Tuning-Math Digests messages 4475 - 4499

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Message: 4475

Date: Sun, 31 Mar 2002 10:53:57

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > i don't like this kind of constraint because it makes 11/7 seem 
> as 
> > > complex as 77/64.
> > 
> > Remember that the purpose of this temperament is to make a 
notation 
> > with a minimum number of symbols (or sagittal flags) that can 
> notate 
> > rational scales so even Johnny Reinhard can't tell the difference, 
> and 
> > notate all ETs below 100-ET and many above it.
> 
> even so.

So what would you suggest?


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Message: 4476

Date: Sun, 31 Mar 2002 11:06:13

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: genewardsmith

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

> So what would you suggest?

What was wrong with the plan of using 2^a 3^b p commas? I'm not at all clear why you want to abandon it.


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Message: 4477

Date: Mon, 01 Apr 2002 05:25:02

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

One of the possibilities of George's new sagittal notation is that one 
can ignore the meaning of the individual flags and simply take it as a 
set of symbols for 2^a*3^b*p commas, one per prime p from 5 to 31, to 
be used in conjunction with # and b and to be simply placed side-by 
side. As such it presently looks something like this (with the symbols 
for the primes above 13 still being negotiated).

Joseph, I'm sorry to have to point out that these symbols bear a much 
greater resemblance to the "European" symbols, than the Sims symbols.

I've also proposed single-ASCII-character substitutes for some of 
them.

[If you're reading this on the yahoogroups website you will need to 
choose Message Index, Expand Messages, to see the following symbols 
rendered correctly.]

5-comma 80;81

 /|
/ |
  |    / \
  |
  |

7-comma 63;64
   _
  | \
  | |
  |    P L
  |
  |

11-comma 32;33

 /|\
/ | \
  |    ^ v
  |
  |

13-comma 1024;1053
   _
 /| \
/ | |
  |    
  |
  |

17-comma 2176;2187

  |
_/|
  |    
  |
  |

19-comma 512;513

  O
  |
  |    * o
  |
  |

23'-comma 16384;16767  (unfortunately not 729;736)

  |\
_/| \
  |
  |
  |

29-comma 256;261
 _
/ |
| |
  |    q d
  |
  |

31-comma 243;248

 /|
/ |\_
  |
  |
  |

For the down versions of these, flip them vertically (don't rotate 
them 180 degrees). 

The smaller 23-comma _can_ be rendered as, the unfortunately 
complicated:

23-comma 729;736

  O
_/O
  |
  |
  |

Notice that lateral confusability only occurs beyond the 23-limit, and 
this might be eliminated by adding blobs to the end of the curved 
strokes of the 29 and 31 commas like this.

29-comma 256;261
 _
/ |
b |
  |
  |
  |

31-comma 243;248

 /|
/ |\o
  |
  |
  |

We also have optional symbols for larger 11, 13 and 17 commas.

11'-comma 704;729
 _ _
/ | \
| | |
  |    ] [
  |
  |

13'-comma 26;27
 _
/ |\
b | \
  |    
  |
  |

Only if you start combining multiple symbols into a single symbol, do 
you begin to assume the vanishing of the following pair of 
sub-half-cent schismas: 4095:4096 (13-limit), 3519:3520 (23-limit 
using large 23-comma).

If you use the symbols for the large 11-comma and large 13-comma as 
well as the small ones, you are also assuming the vanishing of the 
sub-half-cent schisma 5103:5104 (29-limit).

This is all George Secor's work (apart from the liberties I've taken 
with the post 13-limit symbols), I'm just trying to explain one 
delightful aspect of it.


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Message: 4478

Date: Mon, 01 Apr 2002 05:59:04

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- I wrote:
> We also have optional symbols for larger 11, 13 and 17 commas.

The astute reader will have noticed that I did not in fact give a 
symbol for a larger 17-comma (which would have been 4096:4131). That's 
because it can't be done without either adding another type of 
flag, disallowing the smaller 23-comma, or assuming the vanishing of a 
schisma of about 1.6 cents (1543503872:1544898987) which is too big 
for a JI notation.

I don't really see a need for the larger 17 comma. It would merely 
allow one to notate, for example, a 16:17 above C as a slightly raised 
(pythagorean) Db instead of a slightly lowered (pythagorean) C#.

If we disallowed the smaller 23-comma (729:736) the larger 17-comma 
could have a symbol like this. [Message Index, Expand messages]

17'-comma 4096:4131

  |
_/O
  |
  |
  |


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Message: 4479

Date: Mon, 01 Apr 2002 07:42:04

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Joseph, I'm sorry to have to point out that these symbols bear a 
much 
> greater resemblance to the "European" symbols, than the Sims 
symbols.
> 

The one thing I've always found unjustifiable and now find 
irredeemable about the Sims notation is the use of arrows with full 
heads to indicate something smaller than the arrows with half heads. I 
could almost make a version of this notation that is compatible with 
the Sims notation, if it wasn't for the twelfth-tone arrows.

Joseph, remind me what you don't like about slashes again, assuming 
the up slash has a short vertical stroke thru the middle of it and the 
down slash doesn't?

George, remind me why the 5-flag is on the left and the 7-flag on the 
right, and why the 5-flag is straight and the 7-flag curved? Why 
couldn't either of these properties be switched between 5 and 7?

Here's a more complete "single-ASCII-character substitutes" proposal.

Symbol
dn up  Comma                  Abbrev. descr. of actual symbol
-------------------------------------------------------------
\  /   5-comma      80;81     sL
L  P   7-comma      63;64     xR
v  ^   11-comma     32;33     sL+sR
[  ]   11'-comma   704;729    xL+xR
{  }   13-comma   1024;1053   sL+xR
;  |   13'-comma    26;27     xL+sR
j  f   17-comma   2176;2187   vL
*  o   19-comma    512;513    cO
w  m   23-comma    729;736    vL+cO+cI
W  M   23'-comma 16384;16767  vL+sR
q  d   29-comma    256;261    xL
y  h   31-comma    243;248    sL+vR

If it turns out we allow the large 17-comma instead of the small 
23-comma then this could be:

J  F   17'-comma  4096;4131   vL+cI

The abbreviated descriptions above refer to single-shaft arrows with 
various "flags" making up the arrow head:

sL straight left
sR straight right
xL convex   left
xR convex   right
vL concave  left
vR concave  right
cO small filled circle, outer (on tip of arrow)
cI small filled circle, inner (away from tip of arrow)


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Message: 4480

Date: Mon, 01 Apr 2002 07:54:33

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

By the way, we can actually notate 311-ET with combinations of these 
flags, so that no note has more than one arrow next to it in addition 
to a sharp or flat. Not that this is of any particular importance. The 
values of the flags in steps of 311-ET are:

sL 6
sR 8
xL 9
xR 7
vL 3
vR 3
cO 1
cI 1


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Message: 4481

Date: Mon, 01 Apr 2002 00:14:25

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > So what would you suggest?
> 
> What was wrong with the plan of using 2^a 3^b p commas? I'm not at 
all clear why you want to abandon it.

We're not abandoning that plan at all. Sorry I didn't explain. These 
sub-half-cent thingies are better thought of, not as commas, but as 
schismas, where a schisma is defined (for our purposes here at least) 
as a small difference between commas (just as _the_ schisma is the 
difference between the syntonic and pythagorean commas).

We want to reduce the number of symbols to less than the number of 
primes, if possible, so we're interested in sub-half-cent schismas 
which correspond to a very simple relationship between commas like x + 
y ~= z, where x,y and z are some of our 2^a*3^b*p commas.

The most striking schisma proposed so far, found by George Secor, is 
4095:4096 which says that the 13-comma is the 5-comma plus the 
7-comma.


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Message: 4484

Date: Mon, 01 Apr 2002 21:50:41

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
> wrote:
> 
> > The most striking schisma proposed so far, found by George 
> Secor, is 
> > 4095:4096 which says that the 13-comma is the 5-comma 
> plus the 
> > 7-comma.
> 
> how do you get 13 from 5 and 7?

5-comma 80:81 21.51 c
7-comma 63:64 27.26 c
5-comma + 7-comma = 48.77 c
13-comma 1024:1053 48.35 c

It doesn't matter for notating ETs less than 100 that the 4095:4096 
doesn't always vanish, because one never needs to use all three 
notational commas together (5, 7 & 13).


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Message: 4486

Date: Tue, 02 Apr 2002 13:18:38

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Here's a more complete "single-ASCII-character substitutes" 
proposal.
> > 
> > Symbol
> > dn up  Comma                  Abbrev. descr. of actual symbol
> > -------------------------------------------------------------
> > \  /   5-comma      80;81     sL
> > L  P   7-comma      63;64     xR
> > v  ^   11-comma     32;33     sL+sR
> > [  ]   11'-comma   704;729    xL+xR
> > {  }   13-comma   1024;1053   sL+xR
> > ;  |   13'-comma    26;27     xL+sR
> > j  f   17-comma   2176;2187   vL
> > *  o   19-comma    512;513    cO
> > w  m   23-comma    729;736    vL+cO+cI
> > W  M   23'-comma 16384;16767  vL+sR
> > q  d   29-comma    256;261    xL
> > y  h   31-comma    243;248    sL+vR
> 
> I don't know if anyone cares about 12-et-compatibility up to the 
> 31-limit, but in case they do here is what you get if you enforce 
compatibility with the "standard" h12 and h7 mappings:
> 
> 2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096, 
> 513/512, 16767/16384, 261/256, 67797/65536

I'm not sure I know what you mean by "enforce compatibility with the 
"standard" h12 and h7 mappings", because I can't see why you would get 
67797/65536 as the 31-comma when this contains 3^7 and 248/243 only 
contains 3^-5. This is the difference between a 1:31 from C being a Cb 
or a B respectively (which are of course the same thing in 12-tET).

I'm also unclear how this method chooses 729/704 over 8192/8019 for 
the 11-comma, and 16767/16384 over 736/729 for the 23-comma. Since 
both these choices involve 3^6 versus 3^-6.

I'm guessing these have something to do with h7 or 7-ET. Could you 
please explain your method in more detail?

Perhaps instead of 7-ET it would make more sense to use the C major 
scale in 12-ET.

It seems there is no argument over which commas to use for 3,5,7,19,29 
- the same choices for these can be arrived at in any number of ways - 
but 11,13,17,23,31 are not yet settled. Even if our final system has 
symbols for more than one comma per prime, we will still want to 
specify a "preferred" comma for each prime.

I believe we decided long ago that we were basing this notation on 
pythagorean (i.e a chain of Just fifths) not 12-ET, although we might 
favour 12-ET to the extent that all the odd numbers, up to an 
odd-limit determined by our highest prime, should be expressible 
without requiring enharmonics such as G# and Ab to be used 
simultaneously, and more strictly, without requiring anyone to go 
outside of a chain of fifths having 12 notes, e.g. Eb to G# or Ab to 
C#. And further, not to use any commas (at least as "preferred" 
commas) that fail to vanish in 12-ET, such as 26:27, 16:17, 18:19 or 
23:24. 32:33 is only considered to pass this test because it vanishes 
in the case of 3:11 and 9:11 even though it doesn't vanish in the case 
of 1:11.

This last requirement is more accurately expressed as requiring all 
preferred commas to be smaller than half an apotome (i.e. smaller than  
56.84 cents).

It might seem obvious that we should limit the 12 base notes to which 
the commas must be applied, and hence the range of exponents of 3 in 
these commas, to a range of either -6 to +5 fifths from the 1/1, or a 
range of -5 to +6. However 25 is within our odd-limit and presumably 
we want the "25-comma" to simply be two 5-commas, which means two 
syntonic commas (6400:6561), which means 3^8, which means our range of 
allowed-exponents-of-3 must be shifted at least as far as -3 to +8.

The only alternatives to this are either to go outside a 12 note chain 
of fifths within our odd-limit, or to invoke a different comma for 25, 
requiring its own symbol, such as the diaschisma (2025:2048) which has 
3^-4. A separate 25-comma symbol seems like a bad idea to me, unless 
it is obviously made up of two 5-comma symbols, in which case it 
should _be_ two 5-commas.

Here's a table showing all the 2,3,prime commas (and 2,3,25 commas) 
that could possibly be of any interest in this notational endeavour.

[On the website, use Message Index, Expand Messages to see the columns 
formatted correctly.]

3  Note
exp   Odd  Comma      Cents  Odd  Comma      Cents
--------------------------------------------------
-7 Cb 29 65536:63423 -56.74  31 65536:67797  58.72
-6 Gb 11 8192:8019   -36.95  23 16384:16767  40.00
-5 Db 17 4096:4131    14.73				
-4 Ab 13 1024:1053    48.35  25 2048:2025   -19.55
-3 Eb 19 512:513       3.38				
-2 Bb  7 64:63       -27.26  29 256:261      33.49
-1 F  11 32:33        53.27  21			
 0 C   1							
 1 G   3							
 2 D   9							
 3 A  27                     13 27:26       -65.34
 4 E   5 81:80       -21.51				
 5 B  31 243:248      35.26  15			
 6 F# 23 729:736      16.54  11 729:704     -60.41
 7 C# 17 2187:2176    -8.73				
 8 G# 13 6561:6656    24.89  25 6561:6400   -43.01

Within a range of -3 to +8 fifths we have the following commas for the 
disputed primes, smaller than half an apotome.

11 32:33
13 6561:6656 (note: not 1024:1053)
17 2187:2176
23 729:736
31 243:248

I would be very sorry not to have 1024:1053 as the 13-comma because:
(a) it has that neat sub-half-cent schisma with the 5 and 7 commas 
(4095:4096).
(b) 8:13 makes more sense as a neutral sixth than a superaugmented 
fifth.

There is a completely unrelated reason why we should go to at least a 
3-exponent of 7, and that is so that we have 2187:2176 as the 
preferred 17-comma. We need a comma of about this size (8.7 cents) to 
help us in notating the larger ETs, because it fills in a huge gap 
between the 19-comma of 3.4 cents and the 5-comma of 21.5 cents. I 
expect the other 17-comma (4096:4131) at 14.7 cents would be nowhere 
near as useful.

Within a range of -4 to +7 fifths we would have the same commas as -3 
to +8 fifths, except for the 13-comma, which would become the beloved 
1024:1053. But rather than introduce the disachisma as a 25-comma I'm 
inclined to allow the 3-exponents to range from -4 to +8 so that, from 
C, an 8:11 is a variety of Ab (or A, using the non-preferred 13-comma) 
and a 16:25 is a variety of G#. Too bad about the possibility of 
simultaneous enharmonics. How do others feel about this?


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Message: 4487

Date: Tue, 02 Apr 2002 13:27:35

Subject: Re: 31-limit microtemperament challenge (was: _The_ 31-limit temperament?)

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> 
> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> 
> wrote:
> > > how do you get 13 from 5 and 7?
> > 
> > 5-comma 80:81 21.51 c
> > 7-comma 63:64 27.26 c
> > 5-comma + 7-comma = 48.77 c
> > 13-comma 1024:1053 48.35 c
> > 
> > It doesn't matter for notating ETs less than 100 that the 
> 4095:4096 
> > doesn't always vanish, because one never needs to use all 
> three 
> > notational commas together (5, 7 & 13).
> 
> i'm not following.

I'm not sure what it is you're not following. Do you follow how to get 
13 from 5 and 7 (within 0.5 cents)? Please ask a specific question, or 
put up a specific objection.


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Message: 4488

Date: Tue, 02 Apr 2002 13:32:56

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Mon, 01 Apr 2002 07:42:04 -0000, "dkeenanuqnetau" <d.keenan@u...>
> wrote:
> 
> >Here's a more complete "single-ASCII-character substitutes" 
proposal.
> >
> >Symbol
> >dn up  Comma                  Abbrev. descr. of actual symbol
> >-------------------------------------------------------------
> >\  /   5-comma      80;81     sL
> >L  P   7-comma      63;64     xR
> >v  ^   11-comma     32;33     sL+sR
> >[  ]   11'-comma   704;729    xL+xR
> >{  }   13-comma   1024;1053   sL+xR
> >;  |   13'-comma    26;27     xL+sR
> >j  f   17-comma   2176;2187   vL
> >*  o   19-comma    512;513    cO
> >w  m   23-comma    729;736    vL+cO+cI
> >W  M   23'-comma 16384;16767  vL+sR
> >q  d   29-comma    256;261    xL
> >y  h   31-comma    243;248    sL+vR
> 
> On Windows systems, the | character appears as an unbroken vertical 
line.
> It might make more sense to use : for the 13'-comma up (Windows does 
have a
> broken vertical line character, but it's not ASCII).

Thanks Herman,

I had forgotten that the vertical bar is in two pieces in some fonts 
(where?). I proposed that pair of characters ; | for compatibility 
with Scala's JI notation.


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Message: 4489

Date: Tue, 02 Apr 2002 14:09:23

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> But rather than introduce the disachisma as a 25-comma I'm 
> inclined to allow the 3-exponents to range from -4 to +8 so that, 
from 
> C, an 8:11 is a variety of Ab (or A, using the non-preferred 
13-comma) 
> and a 16:25 is a variety of G#.

That should have been "... an 8:13 is a variety of Ab ...".


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Message: 4490

Date: Tue, 2 Apr 2002 18:03:05

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Dave wrote:
>v  ^   11-comma     32;33     sL+sR
>[  ]   11'-comma   704;729    xL+xR

Ai, I had recently removed v ^ from the JI
notation systems to make them exclusively
denote the diaschisma, and replaced them with 
[ ] for the undecimal comma 33/32. They are also
used in the new E217 system which is probably
the highest ET system I'm going to support.
33/32 is smaller anyway so I feel it's more
logical to have the symbols exchanged.

Manuel


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Message: 4492

Date: Tue, 02 Apr 2002 21:40:34

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> Dave wrote:
> >v  ^   11-comma     32;33     sL+sR
> >[  ]   11'-comma   704;729    xL+xR
> 
> Ai, I had recently removed v ^ from the JI
> notation systems to make them exclusively
> denote the diaschisma, and replaced them with 
> [ ] for the undecimal comma 33/32.

I'm sorry. I wasn't aware of this. Maybe it's a good idea, but I would 
have used u and n for the diaschisma since Rappoport's symbols are 
convex arrow-heads (with no shaft).

> They are also
> used in the new E217 system which is probably
> the highest ET system I'm going to support.

How many steps of E217 are they? I assume by "they" you mean [ ].

> 33/32 is smaller anyway so I feel it's more
> logical to have the symbols exchanged.

I don't follow this. v ^ seem like smaller symbols to me than [ ].

When considering the ASCII symbols alone, and other notation systems, 
I wouldn't have a problem with using [ ] for 32:33 and v ^ for 
704:729. It would be nice to have the preferred (smaller) 11 and 13 
commas (dieses) be [] and {}. The trouble is that George's symbol for 
32:33 is an arrow with straight head-flags (an ordinary arrow) and the 
symbol for 704:729 has convex head-flags (like Rappoport's diaschisma 
symbols with a vertical shaft added).

Anyway, things are still in flux, so lets not worry about it too much 
just yet.


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Message: 4493

Date: Tue, 02 Apr 2002 21:54:17

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > That should have been "... an 8:13 is a variety of Ab ...".
> 
> i'm much more comfortable with 8:13 as a variety of C:A. the 
dominant 
> 13th chord in 12-equal definitely gains a lot of tonalness because 
> 13:11 and even 13:7 are represented by their best approximations in 
> 12-equal. and simpler chords in 12-equal simply fail to evoke the 
13-
> limit, under any circumstances.

Paul,

Don't panic. I took that on board some time ago, thanks to you. The 
complete corrected sentence reads.

"But rather than introduce the disachisma as a 25-comma I'm 
inclined to allow the 3-exponents to range from -4 to +8 so
that, from C, an 8:13 is a variety of Ab (or A, using the 
non-preferred 13-comma) and a 16:25 is a variety of G#."

Which is one reason we have both 26:27 ; |, and 1024:1053 { } as 
13-commas. Are you saying we shouldn't have 1024:1053 at all? The 
notation is pythagorean-based not 12-ET based and so to avoid multiple 
accidentals one might wish to notate an 8:13 from A as A:F} rather 
than A:F#;


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Message: 4495

Date: Tue, 02 Apr 2002 06:27:51

Subject: Re: A common notation for JI and ETs

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> Here's a more complete "single-ASCII-character substitutes" proposal.
> 
> Symbol
> dn up  Comma                  Abbrev. descr. of actual symbol
> -------------------------------------------------------------
> \  /   5-comma      80;81     sL
> L  P   7-comma      63;64     xR
> v  ^   11-comma     32;33     sL+sR
> [  ]   11'-comma   704;729    xL+xR
> {  }   13-comma   1024;1053   sL+xR
> ;  |   13'-comma    26;27     xL+sR
> j  f   17-comma   2176;2187   vL
> *  o   19-comma    512;513    cO
> w  m   23-comma    729;736    vL+cO+cI
> W  M   23'-comma 16384;16767  vL+sR
> q  d   29-comma    256;261    xL
> y  h   31-comma    243;248    sL+vR

I don't know if anyone cares about 12-et-compatibility up to the 
31-limit, but in case they do here is what you get if you enforce compatibility with the "standard" h12 and h7 mappings:

2187/2048, 256/243, 81/80, 64/63, 729/704, 1053/1024, 4131/4096, 
513/512, 16767/16384, 261/256, 67797/65536


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Message: 4496

Date: Wed, 03 Apr 2002 08:22:51

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Which is one reason we have both 26:27 ; |, and 1024:1053 { } as 
> > 13-commas. Are you saying we shouldn't have 1024:1053 at all?
> 
> so the rule is that every comma and its 2187:2048 complement has a
> unique symbol?

No, not every comma. So far George and I have only agreed on the 
desirability of apotome complements of those commas which are close to 
the half-apotome, say those between 1/3 and 2/3 apotome. So far that's 
only the 11 and 13 commas (dieses).

> if so, then the symbols should reflect that in a
> natural way . . .

Read George Secor's two most recent posts to this forum (and mine with 
the ASCII graphics of his symbols) and let us know whether you find 
the system sufficiently natural.

The two 17-commas that have been mentioned are pythagorean comma 
complements, and of the two 23-commas mentioned, one is a pythagorean 
comma larger than the other; similarly the two 31-commas mentioned ; 
although I'm still waiting to hear from Gene (or anyone) why anyone 
would want 65536:67797 (2^16:3^7*31) 59.7 c as a 31-comma when we have 
243:248 (3^5:2^3*31) 35.3 c.


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