Tuning-Math Digests messages 6550 - 6574

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Message: 6550

Date: Thu, 20 Feb 2003 16:45:06

Subject: Re: lattice diagram "levels" of complexity

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

Carl and Paul wrote:
>how about a much simpler example, with much fewer factors? i'm
>looking to understand what you mean by "extending the tones out to a
>EF Genus in all directions"

The only way I know to extend a CPS to a EF genus is to add the
remaining combination counts to it. So if you have 2)5 you need to
add 1)5, 3)5, 4)5 and 5)5.

>> What you never showed is how the definition of stellation on
>> George Hart's site requires the 92-tone, and not the 80-tone
>> structure.
>>
>> -Carl

>i think manuel has got that down.

So if I'm not mistaken the 80-tone structure is the
result of the first-order stellation and the 92-tone one of the
complete stellation.
The former is called "stellated" in Scala and Wilson's stellation
is called "superstellated". It's a repeated stellation so that
there are no more unstellated chords anymore.

Manuel


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Message: 6551

Date: Thu, 20 Feb 2003 02:39:50

Subject: Re: Reduced generators and special commas

From: Carl Lumma

>>*The* special comma -- couldn't there be more than one?  In
>>meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc.  Right?
>
>I defined it so that there is only one.

What's Tenney height?  just i?  Did Tenney propose this?

>12 isn't actually mapped to a 7-limit consonance anyway. 

Oh, bad example. :(

-Carl


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Message: 6552

Date: Thu, 20 Feb 2003 16:50:59

Subject: Re: scala show data

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

Carl wrote:

>By the way, the installer puts a desktop shortcut
>even if I tell it not to.  ;)

Oh, I'll look after that.

>I think it's cool that you are still maintaining it.

Thanks, there actually is one blind user (not on the
lists) who can't use the GUI-version.

>I'm really impressed with the speed of 2.05.  Though it
>is 9 megs of widgets, and one can't copy and paste to
>the Windows clipboard as was possible with the console
>version. . .

Not completely true because, weird enough, you can do it
once. Right click in the main window, select "Enable text
editing", then select some text, press ctrl-C and that
can be pasted then. Any subsequent copy doesn't work unless
you restart the program.

Manuel


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Message: 6559

Date: Thu, 20 Feb 2003 01:34:23

Subject: Re: Reduced generators and special commas

From: Carl Lumma

>For a p-limit linear temperament with octave period, we may call the
>p-limit rational number q, 1 < q < sqrt(2), the reduced generator if
>it defines a generator when approximated by the temperament, and if 
>it has minimal Tenney height given this condition. We may call the 
>comma c, c>1 the "special comma" for this temperament if it is the 
>comma defined by taking the p-limit consonance q^i for i>1 of the 
>temperament, and equating it to the corresponding JI version of this 
>consonace. In other words, it is C/q^i or q^i/C, i>1, where C is a 
>ratio of odd numbers less than or equal to p.

*The* special comma -- couldn't there be more than one?  In meantone,
i=4 gives 81:80, i=12 the Pythagorean comma, etc.  Right?

-Carl


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Message: 6566

Date: Mon, 24 Feb 2003 07:46:05

Subject: Re: Algebraic functions for 5-limit temperaments

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> If b is the "brat" as defined on the tuning group, then the fifth and 
> the third, for any linear temperament, will be algebraic functions of 
> b (and vice versa.) If we choose a b value, and specialize one of the 
> polynomials below, then one of its real roots will be the fifth or 
> third we seek; similarly, inserting a value for a fifth or third 
> allows us to compute b. 

I prefer to use numerical methods since a spreadsheet is all you need
for them. 

e.g. for meantone just set up a column with decimal fractions from
1/11 (.091) to 1/3 (0.333) in increments of 0.001, representing the
fractions of a comma by which the 5th is tempered. Call this x. Then
in the next two columns calculate the corresponding tempering of the
major and minor thirds, y=4x-1 and z=1-3x. Then from those calculate
the three beat ratios in the major triad 3/2*z/y, 5/3*y/x, 5/2*z/x and
the three in the minor triad, z/y, 2*y/x, 2*z/x. Then scan down the
columns until you find the beat ratios you are looking for and note
the corresponding value in the comma fraction column (x).


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Message: 6570

Date: Tue, 25 Feb 2003 00:52:48

Subject: Re: Some well-behaved brats for meantone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" 
> <gwsmith@s...> wrote:
> 
> > I checked integer brat ratios, looking for values where all three 
> > brat ratios were integers or their inverses, and did the same for 
> > inverse integer brats. The result was this list:
> > 
> > [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4],
> > [9, -3, -3], [1/4, 1/2, 1/2]
> 
> If we look for everything with either infinity or a Tenney height of 
> ten or less for all three ratios, we add to the above
> 
> [2/3, 1/3, 2], [-1/6, 2/3, -1/4], [0, 3/5, 0], 
> [infinity, infinity, -5/2], [3/2, 0, infinity].
> 
> The b=2/3 system is a "flattone" system, in the vicinity of the 26 et;
> the two "infinity" systems are of course 1/4-comma meantone and JI.

You mean Pythagorean. JI is not a linear temperament. I made this same
mistake myself recently. Of course you (and I, when I made the
mistake) meant that the fifths are just; so-called "3-limit JI"; but
this is an oxymoron to most people, and in any case we are talking
5-limit here.

Yes. I can see the same results in my spreadsheet. However, where you
find something to be very close to say 5/17-comma, I find it to be
exactly so (to the limit of accuracy of IEEE floating-point numbers).

This is because (and this answers a question of Paul Erlich's on the
tuning list in response to a 4 year old post of mine), like Bob
Wendell, I am using the approximation that frequency deviation is
proportional to comma fraction deviation (i.e. log frequency 
deviation). This is an extremely good approximation in the range from
0 to about 20 cents. The few thousandths of a cent error that result
in the size of the generator are of no interest to anyone but a
mathematician, and it means that one can deal with linear equations
rather than polynomials.

And of course, just as you do with the polynomials, you could solve
these approximate linear equations algebraicly instead of numerically
to get the rational comma fraction solutions that are familiar to people.


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Message: 6573

Date: Wed, 26 Feb 2003 21:04:34

Subject: Re: A Property of MOS/DE Scales

From: Carl Lumma

Kalle wrote...
>I am aware of these more sophisticated (and also more elegant)
>tools and possibilities. So if anyone is interested please read
>the thread in tuning list and continue on tuning-math. 

Anybody else interested in reviving this thread?

Relevant messages...

Yahoo groups: /tuning/message/41371 *
Yahoo groups: /tuning/message/41383 *

Yahoo groups: /tuning-math/message/5132 *
Yahoo groups: /tuning-math/message/5136 *

-Carl


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