Tuning-Math Digests messages 4675 - 4699

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Message: 4675

Date: Thu, 18 Apr 2002 03:54:51

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: 
> I did some more work on the symbols in a second file:
> 
> > Yahoo groups: /tuning- *
> math/files/secor/notation/symbols2.bmp
> 
> which I will put out there, once Yahoo gets over its cranky spell 
and 
> lets me upload it.

I'm still waiting to see that, but in the meantime I've put up my 
latest versions with changes based on several of your suggestions, and 
one innovation.

Yahoo groups: /tuning-math/files/Dave/SagittalSingle217 *
C2DK.bmp
Yahoo groups: /tuning-math/files/Dave/SagittalMulti217C *
2DK.bmp


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Message: 4676

Date: Thu, 18 Apr 2002 04:35:29

Subject: Re: 41 ET 11-tone diatonic

From: genewardsmith

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

> of course, we've discussed not only two-dimensional but also 
> three- and four-dimensional periodicity blocks quite a bit -- we've 
> focused more attention on the ones where there's only one chromatic 
> unison vector and all the rest are commatic, but there's no 'rule' 
> that says there shouldn't be more than one chromatic unison vector.

How would you work it?


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Message: 4677

Date: Thu, 18 Apr 2002 05:05:23

Subject: Re: one from the archives

From: clumma

>It's been a few days, so I'm joining the resending parade; I hope
>Carl still remembers what these are about!
> 
>[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
> 
>edge connectivity = 3

I remember connectivity; 3 isn't so hot for a 12-tone scale in
the 7-limit, eh?

>characteristic polynomial =
>x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
>x^4-8*x^3+304*x^2+96*x

Never did the homework to understand these.

>This tells us there are 29 consonant intervals and 22 consonant
>triads

That doesn't seem like much to write home about either.  What
are some good scales here in your experience, Gene?  The three
winners on my list were:

1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8

1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15

1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5

I ended up using the first for melodic reasons.

>The scale is not epimorphic, but can be extended in various ways,
>for instance to the h15-epimorphic scale
/.../
>I've attached two files which show the graph for each of these
>scales.

Thanks!

>Now I'm wondering what the story is--where does this scale
>arise from?

I came up with it when drawing lattices on paper some time ago,
when looking for good 12-tone 7-limit tunings for my piano.

-Carl


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Message: 4678

Date: Thu, 18 Apr 2002 05:10:16

Subject: Re: one from the archives

From: clumma

I wrote...

> The three winners on my list were:
> 
> 1/1 21/20 9/8 7/6 5/4 4/3 7/5 3/2 14/9 5/3 7/4 15/8

"lester"

> 1/1 16/15 28/25 7/6 5/4 4/3 7/5 112/75 8/5 5/3 7/4 28/15

"prism"

> 1/1 21/20 7/6 6/5 5/4 21/16 7/5 3/2 8/5 42/25 7/4 9/5

"stelhex"

And this one by Wilson/Hahn:

1/1 21/20 35/32 6/5 5/4 21/16 7/5 3/2 25/16 42/25 7/4 15/8

"class"

-Carl


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Message: 4679

Date: Thu, 18 Apr 2002 07:49:59

Subject: Euclidean reduced 9-note scales

From: genewardsmith

Here are the Euclidean reductions for h9 in the 5 and 7 limits:

5-limit: [1, 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8]

This is pretty much of a classic--a rhombic block.

7-limit: [1, 15/14, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 28/15]

This, with steps of size 16/15 and 15/14, as well as 9/8 and
28/25, fairly cries aloud to be tempered via 225/224~1. If we do that of course both scales become the same.

I looked at all of the permutations of the 72-et version of this scale, and the original form turned out to be the clear winner:

[0, 7, 19, 23, 30, 42, 49, 53, 65]

Here are the 5,7, and 9-limit characteristic polynomials:

x^9-16*x^7-16*x^6+57*x^5+84*x^4-34*x^3-84*x^2-8*x+16
x^9-21*x^7-28*x^6+65*x^5+100*x^4-71*x^3-116*x^2+26*x+44
x^9-29*x^7-80*x^6-39*x^5+70*x^4+52*x^3-16*x^2-9*x+2

We have 40 9-limit triads here, which looks pretty good!

Here is a runner-up scale:

[0, 7, 19, 23, 30, 42, 49, 53, 60]

x^9-16*x^7-16*x^6+57*x^5+86*x^4-27*x^3-80*x^2-18*x+6
x^9-19*x^7-20*x^6+67*x^5+76*x^4-79*x^3-84*x^2+30*x+28
x^9-27*x^7-64*x^6+3*x^5+104*x^4+37*x^3-40*x^2-14*x


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Message: 4680

Date: Fri, 19 Apr 2002 11:23:07

Subject: Re: My Approach Generalized Diatonicity

From: Carl Lumma

>Two dissonant chords with no particular rationalisation may be hard to 
 >tell apart.  But there are plenty of dissonances, like 11-limit intervals, 
 >which have their own quality distinct from other dissonances.  And miracle 
 >tuning is specifically optimized for them.
 >
 >I'm certainly not going to reject scales because they don't have enough 
 >unambiguous consonances.  We should be collecting scales that have most 
 >reasonable properties, and see how well they work as diatonics.  That's 
 >going to take a long time, because it means writing fairly complex music 
 >for each.

It would be fun to discuss this, but I'm not sure what it has to do with
where we started from.  Do we agree that a principle thing about the
diatonic scale, that is not a common thing among scales, is the ability
to harmonize a melody inside the scale and have the harmony voice
still sound like the same melody?  If we do, then we can proceed to figure
out which properties have anything to do with it, and which don't.

Regardless of if we agree about this, we can ask if there are any
other principle things that differentiate the diatonic scale from the
wide variety of possible scales.  What might they be?

-Carl


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Message: 4687

Date: Fri, 19 Apr 2002 11:24:35

Subject: Re: one from the archives

From: Carl Lumma

>>>The adjacency matrix of a graph is a square matrix labeled by
 >>>verticies; it has a "1" if they are connected, and a "0" if not
 >>>(counting the vertex to itself as a "0".) The characteristic
 >>>polynomial of this is as above, and is a graph invariant. The
 >>>n-2 term gives the number of edges, and the n-3 term twice the
 >>>number of triads.
 >> 
 >>That's crazy.
 >
 >Why?

What makes anything crazy?  That you don't understand it?  That's
a pretty good def., I guess.

 >> Does it show tetrads?
 >
 >It doesn't show anything. To show tetrads, we would need to count
 >principal minors which were all 1s except along the diagonal, which
 >would be 0. I don't know if that can be done using the coefficients
 >of the characteristic polynomial.

Noted.

-Carl


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Message: 4688

Date: Fri, 19 Apr 2002 00:33:41

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> > > I think there should be a strong family resemblance between the 
> standard
> > > symbols and our new ones, or they will not be found unacceptable 
> on visual
> > > aesthetic grounds.
> > 
> > Yes, that is a very valid point.

Of course you knew I meant to write "will be found unacceptable" or "will
not be found acceptable".

> I did some more work on the symbols in a second file:
> 
> > Yahoo groups: /tuning- *
> math/files/secor/notation/symbols2.bmp
> 
> which I will put out there, once Yahoo gets over its cranky spell 

and 
> lets me upload it.

Got it at last. Thanks.

> The more I look at your symbols, the more I like their style, 

I've just uploaded a MsWord document containing drawings that show how I
conceive of these flags in a resolution-independent manner, so as to
produce that style. You will see how the style is designed to be compatible
with the conventional symbols, in particular the conventional sharp and
flat symbols which the sagittals will most often have to appear next to.

Yahoo groups: /tuning-math/files/Dave/Flags.doc *

I suggest you print it and take a hiliter pen and colour in the parts that
actually make up the flags and stems. I couldn't figure any easy way to do
that in Word. I'm sure you'll figure out what needs colouring. Then turn
the second page upside down and hold each flag in turn, beside the standard
flat and then the standard sharp.

Notice that the prototype convex and concave flags are exact 180 degree
rotations of each other, and wavy is an exact 180 degree rotation of
itself. This was partly intended to help with flag complementation in 217-ET.

> so 
> (assuming that the file is out there) please follow along with me.
> 
> The fifth staff is a synthesis of features from both of our efforts 
> above that.  I made the sesequisharp (|||) and double-sharp (X) 

group 
> of symbols intermediate in width between what each of us had,

Ok. We agree on the line-thickness and overall width of all the tails now,
5 pixels for ||, 7 pixels for both ||| and X. I hope you like the idea of
shortening the middle stem of the ||| by 3 pixels so it's more like |'|. I
think that having that ^ shape in the tail tends to put them
psychologically in the same apotome as the X tails.

We also agree on how far the tail projects away from the centreline of the
corresponding notehead. That's 11 pixels not including the pixel that's
_on_ the centreline. That's the same as a sharp or natural, but two pixels
shorter than a flat. These agreements are good.

But we still don't agree on the height of the X's. Your X's are not
constant. They vary according to what flags they have on them, and are
often not laterally symmetrical. My X's are all the same height as they are
wide (7 pixels) and are laterally symmetrical. They just meet the concave
flags, but for other flag types they are extended by two parallel lines at
the same spacing as the outer two of the |'|. If nothing else, it certainly
simplifies symbol construction, not to have to design a new X tail for
every possible combination of flags. And if we get into using more than one
flag on the same side (e.g. for 25) with these X tails, I figure we're
gonna need those parallel sides.

> while 
> the semisharp (|) and sharp groups (||) are either the same as or 
> very close to your symbols.  The biggest problem I had was with the 
> nubs (which I made rather large and ugly) still tending to get lost 
> in the staff lines.  I tried one symbol (in the middle of the staff) 
> with a triangular nub, which looks a little neater, I think.

Yes it looks neater, but I fear it is out of character with the standard
accidentals. I even think that maybe _any_ nubs are out-of-character. Of
course we have the precedent of the double-sharp symbol, but I tend to
think of _it_ as being out-of-character with the other 3 standard symbols.
I suspect it is more often seen as the unpitched notehead than as an
accidental.

By the way, I'm finding Elton John and Bernie Taupin's 'Goodbye Yellow
Brick Road' songbook to contain examples of just about everything with
regard to accidentals.

My recommendation is to make the nubs 4 x 4 with the corner pixels knocked
out. More round, less square.

  @@@@
@@@@@@@@
@@@@@@@@
  @@@@

> To the right of that I copied three of your symbols so I can comment 
> on them.  In all three of them the concave or wavy flag is 
> significantly lower than the line or space for its note.

Again, I don't understand this statement in regard to the wavy flags. But I
do notice you're missing one pixel from your copy of one of my wavys, which
makes it look a little bit lower. I hope the drawings in the flags.doc file
will help you understand where I'm coming from on the wavy and concave
flags. Unfortunately, in this conception, the concaves do not lend
themselves to the addition of nubs, because they are already quite thick on
the ends.

> I propose 
> using instead the concave style of flag that I described before, for 
> which I prepared a set of symbols on the 7th staff.  (Note that the 
> nubs don't get lost, even though they are quite small.)

I'm not averse to a slight recurve on the concaves, but I'm afraid I find
some of those in symbols2.bmp, so extreme in this regard, that they are
quite ambiguous in their direction. With a mental switch akin to the Necker
cube illusion, I can see them as either a recurved concave pointing upwards
or a kind of wavy pointing down. Apart from any nub, I don't think that
they should go more than one pixel back in the "wrong" direction. Those at
the extreme lower left of the page look ok.

I'm guessing that you need the huge recurve to convince yourself that
concave can represent larger commas than wavy?

I would have agreed that, if you want the set of 3 flag types that are
maximally distinct from one another, (to be used for the lowest primes) it
is probably {concave, straight, convex}. However in typing those curly
brackets above, I had the thought: Isn't it interesting that our character
set includes brackets that correspond to some of our flags (in like pairs
turned sideways). It has
(-- convex,
<-- straight, and
{-- wavy,
but _not_ concave. 

And of course we also have
[-- convex right angle, but my feeling is that it would be hard to make
those fit the style of the standard accidentals.

> I like your wavy flag, but I would propose waving it a little 

higher, 
> as I did in the symbols just to the right of yours (back on the 5th 
> staff); I seemed to be getting a better result with a thinner flag, 
> which would also serve to avoid confusion of the wavy with the 

convex 
> flag.

It seems to me that you have increased the possibility of confusion of wavy
with convex, by waving it higher.

> Perhaps the wavy flag would now be most appropriate for the 
> smallest intervals.

I'd need to know what they all mean re commas or see a complete set, in
order, for the first 12 degrees of 217-ET. I thought it made sense for
apotome-complements, that wavy should be its own complement, and convex and
concave should be complements, when they _have_ complements (which is only
on the right).

> I put a set of convex flag symbols on the 6th staff, which (like the 
> straight-flag symbols) combines features from both of our previous 
> efforts).

These all look fine to me, except I'd leave off the nubs for those with two
flags of the same type.

I don't have a clear preference yet for nubs versus change-of-width, for
indicating relative size while reducing lateral confusability. Maybe we can
perfect both, then present them and ask folks to vote on them.

On a single shaft I made sL 5 pixels (including the shaft) and sR 7 pixels.
But when I combine the two I make them both 6 pixels wide for a total
symbol width of 11. I did the same thing with xL 7 pixels (not shown) and
xR 5 pixels. I made wL 4 pixels and wR 5 pixels. wR represents a smaller
comma than sL, so it couldn't be more than 5. A 3 pixel wavy wouldn't work,
but when they are both on the same shaft I would make them both 4 pixels.
Both vL and vR would be 4 pixels because they represent smaller commas than
the wavys, and I figure you just can't go narrower than 4 pixels. I want to
make all the flags as narrow as is reasonable so that the double flag
symbols are not getting too wide and out-of-character with the standard
symbols.

> At the top right (under altitude considerations) I put my latest 
> version of the symbols in combination with conventional sharps and 
> flats, with single-symbol equivalents included (above the staff).

As I said before, it would be good to show some down-pointing ones with
conventional flats.

> Let me know what you think.

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


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Message: 4689

Date: Fri, 19 Apr 2002 07:45:47

Subject: Re: 41 ET 11-tone diatonic

From: genewardsmith

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

> what do you mean, how would you work it? and btw, didn't you produce 
> a 9-tone scale that is a perfect example of this, having two 
> chromatic unison vectors?

You tell me, starting with the scale and passing on to the unison vectors.


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Message: 4690

Date: Fri, 19 Apr 2002 10:20:23

Subject: Re: 41 ET 11-tone diatonic

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

Yahoo wrote:
>------------------------ Yahoo! Groups Sponsor ---------------------~-->
>Kwick Pick opens locked car doors,
>front doors, drawers, briefcases,
>padlocks, and more. On sale now!

Kwick Pick? They're carrying advertisements for the 
burglar's guild now?

I'm in favour of moving to the Columbia list too.

Manuel


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Message: 4691

Date: Fri, 19 Apr 2002 07:50:41

Subject: Re: one from the archives

From: genewardsmith

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

>  >The adjacency matrix of a graph is a square matrix labeled by
>  >verticies; it has a "1" if they are connected, and a "0" if not
>  >(counting the vertex to itself as a "0".) The characteristic
>  >polynomial of this is as above, and is a graph invariant. The
>  >n-2 term gives the number of edges, and the n-3 term twice the
>  >number of triads.
> 
> That's crazy.

Why?

> What happened to the n-1 term, above?

Since the diagonal terms are all 0, the trace is 0 and so the trace term is 0.

> Does
it show tetrads?

It doesn't show anything. To show tetrads, we would need to count
principal minors which were all 1s except along the diagonal, which
would be 0. I don't know if that can be done using the coefficients of
the characteristic polynomial.


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Message: 4692

Date: Fri, 19 Apr 2002 11:40 +0

Subject: Re: My Approach Generalized Diatonicity

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <4.2.2.20020418115051.01e58740@xxxxx.xxx>
Me:
>  >That looks like the heart of the issue.  I find it much easier to tell
>  >two dissonances apart than a dissonance from a consonance.

Carl:
> You do?  Have you got that backward?  Anyway, that isn't the question.
> The question is: if I randomly play you either 6:5 or 5:4, harmonically,
> and ask you to identify them, would you perform better than if I had
> used 11:9 and 9:7?

Um, yes, wrong way round.  I don't have very good relative pitch at all.  
The real test would be 9:7 and 11:8, because they're close enough to be 
confused.  The easiest way of telling intervals apart is by how consonant 
they are, so I should easily be able to differentiate 5:4 from either 11:9 
or 9:7 if they're tuned well enough.

I'm sure I can hear the in-tune-ness of 11:8 as well, in the right 
circumstances.  So an interval class containing both 9- and 11-limit 
"consonances" should have more audible variety than one with only 5-limit 
consonances.

>  >With my schismic keyboard setup I found it remarkably difficult to
>  >distinguish 5:4 and 6:5 what with them both being well tuned
>  >consonances and not far apart.
> 
> Really?  Maybe I just find consonances easier to recognize because
> I've trained myself to do it... maybe it isn't innate.

Really, I think it's your training at work here.  Fifths and thirds are 
easy to tell a part, of course, because of the change in consonance.  
Thirds can also be distinguished in 12- or 31-equal because major thirds 
are better tuned.  I was surprised at how the nature of a minor third 
changes in 19-equal, and how similar major and minor triads (to be 
specific, rather than thirds) become when the two thirds are well tuned, 
and slightly closer in pitch than they would be in JI.

Two dissonant chords with no particular rationalisation may be hard to 
tell apart.  But there are plenty of dissonances, like 11-limit intervals, 
which have their own quality distinct from other dissonances.  And miracle 
tuning is specifically optimized for them.

I'm certainly not going to reject scales because they don't have enough 
unambiguous consonances.  We should be collecting scales that have most 
reasonable properties, and see how well they work as diatonics.  That's 
going to take a long time, because it means writing fairly complex music 
for each.


                 Graham


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Message: 4693

Date: Fri, 19 Apr 2002 11:40 +0

Subject: Re: scala stability logic

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a9n8mm+abjt@xxxxxxx.xxx>
emotionaljourney22 wrote:

> i, for one, am opposed to defining an "extra-sensory" chromatic, as i 
> complained before in reference to balzano and clough. i'm glad to 
> hear mark gould is with me on this -- sorry if i implied otherwise, 
> mark.

I'm not suggesting anything "extra-sensory" either.  If your target 
listener senses the 612 cent interval as belonging to a distinct interval 
class, rather than being a tuning of a "tritone" of around 600 cents, 
we'll consider the Pythagorean diatonic independent of the 12 note 
chromatic.  If they can hear a 612 cent interval as being wider than a 588 
cent interval, when they're at unrelated pitches, we can even call the 
Pythagorean diatonic improper.


                           Graham


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Message: 4694

Date: Fri, 19 Apr 2002 11:40 +0

Subject: Re: scala stability logic

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <4.2.2.20020418114756.01ece7b0@xxxxx.xxx>
Carl Lumma wrote:

>  >For Rothenberg stability and efficiency to work properly, you need to 
>  >define each diatonic on a chromatic.
> 
> ?  What terminology is this?

Mark Gould defines diatonics, pentatonics and chromatics.  I'm removing 
the distinction between diatonics and pentatonics.  Treating the chromatic 
as an equal temperament, even if it's tuned differently, is needed for 
Rothenberg's conclusions about the ambiguity of the tritone to be valid in 
meantone.

                   Graham


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Message: 4695

Date: Fri, 19 Apr 2002 11:07:02

Subject: Re: one from the archives

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
> > 
> >edge connectivity = 3
> >characteristic polynomial =
> >x^12-29*x^10-44*x^9+192*x^8+500*x^7-32*x^6-1076*x^5-968*
> >x^4-8*x^3+304*x^2+96*x

Let's compare this to some other possibilities:

[1, 21/20, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

This is a sentimental favorite--my first scale. 

x^12-27*x^10-38*x^9+168*x^8+366*x^7-206*x^6-950*x^5-474*x^4+400*x^3+437*x^2+130*x+12

connectivity = 2

Not quite up to your numbers, but a good extension of JI diatonic.


[1, 15/14, 9/8, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 7/4, 15/8]
x^12-26*x^10-36*x^9+156*x^8+334*x^7-176*x^6-768*x^5-266*x^4+370*x^3+190*x^2-36*x-15
2

Another verion of the above.

[1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15]

This is Euclidean reduced, vertex centered.

x^12-27*x^10-38*x^9+168*x^8+368*x^7-172*x^6-792*x^5-232*x^4+368*x^3+96*x^2-64*x
2


[1, 15/14, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

This is Euclidean reduced, tetrad (shallow hole) centered.

x^12-28*x^10-42*x^9+175*x^8+430*x^7-70*x^6-812*x^5-396*x^4+374*x^3+302*x^2+
32*x-3
2


[1, 15/14, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 7/4, 15/8]

This is Euclidean reduced, hexany (deep hole) centered. Finally something to beat your numbers; I think your scale is actually
quite good. This has 30 intervals and 25 triads.

x^12-30*x^10-50*x^9+189*x^8+506*x^7-103*x^6-1118*x^5-487*x^4+772*x^3+508*x^2-128*x-96
3

[1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

A modified version of the shallow hole reduction.

x^12-30*x^10-48*x^9+193*x^8+498*x^7-96*x^6-1096*x^5-735*x^4+238*x^3+278*x^2-8*x-27
2

[1, 21/20, 35/32, 7/6, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]

Another modified shallow hole reduction.

x^12-29*x^10-44*x^9+183*x^8+432*x^7-148*x^6-924*x^5-444*x^4+276*x^3+177*x^2-22*x-15
2


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Message: 4696

Date: Fri, 19 Apr 2002 12:07:41

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

See my latest attempt at the variable-width nub-free style. I've 
included a symbol for every one of our prime commas up to 47, and 
shown the purely sagittal double-symbol notation between sharp and 
double-sharp.

Yahoo groups: /tuning-math/files/Dave/SymbolsDK.bmp *


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Message: 4697

Date: Sat, 20 Apr 2002 01:28:16

Subject: Re: A common notation for JI and ETs

From: dkeenanuqnetau

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> >          |  Left    Right
> > ---------+---------------
> > Convex   |  29        7
> > Straight |   5     (11-5)
> > Wavy     |  17       23
> > Concave  |  19     (17'-17)
> 
> This looks very workable, and I am about 99 percent sold on it.  
> (Just give me some more time.)

Sure. We want to be sure we've explored every option thoroughly.

> In your table of symbols:
> 
> Symbol  Left      Right
> for     flags     flags
> ------------------------------
> 
> 23' =      17  +  (11-5)
> 
> 31' =       5  +  (17'-17) + 7
>    or       5  +  23 + 23
> 
> 37  =  29 + 17
> 
> options can be added for the following:
> 
> 23' =      17  +  (11-5)
>    or      29  +  (17'-17)
> 
> 31' =       5  +  (17'-17) + 7
>    or       5  +  23 + 23
>    or       7  +  7
> 
> 37  =  29 + 17
>    or   5 + 5
> 
> These 5+5 option for the 37-comma uses a much smaller schisma 
> (6553600:6554439, ~0.222 cents) than what you have.  But the problem 
> with these three options that I have given is that none of the 
> schismas vanish in 1600-ET.
> 
> Should we rethink the question of whether it is really necessary for 
> these schismas to vanish in 1600-ET, because I don't see any good 
> reason.
It doesn't have to be 1600-ET. It doesn't even have to be an ET. It 
might be a linear or planar or whatever temperament.

But I feel it is highly desirable to know that the schismas we are 
using do not somewhere add up to something considerably more than 0.5 
cents. i.e. I want to know what maximum error (over all the intervals 
in our highest odd limit) is implied by our choice of notational 
schismas. If we don't know what temperament it is based on, we may 
happen to have two near 0.5 cent schismas that "pull in opposite 
directions".

> While it is nice to have everything come out exact using 
> 1600 as a frame of reference, do you think anyone is actually going 
> to be able to use it in a performance to produce pitches?

Not at all. But it is significant that (in the simplest example) our 
single symbols for 13 and 35 are identical. We are presenting the 
composer with a choice. Either use a pair of separate symbols for 35, 
or accept that the performer will read it as a 13 diesis (or the 
corresponding number of cents) and introduce a certain error. We're 
trying to keep that error below 0.5 cents, although I think we've 
already got one of 0.6 c. 

> The 
> increments are much smaller than 1 cent, and the pitches can't be 
> related easily to 12-ET, as Johnny Reinhard is doing.  (i.e., not a 
> subdivision, as is 1200-ET), )

Although it may be convenient that its divisions are exactly 3/4 of a 
cent.

>  So if we're trying to accommodate 
him 
> with this notation, all that's really necessary is to keep the 
> schismas small and provide the number of cents somewhere on the 
> score, at least in a table with the symbols.

That's right. But "keep the schismas small" means also the 
effective _combination_ schismas for all the intervals between pairs 
of odd numbers, not just the odd numbers themselves. Actually, I'm 
not sure if I know what I'm talking about here. At least with 1600-ET 
we knew where we stood. The thing may be to find another ET above 
1000 or some other system that accomodates all the schismas we want.

Gene was looking at these for us and he found 1600 was the best 31 
limit unique ET of all those less than or equal to it in size, but  
hasn't gone to higher primes yet. He may have lost interest.

There are 11 odd primes up to 37, if we have 11 independent schismas, 
and express them as prime-exponent vectors and take the determinant of 
the resulting square matrix, I understand we'll get the cardinality of 
the corresponding ET. However I'm pretty sure we don't have 11 
independent schismas and I get a little hazy here about how to find 
generator mappings. I'm hoping Graham Breed or Gene Smith can help us 
here. I think we just need to give the vector for every schisma we're 
interested in. Then ask them to tell us the maximum error implied by 
various sets of these.


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Message: 4698

Date: Sat, 20 Apr 2002 09:07:46

Subject: More 12-tone JI scale comparisons

From: genewardsmith

I put together the ones Carl mentioned with the ones I cooked up, and
compared them using some of my measures and some from Scala. Playing
about with them, I got the impression that high lumma stability,
propriety, and CS (which seemed to go together) were good things for
scales to have, so that the ones with the most harmony did not
necessarily sound the best melodically. I'm still trying to figure out
all the arcane measures Carl and Graham are tossing at each other;
maybe they could explain using these scales as examples.

I also put in "Wille's k value" to get a start on some of the measures
I don't understand; this seemed to be a good place to start since it
makes no sense to me at all. Does anyone have a clue?

Class
[1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 25/16, 42/25, 7/4, 15/8]
triads 26 intervals 31 connectivity 3
improper CS lumma .043920 k 437

Stelhex
[1, 21/20, 7/6, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 42/25, 7/4, 9/5]
triads 26 intervals 30 connectivity 3
improper lumma .081284 k 787

Euchex
[1, 15/14, 8/7, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 12/7, 7/4, 15/8]
triads 25 intervals 30 connectivity 3
strictly proper CS lumma .253235 k 787

Prism
[1, 16/15, 28/25, 7/6, 5/4, 4/3, 7/5, 112/75, 8/5, 5/3, 7/4, 28/15]
triads 24 intervals 30 connectivity 3
strictly proper CS lumma .440966 k 262

Tet-a
[1, 21/20, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]
triads 24 intervals 30 connectivity 2
improper CS lumma .321977 k 262

Tet-b
[1, 21/20, 35/32, 7/6, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]
triads 22 intervals 29 connectivity 2
improper CS lumma .355766 k 262

Lumma
[1, 36/35, 8/7, 6/5, 5/4, 48/35, 10/7, 3/2, 5/3, 12/7, 9/5, 40/21]
triads 22 intervals 29 connectivity 3
improper lumma .081284 k 262

Euctetrad
[1, 15/14, 35/32, 6/5, 5/4, 21/16, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]
triads 21 intervals 28 connectivity 2
improper CS lumma .166834 k 1837

Gene
[1, 21/20, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8]
triads 19 intervals 27 connectivity 2
strictly proper CS lumma .437710 k 112

Eucvert
[1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15]
triads 19 intervals 27 connectivity 2
strictly proper CS lumma .262832 k 367

Lester
[1, 21/20, 9/8, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 15/8]
triads 18 intervals 26 connectivity 2
strictly proper CS lumma .490032 k 337

Gene-a
[1, 15/14, 9/8, 6/5, 5/4, 4/3, 10/7, 3/2, 8/5, 5/3, 7/4, 15/8]
triads 18 intervals 26 connectivity 2
strictly proper CS lumma .333977 k 787


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Message: 4699

Date: Sat, 20 Apr 2002 03:18:50

Subject: Re: A common notation for JI and ETs

From: David C Keenan

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> The symbols get fatter as the alterations become larger, which is 
> only logical.  

Sure.

> And I even put the fattest ones on a diet, and now 
> none of them is wider than its height.  So what is the problem?

I'm ignoring the tails. With the standard symbols the _body_ of the symbol
is never wider than it is high. But hey, I can live with it.

> I believe that shortening the middle line makes it more difficult to 
> see it, thereby making it *more* difficult to distinguish three from 
> two.  This is particularly true when the symbol modifies a note on a 
> line and the middle line terminates at a staff line (so you see only 
> two lines sticking out).

Good point. How about making the middle one only 2 pixels shorter than the
outer ones. That will solve the latter problem.

> In fact, after looking at this again, I 
> think I would be in favor of shorting all of the symbols from 17 to 
> 16 pixels so that no vertical line would terminate at a staff line. 

Then I think the sagittals will look odd with sharps too, not just flats.
And it will worsen the aspect-ratio problem. I believe flats have such long
tails, precisely to give them a similar aspect ratio to sharps and naturals.
 
> (This would also keep symbols modifying notes a fifth apart from 
> colliding.  But you made a comment below regarding how the length of 
> a new symbol looks when placed beside a conventional flat, so I need 
> to evaluate this further.)

I don't see a problem with them colliding. Have you found examples of flats
doing that yet? I have.

> In your latest figures I notice that you are making a noticeable 
> difference in width between the left and right flags, which is very 
> effective with the straight flags.

They were like that from the start. For straight and concave I have 5
pixels wide versus 7 and for wavy I have 4 versus 5, but concave are both 4
pixels.

> Perhaps this will be the best 

way 
> to distinguish left from right.  A very small nub could still be 

used 
> at the end of the larger of each pair of curved flags as a stylistic 
> embellishment.

I agree it would help with the lateral confusability. But from a purely
aesthetic point of view, I think I'd prefer not.

> With your concave flags, half of the length of the curve is 
> coincident with the vertical arrow shaft, which makes it difficult 

to 
> tell that this was intended to be a concave curve.  The portion of 
> the curve with least slope is much thicker, and taken together with 
> the overall lateral narrowness of the flag, it comes out looking 

more 
> like a blob than a curved line.

You're absolutely right. The concaves just don't work at only 4 pixels
wide. It's interesting how the knowledge of what it's supposed to look like
can blind one to alternative interpretations. That's why it's so good to
cooperate the way we are.

Trouble is, I just can't accept a 19 comma flag that's wider than 4 pixels
(including shaft) since it represents a barely perceptiple comma of about 3
cents. I'd really prefer to make it only 3 pixels, but that seems too low res.

How about we forget abour concave and _make_ it a (circular or semicircular
or triangular) blob? And move it up the shaft as you suggest, to center the
blob on the notehead. Too bad about the convex/concave complementarity.

> As with the concave flag, the top part of the curve is coincident 
> with the arrow shaft, so it (i.e., the version on which I was 
> commenting) tends to look like a smaller and lower convex flag that 
> is modifying a note one staff position lower.  Your latest version 
> (19 April) of the wavy flag is identical to what I now have, except 
> that I have made the (vertical) extremity of the flag one pixel 
> shorter.  Why shorter?  I think that the concave and wavy flags 
> should be smaller than the convex and straight flags -- both in 
> length and thickness.

Yes. The wavy doesn't work at 4 pixels wide, and apparently you find it
only barely works at 5 pixels. I like your idea of making both concave and
wavy vertically shorter than the others too. And I agree that the vertical
position should be a sort of compromise between centering the flag
_including_ the part coincident with the shaft, and centering it
_excluding_ the part coincident with the shaft.

> I would further like to modify what I have for these by using 
> different lateral widths (left vs. right), so I still have some work 
> to do on the symbols before putting a new file out there.

I look forward to it.

> Okay, I'll try this and let you know what I think.  (But I always 
> thought that the tails of conventional flats were too long anyway.)

I believe flats have such long tails, precisely to give them a similar
aspect ratio to sharps and naturals.

> Slowly, but surely, we are making progress.

Yes indeed. :-)
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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