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Message: 7875

Date: Fri, 31 Oct 2003 23:33:52

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >>5.7.11-kleisma has no advantages over 385/384 that I can see.
> 
> The latter must be factored to see what it's good for, and
> log'ed to give an exact size.  The former gives a size range,
> and with the addition of the 3 exponent tells you what it's
> good for

So these are two advantages of "5.7.11-kleisma" over "385/384".

> (otherwise how'reyou going to say what pythagorean
> commas are good for?). 

There aren't too many of them that have come to my attention so far.
These are well known and have common names: Pythagorean-limma,
apotome, Pythagorean-comma. If the previously described naming system
was simply applied they would all be 1-<whatevers> where "<whatever>"
stands for the correct size category. But I would change this to
3-<whatevers> or even better, Pythagorean-<whatevers>. So the limma
and comma would be the same as their common names and the apotome
would probably have the systematic name: Pythagorean-semitone. The
3-exponent can be extracted from the size category if necessary.

I think that in any list of common commas you will find that more than
90% of them have a 3-exponent other than zero. So it is fairly
unnecessary to include, up front, the fact that they have 3's in them;
and it would be technically redundant since the 3-exponent can be
extracted from the size category in conjunction with the numbers that
are supplied.

> But with the addition of the 3 exponent,
> we loose the ability to draft size ranges.  What say you to
> this, Dave? 

I'm not sure what you mean by "to draft size ranges"? If you mean "to
decide the boundaries of size ranges", I don't understand why you
would lose that ability. You could keep the same ranges and the up
front information about the 3-exponent would simply be redundant.

I don't want to call 81/80 the 5:3^4-comma, but just the 5-comma. And
64/63 would be simply the 7-comma, not the 7.9-comma. That's the whole
point of setting the size-category boundaries so carefully, to
eliminate the need to have the power of 3 explicit in the systematic name.


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Message: 7876

Date: Fri, 31 Oct 2003 05:03:32

Subject: Re: UVs for 46-ET 11-limit PB

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> i've uploaded a graphic to tuning_files, showing both my
> original pseudo-PB and paul's latest PB, for 46-tone 11-limit:
> 
> 
> Yahoo groups: /tuning_files/files/monz/compact... * [with cont.] 
> et_pb.gif
> 
> or
> 
> Sign In - * [with cont.]  (Wayb.)
> 
> 
> i know that it's too small for the numbers and letters to
> be legible, but the point is simply to see by the colors
> which notes are in the PB and which are not.
> 
> in both diagrams, grey shading indicates notes which occur
> only one time in the PB.
> 
> my original pseudo-PB, blue indicates duplicate notes and
> green indicates triplicate, which are the same number of
> (rectangular metric) steps away from 1/1.  ... the brown 
> shading was only used to keep track of notes and can be
> ignored.



sorry ... i was in a hurry when i posted that.
i should have added:


the lattice only show 2 dimensions at a time, those of
prime-factors 3 and 5.  the horizontal axis is 3, the
vertical is 5.

the big diagrams on the left show only 3 and 5.  the smaller
ones on the right show, from the top down respectively,
7^1, 7^-1, 11^1, and 11^-1.




-monz


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Message: 7877

Date: Fri, 31 Oct 2003 23:40:57

Subject: A remarkable property of 270

From: Gene Ward Smith

If we take the 12 smallest 13-limit superparticular commas, namely

1001/1000, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, 
4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647, 123201/123200

we find that they all have the property of being 270-et commas.
Moreover, if we take this comma list five at a time, we get the 270 et
in all cases where the five commas are linearly independent. Sometimes
we get what we might regard as 540, but this is just a doubling of 270.


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Message: 7878

Date: Fri, 31 Oct 2003 05:11:06

Subject: Re: Eponyms

From: Dave Keenan

I realise I missed this the first time:

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> In your scheme, the term kleisma tells us that the denominator must
> be 384, and not 383?

Yes! But not directly of course. All it tells us directly is the size
range. From that, and the 385, we can get the factors of 2 and 3, as I
explained in the previous message. It's such a neat trick, I guess
it's hard to believe it works.

But note that the number given in the name is not necessarily the
numerator or demominator of the comma ratio, it's the comma ratio with
the 2's and 3's removed, (and inverted if it's less than one).

> >or "5.7.11-kleisma".
> 
> ...tells us how to combine factors of 5, 7, and 11 to get the
> right ratio?

Yes. When the dots are read as multiplication, this contains no more
and no less information than "385-kleisma" since prime factorisations
are unique.


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Message: 7879

Date: Fri, 31 Oct 2003 07:00:34

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> On the subject of eponyms:
> 
> Manuel, I'd prefer it if Scala did not refer to 384:385 as Keenan's
> kleisma, although I thank Paul for his sentiments in proposing it.
> 
> Now that I've found what I think is a good system for naming kommas,
> I'd prefer it to be called "385-kleisma" or "5.7.11-kleisma". I 
think
> I prefer the latter, and would pronounce it "five seven eleven 
kleisma".
> 
> Does anyone have any objection to this, or want to propose another 
name?

There isn't really much point in naming commas like 385/384 in the 
first place, but if you do, the name should be easier than the thing 
it is naming. 5.7.11-kleisma has no advantages over 385/384 that I 
can see.


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Message: 7880

Date: Fri, 31 Oct 2003 07:48:58

Subject: Re: Eponyms

From: monz

hi Gene,


--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
>
> > On the subject of eponyms:
> > 
> > Manuel, I'd prefer it if Scala did not refer to 384:385
> > as Keenan's kleisma, although I thank Paul for his sentiments
> > in proposing it.
> > 
> > Now that I've found what I think is a good system for
> > naming kommas, I'd prefer it to be called "385-kleisma"
> > or "5.7.11-kleisma". I think I prefer the latter, and would
> > pronounce it "five seven eleven kleisma".
> > 
> > Does anyone have any objection to this, or want to propose
> > another name?
> 
> There isn't really much point in naming commas like 385/384
> in the first place, but if you do, the name should be easier
> than the thing it is naming. 5.7.11-kleisma has no advantages
> over 385/384 that I can see.



i disagree quite strongly.

to me, the only thing the ratio shows is that it's 
superparticular (or epimoric, if you prefer Greek over Latin).

i think 5.7.11-kleisma is a much better name ... altho i
think my first choice would be to use the monzo and call it
the [-7 -1 1 1 1]-kleisma, or if you can do without the 2
(which i also always prefer if possible), [-1 1 1 1]-kleisma.

:)



-monz


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Message: 7881

Date: Fri, 31 Oct 2003 17:36:58

Subject: Re: Eponyms

From: Carl Lumma

>> >If you want to make this systematic, why not simply monzo-size range?
>> 
>> Example?
>
>225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
>[-7,-1,1,1,1]-kleisma.

That works.

-Carl


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Message: 7882

Date: Sat, 01 Nov 2003 12:10:44

Subject: Re: I get the message!

From: monz

hi Dave (and Gene and paul),


--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> Gene,
> 
> I apologise again for the personal insult.
> 
> You said you did not call me names, and that's technically true.
> 
> However I pride myself on my teaching and writing ability (perhaps
> mistakenly) as well as my system design ability, and so I'm afraid I
> do take it as a personal insult when someone who hasn't even read my
> article makes assumptions about what my method of exposition will be
> and declares them "sloppy", using the word three times, no less.
> 
> You also said I was "going about it in a very, very bad way", and
> implied that my readers would need a "secret decoder ring" to
> understand my article. All without having read any of it.
> 
> Yahoo groups: /tuning-math/message/7236 * [with cont.] 
> 
> I hope you can now understand why I found all this far more hurtful
> even than being called an anal retentive. 



i must say that i found Gene's comment about the "secret
decoder ring" very funny and amusing.  i realize it was
at your expense, Dave ... sorry, but i have to be honest
above all else, and i did get a laugh from it.


 
> But even if I felt insulted, I should not have responded in kind. 
> I'm sorry.


that's beautiful.  i really admire you for posting it
publicly here.



> Paul,
> 
> I certainly don't want you to use letters instead of names
> in those wonderful diagrams. I'm just saying I think it has
> gone far enough, and besides there is at least _some_ kind
> of logic to most of those names.



one side of me really enjoys the whimsy of the names paul
has already coined, and the other side of me agrees with
you, Dave.  

and i guess i also sense "some kind of logic", because i
know paul well enough to know that he wouldn't simply vent
his imagination on something like this without exercising
his powerful logical abilites too.

but i'm also anal-retentive enough to desire a nice
systematic naming for everything.  (remember? ... i'm 
the guy who wanted a tuning dictionary so badly that he
simply created it.)



-monz


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Message: 7883

Date: Sat, 01 Nov 2003 01:45:50

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > >If you want to make this systematic, why not simply monzo-size range?
> > 
> > Example?
> 
> 225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
> [-7,-1,1,1,1]-kleisma.

I agree with Monz. There's definitely no need to include the
2-exponents here. They're musically irrelevant in most cases, and if
you do need them, the mere fact that these are commas of some kind,
and hence smaller than 600 cents, is enough to give you their
2-exponents. 

I wouldn't want to start using monzos in this role until things got
really complex, like if you would otherwise have more than say 12
characters in the numeric part.

There's no need for systematic names to be so unfriendly as to call
81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name
"5-comma" can be generated and decoded systematically, as I've shown.


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Message: 7884

Date: Sat, 01 Nov 2003 14:37:52

Subject: Re: Eponyms

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:


> 81/80 is the 5-comma. 


using the convention Gene just proposed which i accept
(OK, i'll keep the comma punctuation) :

<3, 5>-monzo: <4, -1>-comma.



> 32805/32768 is the 5-schisma.

<3, 5>-monzo: <8, 1>-schisma.



> 64/63 is the 7-comma

<3, 5, 7>-monzo: <-2, 0, -1>-comma.



> 59049/57344 is the 7-medium-diesis or 7-M-diesis

<3, 5, 7>-monzo: <10, 0, 1>-diesis.



> 28/27 is the 7-large-diesis or 7-L-diesis

<-3, 0, 1>-diesis.


 
> 2048/2035 is the 25-comma

Dave, here i'm not sure if your ratio is correct,
because 2035 = 5 * 11 * 37 .



> 6561/6400 is the 25-small-diesis or 25-S-diesis

<8, -2>-diesis.

 

> 128/125 is the 125-small-diesis

<0, -3>-diesis


> 250/243 is the 125-medium-diesis

<-5, 3>-diesis.



> 531441/512000 is the 125-large-diesis

<12, -3>-diesis.


 
> 5120/5103 is the 5:7-kleisma

<-6, 1, -1>-kleisma.



> 3645/3584 is the 5:7-comma

<6, 1, -1>-comma.





-monz


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Message: 7885

Date: Sat, 01 Nov 2003 16:52:50

Subject: Re: Eponyms

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> <snip>
>
> > 59049/57344 is the 7-medium-diesis or 7-M-diesis
> 
> <3, 5, 7>-monzo: <10, 0, 1>-diesis.


oops ... my bad.  missed a sign.

that should be <10, 0, -1>-diesis.



> > 2048/2035 is the 25-comma
> 
> Dave, here i'm not sure if your ratio is correct,
> because 2035 = 5 * 11 * 37 .


Dave made a typo here and the ratio should be 2048/2025,
in monzo form: the <-4, -2>-comma.



here's the whole list of Dave's "kommas", from largest
to smallest, with the [2, 3, 5, 7]-monzos and cents:

(if viewing on the Yahoo web interface, you'll have
to forward it to your email account to see it properly.)


   2   3   5   7             cents          ratio

[-12, 12, -3,  0]-diesis   64.51886879  531441 : 512000 
[  2, -3,  0,  1]-diesis   62.96090387      28 : 27  
[-13, 10,  0, -1]-diesis   50.72410218   59049 : 57344 
[  1, -5,  3,  0]-diesis   49.16613727     250 : 243 
[ -8,  8, -2,  0]-diesis   43.01257919    6561 : 6400 
[  7,  0, -3,  0]-diesis   41.05885841     128 : 125 
[ -9,  6,  1, -1]-comma    29.21781259    3645 : 3584 
[  6, -2,  0, -1]-comma    27.2640918       64 : 63 
[ -4,  4, -1,  0]-comma    21.5062896       81 : 80 
[ 11, -4, -2,  0]-comma    19.55256881    2048 : 2025 
[ 10, -6,  1, -1]-kleisma   5.757802203   5120 : 5103 
[-15,  8,  1,  0]-schisma   1.953720788  32805 : 32768 



or if you prefer <3, 5, 7>-monzos:


   3   5   7             cents          ratio

< 12, -3,  0>-diesis   64.51886879  531441 : 512000 
< -3,  0,  1>-diesis   62.96090387      28 : 27  
< 10,  0, -1>-diesis   50.72410218   59049 : 57344 
< -5,  3,  0>-diesis   49.16613727     250 : 243 
<  8, -2,  0>-diesis   43.01257919    6561 : 6400 
<  0, -3,  0>-diesis   41.05885841     128 : 125 
<  6,  1, -1>-comma    29.21781259    3645 : 3584 
< -2,  0, -1>-comma    27.2640918       64 : 63 
<  4, -1,  0>-comma    21.5062896       81 : 80 
< -4, -2,  0>-comma    19.55256881    2048 : 2025 
< -6,  1, -1>-kleisma   5.757802203   5120 : 5103 
<  8,  1,  0>-schisma   1.953720788  32805 : 32768 



-monz


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Message: 7886

Date: Sat, 01 Nov 2003 01:57:57

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> The idea of leaving out the 3's is clever but not beneficial, in
> my opinion.

I'm a little confused here. On the one hand you'd apparently be quite
happy to call something the "fartisma", because all you need is a
"hook" to hang the meaning on, and names with numbers in them are
boring, but when I show you a way to eliminate some of the numbers
while still remaining systematic and unambiguous, you want to keep all
the numbers, even redundant ones.

So what would your systematic names for 81/80 and 64/63 look like?

> With the wrong ranges, you wouldn't be able to extract the 3 exponent,
> I assumed.

That's true. But I don't understand what point you're making here.


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Message: 7887

Date: Sat, 01 Nov 2003 17:17:32

Subject: Re: The supertemperament

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >For any odd prime p, there is a finite list of superparticular 
ratios
> >which belong to the p-limit.
> 
> Here's something I can believe but which isn't immediately obvious.
> Can you prove it?

Been there, done that. It follows from Baker's Theorem.

> Cool.  Howabout moving a fixed n down the list (or n's which, for 
each
> starting point in the list, uniquely define a val)?

There's a thought, but first I've got to write up the capstone 
temperament (same thing but for linear temperaments.)


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Message: 7888

Date: Sat, 01 Nov 2003 02:34:24

Subject: Re: Eponyms

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> I think that in any list of common commas you will find
> that more than 90% of them have a 3-exponent other than
> zero. So it is fairly unnecessary to include, up front,
> the fact that they have 3's in them; 



i can pretty much agree with that, with one very important
comma residing in that other 10%: the enharmonic diesis,
ratio 128/125, [3,5]-monzo version: the [ 0 -3]-diesis.



> I don't want to call 81/80 the 5:3^4-comma, but just 
>the 5-comma. 


[3,5]-monzo version: the [4 -1]-comma.



> And 64/63 would be simply the 7-comma, not the 7.9-comma.


[3,5,7]-monzo version: the [-2 0 -1]-comma.



> That's the whole point of setting the size-category
> boundaries so carefully, to eliminate the need to have
> the power of 3 explicit in the systematic name.


so if they're being described as monzos, just leave out
the first exponent of the vector and the first prime-factor
of the label.


... looks like Gene and i support each other on this method 
of description.



-monz


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Message: 7889

Date: Sat, 01 Nov 2003 17:20:35

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> > 
> > > Have you actually read any of the several descriptions I've 
given of
> > > the proposed komma naming algorithm and its inverse? Are they 
all
> > > really that unclear?
> > 
> > I don't buy kommas.
> 
> Do you mean you don't like spelling it with a "k" when it's being 
used
> as a generic term. That's fine. That's not part of the naming
> algorithm. That's just me. 

(1) I don't like having two words which sound the same and with 
related meanings

(2) I think the goofy spelling, if you do this, should be for your 
new meaning, and not imposed on an established one. "Comma in a 
particular range of cents"="komma", in other words.


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Message: 7890

Date: Sat, 01 Nov 2003 02:34:55

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Another vague idea: The order of mention of primes could be different
> >depending whether they are being multiplied (dot) or divided (colon).
> 
> Again a cool idea, and I find these sort of inquiries fascinating, but
> I try to avoid them when I can't see them being very useful.  YMMV.

Seems useful to me.

I think we're now past the point where all the common comma's and
temperaments have been named.

My purpose in proposing systematic naming methods for both linear
temperaments and commas is to avoid drowning in lots of meaningless
names where, if you haven't been following the tuning-math list
religiously for the past x years the only way you have of figuring out
what someone's talking about is to look the names up in a database
somewhere.

> >Tanaka's kleisma (_the_ kleisma) has the systematic name of
> >5^6-kleisma (five-to-the-six-kleisma)
> 
> I've so far tried my best not to mention the term "anal retentive".
> :)

It takes all kinds. That's funny. Gene says I'm sloppy, and you say
I'm anal-retentive. A contradiction, wot?

Actually, I think I'm just your classic engineer/architect type. I
design systems. I'm good at it. I make my living designing systems of
several different kinds. I'm apparently genetically predisposed to it.
You, presumably, have other wonderful and complementary qualities.

Would you care to explain what your objection's are to the proposal,
as opposed to your objections to my online personality?


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Message: 7891

Date: Sat, 01 Nov 2003 19:12:08

Subject: Capstone temperament

From: Gene Ward Smith

Before we get to the range which defines the supertemperament (where
the nullspace of the matrix of monzos has dimension one, defining a
val, and thus an equal temperament) we have a range where the
nullspace is dimension two, and defines a linear temperament--which
I've mentioned before, under the name of the capstone temperament.
Here are mappings for capstones up to the 19-limit, and generators in
terms of the corresponding supertemperament. (This last is a little
crude, but does get us into the ballpark.)

5 limit: 81/80 meantone

7 limit: [[9, 15, 22, 26], [0, -2, -3, -2]] ennealimmal
generators: [1/9, 1/24]


11 limit: [[18, 28, 41, 50, 62], [0, 2, 3, 2, 1]] hemiennealimmal
generators: [1/18, 1/72]


13 limit: [[2, 7, 13, -1, 1, -2], [0, -11, -24, 19, 17, 27]]
generators: [1/2, 47/270]


17 limit: [[1, 35, 221, 161, -5, 197, 367], 
[0, -79, -517, -374, 20, -457, -858]]
generators: [1, 637/1506]


19 limit: [[1, 250, 324, 62, -178, -481, 1579, 258], 
[0, -512, -663, -122, 374, 999, -3246, -523]]
generators: [1, 4143/8539]


The 13-limit capstone has reduced basis

[1716/1715, 2080/2079, 3025/3024, 4096/4095]

and wedgie

[22, 48, -38, -34, -54, 25, -122, -130, -167, -223, 
-245, -303, 36, -11, -61].

It is well-covered by 494-et, and can be thought of as the 
270&494 13-limit linear temperament. I don't want anyone to pitch a
fit, but I suppose Cap13 is one possible name.


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Message: 7892

Date: Sat, 01 Nov 2003 02:46:03

Subject: Re: Eponyms

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:


> There's no need for systematic names to be so unfriendly
> as to call 81/80 the [-4, 4, -1]-comma, or even the 
> [4, -1]-comma. The name "5-comma" can be generated and
> decoded systematically, as I've shown.



i'm sorry to respectfully disagree with you, Dave, but i
don't see anything unfriendly about "[4 -1]-comma".

(note that i don't consider the comma punctuation necessary.)


admittedly, "5-comma" is a whole lot easier and, yes,
i'll admit, friendlier.


but for me, so used to visualizing tunings on a lattice, 
"[4 -1]-comma" tells me exactly what i need to know.

if i'm picturing the whole prime-space on a lattice,
[4 -1] helps me to *immediately* set a boundary in my
mind which filters out a large number of redundant 
lattice-points.


in fact, when i hear or read the word "syntonic" the
first thing i think about is the vector on the lattice
which would describe it in a prime-space ... and then
the second thing i think about very quickly after that
is 3^4 / 5^1 .



and as i've already been arguing with the ratios, forget it.

there's almost always nothing valuable about retaining the
data for prime-factor 2, unless it need be considered for
(to cite two examples i can think of quickly):

- actual orchestral scoring where the 8ve-register must
be considered, or 

- analyzing ancient Greek and Roman theory, which was 
based on 4:3 "perfect-4ths" and always specified 8ves,
and gave different names to notes an 8ve apart.





-monz


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Message: 7893

Date: Sat, 01 Nov 2003 19:34:36

Subject: Re: The supertemperament

From: Carl Lumma

>>>For any odd prime p, there is a finite list of superparticular
>>>ratios which belong to the p-limit.
> > 
> > Here's something I can believe but which isn't immediately
> > obvious.  Can you prove it?
> 
> Been there, done that. It follows from Baker's Theorem.

There are no results for "Baker's theorem" or "bakers theorem"
at mathworld, but the 2nd google result for "baker's theorem"
is this post of yours...

Yahoo groups: /tuning-math/message/1108 * [with cont.] 

Hooray again for google.  I wonder how much of these lists
are google-searchable?

-Carl




> 
> > Cool.  Howabout moving a fixed n down the list (or n's which, for 
> each
> > starting point in the list, uniquely define a val)?
> 
> There's a thought, but first I've got to write up the capstone 
> temperament (same thing but for linear temperaments.)


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Message: 7895

Date: Sat, 01 Nov 2003 11:48:40

Subject: Re: I get the message!

From: Carl Lumma

>However I pride myself on my teaching and writing ability (perhaps
>mistakenly) as well as my system design ability,

I think you're a great teacher, writer, and system designer.

-Carl


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Message: 7897

Date: Sat, 01 Nov 2003 22:09:29

Subject: Re: A remarkable property of 270

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> If we take the 12 smallest 13-limit superparticular commas, namely
> 
> 1001/1000, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095, 
> 4225/4224, 4375/4374, 6656/6655, 9801/9800, 10648/10647, 
123201/123200
> 
> we find that they all have the property of being 270-et commas.

much like 72-equal in the 11-limit.

> Moreover, if we take this comma list five at a time, we get the 270 
et
> in all cases where the five commas are linearly independent.
Sometimes
> we get what we might regard as 540, but this is just a doubling of 
270.

in other words, torsion?


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Message: 7898

Date: Sat, 01 Nov 2003 03:18:00

Subject: Re: Eponyms

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> in fact, when i hear or read the word "syntonic" the
> first thing i think about is the vector on the lattice
> which would describe it in a prime-space ... and then
> the second thing i think about very quickly after that
> is 3^4 / 5^1 .


actually that's not true.  i don't visualize the numbers as
3^4 / 5^1 , but rather as 3^4 * 5^-1 , since that's exactly
how the lattice works.  and that visualization agrees
exactly with the monzo of the syntonic comma.



> and as i've already been arguing with the ratios, forget it.
> 
> there's almost always nothing valuable about retaining the
> data for prime-factor 2, unless it need be considered for
> (to cite two examples i can think of quickly):
> 
> - actual orchestral scoring where the 8ve-register must
> be considered, or 
> 
> - analyzing ancient Greek and Roman theory, which was 
> based on 4:3 "perfect-4ths" and always specified 8ves,
> and gave different names to notes an 8ve apart.


but what i forget to emphasize here again is: even in
these cases where 8ves must be considered, it's easier
to use the monzo including 2's exponent, instead of the 
actual ratio.



-monz


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Message: 7899

Date: Sat, 01 Nov 2003 14:13:06

Subject: ennealimmal

From: Carl Lumma

Am I correct that the first ennealimmal scale with an octave
is simply 9-equal, and the next is this 17-tone one...

 50.
 133.3
 183.3
 266.7
 316.7
 400.
 450.
 533.3
 583.3
 666.7
 716.7
 800.
 850.
 933.3
 983.3
 1066.7
 1116.7

...?

Manuel, is there a convenient way to get MOS-like scales with
non-octave periods in Scala?

-Carl


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