Tuning-Math Digests messages 4275 - 4299

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 5

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4250 - 4275 -



top of page bottom of page down


Message: 4275

Date: Thu, 14 Mar 2002 19:15:43

Subject: Ferguson WHO??? (Re: [tuning] Re: "wolf" pack, rat pack [Isacoff]))

From: genewardsmith

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

> I searched tuning AND tuning math and I didn't find either Ferguson or
> Forcade.
 What exactly is this algorithm?  

It's an integer relation finding algorithm. There is even an integer
relation finding web site now where you can email and get the relation
(using PSLQ, I think.) Here's mathworld on it:

Ferguson-Forcade Algorithm -- from MathWorld *

I've not found these useful for searches, but maybe I should consider
the matter; however brute force seems to suffice.


top of page bottom of page up down


Message: 4277

Date: Thu, 14 Mar 2002 20:12:51

Subject: Re: Ferguson found

From: genewardsmith

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

> Thanks for the link to Harmonic Entropy, that was from a couple weeks before
> I joined the online tuning world.  I'm trying to find my way around the
> weblinks and so far other than this 26 page .pdf file from NASA that's
> taking forever to load I haven't been able to get hold of a digestible
> definition as yet.

That's got to be Bailey. :)

> Is anyone here familiar enough with the PLSQ to sum it up in a few dozen
> words and/or ascii? 

It's an integer relation algorithm.


top of page bottom of page up down


Message: 4279

Date: Thu, 14 Mar 2002 21:39:35

Subject: Starling temperament mapping

From: David C Keenan

Here's an opportunity to describe a planar temperament mapping in the
canonical form I've just proposed in 'Standardising temperament mappings'.

Starling temperament
discovered and named by Herman Miller, 1999

Target
[lg(2) lg(3) lg(5) lg(7)]
Generators (cents)
[1200 498.0 107.1]
Mapping
[1  2  2  1]
[0 -1  1  5]
[0  0 -1 -3]

Which means for example that lg(7) ~= 1*1200 + 5*498.0 + -3*107.1

Can someone please check that that agrees with the fact that the 125:126
(2^1 * 3^2 * 5^-3 * 7^1) is distributed, and remind me how you do that?

Gene,

Can you please give your proposed canonical form if it differs from this,
and explain the conventions involved.

Regards,
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


top of page bottom of page up down


Message: 4280

Date: Thu, 14 Mar 2002 00:37:11

Subject: Periods and generators

From: genewardsmith

My previous example with a title like this hasn't shown up, and now I'm
trying to follow it up anyway. Nuts!

I was going to add to a previous example of 720 as a period for
meantone one of 16/15 as meantone. Assuming the posting does finally
show up, continuewith this point: I want 16/15 to be my period, so I
take 16/15^81/80 = h5. Now one solution to h5^g = 81/80 is g=h7, which
gives us [h5,h7] as our map. We now have a "period" of about 16/15,
inside of which is a "generator" of about 25/24. But a more sensible
way of looking at it is that 16/15 and 25/24 are both generators, and
we do 5-limit meantone by means of a two-dimensional generator
lattice.


top of page bottom of page up down


Message: 4281

Date: Thu, 14 Mar 2002 02:33:27

Subject: Re: Systematic naming of new temperaments (was: amt)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> > > How do you differentiate two different mappings with the same
> > > generator?
> > 
> > I just say 5-limit whatever versus 7-limit whatever, and so on.
> 
> that won't necessarily do the trick.
> 
> look at the fourth and fifth entries here:
> 
> 404 Not Found * Search for http://www.microtonal.co.uk/limit7.txt in Wayback Machine

You'll have to do better than that.

270.8 cent generator, [-2, -3, -1] mapping, 45 cent minimum MA error

The generator here must function as 7:8, 6:7 and 5:6. Look at the 
minimum error. As a 7-limit temperament it is garbage. But if forced 
to find a meaningful name for it I would call it "7:8 = 5:6" or 
"supermajor second minor thirds" or "diminished third minor thirds", 
after Gene's naming of "fourth thirds".

271.3 cent generator, [7, -3, 8] mapping, 4.5 cent minimum MA error

This is of course 7-limit subminor thirds or 7-limit orwell.

I must point out that Gene invented the "generator as fraction of 
consonant interval" nomenclature, with "quadrafourths". I'm just 
arguing that we should 
(a) extend its use, 
(b) refine the prefixes to more clearly indicate fractions as opposed 
to multiples, 
(c) use latin consistently (not mix it with greek), and 
(d) apply it always to the interval that is made up of the fewest 
generators (and doesn't contain any periods except as whole octaves).

Regardsing (c), I wouldn't mind if they were consistently greek except  
I feel the latin ones work better regarding criterion (b) (fractions 
versus multiples).

Regarding (d): In the case where there is more than one consonant 
interval which is made up of the minimum number of generators, then 
the largest and smallest in size should both be mentioned, as above. 
This was Gene's invention too, with fourth thirds.

I also think that when a temperament _does_ extend to being good at a 
higher limit, fewer generators map to a consonant interval at the 
higher limit, then it fine to use the higher limit name prefixed by 
the limit. So 5-limit Orwell can also be called 5-limit subminor 
thirds and we can forget about calling it tertiminorsixths.

Kleismic and parakleismic is a better example of the problem you were 
setting up for me. I approve of these names, and para (presumably 
meaning "near to") might generally be used for the badder of the two 
since para can also mean "improper". 

So the horrible 270.8 cent temperament above could also be para-orwell 
or para-subminor-thirds.

However I remember Gene using cata- meaning "down". Its opposite is 
ana- (up). I quite like these two, and they (or para) would remove the 
temptation to use semi to mean "almost", and leave it to mean "half", 
as in semitone or semicircle.

So the horrible 270.8 cent temperament above could also be cata-orwell 
or cata-subminor-thirds. And parakleismic could be catakleismic.

Gene, What was it that you used cata- for?


top of page bottom of page up down


Message: 4282

Date: Thu, 14 Mar 2002 03:17:59

Subject: Re: Systematic naming of new temperaments (was: amt)

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Gene, What was it that you used cata- for?

It had a generator in the 7 and 11 limit a whisker below the
kleismic 5-limit generator, but I really shouldn't have called it that, since it is a whisker *above* the h19^h15 = [6,5,3,-7,12,-6] 
7-limit generator, which I was taking to be the 7-limit kleismic.
So it's a misnomer. The other reason is that it sounds like
"cataclysmic"--ha ha. Get it? I don't either. :)


top of page bottom of page up down


Message: 4283

Date: Thu, 14 Mar 2002 04:30:07

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > what do you mean (who and what limits)?
> 
> I don't want to do another search, this time weighted, unless you 
>tell me what you want to look at.

i think dave needs to answer this question (or repeat his answer).


top of page bottom of page up down


Message: 4284

Date: Thu, 14 Mar 2002 04:37:46

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > 
> > > what do you mean (who and what limits)?
> > 
> > I don't want to do another search, this time weighted, unless you 
> >tell me what you want to look at.
> 
> i think dave needs to answer this question (or repeat his answer).

well, failing that, just do it with the same bounds as before, or 
still more inclusive bounds if you prefer, and with the applied 
badness measure given for each entry.

after that's done, we should strive for consensus and move on to the 
7-limit.


top of page bottom of page up down


Message: 4285

Date: Thu, 14 Mar 2002 06:37:14

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> > > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > 
> > > > what do you mean (who and what limits)?
> > > 
> > > I don't want to do another search, this time weighted, unless 
you 
> > >tell me what you want to look at.
> > 
> > i think dave needs to answer this question (or repeat his answer).

Yes I have already posted them, and they are also in my spreadsheet. 
But here they are again. 35 cents rms, 12 weighted-rms generators (or 
10 for the shorter list) and max 625 for Gene's badness (using 
weighted complexity).


top of page bottom of page up down


Message: 4286

Date: Thu, 14 Mar 2002 09:54:28

Subject: Re: Weighting complexity (was: 32 best 5-limit linear temperaments)

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes I have already posted them, and they are also in my spreadsheet. 
> But here they are again. 35 cents rms, 12 weighted-rms generators (or 
> 10 for the shorter list) and max 625 for Gene's badness (using 
> weighted complexity).

Here's some examples of weighted badness, using weighted complexity. Can you check your limits in terms of this?

81/80: 103.257
15625/15552: 187.010
250/243: 463.641
78732/78125: 525.371


top of page bottom of page up down


Message: 4287

Date: Thu, 14 Mar 2002 13:29:51

Subject: Re: some output from Graham's cgi

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

Graham,

When I enter these numbers in the linear temperament finder
13
28
12.0
25
10
0.5

the first one is:

13-limit
31/75, 495.6 cent generator
basis:
(1.0, 0.41300690788)
mapping by period and generator:
[(1, 0), (2, -1), (11, -21), (9, -15), (8, -11), (7, -8)]
mapping by steps:
[(46, 29), (73, 46), (107, 67), (129, 81), (159, 100), (170, 107)]
highest interval width: 21
complexity measure: 21  (29 for smallest MOS)
highest error: 0.009422  (11.306 cents)

But when I do an RMS approximation I'm able to get a lower
highest error of 10.868 cents (for 13/9), with a generator of 495.708356 
cents.
Did you exclude some consonant intervals?
With many consonant intervals to approximate, minimax makes more sense
than RMS I think.

(29 for smallest MOS)

This means 29 tones in the smallest MOS, doesn't it?

Manuel


top of page bottom of page up down


Message: 4288

Date: Thu, 14 Mar 2002 13:33:36

Subject: Re: Ekmelische Musik

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

Aaron,

Thanks, I wasn't aware that it referred to the harmonic
series.

>Anyone know what's going on over there nowadays?

I don't; they still give courses. In 1999 a CD titled 
"Ekmelische Musik" came out.

Manuel


top of page bottom of page up down


Message: 4289

Date: Thu, 14 Mar 2002 13:52 +0

Subject: Re: some output from Graham's cgi

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <OFC5246248.AC822B6F-ONC1256B7B.00535BCC@xxxxxx.xxxxxxxxx.xx>
Manuel wrote:

> But when I do an RMS approximation I'm able to get a lower
> highest error of 10.868 cents (for 13/9), with a generator of 
> 495.708356 cents.
> Did you exclude some consonant intervals?

No, I only exclude intervals that don't depend on the generator.  I'm not 
sure if that's right or not, but it doesn't matter here because the period 
is an octave.

> With many consonant intervals to approximate, minimax makes more sense
> than RMS I think.

Yes, but the RMS optimum is more efficient to calculate.

> (29 for smallest MOS)
> 
> This means 29 tones in the smallest MOS, doesn't it?

Yes, the smallest one that contains a complete 13-limit otonality.


                       Graham


top of page bottom of page up down


Message: 4290

Date: Thu, 14 Mar 2002 15:24:56

Subject: Re: some output from Graham's cgi

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

>No, I only exclude intervals that don't depend on the generator.

O! So you didn't include 7/5 in the given example? Wouldn't
you, generally speaking, get better results if you did?

>I'm not sure if that's right or not, but it doesn't matter here 
>because the period is an octave.

I don't understand why it doesn't matter?

Manuel


top of page bottom of page up down


Message: 4291

Date: Thu, 14 Mar 2002 14:59 +0

Subject: Re: some output from Graham's cgi

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <OFBF3E9787.74F450EC-ONC1256B7C.004EAB50@xxxxxx.xxxxxxxxx.xx>
Manuel wrote:

> O! So you didn't include 7/5 in the given example? Wouldn't
> you, generally speaking, get better results if you did?

What?  No, 7:5 is 20 generator steps.  It would be excluded in Twintone, 
where it's always a half-octave.  On thinking about it more, this doesn't 
affect the optimum because it's only removing a constant from the 
RMS-by-generator-size function that's being minimized.  So I was right to 
exclude such intervals.

I don't know why we're getting different results.

> >I'm not sure if that's right or not, but it doesn't matter here 
> >because the period is an octave.
> 
> I don't understand why it doesn't matter?

An interval would have to be equal to a unison to be excluded, which is 
much less likely than being equal to some division of an octave.


                   Graham


top of page bottom of page up down


Message: 4292

Date: Thu, 14 Mar 2002 17:43:16

Subject: Re: some output from Graham's cgi

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

Graham wrote:
>I don't know why we're getting different results.

Found the problem, I was using a wrong mapping
somewhere. I confirmed your result using all 
intervals.

Anyway, the minimax solution is 495.662963 cents.
Highest error 10.595435 cents.

Manuel


top of page bottom of page up down


Message: 4294

Date: Fri, 15 Mar 2002 16:01:09

Subject: Re: Standardising temperament mappings

From: dkeenanuqnetau

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a6t29j+cdq0@e...>
> dkeenanuqnetau wrote:
> 
> > I don't understand what you mean by "more robust". Whatever 
> > conventions we settle on to give a unique form will allow an 
> > octave-based temperament to be identified by the 
mapping-by-generators 
> > and the number of periods to an octave. Why would my proposal of 
> > generator less than half period, fail to do this.
> 
> The generator might get larger, in which case it would be 
incorrectly 
> described.  If you set the first non-zero element of the generator 
mapping 
> to be positive, the mapping is uniquely described regardless of the 
size 
> of the generator.

You seem to be talking about what your program is doing internally. 
I don't care about that. I'm talking about what we present to people, 
irrespective of how it is generated. The important thing to me, is 
what will be most immediately meaningful or useful to musicians or 
composers. If your optimiser comes out with a generator that is 
between 0.5 and 1 period you can always subtract it from the period, 
or if it is between 1 and 1.5 periods you can subtract a period from 
it etc., and adjust the matrix accordingly.

But the questions is, "Is this the most meaningful generator?". It 
could be argued for example in Pajara/Paultone that a generator which 
is a fourth or fifth may be more meaningful than the one of 108 cents. 
But it isn't clear to me how you would generalise that to cases where 
there isn't a fourth or fifth among the possible generator values. It 
just looks too messy to me.

However Gene seems like he might be favouring a fourth when one exists 
in such cases. What rule are you using there Gene?

> The one for generating linear temperaments from equal temperaments 
does.  
> But the one for generating linear temperaments from unison vectors 
> doesn't.  Some generators are a lot bigger than the period because 
they 
> become larger on optimization.  So I'll have to change that program 
to 
> adjust the mapping after optimization.

OK.

> > I assume you understood I meant "smallest with GCD(row) = 1", so 
no 
> > degeneracy.
> 
> Um what if there's contorsion?

I'm still confused about that.
 
... 
>So I won't be changing the library's 
internal 
> format, but can change the display the CGI uses.

Sure.

But don't bother changing anything yet. One point is to have the same 
format from both you and Gene, and Gene is still thinking about it.


top of page bottom of page up down


Message: 4295

Date: Fri, 15 Mar 2002 16:23:46

Subject: Re: Wedge product definitions from mathworld

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here is exterior algebra:
> 
> http://mathworld.wolfram.com/exterioralgebra.html *
> 
> Here is wedge product:
> 
> http://mathworld.wolfram.com/wedgeproduct.html *
> 
> This could and should be made a lot simpler for musical purposes.

Gene,

If you have attempted that somewhere, could you please point me to it.

The mathworld definition makes no sense to me whatsoever, since it is 
recursive and I can't find where the recursion bottoms out. Can you 
please give a definition that only applies to matrices or vectors 
(whatever makes sense for temperaments) and not general k-forms, and 
that eventually bottoms out into ordinary scalar multiplication *?

Does it take two vectors and give another vector or what?


top of page bottom of page up down


Message: 4296

Date: Fri, 15 Mar 2002 16:42 +0

Subject: Re: Wedge product definitions from mathworld

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a6t76i+vkbe@xxxxxxx.xxx>
dkeenanuqnetau wrote:

> The mathworld definition makes no sense to me whatsoever, since it is 
> recursive and I can't find where the recursion bottoms out. Can you 
> please give a definition that only applies to matrices or vectors 
> (whatever makes sense for temperaments) and not general k-forms, and 
> that eventually bottoms out into ordinary scalar multiplication *?

My Python source code includes an implementation of wedge products.  
Whether that's easier to understand than the mathworld link I don't know.  
Their description looks clear enough, but not the source code.  For mine, 
I think you need the __setitem__ and __getitem__ methods of Wedgable, as 
well as the obvious __mul__ which does the product.

> Does it take two vectors and give another vector or what?

It can take vectors, implemented as a list of real (or maybe even complex, 
but scalar) coefficients of a set of unison vectors.  The wedge product of 
a pair of such vectors will be a list of coefficients of a combination of 
unison vectors.  So if you start with 3-D vectors defined on i, j and k 
the wedge product will be defined on ij, ik and jk.  My Wedgable class is 
to implement this generalisation of vectors, and the product of two 
wedgables is always another wedgable.


                    Graham


top of page bottom of page up down


Message: 4297

Date: Fri, 15 Mar 2002 03:13:23

Subject: Re: Standardising temperament mappings

From: genewardsmith

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:

> Gene and Graham are using different ways of presenting both the basis
> vector
for a temperament and its mapping matrix.

I wish I had checked what Graham did before coding, but I am inclined
to think I want a different standard mapping anyway, since I have what
seems like a good, general method convention suitable for planar and
higher temperaments also.

> I see the following issues which need to be addressed in the
following order.
> 1. Standardising the basis vector

What's the basis vector?


top of page bottom of page up down


Message: 4298

Date: Fri, 15 Mar 2002 17:05 +0

Subject: Re: Wedge product definitions from mathworld

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <memo.707336@xxx.xxxxxxxxx.xx.xx>
I wrote:

> My Python source code includes an implementation of wedge products.  
> Whether that's easier to understand than the mathworld link I don't 
> know.  Their description looks clear enough, but not the source code.  
> For mine, I think you need the __setitem__ and __getitem__ methods of 
> Wedgable, as well as the obvious __mul__ which does the product.

Oops, sorry.  __mul__ is for multiplication by a scalar.  You want the 
standalone wedgeProduct function.


top of page bottom of page up down


Message: 4299

Date: Fri, 15 Mar 2002 03:26:30

Subject: Re: Standardising temperament mappings

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote:
> 
> > Gene and Graham are using different ways of presenting both the 
basis
> > vector for a temperament and its mapping matrix.
> 
> I wish I had checked what Graham did before coding, but I am 
inclined to think I want a different standard mapping anyway, since I 
have what seems like a good, general method convention suitable for 
planar and higher temperaments also.
> 
> > I see the following issues which need to be addressed in the 
following order.
> > 1. Standardising the basis vector
> 
> What's the basis vector?

The list of generators. Sorry.


top of page bottom of page up

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4250 - 4275 -

top of page