Tuning-Math Digests messages 4950 - 4974

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Message: 4950

Date: Sat, 01 Jun 2002 09:40:22

Subject: Re: A 1029/1024 (385/384) planar temperament scale

From: Carl Lumma

>>10-tone scale, e=24 c=4, in 72-tET
 >>(0 5 14 19 28 33 42 49 58 63)
 >>
 >>Connectivity seems so good, I'm not sure why we're not using it
 >>more often.
 >
 >It's also h10-epimorphic. This is clearly an important scale, and needs
 >a name.

You've discovered it twice now, so the ball is clearly in your court.
I have it as sa118.

Does this address why we haven't just been using connectivity?  Do
you have code which searches for high c, or just code that measures c?

-Carl


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Message: 4951

Date: Sun, 02 Jun 2002 07:15:26

Subject: Re: A 1029/1024 (385/384) planar temperament scale

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> Does this address why we haven't just been using connectivity?  Do
> you
have code which searches for high c, or just code that measures c?

Maple will measure c, but it also will find intervals and triads even
more quickly using the characteristic polynomial. The latter seemed to
me more relevant, so I quit showing c--do you think I should
reintroduce it?


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Message: 4952

Date: Sun, 02 Jun 2002 07:27:12

Subject: Re: A 1029/1024 (385/384) planar temperament scale

From: genewardsmith

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> You've discovered it twice now, so the ball is clearly in your court.
> I have it as sa118.

How about gamelion?


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Message: 4953

Date: Sun, 02 Jun 2002 00:45:55

Subject: Re: A 1029/1024 (385/384) planar temperament scale

From: Carl Lumma

>>Does this address why we haven't just been using connectivity?  Do
 >>you have code which searches for high c, or just code that measures c?
 >
 >Maple will measure c, but it also will find intervals and triads even more 
 >quickly using the characteristic polynomial. The latter seemed to me more 
 >relevant, so I quit showing c--do you think I should reintroduce it?

I don't know... why don't we keep our eye on it for a while.  (that's a
yes).

-Carl


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Message: 4954

Date: Sun, 02 Jun 2002 00:46:59

Subject: Re: A 1029/1024 (385/384) planar temperament scale

From: Carl Lumma

>> You've discovered it twice now, so the ball is clearly in your court.
 >> I have it as sa118.
 >
 >How about gamelion?

Good one!

-Carl


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Message: 4955

Date: Sun, 2 Jun 2002 04:18:20

Subject: Scales for Starling (the 126/125 planar temperament)

From: Gene W Smith

Both star and nova are epimorphic for h8+v7, but I didn't find anything
to match them among other scales of that sort I checked. However, I did
find two interesting twelve note scales.

I've given these in the 108-et; this is because while there are
advantages to 46, etc, the true intonational exactness isn't reached
there, and I at least find that the difference is noticable. I haven't
heard about 108, which is sort of the crazy uncle of the 12-multiple
family kept locked in the attic. It shares its {2,3,7} subgroup with 72,
both of them having the 
36-etvalues, but its different third makes it cover 126/125 instead of
225/224 (as well, of course, as 1029/1024, a comma of {2,3,7} it shares
with 72.) Dave and George can ponder it, and see if they think it is
garbage. :)



[0, 7, 21, 28, 35, 45, 56, 63, 73, 80, 91, 101]

5: 23, 12
7: 36, 36
9: 42, 58

[0, 11, 17, 28, 35, 45, 52, 63, 73, 80, 91, 97]

5: 21, 10
7: 31, 28
9: 44, 65

star108 := [0, 7, 28, 35, 56, 63, 80, 91]

5: 13, 6
7: 21, 20
9: 23, 28

nova108 := [0, 11, 28, 35, 56, 63, 80, 91]

5: 13, 6
7: 20, 18
9: 24, 32


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Message: 4956

Date: Sun, 02 Jun 2002 00:06:12

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > It also seems (from the to and fro between Gene and Paul in that 
> > thread) that the only justifications for using _log_-flat (and not 
> > something stronger) are that
> > (a) it's easy to deal with mathematically, and
> > (b) Gene likes it.
> 
> (a) It has a rational basis; what else does?

Plenty else. Merely having a rational basis is no guarantee of 
anything. It might be the _wrong_ rational basis. 

I think what we really want is a human perceptual/cognitive/motor 
basis; empirical, not purely mathematical. What's the point 
otherwise? To be done properly that would have to be done 
statistically by surveying lots of musicians who have tried different 
temperaments. But we don't have that luxury and must rely on the 
experience of the few of us on this list, and what we've learnt by 
listening to others. To me, that seems quite rational too. 

> (b) I tried g^3, leading to the grooviest 7-limit thread. I thought 
it showed a decided bias in favor of low complexities.

This is a valid consideration. It's good that you expressed it _as_ an 
opinion and I suppose your opinion should carry as much weight as 
anyone else's.

> (c) It works.

That also is a matter of opinion.

I must apologise for not remembering, and not checking the archives, 
regarding the use of complexity^3 for 5-limit and complexity^2 for 
7-limit.

OK. Continue using complexity^2 for 7-limit, but _please_ can we have 
a list with a higher badness cutoff and _weighted_ complexity.

I expect you dislike using weighted complexity because the 
justification for it is not purely mathematical?

Badness should be high enough to include:

twin meantone (injera) [[2 . . .] [0 1 4 4]]
shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
porcupine [[1 . . .] [0 3 5 -6]]
diminished [[4 . . .] [0 1 1 4]]
diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]

One per line, no blank lines between please.

Surely this list won't be too big if you use a weighted complexity 
cutoff of 17 generators and an error cutoff of 25 cents.

This is in no way intended to be the final list, this is just to make 
sure we've seen everything worth seeing before deciding on the final 
cutoffs.

Regards,
-- Dave Keenan


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Message: 4958

Date: Tue, 04 Jun 2002 06:57:26

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > In my opinion Porcupine is of far more interest at the 7-limit 
than 
> > double meantone.
> 
> far more interest? how do you figure?

Porcupine has about the same weighted complexity but a significantly 
smaller error.

Porcupine     [3,5,-6],                        5.84 gens, 6.81 cents
Twin meantone [1,4,4] with half-octave period, 5.34 gens, 11.22 cents

> > Diminished and the other 
> > diashismic don't make it.
> 
> none of the ways of extending diminished to 7-limit suggested by 
> herman miller here:
> 
> Diminished temperament *
> 
> make it? can you detail why that is?

I didn't say that. I was only referring to the one in my message, in 
my list of temperaments that weren't in Gene's list.

The [1,1,1] mapping (which was in Gene's list) will make it. The 
[1,1,4] mapping won't because of its combination of 8.26 gens and 9.15 
cents.

Herman's page referred to above, only gives the [1,1,1] mapping. Are 
there really any others worth considering?


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Message: 4959

Date: Tue, 04 Jun 2002 01:38:49

Subject: Re: A 7-limit best list

From: David C Keenan

Pending Gene's next list with badness up to 350, based on weighted
complexity, here's a list of 27 we might agree on. It's given in order of
increasing complexity.

It consists of all those I _know_of_ with 

  weighted-complexity < 13 gens, 
  error < 23 cents, 
  Gene's-weighted-badness < 343,

but there may well be some we don't know about.

So are there any others that anyone thinks should be included?

                 Periods  Gens per            Gen   Wtd Compl  RMS err
Name             per oct  1:3    1:5    1:7  (cents)  (gens) (cents)
----------------------------------------------------------------------
dominant 7th		1	1	4	-2	498	3.13	20.16
 (meantone with inaccurate 7s)
diminished		4	1	1	1	85	3.14	19.14
quintuple thirds	5	0	1	0	91	3.29	15.82
 (Blackwood's decatonic)
augmented		3	1	0	2	111	3.68	16.60
 with inaccurate 7s
diaschismic (pajara)	2	1	-2	-2	108	3.94	10.90
tertiathirds		1	4	-3	2	126	4.31	12.19
 (Negri's system)
kleismic		1	6	5	3	317	4.37	12.27
 with inaccurate 7s
augmented		3	1	0	-2	89	4.63	8.10
meantone		1	1	4	10	504	5.32	3.67
twin meantone		2	1	4	4	94	5.34	11.22
 (injera)
twin wide		2	4	3	3	274	5.75	10.13
 subminor thirds
porcupine		1	3	5	-6	162	5.84	6.81
superpythagorean	1	1	9	-2	490	6.21	6.41
neutral thirds	1	2	-9	-4	356	6.52	6.25
 with complex 5s
magic			1	5	1	12	381	6.79	4.14
narrow minor thirds	1	10	9	7	310	7.40	3.32
semisixths		1	7	9	13	444	7.42	5.05
 (tiny diesic)
subminor thirds	1	7	-3	8	271	7.42	2.59
 (orwell)
miracle		1	6	-7	-2	117	7.61	1.64
quartaminorthirds	1	9	5	-3	78	7.66	3.07
supermajor seconds	1	3	12	-1	232	7.74	3.58
schismic		1	1	-8	-14	498	8.61	2.86
diaschismic		2	1	-2	-8	104	9.47	3.82
 (15-limit variant)
octafifths		1	8	18	11	88	10.50	2.06
 (semi minimal-diesic)
shrutar		2	2	-4	7	53	11.19	2.25
 (double diaschismic)
kleismic		1	6	5	22	317	11.41	1.61
 with complex 7s (catakleismic)
half wuerschmidt-	1	16	2	5	194	11.75	0.88
 thirds
-------------------------------------------------------------------

I tried hard to get a list that included ennealimmal, and that could be
made to agree between our two methods. This is not because I think
ennealimmal is particularly interesting, but because I know Gene does.

So far, no matter how I tinker with my two rolloff parameters and Gene's
complexity and error cutoffs, I can't make two lists that agree while
including ennealimmal, even if I'm willing to include stuff that I would
normally consider junk.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4960

Date: Tue, 04 Jun 2002 02:10:23

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Badness should be high enough to include:
> > 
> > twin meantone (injera) [[2 . . .] [0 1 4 4]]
> 
> yup.
> 
> > shrutar (double diaschismic) [[2 . . .] [0 2 -4 7]]
> > porcupine [[1 . . .] [0 3 5 -6]]
> > diminished [[4 . . .] [0 1 1 4]]
> > diaschismic (15-limit variant) [[2 . . .] [0 1 -2 -8]]
> > diaschismic (56-ET variant) [[2 . . .] [0 1 -2 9]]
> 
> i'm not sure about the rest of these. maybe diminished, but the rest 
> have never been advocated as 7-limit temperaments.

Apparently you mean they have only been advocated at _higher_limits 
than 7. But that doesn't disqualify them from consideration at the 
7-limit.

So I've been forced to find their minimum rms errors myself, so I can 
calculate Gene's badness (and my own) for them.

In my opinion Porcupine is of far more interest at the 7-limit than 
double meantone. Shrutar and the [1, -2, -8] diaschismic are of about 
equal interest, just slightly less interest than double-meantone, but 
are at (or just off) the bottom of the list. Diminished and the other 
diashismic don't make it.
 
> gotcha. i'm just worried that there will be way too much to consider 
> here. why not just insist that injera makes it in and go from there?

Well hey we could at least take a look at how many are on such a list 
instead of refusing to look 'cause we're worried. I don't understand 
yours or Gene's attitude in this regard.

I guess all I can say is, I won't be at all confident that we haven't 
missed something unless we look at everything with complexity 
(weighted or otherwise) less than 17 gens and rms error less than 25 
cents and Gene's badness less than 500 if unweighted rms complexity is 
used or less than 350 if weighted rms complexity is used. Of course 
weighted should be used.

This is only slightly more than is required to ensure that 
double-meantone appears. Double meantone has a Gene's-badness of 320. 
The [1, -2, -8] diaschismic is 343. These are using weighted 
complexity.


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Message: 4962

Date: Wed, 05 Jun 2002 12:41:28

Subject: "The hypothesis" from a wedge product point of view

From: genewardsmith

In the prime limit with k primes, take k commas in order from largest
to smallest,[c1, c2, ... , ck], defining a square matrix with
determinant +-1 (i.e., a "notation".) Leave off c1 and wedge the rest,
and you have a val "h" defining an et mapping, corresponding to a
Fokker block with h(2) notes to the octave. Suppose h linearly orders
the block; then leave off *both* c1 and c2 and wedge the rest, and you
have a linear temperament, which when applied to the block gives (when
the period is an octave) n contiguous
(in the sense of the generator of the temperament) notes of a MOS, and
similarly with periods which are a fraction of an octave.


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Message: 4963

Date: Wed, 05 Jun 2002 23:12:38

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>It's taken me a little time to appreciate the value of your proposal 
>for dual roles for (| and the 19'-19 role of |~.  However, I believe 
>that a dual role should be retained for |~ also; it is quite useful 
>as the 23 comma for notating 135, 147, 159, 198, and 224-ET 
>(particularly 198).

I suppose if the 23-comma interpretation is confined to such large ETs it
might be ok, but I'd need to be convinced that there was no other way to do
it and the ET is actually 1,3,9,23-consistent and preferably
1,3,9,...,23-consistent where "..." are the other primes used for its
notation.

135, 147, 159, and 224-ET are all 17-limit notatable, although you'd
probably go to 19 limit for 224-ET.
198-ET is not 19-limit notatable, but why do you feel any need to notate
it? And is it 23-limit notatable anyway?

>> Here's my proposal for notating 74-ET using its native fifth (since 
>it's a
>> meantone), despite the 1,3,9 inconsistency.
>
>> Steps  Symbol  Comma
>> ----------------------
>> 1      )|)     19+7
>> 2      )|\     19+(11-5)
>> 3      /|\     11
>> 4      )||\
>> 5      /||\
>> 
>> The )| flag actually has a value of -1 steps, but it never occurs 
>alone, so
>> it doesn't really matter.
>
>While I was away, I worked on the notation for a number of ET's.  I 
>decided to tackle 74 on my own, since it seemed to be a challenge.  
>The solution I came up with minimizes the use of flags with non-
>positive values:
>
>74:  )|)  /|)  (|\  /||)  /||\
>
>Your solution is simpler in that it uses fewer flags and has no 
>lateral confusability, so it would probably be preferable on that 
>basis.  However, I mention below that I would rather not use /|\ for 
>anything greater than half of /||\ unless absolutely necessary.  On 
>the other hand, the 11 factor is almost exact in 74, so it would be a 
>shame not to represent it in the notation.

I'll go with your solution, for the single-shaft symbols at least. The
lateral confusability is addressed by the large difference in widths, and
the user will just have to learn that 11 = 13' and 11' = 13.

I propose (||( for 4deg74 purely because it is the mirror image of )|)
(plus a shaft). More about this later.

>> >Would you also now prefer my selection of the /|) symbol for 
>[6deg152] 
>> >to your choice of (|~ on the grounds that it is a more commonly 
>used 
>> >symbol, particularly in view of the probability that you might 
>want 
>> >to use (|\ instead of )|| or ||( for 9deg as its complement?
>> 
>> Yes, but not on those grounds.
>
>Then we agree on the following (cf. below):
>
>152:  )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  
>||\  /||~  /||)  /||\

Yes, for the single-shaft symbols at least.

>Perhaps "guideline" would be a better term than "requirement."  
>Applying this notation to different systems is as much an art as a 
>science in that you need to decide which guidelines take priority 
>over the others to achieve the most user-friendly result.

You know, I won't really be happy until I have a spreadsheet that generates
the notation for every ET, based on a bunch of rules, because that's the
only way I'll be sure we're being consistent. The rules may of course end
up being very complicated, but I wouldn't want to see any rule that only
applied to a single ET.

By the way, perhaps we should use >| and |< instead of )| and |( to make
the concave flags more distinct from the convex ones in ASCII. What do you
think?

>> >17, 24, 31: s|s  s||s
>> 
>> 17 and 24 agreed. I guess you want (|) for 1deg31 because it is 
>closer in
>> cents than |), but I feel folks are more interested in its 
>approximations
>> of 7, than 11.
>
>I think you meant /|\ instead of (|).

Yes, I did. Sorry.

>As with 17 and 24, I think it's more intuitive to use /|\ (semisharp) 
>for half of /||\ (sharp) where it's exactly half the number of 
>degrees.  Anyone who has used the Tartini/Fokker notation already 
>calls an alteration of 1deg31 a semisharp or semiflat and would 
>expect to see this symbol used.
>
>Besides, if there is no problem with lateral confusability, I think 
>that straight flags are the simplest way to go.

I guess if there is already a popular notation in use for some ET, and
there is a fairly direct correspondence to that notation available in
sagittal, then we should use it.

So on that basis at least, I can tentatively agree with your proposal for
31. I reserve the right to change my mind on this on further investigation.
:-)

31: /|\  /||\

>> >22:  s|  ||s  s||s
>> 
>> I agree, but how come you didn't want s|s for 1deg22? 
>
>If you did that, then you wouldn't have the comma-up /| /|| and comma-
>down \! \!! symbols that are one of the principal features of this 
>notation; 

I understand wanting comma up and down symbols in 22-ET. That's what I want
too. But what do you mean by /|| and \!! as comma up and down symbols?
We're proposing _not_ to have those those double-shafters with a _left_
flag. I assume you meant the double-shafters to have right flags. In which
case 22 is agreed.

22: /|  ||\  /||\

>this is something that I would want to have in every ET in 
>which 80:81 does not vanish, even if that doesn't result in a 
>completely matched sequence of symbols in the half-apotomes.  I 
>believe the matched sequence is more imporant once the number of 
>tones gets above 100, by which point /| and |\ are usually a 
>different number of degrees.

In fact, I'm very happy that /| and ||\, |\ and /|| should always be
complements and to _always_ have a mirrored sequence rather than
_sometimes_ have a matched one.

>Also, with the apotome divided into fewer than 5 parts, I would want 
>to use /|\ only when it is exactly half of /||\.

Fair enough.

>In essence, what I am proposing here is that, for the lower-numbered 
>ET's, we should place a higher priority on the use of rational 
>complements than on a matching sequence of symbols.  (Note that 
>virtually everything that we agree on below follows this principle.)

I am instead inclined to totally ignore rational complements with regard to
ETs, especially for the lower numbered ones. One reason is that I feel that
the choice of double-shaft symbols cannot in any way be allowed to
influence the choice of single-shaft. One must first choose the best set of
single shaft symbols (ignoring complements) since some users will have no
interest in the double-shaft symbols and should not be penalised for it.

In fact, (and I've been making gentle noises about this possibility for a
some time now), I'm willing to throw away everything we agonised over with
regard to rational complements and instead adopt a simple system that
applies automatically to all ETs and rational tunings.

I propose that the complement of a|b is always b||a, except that the
complement of |//| (natural) is /||\ and the complement of /|\ is /|\ if it
represents the same number of steps and (|) if it represents a different
number of steps.

You will notice that this would require no change to the 72-ET version 3
notation, nor any change to most of the smaller ETs we've agreed on such as
12, 17, 19, 22, 24, 26, 31, 46, 53.

By the way, are you collecting all those we've agreed on into one place? I
haven't been.

Why do I want to do this despite some obvious disadvantages? Because I
realised while trying to consistently notate the whole n*12-ET family, that
it required us to repeat the whole somewhat arbitrary process we went thru
for rational tunings, to find complements with minimum offsets. And what's
more, that this process would have to be repeated for every such family or
small range of fifth-sizes across the whole range of ETs. For example the
n*29-ET family is the next largest, followed by n*17. And every such
family, or small range of fifth sizes, would have a completely different
complement mapping. The cognitive load for anyone who uses more than two
such systems would be enormous.

Now for those obvious disadvantages:
1. The second shaft does not have a fixed comma value.

This doesn't seem very important to me? 

2. We lose the association of flag size with rational comma size in the
second half-apotome.

This is the biggie. It can be remedied to some degree by redesigning the
double-shaft (and X-shaft) symbols so their concave flags are wider than
their wavy flags which are wider than their straight and convex flags.
However it will be difficult to make single flag symbols bigger than
double-flag ones.
 
What other disadvantages have I omitted?

Advantages:

Simple to remember. 
Covers all tunings. 
Flags are more strongly associated with particular primes because the flags
don't change when the comma is complemented.
No new flag types ever need to be introduced merely to handle complements.
Doesn't require /| and ||\ as a special case.

>I would rather not use /|\ for anything greater than half of /||\ 
>unless absolutely necessary.  How about using
>
>36, 43:  |)  (|\  /||\
>
>for both?  Since I re-evaluated 72-ET, I changed my mind about 36, 
>which hinges on how 72 is done (see below).

I agree with the above for 43-ET, but 36-ET can be notated as every second
note of 72-ET, which means I want:
36:  |)  (||  /||\
43:  |)  (|\  /||\

Do you have some argument as to why 36 and 43 should be the same? I don't
see it.


>> >29:  w|x  w||v  s||s
>> 
>> Why wouldn't you use the same notation as for 22-ET? There's no 
>need to
>> bring in primes higher than 5.
>
>I was making it compatible with my non-confusable version of 58, 
>which I no longer favor.  When I discuss 58 (below), I will give 
>another version, which would result in this:
>
>29:  /|)  (|\  /||\
>
>But if you prefer version 1 of 58 (with all straight flags), then we 
>might as well do 29 like 22-ET.

I now realise I will need to consider the entire n*29 family
29 58 87 116 145 174 203 232 261, before agreeing on either 29 or 58.

>> >27:  s|  x|s  ||s  s||s
>> 
>> Why do you prefer (|\ to /|)?
>
>2deg27 is almost 90 cents, so (|\ is nearer in size than /|).  
>Otherwise, it's a tossup.

I prefer /|) because it introduces only one new flag where (|\ introduces
two, for single-shaft-only users.

>> >48:  |x  s|s  ||x  s||s
>> 
>> In 48-ET, {1, 3, 7, 9, 11} has only slightly lower errors than {1, 
>3, 5, 9,
>> 11}, 10 cents versus 11 cents. Why prefer the above to the lower 
>prime scheme
>> 48: /|   /|s  ||\  /||\ ?
>
>To make 48 compatible with 96-ET (see below).

I now agree that 48 should be every second note from 96 and will address
all n*12-ETs elsewhere.

>> >58, 72:  s|  |s  s|s  s||  ||s  s||s  (version 1 -- simpler, but 
>more confusability)
>> >72:  s|  |x  s|s  ||x  ||s  s||s  (version 2 -- more complicated, 
>but less confusability)
>> 
>> Of course, I prefer version 2 for 72-ET, since I started the whole
>> confusability thing. It isn't significantly more complicated.
>
>To further confuse the issue, I now have even more options for 72-ET:
>
>72:  /|  |\  /|\  /||  ||\  /||\  (simplest, but most confusability)
>72:  /|  |)  /|\  ||)  ||\  /||\  (version 2 -- more complicated, no 
>confusability, inconsistent)
>72:  /|  |)  /|\  (||  ||\  /||\  (version 3 -- simpler, no 
>confusability, but (|| < ||\  )
>72:  /|  |)  /|\  (|\  ||\  /||\  (version 4 -- simple, no 
>confusability, consistent, harmonic-oriented)
>
>The symbol arithmetic in version 2 is inconsistent:
>
>/|\ minus |) equals 1deg72, but
>/||\ minus ||) equals 2deg72
>
>This is remedied in version 3, which also has a problem in that the 
>symbol for 4deg72 is a larger rational interval than that for 5deg72, 
>something I would rather not see in a division as important as 72, 
>although the difference between (|| and ||\ is rather small.
>
>This leaves me with version 4 as my choice.  Notice that the first 4 
>symbols are, in order, the 5 comma, the 7 comma, the 11 diesis, and 
>the 13' diesis, all of which are the rational symbols used for a 13-
>limit otonal scale:  C  D  E\!  F/|\  G  A(!/  Bb!)  or  B!!!)  C.
>
>This option should also be considered in connection with our 
>discussion of 36 and 43 above.

I can only agree to your version 3.

72: /|  |)  /|\  (||  ||\  /||\

I prefer the above to version 4, with (|\ as 4 steps, because I think that
in any given ET, (| and |) flags should either both be 13-based or both be
7-based, so that (|) is always 11' (whether it is used or not). Otherwise
we have 3 different possible values for (|), the largest and smallest of
which differ by 0.84 cents in rational tuning.

It's bad enough that folks have to know whether the convex flags refer to 7
or 13 in ETs where the tridecimal schisma doesn't vanish. I wouldn't want
them to have to worry about the two convex flags _independently_.

Also, 72-ET is not terribly good at the 13-limit, the error hikes from 3.9
cents at the 11-limit to 7.2 cents at the 13-limit, and in any case folks
can learn that the 7-comma symbol doubles as the 13-comma symbol in 72-ET,
just as they must learn that the 11-comma symbol doubles as the 7-comma
symbol in 31-ET.

>> >58:  s|  w|x  s|s  w||v  ||s  s||s  (version 2 -- more 
>complicated, but less confusability)
>> 
>> I'm inclined to go with version 1 despite the increased lateral
>> confusability, rather than introduce 17-flags. Version 2 is a _lot_ 
>more
>> complicated. 
>
>These are my latest options for both 58 and 65-ET:
>
>58, 65, 72:  /|  |\  /|\  /||  ||\  /||\  (simplest, but most 
>confusability)
>58:  /|  ~|)  /|\  ~||  ||\  /||\  (version 2 -- more complicated, no 
>confusability)
>65:  /|  /|~  /|\  ||~  ||\  /||\  (version 2 -- more complicated, no 
>confusability)
>58:  /|  /|)  /|\  (|\  ||\  /||\  (version 3 -- simpler, some 
>confusability)
>
>Version 3 could offer 29/58 compatibility, but the straight flags of 
>version 1 are the simplest.

I don't see any need for 29 to be every second of 58, but I do want to look
at the whole huge family first.

>I also threw 65-ET in there.  Below I have a proposal for 130-ET, 
>which results in 65 having all straight flags (as in the first 
>version above), so I believe I would prefer that.
>
>> >96:  s|  |x  |s  s|s  s||  ||x  ||s  s||s  (version 1 -- simpler, 
>but more confusability)
>> >96:  s|  |x  w|s  s|s  w||  ||x  ||s  s||s  (version 2 -- more 
>complicated, but less confusability)
>> 
>> The only maximal 1,3,9-consistent 19-limit set for 96-ET is {1, 3, 
>5, 9,
>> 11, 13, 15, 17}. It is not 1,3,7-consistent so the |) flag should be
>> defined as the 13-5 comma (64:65) if it's used at all. The 17 and 
>19 commas
>> vanish, so we should avoid )| |( ~| and |~. So I end up with
>> 96:  /|  |)  /|)  /|\  /||  ||)  /||)  /||\
>> Simple _and_ non confusable.
>
>My latest proposal for 96 is:
>
>96:  /|  |)  /|)  /|\  (|\  ||)  ||\  /||\
>
>As I mentioned above, I would like to see both /| and ||\ used 
>whenever possible.
>
>At least we agree on 48, if that is to be notated as a subset of 96.

I changed my mind on 96, as you will have seen in other posts, but might
end up changing it back.

>> >94:  w|  s|  w|s  s|s  x|x  w||  ||s  w||s  s||s
>> 
>> Why do you prefer that to
>> 
>> >94:  ~|  /|   |)  /|\  (|)  ~||  ||\   ||)  /||\
>> 
>> Surely we're more interested in the 7-comma than the 17+(11-5) 
>comma.
>> 
>> Also, it makes sense that /| + ||\ = /||\, but it makes the second 
>half
>> apotome have a different sequence of flags to the first. Which 
>should we
>> use, /|| or ||\ ?
>
>My proposal above for a matched sequence being subordinate to having 
>||\ and rational complements would apply here.  While ~| and ~||\ are 
>not rational complements, they are the 217-ET complements -- the 
>nearest we can get to a rational complement for 1deg94.
>
>I calculate both |) and |\ as 2deg94, so I needed something else for 
>3deg.  The best possibilities were (| and ~|\ -- neither one uses a 
>new flag.  My choice was:
>
>87, 94:  ~|  /|  ~|\  /|\  (|)  ~||  ||\  ~||\  /||\
>
>The symbol sequence is fairly simple, particularly in the second half-
>apotome.  Or is the other option:
>
>87, 94:  ~|  /|  (|  /|\  (|)  ~||  ||\  (||  /||\
>
>better?  (Perhaps this is what you meant?)

Yes. That's what I meant. Sorry.

I now want one of

94:   ~|  /|   (|   /|\  (|)   ||)  ||\   ||~  /||\
94:   ~|  /|   ~|\  /|\  (|)  /||~  ||\   ||~  /||\

and need to look at 94 188 282 to decide.

>> >111 (37 as subset):  w|  s|  |s  w|s  s|s  x|s  w||  s||  ||s  
>w||s  s||s
>> 
>> Dealt with above. I'd prefer (|) for 6deg111.
>
>Yes.  Agreed!

Only now I want the fully-mirrored half-apotomes: 

111 (37):  ~|  /|  |\  ~|\  /|\  (|)  /||~  /||  ||\  ||~  /||\

>140:  )|  |~  /|  )|\  /|~  /|)  (|~  (|\  )||  ||~  ||\  )
>||\  /||~  /||\
>
>This is the simplest set I could come up with that uses both /| and 
>||\.

I'll leave the second half-apotome out of it for now. It seems we have 4
options:
140:  )|  |~  /|  )|\  (|   /|)  (|~    6 flags
140:  )|  |~  /|  )|)  (|   /|)  (|~    5 flags
140:  )|  |~  /|  )|\  /|~  /|)  (|~    6 flags monotonic flags per symb
140:  )|  |~  /|  )|)  /|~  /|)  (|~    5 flags monotonic flags per symb

I prefer the last one, and with mirror complements it would be

140:  )|  |~  /|  )|)  /|~  /|)  (|~  (|\  ~||\  (||(  ||\  ~||  ||( /||\

Note that with mirror complements, (|\ is the same as (||\.

>> >152:  |v  |w  s|  |s  s|w  s|x  s|s  x|x  x|s  ||w  s||  ||s  
>s||w  s||x  s||s
>> 
>> Dealt with elsewhere. I see no reason to use |( which is really 
>zero steps,
>> when )| is 1 step.
>
>Yes.  Agreed!

With mirror complements we have:
152: )|  |~  /|  |\  /|~  /|)  /|\  (|)  (|\  ||~  /||  ||\  ~||  ||(  /||\

>> >171:  |v  w|v  s|  |x  |s  w|s  s|x  s|s  x|s  w||v  s||  ||x  
>||s  w||s  s||x  s||s
>> 
>> Why not ~| for 1 step?
>> 
>> >183:  |v  w|v  s|  |x  |s  w|s  s|x  s|s  x|x  x|s  w||v  s||  
>||x  ||s  w||s  s||x  s||s
>> 
>> Why not use w| for 1deg183, being a simpler comma than |v? 17 vs. 
>17'-17.
>
>After re-evaluating, I would keep what I had above for both 171 and 
>183.
>
>The  choice between |( and ~| is almost a tossup, but I found two 
>reasons to prefer |(:
>
>1) It is closer in size to both 1deg171 and 1deg183; and
>
>2) It is the rational complement of /||).

I'll buy 1), but no longer care about 2). So I agree with the above, as far
as the single-shaft symbols.

>> >181:  |v  w|  w|v  s|  |s  w|x  w|s  s|x  s|s  x|x  w||  w||v  
>s||  ||s  w||x  w||s  s||x  s||s
>> 
>> I don't see how  |) can be 5deg181 or how /|\ can be 9deg181. 
>
>More wishful thinking on my part that /|\ should be half of /||\ -- I 
>guess I was getting tired.
>
>> The only
>> symbol that can give 9deg181 with 19-limit commas is (|~. Here's my 
>proposal.
>> 
>> 181:  |(  ~|   |~  /|  /|(  (|  (|(  /|)  (|~  (|\  ~||  
>||~  /||  /||( (||  (||(  /||)  /||\
>
>And here's my new proposal.
>
>181:  |(  ~|  |~  /|  /|(  ~|)  /|~  /|)  (|~  (|\  ||(  ~||  ||~  
>||\  /||(  ~||)  /||~  /||\
>
>We don't agree on the symbol arithmetic in the second half-apotome.  
>Both /| and |\ are 4deg181, so /||\ minus /| equals /||\ minus |\ 
>equals 4deg.  You have /|| as 5deg less than /||\.
>
>My choice for 6deg ~|) was on the basis of its being the rational 
>complement of 12deg ||~; 7deg /|~ logically followed as 3deg plus 
>4deg.

It is still unclear to me what's best for 181, but you will realise that
rational complements may no longer be of any relevance to me.

>> >217:  |v  w|  |w  s|  |x  |s  w|x  w|s  s|x  s|s  x|x  x|s  w||  
>||w  s||  ||x  ||s  w||x  w||s  s||x  s||s
>> 
>> Agreed.

Except for the mirror complement thingy that we need to thrash out now.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 4967

Date: Wed, 05 Jun 2002 23:29:29

Subject: Re: "The hypothesis" from a wedge product point of view

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> 
> > Okay, I've been waiting for this one... now, could somebody, maybe
> > somebody other than Gene, say the same thing and give some 
examples
> > *slowly*!
> > 
> > 
> > thanks,
> > 
> > --Dan Stearns
> 
> i wish i could. unfortunately i haven't had the time to assimilate 
> the whole wedge product business. i'm pretty sure graham breed and 
> dave keenan have a handle on it. i'd say dave keenan might be the 
> best candidate to study gene's work here and translate it into a 
form 
> lots of us (especially people like you and me, who are familiar with 
> dave's language at least) could understand and visualize (i'm 
> thinking of dave's skill with graphics and diagrams) . . .

Whew! That's flattering, but I don't have the time right now (end of 
semester marking), but anyone who would hope to understand even my 
explanation should first learn what the wedge product (a.k.a exterior 
product) is all about, from another Australian, John Browne.

Gene found this excellent introduction some time ago. See the thread 
starting at
Yahoo groups: /tuning-math/message/3754 *
or go straight to
Grassmann Algebra Book *
You might only need to read the first 6 pages of the introduction, to 
get what it's all about.


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Message: 4969

Date: Thu, 06 Jun 2002 17:02:25

Subject: Fwd: scala questions

From: Carl Lumma

All;

I've created a tuning-math list on freelists.org...

Welcome to FreeLists - Free, No-hassle Mailing Lists *

...It's the closest thing I've seen to my wishlist for a list
service, but it isn't perfect.  For example, there's no database
for all the lists you belong to like yahoo's "my groups"
feature -- you have to remember them.  And you can't post
directly from the site.  And it doesn't seem to assign unique
message numbers to posts.  The messages are archived by thread
and date, though, and are fully searchable.  Completely ad-free.
Check it out, if you like, see what you think!

-Carl

>Date: Thu, 06 Jun 2002 15:48:29 -0700
>To: tuning-math@xxxxxxxxx.xxx
>From: Carl Lumma <carl@xxxxx.xxx>
>Subject: scala questions
>
>Manuel,
>
>() What's the best way to get Scala to represent scales as degrees
>of an et?
>
>() As far as inputting scales as et subsets, I do "equal n" and then
>"select".  Is that the Official Way?
>
>() Think we could get View -> rank-order matrix?  (Yes, I'm using
>2.05 now).
>
>-Carl


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Message: 4970

Date: Thu, 06 Jun 2002 17:03:25

Subject: Re: Fwd: scala questions

From: Carl Lumma

>All;
>
>I've created a tuning-math list on freelists.org...
>
>Welcome to FreeLists - Free, No-hassle Mailing Lists *

Oh yeah, and I haven't tested digest mode, so if anybody
wants to volunteer for that...

-C.


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Message: 4971

Date: Thu, 06 Jun 2002 01:28:35

Subject: Re: "The hypothesis" from a wedge product point of view

From: genewardsmith

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Okay, I've been waiting for this one... now, could somebody, maybe
> somebody other than Gene, say the same thing and give some examples
> *slowly*!

I cancelled that about five minutes after posting, and I don't think
it's a very good place to start. I'm thinking about how to remove the
errors and get the basic idea to work, but I'm not there.


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Message: 4972

Date: Thu, 06 Jun 2002 17:31:31

Subject: Re: Fwd: scala questions

From: Carl Lumma

>> I've created a tuning-math list on freelists.org...
>> 
>> Welcome to FreeLists - Free, No-hassle Mailing Lists *
>
>All I get is an error message.

My mistake -- no slash allowed:

Welcome to FreeLists - Free, No-hassle Mailing Lists *

-Carl


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Message: 4973

Date: Fri, 7 Jun 2002 11:08:24

Subject: Re: scala questions

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

Carl,

() What's the best way to get Scala to represent scales as degrees
of an et?

set attribute log_fract [octave]

Maybe the format is not ideal. I could add it to show data 
if the scale is an ET subset.

() As far as inputting scales as et subsets, I do "equal n" and then
"select".  Is that the Official Way?

That's how I do it too. But if it's expressed in steps, I do
mode/equal

() Think we could get View -> rank-order matrix?  (Yes, I'm using
2.05 now).

Could you explain what exactly you have in mind? Thanks,

Manuel


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