Tuning-Math Digests messages 4776 - 4800

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Message: 4776

Date: Wed, 08 May 2002 20:12:06

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I've added one more rational complement, for vw|w, which may be of use as
an alternate 7/5 comma.

Symbol Comp   Comma name  Comments
------------------------------------
 v|    x||w   19
  |v   s||x   (17'-17)
 w|   ww||x   17
 v|v  vw||s
 w|v   x||v   17'
  |w   s||w   23
 v|w   x||    19'
 s|     ||s   5
ww|v   v||x   pythag comma (comp probably not required)
  |x    ||x   7
vw|w   w||w   (7/5)'
 v|x  ww||v      (probably not required)
  |s   s||    (11-5)
 x|    v||w   29 or (11'-7)
 v|s  vw||v   31 (comp probably not required)
 w|x  ww||       (probably not required)
 s|w    ||w      (prob not required, 5 comma + 23 comma)
vw|x   none   11/5 (hope comp is not required)
 x|v   w||v   alt 23' (comp is good reason to make this standard 23')
 w|s  vw||    23'
ss|     ||vv  25
 v|wx vv||    37' (comp probably not required)
 s|x    ||v   13
 s|s   x|x    11
sx|     |sx   31'

The above complements correspond to flags being the following numbers of
steps of 665-ET.

v|  2       3  |v
w|  5       9  |w
s|  12     15  |x
x|  19     18  |s

I note that, apart from a few exceptions below the resolution of 665-ET, we
have the following complementary pairs of flags on the same side.

Left side
v     ww
w     vw
s     (blank)
x     (none)

Right side
v     x
w     w
s     (blank)

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4777

Date: Wed, 08 May 2002 03:38:05

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote:

"My current thinking is that the rational complements should be based on
665-ET, an ET with an extremely good 1:3 so there is no danger of any size
cross-overs with any pairs of symbols, existing or future. 

We only need to introduce a |vv symbol (instead of my earlier proposed vw|)
as the complement to ss|, the 25 comma symbol."

The last paragraph was wrong. It seems that at least one other 3 flagger
must be introduced as the complement of the 17 flag, and possibly some
others, as shown below.

Here's my latest proposal for rational apotome complements.

Symbol Comp   Comma name  Comments
------------------------------------
 v|    x||w   19
  |v   s||x   (17'-17)
 w|   ww||x   17            (or vw||s, less preferred)
 w|v   w||s   17'
  |w   s||w   23
 v|w   x||    19'
 s|     ||s   5
ww|v   v||x   pythag comma (comp probably not required)
  |x    ||x   7
  |s   s||    (11-5)
 x|    v||w   29 or (11'-7)
 v|s  vw||v   31 (comp probably not required)
 s|w    ||w      (prob not required, 5 comma + 23 comma)
vw|x   none   11/5 (hope comp is not required)
 w|s   w||v   23'
ss|     ||vv  25
 v|wx vv||    37' (comp probably not required)
 s|x    ||v   13
 s|s   x|x    11
sx|     |sx   31'

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4778

Date: Wed, 08 May 2002 21:08:50

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> Dave,
> 
> I've put out a file containing my latest proposal for symbols for 
> alterations above the half-apotome.
> 
> Yahoo groups: /tuning- *
> math/files/secor/notation/Symbols3.bmp

The | and || shaft symbols look great, but I'm afraid the whole 

concept of ||| and X shaft symbols will have to be a minority report. 

I'd rather just stack a s||s beside the | and || symbols.

What did you think of my suggestion to use V tails or single and 

double wavy tails?

> I've paid particular attention to scaling the width of the 2 & 3-
> shaft and X symbols.  For these I didn't think it was appropriate to 
> make the concave flags as small as you did in your examples 
> (comparing the size of the symbols at the left extreme with those at 
> the right extreme of the line above), and I find that these and the 
> wavy flags are quite readable this way. 

Agreed. I may want to fiddle with a pixel here and there if I get 

time. But otherwise I think they are great.

> I realize that a few of 
> these symbols won't be used the way we presently have things figured 
> out, but I did all of these just to get a sense of continuity in the 
> progression of size moving vertically.
> 
> I also tried my hand at ss||, ss|||, and ssX symbols at the far 
> right, just to see how those might look. 

ss|| looks OK, but maybe you should try omitting the part of one shaft 

that appears between the two flags.

Wanna try some of the other two-flags-on-one-side symbols I've 

proposed as rational complements? e.g. ww|x (17), vw||s (other 

possibility for 17), vw|| (23'), ||vv (25).

> (I hope that the meanings 
> of x and X don't get too confusing.)

I read them just fine.

I'm keen to finalise the rational complement relationships and the 

single 11/5 comma symbol if any.

Should we look at possible single symbols for 13/5, 13/7, 13/11 commas too,
or is this getting too silly?

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4782

Date: Thu, 09 May 2002 21:58:15

Subject: Re: 3D geometry

From: dkeenanuqnetau

--- In tuning-math@y..., "robstrange66" <robstrange@n...> wrote:
> Hi-
> I am very interested in 3D & geometry. Anyone here study 
> Platonic solids, polyhedra, etc?
> 
> Regards,
> Rob

Yes, and unless it relates to music, we should probably talk about it 
in the ZomeWorld yahoogroup.


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Message: 4784

Date: Thu, 09 May 2002 04:15:21

Subject: I found it!

From: genewardsmith

I couldn't find this group, and wondered what had happened to it. It
turns out that for some reason I'm now a moderator, and it moved
positions in my list of groups.


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Message: 4786

Date: Thu, 09 May 2002 04:18:29

Subject: Re: Microtonal Fiction (Was: A common notation for JI and ETs)

From: genewardsmith

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

> So, Dave, where and when do we get started?  (Off-list, I would say.  
> Tuning-math just doesn't seem to be the right place.)

What about spiritual-tuning?


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Message: 4792

Date: Fri, 10 May 2002 06:20:53

Subject: Re: 3D geometry

From: genewardsmith

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

Him Paul--I was afraid you'd run off somewhere. :)


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Message: 4793

Date: Fri, 10 May 2002 00:56:39

Subject: Another Blackjack detempering

From: Gene W Smith

Here is the symmetrical JI scale I get from the Fokker block using
<36/35,225/224,1029/1024> as commas. Does this correspond to any of your
(Paul's) detemperings?

Scale:

1, 49/48, 16/15, 35/32, 8/7, 7/6, 128/105, 5/4, 21/16, 4/3, 7/5, 10/7,
3/2, 32/21, 8/5, 105/64, 12/7, 7/4, 64/35, 15/8, 96/49

Steps:

49/48, 256/245, 525/512, 256/245, 49/48, 256/245, 525/512, 21/20, 64/63,
21/20, 50/49, 21/20, 64/63, 21/20, 525/512, 256/245, 49/48, 256/245,
525/512, 256/245, 49/48


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Message: 4794

Date: Sun, 12 May 2002 11:24:53

Subject: Re: graham's linear temperament page

From: monz

hi Paul,

> From: "emotionaljourney22" <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, May 10, 2002 3:25 PM
> Subject: [tuning-math] graham's linear temperament page
>
>
> this page is great and very important (by contrast, check out joe 
> monzo's definition of linear temperament if you want to turn red).


how about a more constructive criticism?  post something that
i can use to replace or supplement my definition to make it better.
(and also please specify whether i should be replacing or supplementing!)
thanks.

(BTW, i'm only checking in on the lists sporadically these days.
too much other "life" stuff happening...)


-monz


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Message: 4796

Date: Sun, 12 May 2002 15:16:15

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 22:17 9/05/02 -0000, you wrote:
>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>I have a serious problem with using 665-ET as a basis for anything.  
>It is only 9-limit consistent -- the 11 factor falls almost midway 
>between degrees.  Among other things, this causes xL+sR to be 37 
>degrees, whereas it should be 36.

True, but an error of a whole step of 665-ET is still 33% smaller than an
error of a half step of 217-ET. Consistency relates to accuracy relative to
step size, but surely absolute accuracy is more relevant here? I wasn't
intending to notate 665-ET, it was just a way of ensuring that the flags
(and the second shaft) could consistently be assigned fixed values
(different kind of consistency) while minimising offsets.

But you'll be pleased to know that I've abandoned 665-ET (and all ETs) as a
basis for rational complements.

I agree 306-ET is not very enticing.

Thanks for those spreadsheets. I like the idea of ignoring ETs and just
trying to minimise the offsets. 

We can take a set of symbol complements and treat them as a system of
linear equations which can then be solved to obtain values in cents for the
individual flags. It is possible to make a set that has no solution. This
is a different (and more serious) kind of inconsistency than the kind we
talk about when we say a certain ET is n-limit inconsistent. I think it is
important that they be consistent in this sense.

Of course the glaring problem with your recent proposal is the 4 cent
offset for w|  <->  w||s. All the others are less than half that. I'd be
much happier if we could keep the max offset to 1.5 cents or less. But I
also agree that the use of symbols with two-flags-a-side should be a last
resort.

The 25 comma symbol really does need a complement, e.g. C:G#\\. I don't
think there's any problem with _its_ complement having two flags a side, in
fact I think it would be expected. My favourite complement for 
ss|  is  ||vv. This works in your system as well as mine.

For our system of linear equations, we can write this as
ss|  +  ||vv  =  113.685 cents

If we insist on consistency (as in having a solution for the flag sizes)
then the above implies
s|v  +  s||v  =  113.685

It seems very desirable to have
 |x  +   ||x  =  113.685
We agree on that.

Taken with the above, that implies 
s|v  =  |x
and means that s||v is not available as a complement for anything else. I
assume we want our rational complements to be uniqu. I would simply outlaw
s|v and s||v. You have s||v as the complement of w|w. I don't think we
actually need a complement for w|w. Do we?

The above implies that 
s|v  +   ||x  =  113.685
which implies
 |v  +  s||x  =  113.685
another complement that we agree on.

Of course there's no question that
s|   +   ||s  =  113.685 

Another equivalent pair that we agree on is
v|   +  x||w  =  113.685
v|w  +  x||   =  113.685
with its 0.14 cent offset.

I set up a speadsheet that allows one to enter these equations and then
solves them for the size of each flag in cents. I've made the value of the
second shaft (the difference between s|s and s||s) a free variable too,
which is the equivalent of allowing the top and bottom parts of my
complement.bmp diagrams to slide against each other to minimise offsets.

Based on the solution for all the flags, I calculate the errors in all the
commas (including some alternate symbols for 23', 31 and 37') and find the
maximum error over all of them.

I find that with all the complement equations I want, I still have two
degrees of freedom left over. I use these to specify the values in cents of
the 5 and 7 comm flags. I adjust these to minimise the maximum-absolute
error. I found a set of complement relationships, which is a mixture of
yours and my earlier ones, that lets me get the maximum error in any comma
(i.e. sum of flag values minus comma value) down to 1.12 cents. This
includes every comma up to 41 and the ones for 11/7 and 7/5. 

The best I can do with your complements is a max error of 2.0 cents. It
makes sense that the max error would be half the max offset.

Unfortunately mine requires that the complement of w| has 3 flags, x||vv.
This is a consequence of the complement of w|v being x||v. I'm hoping you
can find a set of complements that either have a lower max error, or a
similar max error without needing a 3 flag symbol as the complement of a
one-flag symbol, but it doesn't look too hopeful.

Here's the system I'm talking about. I've put an asterisk against those
that differ from yours.

symbol complement comma     offset
---------------------------------------
  v|     x||w     19        -0.14 cents
   |v    s||x     (17'-17)  -1.50
* w|     x||vv     17        
  v|v    none
* w|v    x||v     17'        
   |w    w||x     23         0.73
  v|w    x||      19'       -0.14
  s|      ||s     5          0.00
* w|w    none
   |x     ||x     7         -1.26
* s|v  not used                     equiv to |x and so not used
  v|x    none
   |s    s||      (11-5)     0.00
  x|     v||w     29        -0.14
  v|s    none     31
  w|x     ||w     alt 31     0.73
* s|w    none
* x|v    w||v     23'               was alt 23'
* w|s    none                       was 23'
*ss|      ||vv    25
  s|x     ||v     13        -1.50   alt 37' (replacing 3-flag symbol)
  x|w    v||                -0.14

>You passed up some nice small-offset complements that are 665-
>inconsistent.  The only ones that I was forced to pass up in 217-ET 
>have an offset of over 2.6 cents.  And you can see that 653-ET also 
>has a number of inconsistent complements.  This is due in large part 
>to the fact that 653 and 665 are much finer divisions, so this is not 
>surprising.

Yes. Consistency is irrelevant here. It's the offsets (or errors) in cents
that matter.

>However, this is a good reason not to base rational complementation 
>on a particular division of the octave, but rather on the basis of a 
>small offset.

Agreed.

The spreadsheets I used for solving the two sets of equations are at
Yahoo groups: /tuning-math/files/Dave/DKCompSolve.xls *
Yahoo groups: /tuning-math/files/Dave/GSCompSolve.xls *

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4797

Date: Sun, 12 May 2002 17:21:30

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote:
"Consistency is irrelevant here. It's the offsets (or errors) in cents that
matter."

Of course I meant n-limit consistency of ETs is irrelevant here. I think
consistency of the system of linear equations representing the complement
rules is very important.

Another problem I have with your proposal is a crossover of symbol sizes
between single and double shafts when one includes the obvious complement
for the 25 comma symbol. It is related to the 4 cent offset. Assuming we
take the complement of ss| to be ||vv, and you have w| complement is w||s,
then in order of increasing size (of commas represented, not of solutions
to equations) we have 
w|      |vv  ...   w|s  ss|
but in order of increasing size, the complements go
 ||vv  w||   ...  ss||  w||s

Maybe what this is trying to tell us is that we should consider making ss||
the complement of w| (17 comma), and w|| the complement of ss| (25 comma).
When I substitute that for the w| rule in your system I can get the max
error down to 1.21 cents, provided I use w|x for 31 and s|x for 37'.

I've put up the spreadsheet as 
Yahoo groups: /tuning-math/files/Dave/GS2CompSolve.xls *

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4798

Date: Mon, 13 May 2002 09:19:55

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I realised that the system of my previous message is no good because it
didn't give a complement for the 23' comma with either the standard or
alternate symbol. But now I think I've cracked it. 

I've found a system where every comma that needs a complement has one, and
no new symbols are required, and the maximum error is 1.23 cents according
to my spreadsheet. The maximum offset is 1.53 cents according to your
spreadsheet. The system happens to be consistent with 494-ET. You can see
it in

Yahoo groups: /tuning-math/files/Dave/DK2Compls.xls *
and
Yahoo groups: /tuning-math/files/Dave/DK2CompSolve.xls *

The 23' comma symbol is now x|v, not w|s, because w|s has no
one-flag-per-side complement in this system. This involves a 0.52 cent
schisma. It also frees w|s to be used as purely a 125-diesis symbol if we
want (0.56 cent schisma).

The 31 comma symbol is now w|x, not v|s, and the 37' symbol is s|x, not
v|wx. These involve schismas of 0.73 cents and 0.88 cents respectively, but
I consider these a price worth paying for the reduced number of symbols and
the complete rational complements without any 3-flag symbols.

As a bonus, all the symbols that do not have complements are not needed at
all. We could take w|w and s|v as complementary but I don't think we need
either of them. I believe we only need 20 single-shaft symbols and 16
double-shaft symbols. None of these symbols have more than 2 flags and only
4 have two flags on the same side (the 25 and 31' symbols ss| and sx| and
the complements of the 17 and 31' symbols ss|| and |xs).

Here it is. The differences from your most recent proposal are shown with
asterisks.

symbol complement comma     offset (cents)
---------------------------------------
natural  s||s     apotome    0.00
  v|     x||w     19        -0.14
   |v    s||x     (17'-17)  -1.50
* w|    ss||      17         1.53
* w|v    x||v     17'       -1.03
   |w    w||x     23         0.73
  v|w    x||      19'       -0.14
  s|      ||s     5          0.00
   |x     ||x     7         -1.26
   |s    s||      (11-5)     0.00
  x|     v||w     29        -0.14
  w|x     ||w     31         0.73
* x|v    w||v     23'       -1.03
*ss|     w||      25         1.53
  s|x     ||v     13        -1.50
  s|x    x|s      13        -1.50 alternative single-shaft complement
  x|w    v||                -0.14
  s|s    x|x      11         0.00
 sx|      |sx     31'        0.00

In 494-ET the flags correspond to the following numbers of steps.

v|  1       2  |v
w|  4       7  |w
s|  9      11  |x
x|  14     13  |s

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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