet
> that might satisfy us both, so I'm just going to put it in the too 
hard
> basket for a while, or let it churn away in my subconscious.

I apologize for having responded to your apotome-complement symbol 
proposal yesterday without having given it sufficient thought, but I 
hope that today's messages clear things up a bit so we can continue 
working on this without too much discouragement.

--George

Message: 4557 - Contents - Hide Contents

Date: Thu, 11 Apr 2002 19:08:10

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:
>> Hi George,
>> 
>> -----------

>> The 19 flag
>> -----------
>> I don't require that the new type of flag be small irrespective of 
> what it

>> is used for. I only want the flag for the 3.3 cent 19 comma to be 
> smaller

>> than all the others, because it is less than half the size of any 
> other

>> flag comma and less than 1/6th of the size of all but the 17 

comma. 
> If this

>> is allowed, then it follows that it must be a new kind of comma, 

not
>> convex, striaght or concave.
> 

> Why don't you look at this way:  The four straight and convex flags 
> are all greater than 20 cents.  The 17 and 19 flags are both less 
> than 10 cents.  The 23 flag, 729:736, is ~16.5 cents, which is 

closer 
> in size to the four larger flags than the two smaller ones.  So, as 

I 
> see it, *both* the 17 and 19 flags should be smaller than the 

others, 

Here are the relative sizes of the flag commas (shown using steps of 1600-ET).

_19__
_____17_____
__________23__________
______________5______________
_________________7__________________
___________________11-5___________________
_____________________29______________________

You are right that the largest difference in the series occurs between 17
and 23. But the next largest difference occurs between 19 and 17, and
between 23 and 5. So it seems that following your argument to its
conclusion would require that the 19, 17 and 23 flags should be of _three_
different types. I think it will be too hard to find 3 types that look
smaller than convex and straight. We're having enough trouble agreeing on
two. Although the 19 to 17 and 23 to 5 differences are the same in steps of
1600-ET, the 19 to 17 difference is slightly greater in cents.

But what if we look not at the _differences_ between sucessive comma sizes
but at the _ratios_ between them? This is equivalent to looking at the
differences of their logarithms. I don't think we can expect the apparent
flag size to correspond to comma size anyway, the largest being nearly 10
times the smallest, so why not consider making flag size roughly
proportional to the log of comma size.

Here are the logs shown graphically (natural log of cents, times 17).

________19___________
_________________17__________________
_______________________23_______________________
__________________________5_________________________
____________________________7___________________________
____________________________11-5___________________________
_____________________________29_____________________________

In this case it is clear that 19 to 17 is a more significant step than 23
to 5.

Maybe log is a bit too strong. What about square root, which could be
interpreted as making areas correspond?

Square root of cents, times 12.

_________19___________
________________17_________________
______________________23_________________________
__________________________5_____________________________
______________________________7________________________________
_______________________________11-5_________________________________
________________________________29___________________________________

19 to 17 is still more significant than 23 to 5, but about the same as 17
to 23.

> which is why I have been making concave flags laterally narrower 

than 
> the others in my bitmap diagrams for the past several weeks.  You 
> haven't seen them this way yet, but I think they look great.  In 
> particular, the symbol with two concave flags is noticeably smaller 
> than ones with two straight/convex flags -- more similar in size to 
> one with a single straight or convex flag.

That's good. I look forward to seeing them.

> So I don't think there is any real problem here.  We could make the 
> new 23 flag intermediate in size between a (smaller) concave flag 

and 
> a (larger) straight flag.  All we need to do is to decide on a 

shape.

I am completely stumped as to what could possibly be intermediate between
small-concave and straight and still be sufficiently distinct from both. I
don't think it is possible. 

However this isn't really your requirement. It's too strict. All you really
need, to acheive this (which I'm not necessarily agreeing is a good idea),
is two sizes of flag smaller than straight and convex, where 17 and 19 are
of the smallest type and 23 alone is of the second smallest type. 

I've already proposed a fourth type, with right-angle being smaller than
concave. So you could make both 17 and 19 right angle flags and 23 would be
the sole concave flag.

Note that right-angle-ness can happen in the other direction too. We
actually have 5 types of flag. From smallest to largest they are:

concave right-angle
concave quarter-circle
straight
convex quarter-circle
convex right-angle

If these are all imagined to fit in a square of side 2, then ignoring the
thickness ofthe lines, the area enclosed by each (with the shaft and an
invisible horizontal line as the other boundaries) is respectively 0, 1, 2,
3, 4 (taking pi to be 3).

Perhaps the best way to take line thickness into account is to try to
render them in an extremely small bitmap. If try to put the flags into a
2x3 bitmap, which could well be required of a 9 or 10 point version of the
symbol when displayed on a computer screen, we find that the two smallest
ones are not distinct as described above, which is why I said the concave
right-angle needed to be smaller, as well as being a right angle. We can do
these:

@@@@@@
@@  @@
@@  @@
    @@
    @@
    @@
    @@

  @@@@
@@  @@
@@  @@
    @@
    @@
    @@
    @@

    @@
  @@@@
@@  @@
    @@
    @@
    @@
    @@

    @@
    @@
@@@@@@
    @@
    @@
    @@
    @@

    @@
  @@@@
    @@
    @@
    @@
    @@
    @@

There are of course only 64 possible 2x3 bitmaps. I generated them all
below and then deleted those that were 
(a) not connected,
(b) pointed in the wrong direction
(c) apeared as two flags overlaid. i.e. had two or more ends

I've left those whose direction is ambiguous because they can be made
unambiguous at higher resolution and their nearness to one end of the arrow
disambiguates them.

There are 29 left.

  @@@@
    @@
    @@
    @@
    @@
    @@
    @@

@@@@@@
    @@
    @@
    @@
    @@
    @@
    @@

  @@@@
@@  @@
    @@
    @@
    @@
    @@
    @@

@@@@@@
@@  @@
    @@
    @@
    @@
    @@
    @@

    @@
  @@@@
    @@
    @@
    @@
    @@
    @@

  @@@@
  @@@@
    @@
    @@
    @@
    @@
    @@

    @@
@@@@@@
    @@
    @@
    @@
    @@
    @@

  @@@@
@@@@@@
    @@
    @@
    @@
    @@
    @@

@@@@@@
@@@@@@
    @@
    @@
    @@
    @@
    @@

  @@@@
@@  @@
@@  @@
    @@
    @@
    @@
    @@

@@@@@@
@@  @@
@@  @@
    @@
    @@
    @@
    @@

    @@
  @@@@
@@  @@
    @@
    @@
    @@
    @@

  @@@@
  @@@@
@@  @@
    @@
    @@
    @@
    @@

    @@
@@@@@@
@@  @@
    @@
    @@
    @@
    @@

  @@@@
@@@@@@
@@  @@
    @@
    @@
    @@
    @@

@@@@@@
@@@@@@
@@  @@
    @@
    @@
    @@
    @@

    @@
    @@
  @@@@
    @@
    @@
    @@
    @@

  @@@@
@@  @@
  @@@@
    @@
    @@
    @@
    @@

    @@
  @@@@
  @@@@
    @@
    @@
    @@
    @@

  @@@@
  @@@@
  @@@@
    @@
    @@
    @@
    @@

  @@@@
@@@@@@
  @@@@
    @@
    @@
    @@
    @@

    @@
    @@
@@@@@@
    @@
    @@
    @@
    @@

  @@@@
@@  @@
@@@@@@
    @@
    @@
    @@
    @@

@@@@@@
@@  @@
@@@@@@
    @@
    @@
    @@
    @@

    @@
  @@@@
@@@@@@
    @@
    @@
    @@
    @@

  @@@@
  @@@@
@@@@@@
    @@
    @@
    @@
    @@

    @@
@@@@@@
@@@@@@
    @@
    @@
    @@
    @@

  @@@@
@@@@@@
@@@@@@
    @@
    @@
    @@
    @@

@@@@@@
@@@@@@
@@@@@@
    @@
    @@
    @@
    @@

> I apologize for having responded to your apotome-complement symbol 
> proposal yesterday without having given it sufficient thought, but I 
> hope that today's messages clear things up a bit so we can continue 
> working on this without too much discouragement.

Certainly. No apology required. I'm very glad that you found the same thing
but by completely different means.

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

Message: 4558 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 00:36:46

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi George,

Of all those 2x3 bitmaps I gave in the preceding post, the only one that
leaps out at me as being sufficiently arrow-like and sufficiently distinct
from the other 5, is

    @@
@@@@@@
@@  @@
    @@
    @@
    @@
    @@

Which may be interpreted as concave followed by convex (concavoconvex). It
helps to stand back and squint :-)

I find this to be between convex and straight in apparent size, but more
similar to straight. If one counts pixels (whether black or white), within
the 2x3 space of the flag, which are either black or are on the same
horizontal line as, and to the right of, a black pixel, then one gets the
following sizes.

@@@@@@
@@  @@    6 (the maximum possible)
@@  @@
    @@    convex right-angle
    @@
    @@
    @@

  @@@@
@@  @@    5
@@  @@
    @@    convex quadrant
    @@
    @@
    @@

    @@
@@@@@@    4
@@  @@
    @@    concavoconvex
    @@
    @@
    @@

    @@
  @@@@    3
@@  @@
    @@    straight
    @@
    @@
    @@

    @@
    @@    2
@@@@@@
    @@    concave quadrant
    @@
    @@
    @@

    @@
  @@@@    1 (the minimum possible)
    @@
    @@    small concave right-angle
    @@
    @@
    @@

Notice that doing it the other way, convex followed by concave
(convexoconcave), doesn't work in 2x3 and has the same "size" as
concavoconvex.

  @@@@
  @@@@ 4
@@@@@@
    @@ convexoconcave?
    @@
    @@
    @@

Here are sharp # and flat for comparison

  @@  @@
  @@  @@
@@@@@@@@@@
  @@  @@
@@@@@@@@@@
  @@  @@
  @@  @@

  @@
  @@
  @@
  @@@@
  @@  @@
  @@@@
  @@

As far as I can tell there can be no others besides these 6.

Here they are in 3x4. If the previous was 10 point, this would be 12 point.

@@@@@@@@
@@    @@   12
@@    @@
@@    @@   convex right-angle
      @@
      @@
      @@
      @@
      @@

    @@@@
  @@  @@    9
@@    @@
@@    @@    convex quadrant
      @@
      @@
      @@
      @@
      @@

      @@
  @@@@@@    8
@@    @@
@@    @@    concavoconvex
      @@
      @@
      @@
      @@
      @@

      @@
    @@@@    6
  @@  @@
@@    @@    straight
      @@
      @@
      @@
      @@
      @@

      @@
      @@    5
  @@@@@@
@@    @@    alternative concavoconvex
      @@
      @@
      @@
      @@
      @@

      @@
      @@    4
    @@@@
@@@@  @@    concave quadrant
      @@
      @@
      @@
      @@
      @@
      @@

      @@
      @@    2
  @@@@@@
      @@    small concave right-angle
      @@
      @@
      @@
      @@
      @@

        @@
  @@    @@
  @@    @@@@
  @@@@@@@@
@@@@    @@@@
  @@@@@@@@
@@@@    @@
  @@    @@
  @@

  @@
  @@
  @@
  @@
  @@@@
  @@  @@
  @@  @@
  @@@@
  @@

I figure, by measuring some sharps and flats on scores, that at full size
the symbols will be 11 (computer-screen) pixels high, which would be 14
point the way I've been calling them. But maybe what I've called 10, 12 and
14 point, should be called 8, 10 and 12 point. Anyway, here they are:

@@@@@@@@@@
@@      @@   20
@@      @@
@@      @@   convex right-angle
@@      @@
        @@
        @@
        @@
        @@
        @@
        @@

    @@@@@@
  @@    @@    17
@@      @@
@@      @@    convex quadrant
@@      @@
        @@
        @@
        @@
        @@
        @@
        @@

      @@@@
    @@  @@    14
  @@    @@
@@      @@    alternative convex quadrant
@@      @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
        @@    11
  @@@@@@@@
@@      @@    concavoconvex
@@      @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
      @@@@    10
    @@  @@
  @@    @@    straight
@@      @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
        @@    7
      @@@@
    @@  @@    concave quadrant
@@@@    @@
        @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
        @@    5
        @@
      @@@@    alternative concave quadrant
@@@@@@  @@
        @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
        @@    3
        @@
  @@@@@@@@    small concave right-angle
        @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
        @@    2
    @@@@@@
        @@    alternative small concave right-angle
        @@
        @@
        @@
        @@
        @@
        @@
        @@

        @@
  @@    @@
  @@    @@@@
  @@@@@@@@
@@@@    @@
  @@    @@
  @@    @@@@
  @@@@@@@@
@@@@    @@
  @@    @@
  @@

  @@
  @@
  @@
  @@
  @@
  @@@@@@
  @@    @@
  @@    @@
  @@  @@
  @@@@
  @@

It's tempting to eliminate the two extremes, the right angles, and use the
middle four, but that only gives us one type that is smaller than straight,
and we don't really want to use convexoconcave for the 5 comma just because
we want to have two types smaller than it.

I suppose if we tried real hard we could convince ourselves that
concavoconvex looks smaller than straight. Or we could deliberately make it
narrower, but only in the two larger point sizes I gave above, like:

      @@
  @@@@@@    4
  @@  @@
      @@    small concavoconvex
      @@
      @@
      @@
      @@
      @@

        @@
        @@    8
    @@@@@@
  @@    @@    small concavoconvex
  @@    @@
        @@
        @@
        @@
        @@
        @@
        @@

And make the concave quadrants smaller too as you suggest.

      @@
      @@    3
    @@@@
  @@  @@    small concave quadrant
      @@
      @@
      @@
      @@
      @@
      @@

        @@
        @@    4
        @@
      @@@@    small concave quadrant
  @@@@  @@
        @@
        @@
        @@
        @@
        @@
        @@
        @@

But I'd still want both 23 and 17 to be concavoconvex and 19 to be concave,
unless we made 23 concavoconvex, 17 concave quadrant and 19 small concave
right-angle.

Another suggestion: In the larger point sizes above, there are two subtly
different alternatives for some of the flags (and others are possible). Why
not make the left and right varieties of the same flag-type use these
subtly different alternatives (one for the right and the other for the
left) according to their relative sizes?

We're really getting into the fine details of font design here. What
software are you using to create the bitmap versions you've been giving?
How many pixels high are they?

Eventually we'd need to give a resolution independent description as in a
True-type or Postscript font. Presumably you'd want to copy the style of
existing sharps and flats, such as making horizontal (or near-horizontal)
strokes much thicker than vertical ones, (as if painted with a brush that
is about 3 times higher than it is wide) to avoid them getting lost against
the staff lines.

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

Message: 4560 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 01:51:36

Subject: Re: Scales, Analysis, Accidentals and Performance

From: Carl Lumma

>Some may ask why I am on this list. Because I had the temerity (as a
 >musician and a composer) to write about a mathematical view on scales and
 >tonality. But only from the viewpoint of a musician. I am still
 >interested in all everyone has to say, but only if it has musical
 >relevance.
 >
 >Mark Gould

Heya Mark,

I for one am glad you're on this list.  So far, I've read all your posts.
I doubt Graham meant anything by it (probably just colorful metaphor, not
uncommon on the other side of the pond), but I'll let him speak to it.
Which isn't to say there isn't a healthy dose of oneupsmanship at work on
these lists, and even some amount of anti-academic sentiment...

I think propriety and the attached model is generally important in music,
but it certainly isn't the whole picture, and there are many valid ways
to approach the subject.  Anyway, I still have the explanatory document
for my diatonicity "shopping list" in my stack of things to do in the
next month...

-Carl

Message: 4561 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 10:47:26

Subject: Re: Scales, Analysis, Accidentals and Performance

From: dkeenanuqnetau

--- In tuning-math@y..., "Mark Gould" <mark.gould@a...> wrote:

> For my own part I find the endless discussion of increasingly 
complex
> analyses of ET scales with many hundreds of tones a pointless 
persuit of
> mathematics beyond the call of music. Who has played music in 311 
ET? Where
> has it been written.  And that is just one scale from the many 
discussed
> here. 

Hey Mark, relax. Did you miss where we were talking about 1600-ET. :-)

Just as when one talks about music in 7-limit JI, one doesn't mean it 
uses the infinity of notes in the 7-limit, one might only use a 
handful of notes, and yet they might meaningfully be said to be in say 
217-ET, such as Paul's adaptive-JI chord-progression request.

And such large ETs are mainly being employed as a mathematical 
convenience in designing a general purpose notation, after which its 
high ET origins will likely be forgotten.

> How does a musician react to the many descriptions of vectors and
> matrices? They are merely tools, not the understanding.

Indeed. I guess that's why we started the tuning-math list. So we 
wouldn't have to put up with the endless whining of musicians who 
didn't want to see all that stuff, but just wanted to know the final 
results.

> As for accidentals. Tell me, how would a string player react to
> 
> 
>     _
>    | \
>  _/| |
>    |
>    |
>    |
> 
> and
>     _
>   _| \
>    | |
>    |
>    |
>    |
> 
> in a passage of semiquavers in Allegro tempo?

In allegro semiquavers, they could simply ignore them and no-one would 
be any the wiser. If they suceeded in recognising that, as kinds of 
full-headed arrows pointing upwards they indicated a sharpening of 
somewhere between an eighth and a quarter of a tone, they would be 
doing exceedingly well.

In a long-sustained large otonal chord however, a more detailed 
knowledge might be of benefit. They would then need to know if the 
piece is in JI or which ET, and preferably the composer would offer a 
legend on the score.

> When you can tell the difference in performance, and that difference 
has a
> musical effect, then it is important. 

I totally agree.

> I'd like to see a decent 
analysis of
> the intonational variance of a string player or vocalist on 
successive
> performances of the same passage, or different performers on the 
same
> passage.

So would I. I have no doubt that we are going to ridiculous lengths to 
design a notation that will suit every conceivable tuning, but so long 
as we are having fun, and doing it in a way that does not complicate 
the notation of the most common tunings, where's the harm?

> This would be a useful correlative to the proliferation of
> accidentals. If was merely the act of playing or singing 'in tune' 
then
> often the ear will guide the performer more than any accidental. 

Don't count on it.

> Only if
> you want to be 'out of tune' do you want to 'alter' the note by some 
micro
> amount. After all, if the ET is designed to duplicate some n-limit 
tuning
> then surely an indication of the desired ratio by number would be 
more
> appropriate?

I think it's been tried. It just gets too cluttered.

> Some may ask why I am on this list. Because I had the temerity (as a
> musician and a composer) to write about a mathematical view on 
scales and
> tonality. But only from the viewpoint of a musician. I am still 
interested
> in all everyone has to say, but only if it has musical relevance.

I feel the same way, believe me. I think we all greatly value your 
contribution to this list.

Regards,

Message: 4562 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 13:18 +0

Subject: Re: Scales, Analysis, Accidentals and Performance

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <3969.194.203.13.66.1018597443.squirrel@xxxxx.xxxxxxx.xx.xx>
Mark Gould wrote:

> I don't think I was trying for propriety, only some measure of whether 
> the
> scale bears similarity to the diatonic scale that I started from.

What you originally said was "...it will show up as intervallically 
incoherent (as defined by Balzano)."  The consensus of the list is that 
Balzano's intervallic coherence is the same as Rothenberg's propriety.  If 
that wasn't what you were trying for, why didn't you say what you meant?

> I still
> think that scales of this form are not 'diatonic' scales. They may be
> perfectly useable, after all the 7ET scale is perfectly useable with 
> all of
> its pitch-classes, so all seven PCs are adjacent there.

Well, yes, there are a whole load of first rate scales that fail 
completely as generalized diatonics.  I don't want to give the impression 
that the criteria we give for diatonics are important for music in 
general.  But even so, I don't see why this three-adjacent-pitch-classes 
rule has anything to do with the important properties of the diatonic 
scale -- the ease of establishing a key centre, clarity of modulation, the 
sense of roughly equal melodic intervals and the clever things you can do 
with symmetric chords and tritone substitutions.

One very promising diatonic that breaks this rule is the 7 from 10 neutral 
third MOS:

0 1 3 4 6 7 9 0
 1 2 1 2 1 2 1

Bill Sethares has a fair amount on it in the chapter on 10-equal music 
theory in his book.  It's also something I've found works well as a fuzzy 
subset of the unequal Decimal scale.  Well enough that it is a priority 
for me to get some music together illustrating this.

It's also similar to the classic diatonic in a number of ways.  It has 7 
notes.  It can be generated by alternating the best approximations to 6:5 
and 5:4 (which happen to be identical).  The half-octave is an ambiguous 
interval.

The three adjacent pitch classes are enough to uniquely specify the key.  
The middle note (pitch class) is the obvious tonic because it has a small 
leading tone from both directions.  I find this to be a very useful 
property and, given that it has to be different to the diatonic somehow, 
it may as well be like this.

> Like all rules it was designed for a specific purpose. Maybe there are
> other ways of analysing scales, but analysis alone does not make music.
> This may be a list for tuning math, but at the end of the day we must be
> sure of our focus.

What purpose was it designed for?  You haven't actually said.  If your 
focus is on music, where's the music?

> My question is why you use 'spectacularly'? Does it mean you take 
> personal
> issue? Does it mean that you feel that your own statements carry more
> weight than any of the other posters to this list? From your exchanges I
> sense a certain garrulous sense of personal injustice should anyone 
> appear
> to suggest anything that you would not. DO you 'own' tuning-math?

I say 'spectacularly' because it is spectacular.  You put forward a rule 
of thumb to identify proper scales (although you now say that wasn't the 
intention at all).  It fails to say that this 21-from-26 "pentatonic" is 
proper, it fails to say that the classic diatonic in a positive 
temperament (eg 17, 22, 41) is improper, and it fails to say that the 
classic pentatonic in 7-equal is proper.

Even the rule you give in your paper about the relative sizes of the 
generalized thirds doesn't fail as spectacularly as this.  Assuming both 
thirds should be the same number of diatonic scale steps, it should only 
give false positives.

A better rule, as it happens, is to consider the relative sizes of the 
diatonic scale steps.  If the larger step is more then twice the size of 
the smaller one, you have an improper scale.  If the large step is exactly 
two small steps, you have a proper scale.  If the large step is less than 
two small steps, the scale is strictly proper.  If the scale is an MOS (as 
all your examples are) I think this relationship is exact, although I 
can't prove it.

Hence I didn't have to go to all the trouble of generating the propriety 
grid to know that the 21 note "pentatonic" is proper.  If I were on a 
personal vendetta, I could have sent an immediate reply asserting such, 
and taken the risk of you disproving it.

Of course my statements don't carry special weight here.  Have you heard 
my music?  I have to compensate for not having any inherent credibility by 
making sure what I say is true.

> If this is a disccusion group, I would keep personally directed comments
> out. And, if anything, one rule is as good as another. Who states one 
> rule
> for analysing scales carries ultimate weight. These 'rules' are entirely
> subjective, from all that I have read. How we lay out lattices is also
> subjective, as I prefer one way of constructing them that is different 
> from
> those that I have seen. That does not make mine 'wrong' nor the
> others 'wrong'. All are equally valid views on the complex problem of
> determining the nature of relationships in music.

Well, you're actually the one who introduced the personal comments here.  
And that's fine with me -- if you have a problem it's best to air it.  The 
big list has definitely suffered from people simmering in disagreement, 
and getting more and more upset, but not saying so.

I don't think one rule is as good as another.  Your next sentence 
mystifies me.

The rules Paul gave in his paper are objective.  The rules Rothenberg gave 
(propriety, efficiency and stability -- reading the original papers is not 
the ideal way of learning about them ;) are also objective.  The rules you 
give in your paper are objective, and correctly applied.

Carl's rules, of which we might soon have a discussion, aren't quite 
objective.  He says things like "high efficiency" without putting a number 
to how high.  Still, they could be made objective if that's what you 
really want.  But perhaps he's putting forward objective rules, but 
allowing use to make subjective judgements about their relative 
importance, which sounds right to me.

The layout of lattices is also objective.

> When you can tell the difference in performance, and that difference 
> has a
> musical effect, then it is important. I'd like to see a decent analysis 
> of
> the intonational variance of a string player or vocalist on successive
> performances of the same passage, or different performers on the same
> passage. This would be a useful correlative to the proliferation of
> accidentals. If was merely the act of playing or singing 'in tune' then
> often the ear will guide the performer more than any accidental. Only if
> you want to be 'out of tune' do you want to 'alter' the note by some 
> micro
> amount. After all, if the ET is designed to duplicate some n-limit 
> tuning
> then surely an indication of the desired ratio by number would be more
> appropriate?

Johnny Reinhard has a large amount of experience performing microtonal 
music.  He insists that the most helpful way of notating such music is 
writing the deviations from 24-equal in cents.  If I'm channelling him 
correctly, his response to the "the ear will guide the performer" argument 
is that the ear can't do any guiding until you've started to sing/play the 
wrong note.  But that time, it may be too late.

Some microtonal music is indeed intended to sound "out of tune" in a 
specific way.  For people who actually write 31-limit music, that's the 
most likely intention.

Johnny's also been very dismissive of the kind of experiments you propose. 
 Although he welcomes a precision of 1 cent, he doesn't claim to reproduce 
arbitrary scores with that level of accuracy.  The performer's intentions 
can't be reduced to a set of numbers.  And perhaps they'll have different 
intentions from one performance to another.

Ratios on scores have a very bad record in performance.  Partch used them, 
the scores are usually translated to some other system for modern 
performance.  The big problem is that the ratio doesn't strongly indicate 
pitch height.  You can work it out, but performers fairly obviously aren't 
going to do that when sight reading.  Compositions based on a single 
harmonic series, like those of LaMonte Young and the European spectralist 
school, are a niche where the numbers are more appropriate, and LaMonte 
does use them.  In this case, they are numbers rather than ratios, and so 
are in pitch order.  The compositional style also makes them the simplest 
way of describing the music.

We'll have to see how this profusion of accidentals is received when it 
reaches the big list.  For now, it seems they're working out the best 
system that fulfils a set of criteria.  It's best to save outside comment 
until the best such system's decided on, rather than a continuous stream 
of comments on the working drafts.

> Some may ask why I am on this list. Because I had the temerity (as a
> musician and a composer) to write about a mathematical view on scales 
> and
> tonality. But only from the viewpoint of a musician. I am still 
> interested
> in all everyone has to say, but only if it has musical relevance.

If you're a musician, where's the music?  I don't remember hearing any, 
but perhaps I wasn't paying attention.  Mine's out there for anybody to 
chuckle over at <Music files * [with cont.]  (Wayb.)>.

                      Graham



Message: 4563 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 10:20:26

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Hi George (and anyone else who is still following this crazy thread),

I wrote that the 29 flag (convex left) should be referred to as the 13'-7
flag in sub-29-limit situations. Of course it's actually the 11'-7 flag.
There is no 13'-7 flag.

Here are some possible accidentals as bitmaps at computer-screen
resolution, shown on the staff, both line-centred and space-centred.
              
            @@
@@@@@@@@@@@@@@@@@@@@
      @@    @@@@
      @@@@@@@@
    @@@@    @@
@@@@@@@@@@@@@@@@@@@@    sharp
      @@    @@@@
      @@@@@@@@
    @@@@    @@
@@@@@@@@@@@@@@@@@@@@
      @@      
        

@@@@@@@@@@@@@@@@@@@@
            @@
      @@    @@
      @@    @@@@
@@@@@@@@@@@@@@@@@@@@
    @@@@    @@
      @@    @@          sharp
      @@    @@@@
@@@@@@@@@@@@@@@@@@@@
    @@@@    @@
      @@    @@
      @@      
@@@@@@@@@@@@@@@@@@@@



@@@@@@@@@@@@@@@@@@@@
       @@  @@
         @@             double sharp
       @@  @@
@@@@@@@@@@@@@@@@@@@@



@@@@@@@@@@@@@@@@@@@@

     @@      @@
       @@  @@
@@@@@@@@@@@@@@@@@@@@    double sharp
       @@  @@
     @@      @@

@@@@@@@@@@@@@@@@@@@@


      @@      
@@@@@@@@@@@@@@@@@@@@
      @@    @@  
      @@@@@@@@
      @@    @@
@@@@@@@@@@@@@@@@@@@@    natural
      @@    @@
      @@@@@@@@
      @@    @@
@@@@@@@@@@@@@@@@@@@@
            @@
        

@@@@@@@@@@@@@@@@@@@@
      @@      
      @@      
      @@    @@
@@@@@@@@@@@@@@@@@@@@
      @@    @@
      @@    @@          natural
      @@    @@
@@@@@@@@@@@@@@@@@@@@
      @@    @@
            @@
            @@
@@@@@@@@@@@@@@@@@@@@
      
    @@
    @@
@@@@@@@@@@@@@@@@
    @@
    @@@@@@
    @@    @@
@@@@@@@@@@@@@@@@    flat
    @@  @@
    @@@@
    @@
@@@@@@@@@@@@@@@@


      
@@@@@@@@@@@@@@@@
    @@
    @@
    @@
@@@@@@@@@@@@@@@@
    @@    @@
    @@    @@        flat
    @@  @@
@@@@@@@@@@@@@@@@
    @@


@@@@@@@@@@@@@@@@@@
    @@@@@@@@@@
    @@      @@
    @@      @@
@@@@@@@@@@@@@@@@@@   convex right-angle
    @@      @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            @@

@@@@@@@@@@@@@@@@@@


    @@@@@@@@@@
@@@@@@@@@@@@@@@@@@
    @@      @@
    @@      @@       convex right-angle
    @@      @@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@


              
@@@@@@@@@@@@@@@@@@
        @@@@@@
      @@    @@
    @@      @@
@@@@@@@@@@@@@@@@@@    convex quadrant
    @@      @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            @@

@@@@@@@@@@@@@@@@@@
              

        @@@@@@
@@@@@@@@@@@@@@@@@@
    @@      @@
    @@      @@    convex quadrant
    @@      @@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
              


@@@@@@@@@@@@@@@@@@
            @@
          @@@@
        @@  @@
@@@@@@@@@@@@@@@@@@    straight
    @@      @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            @@

@@@@@@@@@@@@@@@@@@
              

            @@
@@@@@@@@@@@@@@@@@@
        @@  @@
      @@    @@        straight
    @@      @@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
              


@@@@@@@@@@@@@@@@@@
            @@
            @@
        @@@@@@
@@@@@@@@@@@@@@@@@@    narrow concavoconvex
      @@    @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            @@

@@@@@@@@@@@@@@@@@@

              
            @@
@@@@@@@@@@@@@@@@@@
        @@@@@@
      @@    @@        narrow concavoconvex
      @@    @@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
              


@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@@    narrow concave quadrant
      @@@@  @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            @@

@@@@@@@@@@@@@@@@@@

              
            @@
@@@@@@@@@@@@@@@@@@
            @@
          @@@@        narrow concave quadrant
      @@@@  @@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@
              


@@@@@@@@@@@@@@@@@@
              
              
            @@
@@@@@@@@@@@@@@@@@@    small concave right-angle
        @@@@@@
            @@
            @@
@@@@@@@@@@@@@@@@@@
            @@
            
              
@@@@@@@@@@@@@@@@@@


              
@@@@@@@@@@@@@@@@@@
            @@
            @@        small concave right-angle
        @@@@@@
@@@@@@@@@@@@@@@@@@
            @@
            @@
            @@
@@@@@@@@@@@@@@@@@@


One problem is that, at this resolution, a narrow concavoconvex flag (on a
single shaft) differs by only one pixel from a flat symbol when it is
right-hand down-pointing and space-centred.

@@@@@@@@@@@@@@@@
    @@
    @@
    @@
@@@@@@@@@@@@@@@@
    @@    @@
    @@    @@        flat
    @@  @@
@@@@@@@@@@@@@@@@
    @@

@@@@@@@@@@@@@@@@
    @@
    @@
    @@
@@@@@@@@@@@@@@@@
    @@    @@
    @@    @@        narrow concavoconvex
    @@@@@@
@@@@@@@@@@@@@@@@
    @@

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

Message: 4564 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 10:19:12

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote:

"I see that the basic difference in our approaches is that you want the
simplest/smallest comma relative to a diatonic scale (Pythagorean-7), which
you then subtract an apotome from if it is bigger than an apotome, while
I'm happy with the simplest/smallest comma relative to a chromatic scale
(Pythagorean-12)."

I should have said "... which you then subtract from an apotome if it is
bigger than a half apotome ..."

I have worked out what I think are the commas for both approaches. In the
chromatic approach I assume the 12 notes are from -4 to +7  fifths, i.e. Eb
to G# when G is 1/1. I assume the two approaches must agree on the commas
for 5 and 7, so the diatonic approach uses 7 notes from -2 to +4 fifths,
i.e. F to B when G is 1/1.

The two approaches agree on the commas for 5,7,11,13,29,41 and differ on
the commas for 17,19,23,31,37.

Here are the commas they agree on

5  80:81
7  63:64
11 32:33
13 1024:1053
29 256:261
41 81:82

and those that differ

   chromatic  diatonic
17 2176:2187  4096:4131
19 512:513    19456:19683
23 729:736    16384:16767
31 243:248    31:32
37 999:1024   36:37

Do you agree?

I'm hoping we can get all the diatonic commas from the chromatic ones using
schismas valid in 1600-ET.

Notice that it is only the chromatic choices for 17 and 19 that enable us
to notate the higher ETs.

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

Message: 4565 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 21:17:44

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3994]:

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

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

>>> The 19 flag
>>> -----------
>>> I don't require that the new type of flag be small irrespective 

of what it

>>> is used for. I only want the flag for the 3.3 cent 19 comma to 
be smaller
>>> than all the others, because it is less than half the size of 
any other
>>> flag comma and less than 1/6th of the size of all but the 17 

comma. If this

>>> is allowed, then it follows that it must be a new kind of 
comma, not
>>> convex, striaght or concave.
>> 

>> Why don't you look at this way:  The four straight and convex 
flags 
>> are all greater than 20 cents.  The 17 and 19 flags are both less 
>> than 10 cents.  The 23 flag, 729:736, is ~16.5 cents, which is 
closer 
>> in size to the four larger flags than the two smaller ones.  So, 
as I 
>> see it, *both* the 17 and 19 flags should be smaller than the 
others, 
> 

> Here are the relative sizes of the flag commas (shown using steps 
of 1600-ET).
> 
> _19__
> _____17_____
> __________23__________
> ______________5______________
> _________________7__________________
> ___________________11-5___________________
> _____________________29______________________
> 
> You are right that the largest difference in the series occurs 
between 17
> and 23. But the next largest difference occurs between 19 and 17, 
and
> between 23 and 5. So it seems that following your argument to its
> conclusion would require that the 19, 17 and 23 flags should be of 
_three_
> different types. I think it will be too hard to find 3 types that 
look
> smaller than convex and straight. We're having enough trouble 
agreeing on
> two. Although the 19 to 17 and 23 to 5 differences are the same in 
steps of
> 1600-ET, the 19 to 17 difference is slightly greater in cents.

We're dealing with two major issues here, distinct, yet related, each 
of which requires a decision:

1) Whether to make the 19 or the 23-comma the new flag; and

2) What will be the shape for the new flag.

How we resolve either one of these depends on how we resolve the 
other one, so debating these one at a time is not really getting us 
very far.

To further complicate this, we're each looking at this with a 
different primary objective in mind:

a) Modifying-12-things/modulo-1600; vs.

b) modifying-7-things/modulo-217.

with the other being secondary.  Both objectives are important, but 
our priorities are different.

What I suggest that we do is to work on two different solutions 
simultaneously:

A) Using the new flag for the 19-comma; and

B) Using the new flag for the 23-comma,

which will almost certainly require different solutions to issue 2).  
In the process of evaluating possibilities for new flags, we can then 
select our best choice(s) for both Plan A and Plan B.

After developing both plans so as to give the best possible outcome 
for each (complete with actual bitmap examples of the symbols), we 
can then discuss the advantages and disadvantages of each.  That way, 
we'll be evaluating the actual products as well as the concepts 
governing them, instead of the concepts alone (as we have been doing 
up to this point).

This is going to cost us more in time and effort (but fortunately not 
money), but I think that it will be well worth the investment.

With the foregoing in mind, I will be deferring a reply to some of 
the things you have brought up in your last three messages and will 
address those things that need to be answered first.

>> which is why I have been making concave flags laterally narrower 
> than 

>> the others in my bitmap diagrams for the past several weeks.  You 
>> haven't seen them this way yet, but I think they look great.  In 
>> particular, the symbol with two concave flags is noticeably 
smaller 
>> than ones with two straight/convex flags -- more similar in size 
to 
>> one with a single straight or convex flag.
> 

> That's good. I look forward to seeing them.

We'll have to create directories in tuning-math/files for each of us 
to put our examples for the other to retrieve and modify.  Your ASCII 
bitmaps are good up to a point, but they're no substitute for the 
real thing.

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3995]:

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

>> Okay, that sounds like a good description of what we are are very 
>> close to achieving.  I might prefer to call the 11-comma a diesis 
>> (although it is plain that you are using the term "comma" in a 
>> broader sense here), which would further justify the introduction 
of 
>> the 11-5 comma that is used in achieving it, just as the 13-
diesis 
> is 

>> also the (approximate) sum of two commas.
> 

> Yes. I like that idea.

There are also other things such as this that we should keep in mind 
when it comes time to write an article formally introducing the 
notation to the rest of the microtonal world.  I presume that we 
should co-author this, inasmuch as you have gotten so heavily 
involved in this project.  (I have more details to discuss, and I 
think we should continue any further discussion about this off-list.)

>> In the preceding sentence it should be obvious to you that I 
meant to 
>> say "defining 2176:2187 as *vL* and 512:513 as *vR*", but just so 
no 
>> one else misunderstands, I am correcting this here.
> 

> To be honest I had just read them as vL and vR without noticing the 
typos.
> That's a worry. :-)

Not to worry.

>>> I have no objection to using multiple flags on the same side, 
to notate
>>> primes beyond 29. However I consider 999:1024 to be the 

standard 37 comma

>>> because it is smaller than 36:37, also because it only requires 
2
>>> lower-prime flags instead of 3. Can you explain why you want 

36:37 to be

>>> the standard 37 comma?
>> 

>> Using primes this high has more legitimacy, in my opinion, in 
otonal 
>> chords than in utonal chords.
> 
> Certainly.
> 

>> If C is 1/1, then 37/32 would be D 
>> (9/8) raised by 37:36.  With 1024:999 the 37 factor is in the 
smaller 
>> number of the ratio, which is not where I need it.
> 

> But you would simply notate it as Eb lowered by 999:1024. The comma 
is in
> the same direction as the flat. I don't see that this has anything 
to do
> with otonal vs. utonal.

Now that you mention it, it does have more to do with the sharp-vs.-
flat (or in this case natural-vs.-flat) issue, in which case you can 
use whatever else you think is appropriate to decide between the 
two.  So, for the reasons you gave, yes, you are entitled to have 
999:1024 as the principal comma.

> ... I see that the basic difference in our approaches is that you 
want the
> simplest/smallest comma relative to a diatonic scale (Pythagorean-
7), which
> you then subtract an apotome from if it is bigger than an apotome, 
while
> I'm happy with the simplest/smallest comma relative to a chromatic 
scale
> (Pythagorean-12).

This is related to your viewing the new symbols as modifying 12 
pitches vs. mine as modifying 7.  As long as the notation is 
versatile enough to do what both of us require, then I think we'll be 
okay.

>> 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.
> 

> You should definitely mention it wrt 217-ET, but make it clear it 
is not
> universal. Whatever application did you or Erv find for the 41st 
harmonic?
> Sounds crazy to me.

Well, you have to be a little crazy to do microtonality in the first 
place, and the more time you spend with it, the crazier you get.  
Both Erv and I used 41 in different isoharmonic chords.  I wrote some 
notes further developing this last year, but I'll have to look for 
them and get back to you about this.

> I think I lost some schismas for alternate 17 and 19 commas you 
found. Can
> you remind me of those? 

You didn't lose them; I never found any.  I was wondering why you 
didn't bring up the fact that (513/512) * (2187/2176) != (4131/4096) 
when you proposed consolidating my 17-limit-in-183 and 23-limit-in-
217 approaches.  Because of that, in notating a given ET I am 
restricted to using only one (or only the other) of the symbols for a 
17-comma if the inequality doesn't vanish in that ET.

> On another matter: Can you tell me why the apotome symbol should 
not be
> x||x instead of s||s?

The simplest ET notations (17, 22, 24, 31, 41, which require only 5 
and 11-comma symbols in their definition) use only straight flags, so 
there is no point in confusing anyone with curved flags for the 
apotome, which is twice s|s in each of these (except 22, where s|s 
isn't used).  Curved flags appear in the notation only when they are 
necessary or helpful, etc., etc.

> When dealing with 217-ET (or limits lower than 29) I think it's ok 
to
> describe x| as the 13'-7 flag rather than the 29 flag. I expect 
you'd
> prefer this.

Yes, inasmuch as it does produce an exact 26:27 diesis.  This (taken 
together with the low numbers in its ratio) is another reason why I 
consider it the primary 13-diesis, as opposed to 1024:1053, which is 
only *approximated* with the 4095:4096 schisma.

>> As it turns out, the complements that you propose are, as a 
whole, 
>> much more intuitive than what I had, with the flag arithmetic 
being 
>> completely consistent for the symbols that we prefer for each 
degree 
>> of 217 (which we will hopefully agree upon within the next few 
days 
>> as the "standard 217-ET set").  This standard set of 217-ET 
symbols 
>> could then be used for determining the notation for any 
sharp/flat 
>> (or other) equivalents that may be required for any JI interval 
(for 
>> which the composer should be strongly encouraged to indicate the 
>> symbol-ratio association in the score).
>> 
>> There are rules for arriving at the new complement symbols 
(including 
>> the nonstandard ones), and they are a bit more convoluted that 
what I 
>> had. But inasmuch as the overall result is much better than what 
I 
>> previously had, I think that this is something that we should 
adopt 
>> without any further hesitation.

The fact that curved flags always convert to curved flags in arriving 
at apotome complements (and never to straight ones) is a major reason 
why I want a curved (concave) flag for the 19-comma.  Its apotome 
complement symbol ( s||x ) makes more sense that way.  This is 
something we can discuss further if we implement both plans A and B 
above.

> I'm sure glad you found them by another route. I think I proposed 
them
> mainly out of ignorance!

(Which is why you didn't take issue with my hasty -- and equally 
ignorant -- rebuttal.)  Well, in this case (your) ignorance is (my) 
bliss.  (And I'm glad your question prompted me to take another look 
at what I had.)

> Would you please describe the symbols you think should be used for 
1 thru
> 21 steps of 217-ET? s||s type notation will do fine.

Assuming Plan B for the moment, in which the 19-comma is vR, the 
standard symbol set is:

degs  symbol  ratio
----  ------  -----
  1     |v    512:513
  2    v|     2176:2187
  3    v|v    1114112:1121931
  4    s|     80:81
  5     |x    63:64
  6     |s    54:55
  6'   x|     715:729, ~256:261 (not normally used separately)
  7    v|x    238:243
  8    v|s    4352:4455, ~16384:16767
  9    s|x    35:36, ~1024:1053
 10    s|s    32:33
 11    x|x    704:729, ~5005:5184
 12    x|s    26:27

The apotome complements:
 13    v||    2048:2187 less 4352:4455
 14    v||v   2048:2187 less 238:243
 15    s||    2048:2187 less 54:55
 16     ||v   2048:2187 less 63:64
 17     ||s   2048:2187 less 80:81
 18    v||x   2048:2187 less 1114112:1121931
 19    v||x   2048:2187 less 2176:2187
 20    s||x   2048:2187 less 512:513
 21    s||s   2048:2187 (the apotome)

The non-standard combinations:
  5'   s|v    40960:41553 (has ||v as complement)
  6'   x|     715:729, ~256:261 (used separately as 29-comma)
  7'   x|v    366080:373977 (has v||v as complement)

The only inconsistent apotome complement:
 15'   x||    2048:2187 less 256:261 (used separately as /29-comma)

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#3996]:

I have imported all of your symbols into MS Word (in courier font) 
and will be looking at them in full-page view, which reduces them 
considerably.

> We're really getting into the fine details of font design here. What
> software are you using to create the bitmap versions you've been 
giving?

It's strictly low-budget, but it does the job: the "Paint" program 
that is found among the "Accessories" that are shipped as a part of 
the Windows operating system.  If you use this, I would caution you 
to save your work under a different name the first time you try 
saving in monochrome format (which results in a considerably smaller 
file size).  In one version of Windows (2000) the block copy doesn't 
work with a monochrome file, so you have to do all the work in full 
color format, then save the final version as monochrome with a 
different name.

> How many pixels high are they?

I set them up for use with single-pixel staff lines 8 pixels apart, 
and each sagittal symbol is 10 pixels high.  The conventional sharp 
symbol that I made for comparative purposes is 14 pixels high, the 
conventional flat 11 pixels, and the natural symbol 13 pixels.  I 
also did the Sims symbols -- the square root sign and its inverse are 
both 16 pixels high.

I have also made a bolder set of sagittal symbols that are wider (but 
no higher), so that they may be easily read in poor light at music-
stand distance; the widest of these, x|||x, is 16 pixels.  In the non-
bold set this symbol is 13 pixels wide.

> Eventually we'd need to give a resolution independent description 

as in a

> True-type or Postscript font. Presumably you'd want to copy the 
style of
> existing sharps and flats, such as making horizontal (or near-
horizontal)
> strokes much thicker than vertical ones, (as if painted with a 
brush that
> is about 3 times higher than it is wide) to avoid them getting lost 
against
> the staff lines.

Yes, but in my bold sagittal symbols I made the vertical lines 
thicker as well, in response to Ted Mook's complaint about the 
difficulty of distinguishing the Tartini fractional sharps from one 
another.

Till next time, keep in tune!

--George

Message: 4567 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 22:02:56

Subject: Re: A common notation for JI and ETs

From: genewardsmith

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

> To further complicate this, we're each looking at this with a 
> different primary objective in mind:
> 
> a) Modifying-12-things/modulo-1600; vs.
> 
> b) modifying-7-things/modulo-217.
> 
> with the other being secondary.  Both objectives are important, but 
> our priorities are different.

I
think you'd better decide which one is more important! I'm also not
clear why you need to introduce modulo anything. Why introduce
approximations which may not be appropriate for the particular system
you end up notating?

Message: 4568 - Contents - Hide Contents

Date: Fri, 12 Apr 2002 18:24:24

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Here's a summary of what we've found re commas and schismas. Here are what
I understand as all the prime commas we'd like to have symbols for.

5  80:81
7  63:64
11 32:33      11' 704:729
13 1024:1053  13' 26:27
17 2176:2187  17' 4096:4131
19 512:513    19' 19456:19683
23 729:736    23' 16384:16767
29 256:261
31 243:248    31' 31:32
37 999:1024   37' 36:37
41 81:82

Taking the primes (and primed primes) as representing their commas, we have
a flag for each of the following commas.

5
7
(11-5)
17
19
23
29

Now I list the effect of all the notationally-useful 1600-ET schismas we've
found. In other words, how to make a symbol for each comma, using the flags
for the above commas.

And by the way, many thanks to Graham Breed for telling me that 1600-ET was
what I was looking for. We can forget my 31-limit challenge now.

symb  lft-flgs  rt-flgs
------------------------
5   =       5
7   =           7
11  =       5 + (11-5)
11' =      29 + 7
13  =       5 + 7
13' =      29 + (11-5)
17  =      17
17' =                             [none except 19 + 19 + 19 + 19]
19  =           19
19' =      23 + 19
23  =      23                     [also 17 + 19 + 19]
23' =      17 + (11-5)
29  =      29
31  =           19 + (11-5)
31' =       5 + 23 + 23           [also 29 + 41]
37  = 29 + 17                     [also  5 + 41]
37' =  5 + 17 + 23                [also  5 + 5 + 19, also 19 + 7 + 23]
41  =                             [none]

So if we want 17' or 41 in the RT (rational tuning) symbols, we'd need to
add new flags for them, which I think would be a bad idea.

Those schismas above, for primes 29 and below, which are not in square
brackets, are also valid in 217-ET. Also the one for 31'.

The following are valid in 217-ET, but not 1600-ET. Note that 217-ET only
gives unique mappings up to the 19-odd-limit. (Are we sure it isn't only 17?)

17' = 23 = 17 + 19
41 = 5 = 19' = 19 + 23

Have I missed anything?

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

Message: 4569 - Contents - Hide Contents

Date: Sat, 13 Apr 2002 06:43:35

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

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

>> To further complicate this, we're each looking at this with a 
>> different primary objective in mind:
>> 
>> a) Modifying-12-things/modulo-1600; vs.
>> 
>> b) modifying-7-things/modulo-217.
>> 
>> with the other being secondary.  Both objectives are important, 
but 
>> our priorities are different.
> 

> I think you'd better decide which one is more important!

We may need to do so eventually, but so far we are doing astoundingly 
well at satsifying both of these. We are at present in complete 
agreement up to the 17 limit. We really only need to agree to the 
19-limit since beyond that is not required for 217-ET or any other ETs  
that we plan to notate.

In fact I'm comitted to first ensuring that the 19-limit and 217-ET 
notation is as good as it can be, ignoring any higher-limits or higher 
ETs.

Our current disagreement re the 19 comma symbol is not a chromatic vs. 
diatonic thing or a 217-ET vs. 1600-ET thing or a 19-limit vs. higher 
limits thing. It is a question of whether it is better (at the 
19-limit and in 217-ET) to

(a) have only 3 styles of flag making up the symbols
and have apotome complement rules that are more intuitive (according 
to George). (I haven't looked at this yet) 
and have more lateral confusability 
and have the 17 and 19 commas represented by same-style flags even 
though one is 2.6 times the size of the other, or

(b) have 4 styles of flag making up the symbols 
and have apotome complement rules which are less intuitive
and have little or no lateral confusability
and have 17 and 19 commas represented by two different styles of flag 
which give some indication of their relative sizes.

> I'm also 

not clear why you need to introduce modulo anything. Why introduce 
approximations which may not be appropriate for the particular system 
you end up notating?

>

This is a very valid point. You have raised it before and I thought I 
had answered it. But I'll take this opportunity to explain it in more 
detail why the problem doesn't exist.

There is no modulo anything forced upon you by the notation. We have 
always kept open the option of a strictly rational 
one-symbol-for-one-comma-per-prime use of the symbols. In this way of 
using it, you get a single-shaft arrow symbol for each prime comma (to 
be used with a chromatic scale, not diatonic). Some have single flags 
(half arrowheads) and some have double flags (full arrowheads). The 
double-flag symbols are generally larger than the single flag. 

1600-ET has merely been used as a logical way of constructing the 
multiflag symbols. It wouldn't matter if no one ever knew this. You're 
just given a bunch of symbols, one per prime. Provided you treat these 
symbols as atomic there is no approximation whatsoever. It is only if 
you start combining multiple symbols into a single symbol that you 
start introducing approximations.

For example if a note is flattened by both a 5-comma and a 7-comma 
then provided you have two separate arrows for the 5 and the 7 there 
is no approximation, but if you choose to combine the 5 flag and 7 
flag on a single shaft, then you find that you have made the symbol 
for the 13-comma and you have introduced an approximation. 

In rational tunings this approximation is kept to inaudible levels by 
basing it on 1600-ET. In some ETs this approximation will of course be 
quite large and quite audible, e.g. a whole step of the ET. But you 
are still not forced to make the approximation at all.

When we look at the ETs where 5-comma + 7-comma =/= 13-comma (among 
those we intend to notate) we find in most cases that we only need to 
use two of the 3 commas in notating the ET. e.g. In 27-ET the 7-comma 
vanishes and we use the 5 and 13 comma symbols. In 50-ET the 5-comma 
vanishes and we use the 7 and 13 comma symbols.

37-ET is a case I'm not too sure about. Here we have the 5-comma being  
2 steps, the 13 comma being 3 steps and the 7-comma vanishing. There 
is no prime comma within the 41-limit that is consistently equal to 1 
step (11-comma is 2 steps, same as 5-comma). We could notate 1 step as 
13-comma up and 5-comma down, but if we insist on single symbols, is 
it ok to use the 7-comma symbol to mean 13-comma - 5 comma? Or should 
we use the 19-comma symbol for one step, even though it's 
1,3,p-inconsistent?

Anyway, I hope you understand now that it is quite possible to use the 
symbols without any approximations at all (except those native to the 
scale being notated).



Message: 4570 - Contents - Hide Contents

Date: Sat, 13 Apr 2002 07:43:04

Subject: Re: A common notation for JI and ETs

From: genewardsmith

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

> Here's a summary of what we've found re commas and schismas. Here are what
> I understand as all the prime commas we'd like to have symbols for.
> 
> 5  80:81
> 7  63:64
> 11 32:33      11' 704:729
> 13 1024:1053  13' 26:27
> 17 2176:2187  17' 4096:4131
> 19 512:513    19' 19456:19683
> 23 729:736    23' 16384:16767
> 29 256:261
> 31 243:248    31' 31:32
> 37 999:1024   37' 36:37
> 41 81:82


Here are the corresponding 5 and 7 et mappings of the non-prime collection of commas:

[5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27]
[7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37]

These can be modifed to give the 41-limit interval the notation, considered as JI notation, would be notating.

Here are the "standard" mappings, by way of comparison:

[5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27]
[7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38]

Message: 4571 - Contents - Hide Contents

Date: Sat, 13 Apr 2002 08:10:20

Subject: Re: A common notation for JI and ETs

From: genewardsmith

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

These
commas don't seem to be h217-unique--was there some 217 mapping which
you were looking at, or do I have the wrong set, or the wrong idea?

Message: 4573 - Contents - Hide Contents

Date: Sat, 13 Apr 2002 08:12:40

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>We're dealing with two major issues here, distinct, yet related, each 
>of which requires a decision:
>
>1) Whether to make the 19 or the 23-comma the new flag; and

There is (at least) another possibility. We could make both 17 and 23 be
the new style of flag. 

>2) What will be the shape for the new flag.
>
>How we resolve either one of these depends on how we resolve the 
>other one, so debating these one at a time is not really getting us 
>very far.

Agreed. So I ask you to consider a system where 19 is concave right (as you
desire), 17 is narrow concavoconvex left and 23 is (not so narrow)
concavoconvex right.

I believe this acheives all our objectives except the one that says you'd
prefer to have only 3 styles of flag in the 19 limit rather than 4. It has
flags that indicate relative size. It reduces lateral confusability. It has
simple and visually intuitive rules for apotome complements. I believe it
is more intuitive even than your current favourite.

>To further complicate this, we're each looking at this with a 
>different primary objective in mind:
>
>a) Modifying-12-things/modulo-1600; vs.
>
>b) modifying-7-things/modulo-217.
>
>with the other being secondary.  Both objectives are important, but 
>our priorities are different.

I'm happy to try to ensure b) is as good as possible before returning to a).

>What I suggest that we do is to work on two different solutions 
>simultaneously:
>
>A) Using the new flag for the 19-comma; and
>
>B) Using the new flag for the 23-comma,
>
>which will almost certainly require different solutions to issue 2).  
>In the process of evaluating possibilities for new flags, we can then 
>select our best choice(s) for both Plan A and Plan B.
>
>After developing both plans so as to give the best possible outcome 
>for each (complete with actual bitmap examples of the symbols), we 
>can then discuss the advantages and disadvantages of each.  That way, 
>we'll be evaluating the actual products as well as the concepts 
>governing them, instead of the concepts alone (as we have been doing 
>up to this point).
>
>This is going to cost us more in time and effort (but fortunately not 
>money), but I think that it will be well worth the investment.

I'm hoping, with this message, to convince you that developing both A) and
B) is unnecessary, because you will prefer C) when you see how well it works.

C) Using new concavoconvex flags for both 17 and 23 commas.

>We'll have to create directories in tuning-math/files for each of us 
>to put our examples for the other to retrieve and modify.  Your ASCII 
>bitmaps are good up to a point, but they're no substitute for the 
>real thing.

Sure.

>There are also other things such as this that we should keep in mind 
>when it comes time to write an article formally introducing the 
>notation to the rest of the microtonal world.  I presume that we 
>should co-author this, inasmuch as you have gotten so heavily 
>involved in this project.  (I have more details to discuss, and I 
>think we should continue any further discussion about this off-list.)

Thanks. Yes I'd be delighted to coauthor it with you. Do you mean that we
should take all further discussion of the notation off-list, or just
discussions re the article?

>Now that you mention it, it does have more to do with the sharp-vs.-
>flat (or in this case natural-vs.-flat) issue, in which case you can 
>use whatever else you think is appropriate to decide between the 
>two.  So, for the reasons you gave, yes, you are entitled to have 
>999:1024 as the principal comma.

OK.

>This is related to your viewing the new symbols as modifying 12 
>pitches vs. mine as modifying 7.  As long as the notation is 
>versatile enough to do what both of us require, then I think we'll be 
>okay.

It's certainly shaping up well, to do just that.

>You didn't lose them; I never found any.  I was wondering why you 
>didn't bring up the fact that (513/512) * (2187/2176) != (4131/4096) 

That's 19 + 17 =/= 17'

>when you proposed consolidating my 17-limit-in-183 and 23-limit-in-
>217 approaches.  Because of that, in notating a given ET I am 
>restricted to using only one (or only the other) of the symbols for a 
>17-comma if the inequality doesn't vanish in that ET.

Can you live with that? Is it a significant problem? Can you see any easy
way around it? As it stands, you won't have a 17' comma symbol for rational
tunings, but you may have various symbols that correspond to it in various
ETs. It may be that wherever it is required in an ET, it always happens to
be the same as the 23 comma (as it is in 217-ET).

There is a similar problem with the 19' comma, except it _can_ be notated
with two flags on the same side, namely 19 + 23.

>> On another matter: Can you tell me why the apotome symbol should 
>not be

>> x||x instead of s||s?
>

>The simplest ET notations (17, 22, 24, 31, 41, which require only 5 
>and 11-comma symbols in their definition) use only straight flags, so 
>there is no point in confusing anyone with curved flags for the 
>apotome, which is twice s|s in each of these (except 22, where s|s 
>isn't used).  Curved flags appear in the notation only when they are 
>necessary or helpful, etc., etc.

I'm convinced.

>> When dealing with 217-ET (or limits lower than 29) I think it's ok 
>to

>> describe x| as the 13'-7 flag rather than the 29 flag. I expect 
>you'd
>> prefer this.
>

>Yes, inasmuch as it does produce an exact 26:27 diesis.  This (taken 
>together with the low numbers in its ratio) is another reason why I 
>consider it the primary 13-diesis, as opposed to 1024:1053, which is 
>only *approximated* with the 4095:4096 schisma.

As I later corrected, it is of course _not_ a (13'-7) flag. It is either an
11'-7 flag or (what's relevant here) a (13'-(11-5)) flag. Yes 26:27 is
certainly the primary 13-diesis from a diatonic point of view.

>The fact that curved flags always convert to curved flags in arriving 
>at apotome complements (and never to straight ones) is a major reason 
>why I want a curved (concave) flag for the 19-comma.  Its apotome 
>complement symbol ( s||x ) makes more sense that way.  This is 
>something we can discuss further if we implement both plans A and B 
>above.

With plan C) you can keep this property, which I agree is desirable, and
you will not need any new complement rules when the 23 flag is introduced.
You can keep the 19-comma as a concave flag.

>Assuming Plan B for the moment, in which the 19-comma is vR, the 
>standard symbol set is:
>
>degs  symbol  ratio
>----  ------  -----
>  1     |v    512:513
>  2    v|     2176:2187
>  3    v|v    1114112:1121931
>  4    s|     80:81
>  5     |x    63:64
>  6     |s    54:55
>  6'   x|     715:729, ~256:261 (not normally used separately)
>  7    v|x    238:243
>  8    v|s    4352:4455, ~16384:16767
>  9    s|x    35:36, ~1024:1053
> 10    s|s    32:33
> 11    x|x    704:729, ~5005:5184
> 12    x|s    26:27
>
>The apotome complements:
> 13    v||    2048:2187 less 4352:4455
> 14    v||v   2048:2187 less 238:243
> 15    s||    2048:2187 less 54:55
> 16     ||v   2048:2187 less 63:64
> 17     ||s   2048:2187 less 80:81
> 18    v||x   2048:2187 less 1114112:1121931
> 19    v||x   2048:2187 less 2176:2187
> 20    s||x   2048:2187 less 512:513
> 21    s||s   2048:2187 (the apotome)
>
>The non-standard combinations:
>  5'   s|v    40960:41553 (has ||v as complement)
>  6'   x|     715:729, ~256:261 (used separately as 29-comma)
>  7'   x|v    366080:373977 (has v||v as complement)
>
>The only inconsistent apotome complement:
> 15'   x||    2048:2187 less 256:261 (used separately as /29-comma)

I believe there are a number of typos in the above: The first 6' line
should be deleted. The 16 line should have ||x, not ||v. The 19 line should
have v||s, not v||x. 5' should say " has ||x as complement".

So we have plan B):

 1     |v      +19
 2    v|     17+
 3    v|v    17+19
 4    s|      5+
 5     |x      +7
 6     |s      +(11-5)
 7    v|x    17+7
 8    v|s    17+(11-5)
 9    s|x     5+7
10    s|s     5+(11-5)
11    x|x    (11'-7)+7
12    x|s    (11'-7)+5   ~=  (13'-(11-5))+(11-5)
13    v||    
14    v||v   
15    s||    
16     ||x   
17     ||s   
18    v||x   
19    v||s   
20    s||x   
21    s||s   

So the complementation rules are:

s  <->    (irrespective of whether it is left or right)
vL <-> vL
vR <-> xR

xL has no complement, or at least its complement is a vL pointing in the
opposite direction (up/down) so it is well that it can be avoided.

Note the inconsistency where a left v is its own complement while a right v
goes to an x.

Note that, if we include the 23 comma (3 steps of 217-ET) as a new type of
right flag nR, we'll need an additional complementation rule.

nR <-> nR

I assume you've noticed that a left flag and its complement must add up to
sL (in this case 4 steps), and a right flag and its complement must add up
to sR (in this case 6 steps).

It should also be noted that, for 5 steps and its complement 16, these
rules give valid alternatives, but they don't give _the_ answer that makes
the second half-apotome the same as the first. I'm guessing we need a
different kind of rule to deal with any degree that falls between sL and sR
in number of steps. This remains the same for Plan C). In many lower ETs
there will be no such degree.

There are 4 pairs of lateral confusables in the above.

Now I'll show plan C) (actually 2 options). I will use "c" to stand for the
new concavoconvex flag type. I would like the left-hand concavoconvex flag
(the 17 flag) to be narrower than the right-hand one (the 23 flag), but
will use "c" for both. Here's option C1)

 1     |v      +19
 2    c|     17+
 3    c|v    17+19
 4    s|      5+
 5     |x      +7
 6     |s      +(11-5)
 7    c|x    17+7
 8    c|s    17+(11-5)
 9    s|x     5+7
10    s|s     5+(11-5)
11    x|x    (11'-7)+7
12    x|s    (11'-7)+5   ~=  (13'-(11-5))+(11-5)
13    c||    
14    c||v   
15    s||    
16     ||x   
17     ||s   
18    c||x   
19    c||s   
20    s||x   
21    s||s   

So the complementation rules are:

s  <->  
c  <-> c
vR <-> xR

Notice that we've eliminated that inconsistency where a left v was its own
complement while a right v went to an x. And we only have 3 sets of lateral
confusables.

Here's option C2). The only change is to use the 23-flag for 3 steps, so
that the number of flags is monotonic with degree size.

 1     |v      +19
 2    c|     17+
 3     |c    23+
 4    s|      5+
 5     |x      +7
 6     |s      +(11-5)
 7    c|x    17+7
 8    c|s    17+(11-5)
 9    s|x     5+7
10    s|s     5+(11-5)
11    x|x    (11'-7)+7
12    x|s    (11'-7)+5   ~=  (13'-(11-5))+(11-5)
13    c||    
14     ||c   
15    s||    
16     ||x   
17     ||s   
18    c||x   
19    c||s   
20    s||x   
21    s||s   

And the complementation rules are:

s  <->  
c  <-> c
vR <-> xR

No new rule is required for the 23 flag!

We appear to have 5 pairs of lateral confusables in this case, but remember
that I want the 23-flag wider than the 17-flag, although they are the same
shape (concavoconvex). This brings us back to 3 pairs. The straight and
convex flags could be given that treatment too, i.e. make the
larger-in-cents of each pair, slightly wider than the other. However, when
they are combined on the same stem they should be the same size so such
symbols are symmetrical.

In those cases where there is lateral confusability between single s's and
c's, there is at least a consistent rule that left flags are smaller than
right flags.

-----------------------------
I can state the algorithm used for the two options above. It should result
in a good set of symbols for any tractable ET.

Calculate the number of steps for each of the 7 flag commas.
Calculate the number of steps for all 12 combinations of a left and a right
flag.
Sort these 19 symbols according to number of steps.
Eliminate any symbol containing xL if it has fewer steps than the 5+(11-5)
symbol (s|s) and there are other options for that number of steps.

Option 1
For each number of steps, choose the symbol that has the lowest prime
limit. If there is more than one with the lowest prime limit, then consider
their second highest primes, etc. For this purpose the 29 flag should be
considered to be the (11'-7) flag.

Option 2
For each number of steps, choose the symbol that has the fewest flags. If
there is more than one with the fewest flags, then take the one with the
lowest prime limit etc.
------------------------------

I currently favour option 2. At least for 217-ET I like the fact that it
doesn't give a double-flag symbol for something as small as 3 steps.

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

Message: 4574 - Contents - Hide Contents

Date: Sat, 13 Apr 2002 15:54:43

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

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

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

>> Here's a summary of what we've found re commas and schismas. Here 
are what
>> I understand as all the prime commas we'd like to have symbols 
for.
>> 
>> 5  80:81
>> 7  63:64
>> 11 32:33      11' 704:729
>> 13 1024:1053  13' 26:27
>> 17 2176:2187  17' 4096:4131
>> 19 512:513    19' 19456:19683
>> 23 729:736    23' 16384:16767
>> 29 256:261
>> 31 243:248    31' 31:32
>> 37 999:1024   37' 36:37
>> 41 81:82
> 

> Here are the corresponding 5 and 7 et mappings of the non-prime 

collection of commas:

> 
> [5, 8, 12, 14, 17, 18, 21, 21, 23, 24, 25, 26, 27]
> [7, 11, 16, 20, 24, 26, 28, 30, 31, 34, 34, 37, 37]

Thanks for looking at this, but I don't understand. Why should we care 
about their 5 and 7-ET mappings, and what do the numbers above mean? I 
don't understand why any of them are greater than 1.

> These can be modifed to give the 41-limit interval the notation, 

considered as JI notation, would be notating.

> 
> Here are the "standard" mappings, by way of comparison:
> 
> [5, 8, 12, 14, 17, 19, 20, 21, 23, 24, 25, 26, 27]
> [7, 11, 16, 20, 24, 26, 29, 30, 32, 34, 35, 36, 38]

Again, I don't understand what this means, but I'd like to, including 
why the standard is a standard. All I can figure out is that the 5-ET 
one disagrees on 13 and 17, and the 7-ET one disagrees on 
17,23,31,37,41, about something.

> These commas don't seem to be h217-unique--was there some 217
> mapping which you were looking at, or do I have the wrong set, or > 

the wrong idea?

No. I understand that 217-ET is only unique up to the 19-limit. But I 
understand they are all unique in 1600-ET. I'd be pleased if you have 
an easy way to check these.

Also, I wonder if there are any other ETs between say 1000 and 1600-ET 
that are 41-limit unique, or 37 limit unique, or even 31-limit unique.

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