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Message: 11175

Date: Thu, 01 Jul 2004 06:14:06

Subject: Re: my paper nears completion

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> Paul Erlich wrote:
> 
> > I'd appreciate any comments or corrections . . . note that it's 
> > incomplete, and the 46 horagrams are not included -- 
> > 
> > 
Yahoo groups: /tuning/files/perlich/coyotepaper1.doc *
> 
> Is the "Sagittal" font available for download? (Be sure to send a 
copy 
> of it with the paper.)
> 
> Augmented should be [7 0 -3> in the table at the end.
> 
> The mathematical parts looked easy enough to understand -- some of 
it 
> seemed a little too obvious to me, but for the average reader who 
hasn't 
> been following the tuning-math list it would be more of a 
challenge. A 
> whole paper could be written just on the musical applications of 
> wedgies, so it's probably just as well that they're not mentioned. 
But 
> you should at least have some description of what a "bivector" is 
if 
> you're going to include them in the tables.

Yes -- as I mentioned on the tuning list, the part of the paper about 
combining commas (as well as temperament comnplexity) hasn't been 
written yet. It would be nice to have a real clear exposition on the 
hows and whys of 2x2 determinants . . . ;)

> A couple of brief notated musical examples would be nice, like a 
typical 
> octatonic chord progression you might find in 12-ET music to 
illustrate 
> the 648;625 comma (the A minor - C minor - Eb minor - F# minor 
cycle on 
> my diminished temperament page for instance).

If you could provide such notated examples for me to include, I'd be 
extremely grateful. Dave Keenan provided the lattices that are in 
there and one more that will be soon. I'll be sure to thank you both, 
and Gene too.


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Message: 11176

Date: Thu, 01 Jul 2004 19:54:35

Subject: Re: my paper nears completion

From: Carl Lumma

>But here's a notated version of my octatonic chord progression in 12-ET:
>
>ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif
>(MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid *)

Nice!

>and the porcupine chord progression in 12-ET, which illustrates the 
>250;243 comma (which of course doesn't vanish in 12-ET, but does in 
>porcupine temperament):
>
>ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif
>(MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid *)

An old fav.  I don't remember hearing it in 12, though.  What
a great comma.

You should cross-post this to MMM.  It might fit with the discush.
Gene and Jon are having.

-Carl


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Message: 11182

Date: Thu, 01 Jul 2004 20:55:58

Subject: Re: my paper nears completion

From: Paul Erlich

Thanks for your comments; everything helps.

--- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> 
wrote:

> 1. Maybe you could make it clearer what exactly the goal of the
>    paper is; specifically if a lay person makes a big effort and
>    blasts thru all the math and lattice diagrams, they will be
>    rewarded with knowing ...?

The end of the introduction says,

"The purpose of this paper is to bring to light a host of alternative 
temperaments alongside the familiar ones. These should not be 
understood merely as lists of pitches to be employed when tuning an 
acoustical or electronic instrument. More importantly, they should be 
seen as models for the conception and notation of new music, 
regardless of the instruments or precise tuning strategies employed 
in its implementation."

The lists of pitches are in the horagrams, which as I said are not 
contained in this .doc file.

How can I make this seem more "rewarding"?

> 2. I would change "musical ideas" to something like " a pattern
>    of notes" on p.1

I changed it to "patterns of notes" -- any objections?

> 3. p.2 and footnote vii Why is enharmonic equivalence now important?

Beethoven, Schubert, etc. would rely on such equivalence in their 
compositions. It's necessary in order to circumnavigate commas like 
128:125 and 32805:32768. Mathieu's book does some explicit analyses 
showing this . . . I guess I should refer the reader to it?

> 4. p.4  Bras and kets?  You could just say row vector and column
>    vector and stick to the formalism linear algebra

I don't think that would work -- if you've been following this list, 
we appear to need Grassmann algebra.

> rather than
>    quantum mechanics.

No, this has nothing to do with quantum mechanics. Read the mathworld 
links I provided. Also, I plan to include a 3-limit lattice earlier 
with level pitch lines and thus motivate the "ket vector" definition 
as a linear operator that mathworld alludes to.

> 5. I think the inner product would be more acceptable to people
>    is you said "a convenient shorthand" rather than "a fancier way".

Right.

> 6. "****Importance made clear below"  Make it clear up front,
>    otherwise people won't make the effort to read it.

Umm . . . thanks, I'll try . . .

> 7. Right before the Temperment section: ************* that looks
>    like somebody's phone number, rather than anything related to
>    the previous calculations (which I didn't do, by the way. sorry)

OOPS!!!! Yikes. Don't call it.

> 8. Middle of Temperment section: "The relevant possibilities here
>    include..."  Why would someone want to temper out these things?
>    What musical goodies does it buy them?

Basically all kinds of Romantic-period harmonic effects. For example, 
being able to re-intepret the diminished seventh chord as the 
dominant-function chord in four different keys, spaced 1/4 octave 
apart from one another. This is one of the things that made Romantic 
harmony able to play with a wider array of expectation/surprise 
effects and to coherently explore much wider harmonic terrain. Let me 
know what you think I should add in the paper and where. Or maybe a 
reference to Mathieu will do?

Thanks a lot,
Paul


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Message: 11183

Date: Thu, 01 Jul 2004 21:18:29

Subject: Re: Wedgies and generators

From: Paul Erlich

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

> Put it all together and you get an antisymmetric matrix, the upper
> right corner of which is the wedgie. I think Herman was the one who
> suggested writing it in the form
> 
> <<1 4 10
>     4 13
>       12||

When I asked you, some time ago, about these triangles of numbers and 
their great similarity to the upper triangle of certain matrix of 
vanishing commas, you had no response.



________________________________________________________________________
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Message: 11187

Date: Fri, 02 Jul 2004 21:10:21

Subject: Re: Gene's mail server

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> > >I care; I find the bival form very much more convenient. If you 
> > don't care, 
> > >why do you insist on not using it?
> >  
> > Because it would add significantly to the length and complexity 
of 
> > the paper.
> 
> Bivals and bimonzos are equally complex.

Not when you've already introduced the "monzos" concept but haven't 
introduced the "vals" concept.

> > >> >Of course
> > >> >switiching from one to the other is not difficult, but why 
> saddle 
> > us with
> > >> > a wholly unnecessary headache?
> > >
> > >> What is this headache, exactly? I think introducing vals would 
> be a
> > >> headache. 
> 
> But you go ahead and use bimonzos!

Yup -- and I just said why, too.

> > >But there they are, in your table, in bimonzo form.
> >  
> > The 'bimonzos' are there, yes. Geometrically they represent the 
> > periodic unit of the lattice, when the 'commas' are applied as 
> > equivalence relations. Relating this to vals is outside the scope 
> of 
> > the paper, as beautiful as they may be mathematically.
> 
> Are you expecting people to read the comma values off of the 
bimonzo?

No. But as long as we're on the subject here, it might be worth 
reviewing here for list memmbers how you do that. Not in the paper.

Anyway, since these names are so ugly, does *anyone* have suggestions 
for renaming them (Dimipent, Dimisept, Negripent, Negrisept, 
Sensipent, Sensisept) that preserves their approximate alphabetical 
location?


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Message: 11191

Date: Fri, 02 Jul 2004 17:36:38

Subject: The Sagittal website is officially open

From: Dave Keenan

I expect there probably isn't anyone who reads this list and doesn't 
also read the main tuning list, but since most of Sagittal was 
developed here, I figure we deserve our own announcement so it is 
there in the archives. Hey Paul, you'd better not go here until 
you've finish your Xenharmonikon paper. ;-)

See http://users.bigpond.net.au/d.keenan/sagittal/ *

A few items will not be up for another few days. But there's
definitely enough there to make a visit worthwhile, and no reason to
delay the announcement further.

Even if you aren't particularly interested in a universal microtonal
notation system, I think you will enjoy the mythology of its
creation. :-)

http://users.bigpond.net.au/d.keenan/sagittal/gift/GiftOfTheGods.htm *

-- Dave Keenan


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Message: 11192

Date: Fri, 02 Jul 2004 17:42:02

Subject: Re: The Sagittal website is officially open

From: Dave Keenan

P.S. Please post any non-mathematical responses on this topic to the 
main tuning list.
Yahoo! - *

Thanks.


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Message: 11193

Date: Fri, 02 Jul 2004 19:44:22

Subject: Re: Gene's mail server

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Hi Gene,
> > 
> > Your mail server rejected my e-mail response to you as "probable 
> > SPAM".
> > 
> > Assuming you disagree with your server's assessment, let me know 
how 
> > I should proceed.
> 
> Considering the huge quantity of spam which makes it through, this 
is
> kind of depressing. Maybe upload it, or post it if it is postable. I
> suppose I should get an alternative email address.

Anyway, here we go:

>> >"Temperament, when implemented in a regular manner, reduces the 
often
>> >bewildering variety of interval sizes in a scale or tuning system 
to a
>> >manageable few."
>> >
>> >It only produces a finite number in the case equal temperaments;
>> > otherwise, the intervals are dense in the real continuum of 
pitches. Why
>> > not say it makes intervals more managable?
>>
>> I should indeed say that in a few places. But here, I'm not 
talking about
>> an *infinite* tuning system, let alone an infinite scale. So isn't 
my
>> assertion true?

>If you have an equal temperament, it is theoretically infinite, but 
finite in 
>practice since only a finite number of tones are audible. If you 
have 
>something with more than one generator, whether JI or temperament, 
you get 
>density. Meantone is finite only in the same way and for the same 
reason that 
>5-limit JI is finite; you can't distinguish an infinite number of 
tones.
 
Are you completely ignoring what I'm saying here, or what? I 
changed "tuning system" to "finite tuning system". Is my assertion 
now true, or not?

>> >"A fancier way of writing this _expression is
>> ><1200 1901.96 2786.31|-4 4 -1> = 21.5;
>> >this is an example of an inner product operation between two 
vectors."
>> >
>> >Technically, it would be better to call this an angle bracket. 
Most of the
>> >time, an inner product means between two vectors in the same 
space.
>
>> Is it an interior product?

>No, though there is a relationship. It's the linear mapping of an 
element of 
>the dual space V* acting on an element of the original vector space 
V.

>Here's what Math World has on dual vector spaces:

>Dual Vector Space -- from MathWorld *

>http://mathworld.wolfram.com/LinearFunctional.html *

Can't I just refer to it as some kind of product? I really don't want 
to burden the reader with abstruse mathematics, and I think 99% of 
them would thank me for this.

>> >and
>> >the mapping to primes can be read off the bival more readily.
>
>> Maybe the generator part of the mapping is a tiny bit more direct. 
But the
>> period part of the mapping, without which the other part does 
little good,
>> doesn't.follow directly from either. The table tells you both, so 
who
>> cares?

>I care; I find the bival form very much more convenient. If you 
don't care, 
>why do you insist on not using it?
 
Because it would add significantly to the length and complexity of 
the paper.

>> >Of course
>> >switiching from one to the other is not difficult, but why saddle 
us with
>> > a wholly unnecessary headache?
>
>> What is this headache, exactly? I think introducing vals would be a
>> headache. 

>But there they are, in your table, in bimonzo form.
 
The 'bimonzos' are there, yes. Geometrically they represent the 
periodic unit of the lattice, when the 'commas' are applied as 
equivalence relations. Relating this to vals is outside the scope of 
the paper, as beautiful as they may be mathematically. However, I 
will be explaining the meaning and role of the "ket vector" a little 
more fully in the 3-limit case, so an opportunity might open up for 
you to provide a very informative footnote.

>>There's a footnote about Graham Breed's "melodic" approach being
>> the algebraic dual to this one. 

>Graham, can you explain (maybe on tuning-math) what the melodic 
approach is?
 
Take a look at Graham's webpages on meantone, schismic, 
diaschismic . . . It seems to be closely wedded to the val approach, 
which is why I suggested we use the term "breeds" to refer to vals 
(as long as we're using "monzos" to refer to lattice vectors).

>> How do you know which is 5-limit and which is 7-limit? 

>If they have the same tuning,
 
They don't. Different tunings, different horagrams. Take another 
look. These systems do *not* have identical tunings in the 5-limit 
and 7-limit (at least according to the 7-limit results you 
furnished). However, for meantone, porcupine, superpyth, and magic, 
I'm already doing this, because the tunings are indeed the same. Did 
you really not see this?

>I guess I *could*
> redo these as long as whatever you propose stays in the same place
> alphabetically . . .
 
Please think about this again. Otherwise you'll be "saddled" with the 
ugly names

>> >Why orson?
>
>> Son of Orwell, or ORson WELLs, or whatever . . .

>Why not just orwell?

Different tuning, different horagrams. "Mork to Orson!" Hello?

:)


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Message: 11194

Date: Fri, 02 Jul 2004 19:44:59

Subject: Re: Wedgies and generators

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > 
> > > Put it all together and you get an antisymmetric matrix, the 
upper
> > > right corner of which is the wedgie. I think Herman was the one 
who
> > > suggested writing it in the form
> > > 
> > > <<1 4 10
> > >     4 13
> > >       12||
> > 
> > When I asked you, some time ago, about these triangles of numbers 
and 
> > their great similarity to the upper triangle of certain matrix of 
> > vanishing commas, you had no response.
> 
> Sorry. I did discuss it fairly extensively when I first introduced 
the
> wedge product, if I recall correctly, in connection with the 
commutator.

If you were teaching a class, I would have failed out a long time ago.


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Message: 11195

Date: Fri, 02 Jul 2004 19:48:39

Subject: Re: my paper nears completion

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> Paul Erlich wrote:
> 
> > If you could provide such notated examples for me to include, I'd 
be 
> > extremely grateful. Dave Keenan provided the lattices that are in 
> > there and one more that will be soon. I'll be sure to thank you 
both, 
> > and Gene too.
> 
> Well, I don't have any good notation software, but I managed to put 
a 
> couple of examples together with Voyetra Digital Orchestrator and 
some 
> cutting and pasting in Paint Shop Pro. Unfortunately I couldn't 
figure 
> out how to tell it to use sharps instead of flats, if it can even 
do 
> that (probably not, since it's a MIDI editor).
> 
> But here's a notated version of my octatonic chord progression in 
12-ET:
> 
> ftp://ftp.io.com/pub/usr/hmiller/music/octatonic.gif
> (MIDI at http://www.io.com/~hmiller/music/ex/dim12.mid *)
> 
> and the porcupine chord progression in 12-ET, which illustrates the 
> 250;243 comma (which of course doesn't vanish in 12-ET, but does in 
> porcupine temperament):
> 
> ftp://ftp.io.com/pub/usr/hmiller/music/porcupine.gif
> (MIDI at http://www.io.com/~hmiller/midi/porcupine-12.mid *)

Thanks Herman. I can't view these in IE, for some reason. Do I have 
to do something special?


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Message: 11196

Date: Fri, 02 Jul 2004 19:57:12

Subject: Re: my paper nears completion

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "jjensen142000" <jjensen14@h...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> >> The end of the introduction says,
> > 
> > "The purpose of this paper is to bring to light a host of 
> alternative 
> > temperaments alongside the familiar ones. These should not be 
> > understood merely as lists of pitches to be employed when tuning 
an 
> > acoustical or electronic instrument. More importantly, they 
should 
> be 
> > seen as models for the conception and notation of new music, 
> > regardless of the instruments or precise tuning strategies 
employed 
> > in its implementation."
> > 
> > The lists of pitches are in the horagrams, which as I said are 
not 
> > contained in this .doc file.
> > 
> > How can I make this seem more "rewarding"?
> > 
> 
> "You are going to see some really erotic horagrams if you make it
> to the end of this paper. No one under 21 admitted"
> 
> I'm kidding :-)  
> 
> I'm just of the opinion that you should maybe put more description
> of the results in the opening paragraph (or abstract) so the general
> reader will get fired up to read the whole thing.

Oh, yes. There's still going to be an abstract added. Thanks.

> That raises a question though: I assumed this paper was for a 
journal
> like Xenharmonikon...

Yes, it's for Xenharmonikon.

>I think you said something like "the editor made
> me cut out a lot of complicated math".

Well, he really didn't want the paper to be mathematical or mainly 
concerned with math.

> If this is not the case, then
> a lot of my comments were not really relevant! (and probably sound
> needlessly nitpicky)

I'm trying to use the minimum of math needed to show where my results 
are coming from, and why.

> > > 3. p.2 and footnote vii Why is enharmonic equivalence now 
> important?
> > 
> > Beethoven, Schubert, etc. would rely on such equivalence in their 
> > compositions. It's necessary in order to circumnavigate commas 
like 
> > 128:125 and 32805:32768. Mathieu's book does some explicit 
analyses 
> > showing this . . . I guess I should refer the reader to it?
> > 
> 
> This is really interesting, to me.  If possible, I would throw in
> more details about this, although maybe that is a different paper...

I'll try to expand on that footnote, thanks. Yes, I'd recommend 
Mathieu's book, though he really ignores the whole "middle path" idea 
(including meantone!) and jumps straight to 12-equal. Speaking 
of "jumps", his theory in that regard is goofy, if I may say so. But 
he does a great job showing how certain great pieces of Western music 
require certain commas to vanish.


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Message: 11197

Date: Fri, 02 Jul 2004 19:58:49

Subject: Re: Gene's mail server

From: Paul Erlich

BTW, quantum mechanics texts do refer to 

<x|y>

as an "inner product". Are you telling me that I need to be *more* 
mathematical than a quantum mechanics text?



________________________________________________________________________
________________________________________________________________________



------------------------------------------------------------------------
Yahoo! Groups Links

<*> To visit your group on the web, go to:
    Yahoo groups: /tuning-math/ *

<*> To unsubscribe from this group, send an email to:
    tuning-math-unsubscribe@xxxxxxxxxxx.xxx

<*> Your use of Yahoo! Groups is subject to:
    Yahoo! Terms of Service *


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Message: 11198

Date: Sat, 03 Jul 2004 17:14:42

Subject: Re: bimonzos, and naming tunings (was: Gene's mail server))

From: monz

hi Paul,


--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

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

> > now that we have so much broader a view of large numbers
> > of tunings, we should subject the whole "tuning universe"
> > to deep review, and come up with a really good and logical
> > system of classification and naming.
> 
> OK, but this was asked in the context of my paper, which 
> has to be submitted very soon. Did you see the draft?



yes, but i only had time to skim it quickly.  actually,
i'm printing it out right now so that i can give it a
proper reading, which i'll do later this morning.

but it does seem to me that the time may be right for
a drastic reconsideration of existing nomenclature ...
considering especially the advances made in tuning theory
since Gene joined the tuning cyber-community, and the
fact that sagittal notation and my own little contribution
(Tonalsoft software) are about to be born to the world.



-monz


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