Tuning-Math Digests messages 9025 - 9049

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Message: 9025

Date: Thu, 08 Jan 2004 05:51:41

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> I understand that functions of the type f(x) -> x^2 + c are 
shaped
> >> like parabolas, but x isn't a generator size here, it's the sum 
of
> >> errors resulting from a generator size.  If I took out the ^2 it
> >> might be shaped like anything;
> >
> >Huh? x + c is shaped like anything?
> 
> Traditionally a line, but in this case x is actually this other
> function, the summed errors from these arbitrary external just
> ratio things.

No, silly goose :), the squaring is done *before* the summing! If 
it's done *after* the summing, it has no effect (since the location 
of the lowest value of a positive function is also the lowest 
possible value of the function squared, and if you don't start with a 
positive function, you're doing something wrong).

> As I move the generator size from 0-600 cents and
> pump it through say the meantone map, the errors could go up and
> down several times for all I know.

The error of each interval will be a straight line. The errors 
squared will be parabolas. The sum of a set of parabolas is a 
parabola, since the sum of any number of functions of order 2 is a 
function of order 2 -- since you're squaring some linear functions, 
then adding, you'll have quadratic terms, linear terms, and constant 
terms to add, and that's all.


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Message: 9026

Date: Thu, 08 Jan 2004 13:31:47

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

From: Carl Lumma

>> > There should be no angles defined here, just as there are none in
>> > the Tenney lattice.
>> 
>> Why "should"?  They have to be, or it won't work.
>
>Why won't it? My Tenney, non-octave-equivalent way doesn't need 
>angles defined. You can choose any set of angles you want, and still 
>embed the result in Euclidean space, but that doesn't even matter -- 
>what matters are the taxicab distances ONLY.

While I'm certainly hoping taxicab proves sufficient, isn't it
possible that you'll need angles when more than one comma is involved?

-Carl


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Message: 9028

Date: Thu, 08 Jan 2004 21:44:51

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> > There should be no angles defined here, just as there are none 
in
> >> > the Tenney lattice.
> >> 
> >> Why "should"?  They have to be, or it won't work.
> >
> >Why won't it? My Tenney, non-octave-equivalent way doesn't need 
> >angles defined. You can choose any set of angles you want, and 
still 
> >embed the result in Euclidean space, but that doesn't even matter -
- 
> >what matters are the taxicab distances ONLY.
> 
> While I'm certainly hoping taxicab proves sufficient, isn't it
> possible that you'll need angles when more than one comma is 
>involved?

Yes, we'll need the *correct* angle-like concept. Then again, I 
attempted to get around the whole angle issue with my heron's formula 
application, which almost worked!


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Message: 9029

Date: Thu, 08 Jan 2004 01:10:56

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > Interesting! And is that truly the only one that matters?
> 
> The size/complexity tells you the best value for the suitably 
weighted 
> minimax in any temperament in which this comma vanishes.  If a 
comma 
> exists such that its size/complexity is equal to the optimim 
minimax 
> error in a given linear temperament, and the comma is in the linear 
> temperament's kernel, then the two temperaments must be identical.

I can't follow that right now.

> I'm not sure if such a comma will always exist,

Wow -- now that's an interesting question to consider.

> but provided it does 
> it's the only one you need for TOPS.

My intuition says it doesn't exist.

> It doesn't even have to be made up 
> of integers, so long as it's a linear combination of commas that 
>are.

???

> It's a generalization of the proof of the TOP meantone being 
stretched 
> quarter-comma.  All the factors that get tempered the same way as 2 
will 
> be stretched by the same amount.  But if the octaves don't get 
tempered 
> at all, some factors will be tempered in one method but not the 
other.

There's something that seems strange about your octave-equivalent 
method. The comma is supposed to be distributed uniformly (per unit 
length, taxicabwise) among its constituent "rungs" in the lattice. 
But it seems that 81:80 = 81:5 would involve 1, not 0, rungs of 5 in 
the octave-equivalent lattice. But the octave-equivalent lattice 
can't be embedded in euclidean space, so this completely falls apart??


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Message: 9030

Date: Thu, 08 Jan 2004 06:55:44

Subject: for gene

From: Paul Erlich

Yahoo groups: /tuning_files/files/Erlich/gene2.gif *


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Message: 9032

Date: Thu, 08 Jan 2004 01:12:18

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >Hence you can impose a weighted 
> >minimax over all intervals within a given prime limit.
> 
> Aha!  So why then isn't the prime limit also superfluous?

It is, unless you want to control the dimension of your temperament.


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Message: 9034

Date: Thu, 08 Jan 2004 22:20:39

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

From: Paul Erlich

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

> > 81:5 involves three rungs of 3:1 and one rung of 3:5.
> 
> Oh yeah.
> 
> > For the 5-odd 
> > limit, these rungs are of equal length,

Wait a minute -- we're obviously talking about different things here! 
I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio of 
5, so the latter should be longer!!

> and so that error has to be 
> > shared between them.  That leaves 3:1 and 3:5 having an equal 
> amount of 
> > temperament, and so 1:5 must be untempered.
> 
> Ha!

I retract that "Ha!" . . .


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Message: 9035

Date: Thu, 08 Jan 2004 01:19:08

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> I never understood this process,
> >> >
> >> >Solving a system of linear equations?
> >> 
> >> Uh-huh.
> >
> >Well, the easiest way to understand is to solve one equation for 
one 
> >variable, plug that solution into the other variables so that 
you've 
> >eliminated one variable entirely, and repeat until you're done.
> 
> I remember this technique from Algebra, but I didn't think it would
> be applicable here, since I assumed the variables wouldn't be
> independent in that way. 

They're not, you actually have an extra equation.

> What do these equations look like?

For meantone,

prime2 = period;
prime3 = period + generator;
prime5 = 4*generator.

You can throw out any equation -- say the first.

so generator = .25*prime5,
prime3 = period + .25*prime5,
period = prime3 - .25*prime5.

> >> Why are you assuming octave repetition, what does this assumption
> >> amount to?
> >
> >That you'll have the same pitches in each (possibly tempered) 
octave.
> >
> >> If 2 is in the map, one of the generators had better well 
generate
> >> it.  If it isn't in the map, assuming octave repetition seems 
like
> >> a bad idea to me.
> >
> >Any recent cases where you'd prefer not to see 2 in the map?
> 
> I'd always prefer to see it, but why assume?

Agreed. But it's a "default", a "convention", that many would assume.


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Message: 9036

Date: Thu, 08 Jan 2004 07:08:29

Subject: another for gene

From: Paul Erlich

Yahoo groups: /tuning_files/files/Erlich/gene3.gif *


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Message: 9037

Date: Thu, 08 Jan 2004 22:28:13

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

From: Graham Breed

Paul Erlich wrote:

> What's the worst comma for 12-equal in the 5-limit?

[0 28 -19> or 22876792454961:19073486328125

TOPping it gives a narrow octave of 0.99806 2:1 octaves.


> Me too -- but the lengths aren't compatible in a Euclidean space. 
> Remember the whole big "wormholes" discussion from years ago?

Yes, I remember all about the wormholes, and they don't have anything to 
do with this.  You only need them for odd limits.

> We can either embed a lattice, with a taxicab distance, into 
> Euclidean space, or we can't. But just because we can, doesn't mean 
> we should use Euclidean distance! NONONONONONO!

You could try taxicab distance, I'm not sure it'd work right.  But you 
can also use Euclidian distance, and it looks like a more 
straightforward way to me.

> Why won't it? My Tenney, non-octave-equivalent way doesn't need 
> angles defined. You can choose any set of angles you want, and still 
> embed the result in Euclidean space, but that doesn't even matter -- 
> what matters are the taxicab distances ONLY.

When did it become *your* way?  The problem that either triangular or 
angular lattices solve doesn't arise in octave specific lattices, as 
we've always known.  But Euclidian metrics can still be useful.  From 
what I remember/understood, geometric complexity was one.


                     Graham


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Message: 9038

Date: Thu, 08 Jan 2004 01:20:31

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> I wrote:
> > The size/complexity tells you the best value for the suitably 
weighted 
> > minimax in any temperament in which this comma vanishes.  If a 
comma 
> > exists such that its size/complexity is equal to the optimim 
minimax 
> > error in a given linear temperament, and the comma is in the 
linear 
> > temperament's kernel, then the two temperaments must be identical.
> 
> Actually, it's more complicated than that.  After finding the 
planar 
> temperament, you then need to adjust intervals that didn't get 
tempered 
> so that they work with the correct linear temperament family.

Phew! I thought I had gone crazy! Thanks for clarifying.


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Message: 9039

Date: Thu, 08 Jan 2004 07:15:45

Subject: Re: TOP and normed vector spaces

From: Paul Erlich

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

> An example of a normed vector space is the p-limit Erlich space,

This is the Tenney space.

> We may change basis in the Erlich space by resizing the elements, 
so 
> that the norm is now
> 
> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|

how are the v's defined?

> and using the same proceedure we use to get 
> a unique minimax we can find a unique minimal distance point TOP at 
> this minimum distance from SIZE

not following . . .

> One neat thing about this is that it generalizes immediately to 
other 
> normed vector spaces containing complete p-limit (meaning, 2 is 
> included as a prime number) lattices. In particular, there is a 
> geometric complexity version of TOP.

What's better about it? I think Tenney complexity is a better guide 
to consonance, to tuning sensitivity, and even to musical complexity 
of a chain of intervals.


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Message: 9040

Date: Thu, 08 Jan 2004 22:25:41

Subject: Reply to Gene (was: Re: non-1200: Tenney/heursitic meantone temperament)

From: Paul Erlich

Gene's proposed canonical meantone:

5-limit: [1200., 1896.578429, 2786.313713]

Let's evaluate:

Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp
2:1........1200.00.....0.............1...............0
3:1........1896.58....5.38..........1.58............3.41

So already you've exceeded the maximum weighted error of my proposal 
by a factor of 2!

> Interval...Approx....|Error|....Comp=log2(n*d)...|Error|/Comp
> 2:1........1201.70....1.70...........1..............1.70
> 3:1........1899.26....2.69..........1.58............1.70
> 4:1........2403.40....3.40...........2..............1.70
> 5:1........2790.26....3.94..........2.32............1.70
> 3:2.........697.56....4.39..........2.58............1.70
> 6:1........3100.96....0.99..........2.58............0.38
> 8:1........3605.10....5.10...........3..............1.70
> 9:1........3798.53....5.38..........3.17............1.70
> 10:1.......3991.96....5.64..........3.32............1.70
> 4:3.........504.13....6.09..........3.58............1.70
> 12:1.......4302.66....0.70..........3.58............0.20
> 5:3.........890.99....6.64..........3.91............1.70
> 15:1.......4689.52....1.25..........3.91............0.32
> 16:1.......4806.79....6.79...........4..............1.70
> 9:2........2596.83....7.08..........4.17............1.70
> 18:1.......5000.22....3.69..........4.17............0.88
> 5:4.........386.86....0.55..........4.32............0.13
> 20:1.......5193.65....7.34..........4.32............1.70
> 8:3........1705.83....7.79..........4.58............1.70
> 24:1.......5504.36....2.40..........4.58............0.52
> 25:1.......5580.52....7.89..........4.64............1.70
> 6:5.........310.70....4.94..........4.91............1.01
> 10:3.......2092.69....8.33..........4.91............1.70
> 30:1.......5891.22....2.95..........4.91............0.60
> 32:1.......6008.49....8.49...........5..............1.70
> 36:1.......6201.92....1.99..........5.17............0.38
> 8:5.........814.84....1.15..........5.32............0.22
> 40:1.......6395.35....9.04..........5.32............1.70
> 9:5........1008.27....9.33..........5.49............1.70
> 45:1.......6588.78....1.44..........5.49............0.26
> 16:3.......2907.53....9.49..........5.58............1.70
> 48:1.......6706.06....4.10..........5.58............0.73
> 25:2.......4378.82....6.19..........5.64............1.10
> 50:1.......6782.21....9.59..........5.64............1.70
> 27:2.......4496.09....9.77..........5.75............1.70
> 54:1.......6899.49....6.38..........5.75............1.11
> 12:5.......1512.40....3.24..........5.91............0.55
> 15:4.......2286.12....2.15..........5.91............0.36
> 20:3.......3294.39...10.03..........5.91............1.70
> 60:1.......7092.92....4.65..........5.91............0.79
> 1296:5.....
> and so on. Thinking about a few of these example spacially should 
> help you see that the weighted error can never exceed
> 
> cents(81/80)/log2(81*80) = 1.70
> 
> for ANY interval.
> 
> Is there a just (RI) interval in this meantone? The idea of duality 
> leads me to guess 81*80:1 = 6480:1 . . .
> 
> 6480:1....15194.10....0.03
> 
> almost, but no cigar.


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Message: 9042

Date: Thu, 08 Jan 2004 22:29:46

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > What's the worst comma for 12-equal in the 5-limit?
> 
> [0 28 -19> or 22876792454961:19073486328125

Wow. How did you find that?

> TOPping it gives a narrow octave of 0.99806 2:1 octaves.

Shall I proceed to calculate Tenney-weighted errors for all (well, a 
bunch of) intervals? I hope you're onto something!

> > Me too -- but the lengths aren't compatible in a Euclidean space. 
> > Remember the whole big "wormholes" discussion from years ago?
> 
> Yes, I remember all about the wormholes, and they don't have 
anything to 
> do with this.  You only need them for odd limits.

I thought that's what you were talking about in the thread where I 
brought them up! Odd limit, right?

> > We can either embed a lattice, with a taxicab distance, into 
> > Euclidean space, or we can't. But just because we can, doesn't 
mean 
> > we should use Euclidean distance! NONONONONONO!
> 
> You could try taxicab distance, I'm not sure it'd work right.  But 
you 
> can also use Euclidian distance, and it looks like a more 
> straightforward way to me.
> 
> > Why won't it? My Tenney, non-octave-equivalent way doesn't need 
> > angles defined. You can choose any set of angles you want, and 
still 
> > embed the result in Euclidean space, but that doesn't even 
matter -- 
> > what matters are the taxicab distances ONLY.
> 
> When did it become *your* way?

Did someone publish it before? It's currently not Gene's way, anyway.

> The problem that either triangular or 
> angular lattices solve doesn't arise in octave specific lattices, 
as 
> we've always known.  But Euclidian metrics can still be useful.  
From 
> what I remember/understood, geometric complexity was one.

I've been trying to convince Gene otherwise, and he said something 
about minor thirds being shorter than major thirds there . . .


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Message: 9043

Date: Thu, 08 Jan 2004 07:20:45

Subject: Temperament agreement

From: Dave Keenan

Continued from the tuning list.
Paul:
>With my (Tenney) complexity and (all-interval-Tenney-minimax) error
>measures?

With these it seems I need to scale the parameters to
k=0.002
p=0.5
and max badness = 75

where badness = complexity * exp((error/k)**p)

I'd be very interested to see how that compares with your other cutoff
lines.

These errors and complexities don't seem to have meaningful units.

Complexity used to have units of generators per diamond and error used
to have units of cents, both things you could relate to fairly directly.


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Message: 9044

Date: Thu, 08 Jan 2004 22:46:34

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

From: Graham Breed

Paul Erlich wrote:

> Wait a minute -- we're obviously talking about different things here! 
> I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio of 
> 5, so the latter should be longer!!

They're both 5-odd limit intervals, and so they each have a 5-odd limit 
complexity of 1.


                      Graham


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Message: 9046

Date: Thu, 08 Jan 2004 22:45:47

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > Wait a minute -- we're obviously talking about different things 
here! 
> > I read this too quickly. 3:1 is a ratio of 3, and 3:5 is a ratio 
of 
> > 5, so the latter should be longer!!
> 
> They're both 5-odd limit intervals, and so they each have a 5-odd 
limit 
> complexity of 1.

Well then we are talking about different things. I'm talking 
about "expressibility" as the distance measure.


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Message: 9047

Date: Thu, 08 Jan 2004 07:51:31

Subject: Re: TOP and normed vector spaces

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > We may change basis in the Erlich space by resizing the 
elements, 
> > so 
> > > that the norm is now
> > > 
> > > || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|
> > 
> > how are the v's defined?
> 
> For a rational number, vp is the p-adic valuation of number q, that 
> is,the exponent in the factorization of q into primes. For other 
> points in the Tenney space it's just a coordinate.

There are no other points in the Tenney space. Anyway, I lost the 
train of thought.

> > > and using the same proceedure we use to get 
> > > a unique minimax we can find a unique minimal distance point 
TOP 
> at 
> > > this minimum distance from SIZE
> > 
> > not following . . .
> 
> Remember, we have a way of measuring distance between tuning maps. 

In the dual space?

> Hence, given a tuning map SIZE and a subspace of tuning maps Null
(C), 
> we can find those at a minimum distance from SIZE.

Hmm . . .

> > > One neat thing about this is that it generalizes immediately to 
> > other 
> > > normed vector spaces containing complete p-limit (meaning, 2 is 
> > > included as a prime number) lattices. In particular, there is a 
> > > geometric complexity version of TOP.
> > 
> > What's better about it?
> 
> What's better about it is that it is Euclidean, which is convenient 
> in many ways.

Do we really need this convenience? Can't we work with the taxicab 
metric?


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Message: 9048

Date: Thu, 08 Jan 2004 22:50:03

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> 
wrote:
> > Paul Erlich wrote:
> > 
> > > Hmm . . . by *all* the errors, I meant for lots and lots of 
> > > intervals, like I did.
> > 
> > Oh, well, here's the 9-limit with a few bonuses:
> > 
> >   3:1  0.002827
> >   5:1  0.000000
> >   5:3  0.001930
> >   7:1  0.000903
> >   7:3  0.000693
> >   7:5  0.000903
> >   9:1  0.002827
> >   9:5  0.002827
> >   9:7  0.002027
> > 15:1  0.001147
> > 27:1  0.002827
> > 27:5  0.002827
> 
> So you're dividing by expressibility here? Interesting . . . !

Graham, it sure doesn't look like you're using Euclidean distance 
here!!!


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