Tuning-Math Digests messages 8925 - 8949

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Message: 8925

Date: Sat, 03 Jan 2004 16:58:15

Subject: Re: name?

From: Carl Lumma

>> >Oh so you're thinking about 13-equal.
>> 
>> Yup.  And I think only multiples of 13 do the job?
>> 
>> -Carl
>
>don't get your question . . .

Which ETs temper this comma out?

-Carl


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Message: 8926

Date: Sat, 03 Jan 2004 16:59:12

Subject: Re: name?

From: Carl Lumma

>> That's funny, it's supposed to be 222 cents.  Crap, so sorry,
>> it should have been [-30 0 13].
>
>Here we see the advantage of leaving the 2s in.

Or in my case, not failing to leave in the 3s!

-Carl


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Message: 8927

Date: Sat, 03 Jan 2004 17:21:27

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

From: Carl Lumma

>> >> Gene and you had an exchange.  Gene suggested what I was
>> >> thinking, you said that each band represented only a
>> >> denominator, not a comma.
>> >> Or are you talking about something else?
>> >
>> >Yes, a later pair of graphs.
>> 
>> D'oh!  Link?  (I even went to tuning-math to find it myself, but
>> this thread is not connected to anything).
>
>Yahoo groups: /tuning-math/files/Paul/com5monz.gif *
>
>and
>
>Yahoo groups: /tuning-math/files/Paul/com5rat.gif *

These are the graphs I thought Gene replied about -- do they look
very similar to those?

>> >> >No need to specify a consonance limit? -- wow that's hot.
>> >> 
>> >> How do you mean?  Isn't this implicit in the dimensionality of 
>> >> the Tenney lattice you're using?
>> >
>> >No, no consonance limit (aka odd-limit) is implicity in the 
>> >dimensionality of the Tenney lattice you're using -- and even the 
>> >latter, aka prime-limit, doesn't need to be specified -- for
>> >example the results for the pythagorean comma will be valid in
>> >both lower and higher prime limits.
>> 
>> I must not be tracking you -- the pythagorean comma is of course
>> a 3-limit comma.
>
>See the last paragraph of
>
>Yahoo groups: /tuning/message/50964 *

I don't see anything new in any of this, I'm afraid.

Let's back up.  We're talking about the badness of commas?  Typically
that's been a function of their size in cents and the number of notes
you can be expected to search to find them (which depends on their
distance on the lattice and the dimensionality of the lattice).
That's Gene's logflat badness, which I've implemented in Scheme.
Now, what exactly are you up to here?

-Carl


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Message: 8933

Date: Sun, 04 Jan 2004 00:14:20

Subject: Re: name?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Oh so you're thinking about 13-equal.
> 
> Yup.  And I think only multiples of 13 do the job?
> 
> -Carl

don't get your question . . .


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Message: 8934

Date: Sun, 04 Jan 2004 00:28:05

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> Gene and you had an exchange.  Gene suggested what I was 
thinking,
> >> you said that each band represented only a denominator, not a 
comma.
> >> Or are you talking about something else?
> >
> >Yes, a later pair of graphs.
> 
> D'oh!  Link?  (I even went to tuning-math to find it myself, but
> this thread is not connected to anything).

Yahoo groups: /tuning-math/files/Paul/com5monz.gif *

and

Yahoo groups: /tuning-math/files/Paul/com5rat.gif *

> >> >No need to specify a consonance limit? -- wow that's hot.
> >> 
> >> How do you mean?  Isn't this implicit in the dimensionality of 
the
> >> Tenney lattice you're using?
> >
> >No, no consonance limit (aka odd-limit) is implicity in the 
> >dimensionality of the Tenney lattice you're using -- and even the 
> >latter, aka prime-limit, doesn't need to be specified -- for 
example 
> >the results for the pythagorean comma will be valid in both lower 
and 
> >higher prime limits.
> 
> I must not be tracking you -- the pythagorean comma is of course
> a 3-limit comma.

See the last paragraph of

Yahoo groups: /tuning/message/50964 *


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Message: 8936

Date: Sun, 04 Jan 2004 01:00:23

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

From: Paul Erlich

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

> What would you call this kind of meantone? The 3:2 is flattened by 
> about 10/49-comma, while the octave is widened by about 3/38-comma.

It's approximated by 19ED2.00196, very well approximated by 
31ED2.001963, or if you need more accuracy, 205ED2.00196315 . . .

12 doesn't even seem to be a convergent here, it skips right from 7 
to 19 . . .


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Message: 8937

Date: Sun, 04 Jan 2004 01:01:45

Subject: Re: name?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >Oh so you're thinking about 13-equal.
> >> 
> >> Yup.  And I think only multiples of 13 do the job?
> >> 
> >> -Carl
> >
> >don't get your question . . .
> 
> Which ETs temper this comma out?
> 
> -Carl

Yes, as long as 4/13 oct. remains your operative (by being 'best' or 
whatever rule you base your mapping on) approximation of 5:4.


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Message: 8938

Date: Sun, 04 Jan 2004 02:39:16

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> No one commented on the graphs I posted around Christmas, but 
I'll 
> >> keep going, if only for myself . . .
> 
> Gene and you had an exchange.  Gene suggested what I was thinking,
> you said that each band represented only a denominator, not a comma.
> Or are you talking about something else?
> 
> >No need to specify a consonance limit? -- wow that's hot.
> 
> How do you mean?  Isn't this implicit in the dimensionality of the
> Tenney lattice you're using?
> 
> It would indeed be hot.
>
> -Carl

Let's try to get a better grasp of what happens in this particular 
meantone, for a start. I could also do this for 7-limit intervals, 
treating the 81/80 temperament as a 'planar' temperament, but 
hopefully it's clear that the extra intervals won't have enough error 
to exceed the bound. Same for any higher limit.

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: 8939

Date: Sun, 04 Jan 2004 02:45:22

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> Gene and you had an exchange.  Gene suggested what I was
> >> >> thinking, you said that each band represented only a
> >> >> denominator, not a comma.
> >> >> Or are you talking about something else?
> >> >
> >> >Yes, a later pair of graphs.
> >> 
> >> D'oh!  Link?  (I even went to tuning-math to find it myself, but
> >> this thread is not connected to anything).
> >
> >Yahoo groups: /tuning-math/files/Paul/com5monz.gif *
> >
> >and
> >
> >Yahoo groups: /tuning-math/files/Paul/com5rat.gif *
> 
> These are the graphs I thought Gene replied about -- do they look
> very similar to those?

Pretty similar.

> >> >> >No need to specify a consonance limit? -- wow that's hot.
> >> >> 
> >> >> How do you mean?  Isn't this implicit in the dimensionality 
of 
> >> >> the Tenney lattice you're using?
> >> >
> >> >No, no consonance limit (aka odd-limit) is implicity in the 
> >> >dimensionality of the Tenney lattice you're using -- and even 
the 
> >> >latter, aka prime-limit, doesn't need to be specified -- for
> >> >example the results for the pythagorean comma will be valid in
> >> >both lower and higher prime limits.
> >> 
> >> I must not be tracking you -- the pythagorean comma is of course
> >> a 3-limit comma.
> >
> >See the last paragraph of
> >
> >Yahoo groups: /tuning/message/50964 *
> 
> I don't see anything new in any of this, I'm afraid.
> 
> Let's back up.  We're talking about the badness of commas?  
>Typically
> that's been a function of their size in cents

No, not really. Rather, it's the error they induce when tempered out. 
I seem to have figured out a simple formula to get this when the 
tempering is Tenney-weighted-minimax optimal over ALL intervals. And 
a simple way to do such tempering.

> and the number of notes
> you can be expected to search to find them (which depends on their
> distance on the lattice and the dimensionality of the lattice).

I'm just looking at distance on the lattice, and dimensionality is 
irrelevant -- it can be assumed to be infinite.


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Message: 8941

Date: Sun, 04 Jan 2004 01:18:00

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

From: Carl Lumma

>No, not really. Rather, it's the error they induce when tempered out.
>I seem to have figured out a simple formula to get this when the
>tempering is Tenney-weighted-minimax optimal over ALL intervals. And 
>a simple way to do such tempering.

What is that formula a way?  Sorry, can we start at the top?  It
seems like your earlier message was either written in a white heat
or addressed to someone who knows things I don't.

-Carl


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Message: 8942

Date: Sun, 04 Jan 2004 12:28:04

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

From: Graham Breed

Paul Erlich wrote:

> No, not really. Rather, it's the error they induce when tempered out. 
> I seem to have figured out a simple formula to get this when the 
> tempering is Tenney-weighted-minimax optimal over ALL intervals. And 
> a simple way to do such tempering.

The odd-limit rule simplifies to

interval size / complexity

where "complexity" is the smallest number of intervals in the relevant 
limit that make up the comma.  The result is the optimum minimax error 
by tempering out only this comma.  So any temperament involving this 
comma must be at least this well tuned.

What you've done is plugged in the Tenney complexity, and got a result 
that'll have something to do with the weighted minimax.  That's not too 
surprising, and should generalize to any weighted complexity measure. 
At least if it gives a result in terms of octaves.

I thought this was all assumed by your hypothesis anyway.  We know that 
the Tenney complexity and odd limits are linked, depending on whether or 
not you enforce octave equialence.  As geometric complexity looks like 
being an octave-specific weighted complexity measure, this may be the 
way to progress.

The problem remains knowing how best to combine these commas to get a 
temperament of a specific dimension.  For that we need a straightness 
measure, as always.  And if you're not enforcing a prime limit to start 
with, you'll need to take a variable number of commas.


Oh, BTW, I've been looking at commas of the form (n*n):(n+1)(n-1).  Do 
they have a name?  They always come from setting two n-integer limit 
intervals to be equivalent.  They're also more likely than arbitrary 
superparticulars to belong to a low prime limit..  The 11-prime limit 
subset gives us

4:3, 9:8, 16:15, 25:24, 36:35, 49:48, 64:63, 81:80,
100:99, 225:224, 441:440, 2401:2400


                     Graham



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Message: 8943

Date: Mon, 05 Jan 2004 10:03:42

Subject: Re: Squarejacks

From: Graham Breed

Gene Ward Smith wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
> 
>>Here is a list  n < 5000 such that n^2/(n^2-1) is within the 23 limit,
>>together with the prime limit.

That's more like it!  I can't find any more with n<100000, so it might 
be complete.


               Graham


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Message: 8946

Date: Mon, 05 Jan 2004 21:25:32

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >No, not really. Rather, it's the error they induce when tempered 
out.
> >I seem to have figured out a simple formula to get this when the
> >tempering is Tenney-weighted-minimax optimal over ALL intervals. 
And 
> >a simple way to do such tempering.
> 
> What is that formula a way?  Sorry, can we start at the top?

For the comma p/q, p>q, the number of cents you need to temper out is 
cents(p/q) = log2(p/q)*1200.

The distance in the Tenney lattice (taxicab, by definition) is log2
(p*q).

So the tempering per unit length in the direction of the comma is 
cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the 
vertical axis in my graphs.

Now, for all primes r,

If p contains any factors of r, the r-rungs in the lattice (which 
have length log2(r)) are shrunk from
cents(r)
to
cents(r) - log2(r)*cents(p/q)/log2(p*q).
If q contains any factors of 2, they are instead stretched to
cents(r) + log2(r)*cents(p/q)/log2(p*q).


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Message: 8947

Date: Mon, 05 Jan 2004 21:33:47

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > No, not really. Rather, it's the error they induce when tempered 
out. 
> > I seem to have figured out a simple formula to get this when the 
> > tempering is Tenney-weighted-minimax optimal over ALL intervals. 
And 
> > a simple way to do such tempering.
> 
> The odd-limit rule simplifies to
> 
> interval size / complexity
> 
> where "complexity" is the smallest number of intervals in the 
relevant 
> limit that make up the comma.  The result is the optimum minimax 
error 
> by tempering out only this comma.  So any temperament involving 
this 
> comma must be at least this well tuned.
> 
> What you've done is plugged in the Tenney complexity, and got a 
result 
> that'll have something to do with the weighted minimax.  That's not 
too 
> surprising, and should generalize to any weighted complexity 
measure. 
> At least if it gives a result in terms of octaves.
> 
> I thought this was all assumed by your hypothesis anyway.

I don't see the relationship.

>  We know that 
> the Tenney complexity and odd limits are linked, depending on 
whether or 
> not you enforce octave equialence.

Yes, but for octave equivalence (pegged to 1200 cent octaves), I'd 
like to eventually be able to use Kees's expressibility measure 
instead of Tenney harmonic distance. Just as there was no 
finitistic 'limit' assumed for my 'optimization' in the Tenney 
lattice, no odd limit will have to be specified in the octave-
equivalent case (if it can work).

> As geometric complexity looks like 
> being an octave-specific weighted complexity measure, this may be 
the 
> way to progress.

What do you mean?

> The problem remains knowing how best to combine these commas to get 
a 
> temperament of a specific dimension.  For that we need a 
straightness 
> measure, as always.

That's why I was asking about heron's formula, etc. But if we have 
some way of acheiving this Tenney-weighted minimax for the relevant 
temperaments, we may be able to skip this step.


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Message: 8948

Date: Mon, 05 Jan 2004 14:45:25

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

From: Carl Lumma

>For the comma p/q, p>q, the number of cents you need to temper out is 
>cents(p/q) = log2(p/q)*1200.
>
>The distance in the Tenney lattice (taxicab, by definition) is log2
>(p*q).
>
>So the tempering per unit length in the direction of the comma is 
>cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the 
>vertical axis in my graphs.
>
>Now, for all primes r,
>
>If p contains any factors of r, the r-rungs in the lattice (which 
>have length log2(r)) are shrunk from
>cents(r)
>to
>cents(r) - log2(r)*cents(p/q)/log2(p*q).
>If q contains any factors of 2, they are instead stretched to
>cents(r) + log2(r)*cents(p/q)/log2(p*q).

Thanks.  I understand this 100%.  But I don't understand what's
new.  Perhaps it has something to do with using this to get
optimum generators for a linear temperament?

And I don't understand your 'limitless' claim -- since p/q contains
the factors it does and no others, one wouldn't expect its vanishing
to effect intervals different factors.

-Carl


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Message: 8949

Date: Mon, 05 Jan 2004 23:35:07

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

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >For the comma p/q, p>q, the number of cents you need to temper out 
is 
> >cents(p/q) = log2(p/q)*1200.
> >
> >The distance in the Tenney lattice (taxicab, by definition) is log2
> >(p*q).
> >
> >So the tempering per unit length in the direction of the comma is 
> >cents(p/q)/log2(p*q). This was (aside from a factor of 1200) the 
> >vertical axis in my graphs.
> >
> >Now, for all primes r,
> >
> >If p contains any factors of r, the r-rungs in the lattice (which 
> >have length log2(r)) are shrunk from
> >cents(r)
> >to
> >cents(r) - log2(r)*cents(p/q)/log2(p*q).
> >If q contains any factors of 2, they are instead stretched to
> >cents(r) + log2(r)*cents(p/q)/log2(p*q).
> 
> Thanks.  I understand this 100%.  But I don't understand what's
> new.

Where have you seen this before?

> Perhaps it has something to do with using this to get
> optimum generators for a linear temperament?

Well, that's exactly what this does (when the dimensionality is 
right), as I've illustrated already in a few cases.

Here's something new -- Top meantone is, it seems, exactly 1/4-comma 
meantone (I get 0.24999999999997, but that's probably just rounding 
error) except a uniform (in cents, or log Hz) stretch of 
1.00141543374547 is applied to all intervals . . .

> And I don't understand your 'limitless' claim -- since p/q contains
> the factors it does and no others, one wouldn't expect its vanishing
> to effect

affect?

> intervals different factors.

intervals with different factors? Well, 5:4 and 5:3 have *some* 
factors differing from those in the Pythagorean comma, yet both 
intervals are affected by its vanishing, in this scheme.


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