Tuning-Math Digests messages 10376 - 10400

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Message: 10376

Date: Wed, 25 Feb 2004 19:40:59

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> >That's not correct. For one thing, it's not equal!
> 
> Yes, sorry, I forgot again how to use my own program!
> The code is correct, but I gave it the wrong parameters.
> It should have been (1200.6171 1900.9770 2801.4398).
> Step is 100.051421.
> 
> Manuel

Incorrect, I'm afraid.


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Message: 10377

Date: Wed, 25 Feb 2004 19:51:51

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> > 
> > 31-equal TOP is this, identical for 5-limit and 7-limit:
> > (1201.6366 1899.3611 2790.8979 3372.3350)
> 
> I get 1201.4675

Yes, to two places I get [1201.47 1899.09 2790.51 3371.86].


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Message: 10378

Date: Wed, 25 Feb 2004 20:01:45

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> Gene wrote:
> >> 31-equal TOP is this, identical for 5-limit and 7-limit:
> >> (1201.6366 1899.3611 2790.8979 3372.3350)
> 
> >I get 1201.4675
> 
> I don't see how that can be correct. Your twelfth will be
> 1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) =
> 1.95 which is not log2(3)/log2(2) = 1.585.
> 
> Manuel

But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 - 
1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure 
what your point is.


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Message: 10379

Date: Wed, 25 Feb 2004 20:04:12

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> Gene wrote:
> >> 31-equal TOP is this, identical for 5-limit and 7-limit:
> >> (1201.6366 1899.3611 2790.8979 3372.3350)
> 
> >I get 1201.4675
> 
> I don't see how that can be correct. Your twelfth will be
> 1899.094. Then (1901.955 - 1899.094) / (1201.4675 - 1200.0) =
> 1.95 which is not log2(3)/log2(2) = 1.585.
> 
> Manuel

But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.6366 -
1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
what your point is.


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Message: 10382

Date: Thu, 26 Feb 2004 17:23:26

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Manuel Op de Coul

Paul wrote:
>But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 -
>1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
>what your point is.

Yes, never mind, I shouldn't be posting when in a hurry.
Today I found your post where you explain the equal tempered case,
so I understand it now. So the method in Scala is only compatible
with TOP when tempering out a single comma.
I could implement it for more than one comma sometime, but it's
more complicated than the existing procedure.

Manuel


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Message: 10383

Date: Thu, 26 Feb 2004 22:41:36

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> >But Manuel, your tuning gives (2790.8979 - 2786.3137)/(1201.4675 -
> >1200.0) = 2.80 which is not log(5)/log(2) = 2.32. So I'm not sure
> >what your point is.
> 
> Yes, never mind, I shouldn't be posting when in a hurry.
> Today I found your post where you explain the equal tempered case,
> so I understand it now.

Not so fast -- the single-comma case is the only one where the TOP 
tempering is motivated geometrically, so I'd be interested if there 
are other methods of tempering you'd come up with for multiple commas.

> So the method in Scala is only compatible
> with TOP when tempering out a single comma.
> I could implement it for more than one comma sometime, but it's
> more complicated than the existing procedure.

If you have a method that doesn't require searching 2^n corners or 
whatever, I'd be most interested in learning it.


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Message: 10384

Date: Thu, 26 Feb 2004 22:59:49

Subject: Re: TOP and Tenney space webpage

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> > Gene,
> > 
> > Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n 
> corners 
> > to a ball there are 2^(n-1) lines.." etc.)
> 
> The 2^n corners come in 2^(n-1) pairs of opposite corners, and 
lines 
> between them pass through the center of the ball, which is the JIP.
> 
> What's your take on the acronym JIP? I think it makes sense, since 
it 
> is the point corresponding to just intonation, but Paul objects for 
> reasons not entirely clear to me.

First of all, it measures pitch, something you fail to note at all on 
your website, but would be the most comprehensible thing about this 
whole business.

Secondly, who discovered the duality between points and linear 
functionals? Poincare? That's very recent in the history of 
mathematics, and far more advanced than what most musicians could 
ever, let alone should be expected to already, comprehend.


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Message: 10387

Date: Fri, 27 Feb 2004 17:53:50

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Manuel Op de Coul

Paul wrote:
>Not so fast -- the single-comma case is the only one where the TOP
>tempering is motivated geometrically, so I'd be interested if there
>are other methods of tempering you'd come up with for multiple commas.

What I have is not so good as TOP, it needs experimenting with the
prime weights to get a good result for more than one comma.

>If you have a method that doesn't require searching 2^n corners or
>whatever, I'd be most interested in learning it.

I had indeed the brute force approach in mind. I thought it was a
linear programming problem, and Gene's TOP and Tenney space page
confirmed it.

Manuel


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Message: 10388

Date: Fri, 27 Feb 2004 20:30:10

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Not so fast -- the single-comma case is the only one where the 
TOP 
> > tempering is motivated geometrically...
> 
> Did you read my TOP tuning page?

Yes, several times.

>Clearly, this is false.

?


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Message: 10389

Date: Fri, 27 Feb 2004 20:33:37

Subject: Re: TOP and Tenney space webpage

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > What's your take on the acronym JIP? I think it makes sense, 
> since 
> > it 
> > > is the point corresponding to just intonation, but Paul objects 
> for 
> > > reasons not entirely clear to me.
> > 
> > First of all, it measures pitch, something you fail to note at 
all 
> on 
> > your website, but would be the most comprehensible thing about 
this 
> > whole business.
> 
> It doesn't measure anything. It is the just intonation mapping, 
just 
> as other points represent other tunings. Hence, "J I Point".

I have no idea how to reconcile this with

Yahoo groups: /tuning-math/message/9797 *

and with the fact that it's obvious that this linear operator, when 
acting on a monzo, is the only one returns its pitch (or interval 
size) in cents.


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Message: 10390

Date: Fri, 27 Feb 2004 20:34:34

Subject: Re: non-1200: Tenney/heuristic meantone temperament

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul" 
<manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> >Not so fast -- the single-comma case is the only one where the TOP
> >tempering is motivated geometrically, so I'd be interested if there
> >are other methods of tempering you'd come up with for multiple 
commas.
> 
> What I have is not so good as TOP, it needs experimenting with the
> prime weights to get a good result for more than one comma.
> 
> >If you have a method that doesn't require searching 2^n corners or
> >whatever, I'd be most interested in learning it.
> 
> I had indeed the brute force approach in mind. I thought it was a
> linear programming problem, and Gene's TOP and Tenney space page
> confirmed it.

Amazingly, Gene's page would have you believe you need to search even 
in the codimension 1 case!


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Message: 10392

Date: Sat, 28 Feb 2004 04:28:44

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>Two 7-limit notes less than 2 apart in the symmetric lattice can be
>harmonized by two tetrads sharing at least one common note, and notes
>2 or more apart cannot be so harmonized. Hence, a scale consisting
>only of such intervals has the property that stepwise progressions can
>be harmonized by tetrads with common notes--though not, of course,
>necessarily tetrads all of whose notes belong to the scale. 
>
>If list all notes reduced to an octave which are at a distance of less
>than <correction>two</correction> from the unison and smaller than 200
>cents in size, we obtain this:
>
>{28/25, 10/9, 35/32, 15/14, 16/15, 21/20, 25/24, 36/35, 49/48}
>
>The possible types of JI scale with the above property, in terms of
>the intervals and their multiplicities, with the above restriction on
>step size are given below. We can obtain tempered versions of these by
>temperaments which equate steps; we have (28/25)/(10/9) = 126/125,
>(10/9)/(35/32) = 64/63, (35/32)/(15/14) = 49/48, (15/14)/(16/15) =
>225/224, (16/15)/(21/20) = 64/63, (21/20)/(25/24) = 126/125, 
>(25/24)/(36/35) = 875/864, (36/35)/(49/48) = 1728/1715. Meantone,
>magic, orwell, pajara, porcupine, blackwood, superpythagorean,
>tripletone, kleismic or nonkleismic would all be reasonable linear
>temperaments to try--or beep if you think that is reasonable.

Rock!

>[28/25, 10/9, 35/32, 36/35] [1, 3, 2, 3]
>[28/25, 10/9, 25/24, 36/35] [3, 1, 4, 3]
>[28/25, 15/14, 16/15, 49/48] [1, 5, 3, 2]
>[28/25, 16/15, 25/24, 36/35] [3, 1, 5, 3]
>
>[10/9, 35/32, 16/15, 36/35] [2, 3, 2, 3]
>[10/9, 35/32, 36/35, 49/48] [4, 1, 5, 2]
>[10/9, 35/32, 21/20, 36/35] [4, 1, 2, 3]
>[10/9, 15/14, 16/15, 21/20] [2, 3, 2, 3]
>[10/9, 15/14, 21/20, 36/35] [4, 1, 3, 2]
>[10/9, 15/14, 36/35, 49/48] [4, 1, 5, 3]
>[10/9, 21/20, 25/24, 36/35] [4, 3, 1, 3]
>[10/9, 25/24, 36/35, 49/48] [4, 1, 6, 3]
>
>[35/32, 15/14, 16/15, 36/35] [3, 2, 4, 1]
>[35/32, 16/15, 25/24, 36/35] [3, 4, 2, 3]
>
>[15/14, 16/15, 21/20, 25/24] [3, 4, 3, 2]
>[15/14, 16/15, 21/20, 49/48] [5, 4, 1, 2]
>[15/14, 16/15, 36/35, 49/48] [5, 4, 1, 3]
>
>[16/15, 21/20, 25/24, 36/35] [4, 3, 5, 3]
>[16/15, 25/24, 36/35, 49/48] [4, 5, 6, 3]
>
>[28/25, 10/9, 15/14, 25/24] [3, 1, 3, 1]
>[28/25, 15/14, 16/15, 25/24] [3, 3, 1, 2]
>[28/25, 35/32, 15/14, 16/15] [1, 2, 3, 3]
>[28/25, 10/9, 15/14, 21/20] [2, 2, 3, 1]
>
>[35/32, 15/14, 16/15, 21/20] [2, 3, 4, 1]

Rad.  No 6- or 7-toners, eh?  I wonder about the "9-limit".
The 9-limit has the further advantage that you can hit more
fifths, and thus improve omnitetrachordality.

Also, could we screen based on which of the above
combinations, and which orderings of those, produce the
most low-numbered ratios in the scale?  Or does such
an approach fail on the grounds that it ignores temperament
(aka TOLERANCE)?

-Carl


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Message: 10395

Date: Sat, 28 Feb 2004 14:37:14

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>> Rad.  No 6- or 7-toners, eh?  I wonder about the "9-limit".
>> The 9-limit has the further advantage that you can hit more
>> fifths, and thus improve omnitetrachordality.
>
>The 9-limit would be different, for sure. The simple symmetrical 
>lattice criterion wouldn't work, but it would be easy enough to
>find what does.

Nobody ever answered me if symmetrical is synonymous with unweighted. 

>> Also, could we screen based on which of the above
>> combinations, and which orderings of those, produce the
>> most low-numbered ratios in the scale?  Or does such
>> an approach fail on the grounds that it ignores temperament
>> (aka TOLERANCE)?
>
>The main problem I see with that is that it is a huge computational 
>job, since in each case you need to find the optimal version of the 
>scale in question. I wouldn't want to try it using Maple.

The thing is to only store one permutation in memory at a time.
Alas, I haven't come up with an easy way to code these kinds of
evaluations in scheme.  They're very natural in C, I think.  The
present problem may still be hard on account of CPU cycles, tho.

-Carl


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Message: 10397

Date: Sat, 28 Feb 2004 14:40:26

Subject: Re: Harmonized melody in the 7-limit

From: Carl Lumma

>By the way, it seems to me the more general property of a scale with 
>a set of steps (generating the temperament) with the number of 
>different sizes of step equal to the number of generators, or rank, 
>or dimension + 1 or whatever you want to call it for a regular 
>temperament might be worth exploring; it generalizes Myhill's 
>property for linear temperaments.

I've always wanted to understand what's going on geometrically (in
terms of block selection on the lattice) with Myhill scales.

-Carl


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