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Message: 1451 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 18:04:53

Subject: Re: 19-consistent goodness of measurement systems

From: Carl

>> >ene, does this have anything to do with the zeta function?
> 

> Only in the sense that the big spikes in the zeta function occur
> at good et's.


That's by definition, right?  (that is, doesn't actually explain
anything).

-Carl

Message: 1453 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 00:17:09

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:


> What if we first stick with the case where the chromatic unison 
> vector is unchanged in size -- so in this case, 2/7-comma meantone.


That would work, but it seems to me you are relying on the fact that 
the 7-et is relatively good. Suppose we take instead 512/375 as our 
chromatic unison vector, and 10/9 as our commatic unison vector. We 
then get the [7, 12, 17] system. If we temper out the 10/9 and keep 
the 512/375 value just, we want to solve the linear system

a = 1200
a-2*b+c = 0
9*a-b-3*c = 1200 * log_2(512/375) = 512/375 expressed in cents.

Solving this gives us 

a = 1200
b = 1980.12820
c = 2760.25604

The value for b is 77 cents flat from the 1200*(12/7) mark, well over 
the allowed amount; we would not have a MOS.

Message: 1454 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 18:05:46

Subject: Re: 19-consistent goodness of measurement systems

From: Carl

>I just tested the Maple zeta function routine, and it *does* seem
>to work for complex arguments--I had thought otherwise. I'm off to
>look at some graphs of et's, and will report anon.

Cool!
-C.

Message: 1455 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 02:17:26

Subject: Re: 19-consistent goodness of measurement systems

From: Carl

> Alas, 311 and 8539 are prime!


Gene, does this have anything to do with the zeta function?
I've long wondered why many good ETs are prime... 5, 7, 19,
31, 41, 53... notable exceptions being 12, 22, 34, 58, and 72...

I know Paul's been asking for your zeta stuff; did I miss it?

-Carl

Message: 1456 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 19:11:05

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

Whoops . . . I meant "at least one step size is larger than the 
chromatic unison vector" . . .

Message: 1457 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 03:43:17

Subject: Re: 19-consistent goodness of measurement systems

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl" <carl@l...> wrote:


> Gene, does this have anything to do with the zeta function?


Only in the sense that the big spikes in the zeta function occur at 
good et's. The zeta function doesn't have the mojo to do this any 
better than we can do in a much less abstruse way in any case.


> I know Paul's been asking for your zeta stuff; did I miss it?


I'd like to find a good program for computing zeta(s+it) and Z(t) 
(which you might say is zeta(1/2+it) made into a real analytic 
function of a real variable.) I don't want to go back and write my 
own again!

I think they may be availble for Mathematica, but I'd like to find a 
Maple version.

Message: 1458 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 19:10:12

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 

>> What if we first stick with the case where the chromatic unison 
>> vector is unchanged in size -- so in this case, 2/7-comma 
meantone.
> 

> That would work, but it seems to me you are relying on the fact 
that 
> the 7-et is relatively good. Suppose we take instead 512/375 as our 
> chromatic unison vector,


Whoa -- that's 539 cents!


> and 10/9 as our commatic unison vector. We 
> then get the [7, 12, 17] system. If we temper out the 10/9 and keep 
> the 512/375 value just, we want to solve the linear system
> 
> a = 1200
> a-2*b+c = 0
> 9*a-b-3*c = 1200 * log_2(512/375) = 512/375 expressed in cents.
> 
> Solving this gives us 
> 
> a = 1200
> b = 1980.12820
> c = 2760.25604
> 
> The value for b is 77 cents flat from the 1200*(12/7) mark, well 
over 
> the allowed amount; we would not have a MOS.


Carl brought this sort of thing up a while back (actually, he showed 
that some PBs are not CS, also using very large unison vectors). I 
replied that there needs to be some notion of "good" PBs. What's the 
weakest such condition we can come up with? If at least one step size 
is smaller than the chromatic unison vector . . . will that work?

Message: 1459 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 03:55:35

Subject: Re: 19-consistent goodness of measurement systems

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., genewardsmith@j... wrote:


> I'd like to find a good program for computing zeta(s+it) and Z(t) 
> (which you might say is zeta(1/2+it) made into a real analytic 
> function of a real variable.) I don't want to go back and write my 
> own again!


I just tested the Maple zeta function routine, and it *does* seem to 
work for complex arguments--I had thought otherwise. I'm off to look 
at some graphs of et's, and will report anon.

Message: 1460 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 19:16:33

Subject: Re: EDO consistency and accuracy tables (was: A little research...)

From: Paul Erlich

--- In tuning-math@y..., BobWendell@t... wrote:

> Gene said:

>> I think microtonalists should use whatever scale suits them; 
> however 

>> it would be well if they understood the structure of the system 
> they 

>> intend to use.
> 
> Bob answers:

> Thanks, Gene, for your complete comments and not just this quote 
from 
> it. Regarding the above quote from you, see my response in post 
#880 
> to Paul Erlich and Paul's comments just below that response. 
> 
> I'm interested in 72-tET because it is reasonably accurate in its 
> approximation of just intervals, is strong in consistency, and 
> reduces the set of pitches available for composition to a finite 
> number that is conceptually manageable with simple arithmetic you 
can 
> do quickly in your head. 
> 
> Given time and experience with it, you could learn to think 
> comprehensively in terms of it and its implications while in the 
act 
> of composing. These criteria comprise the essential motivations for 
> my quest.  


Sounds great! If you look back a few months in the tuning list 
archives, you will see a great deal of discussion on 72-tET, and 
you'll see me advocating it as a certain sort of "standard" . . . I'm 
sure you'll find some useful information in all that.

Message: 1461 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 08:46:46

Subject: The Riemann Zeta function and n-et's

From: genewardsmith@xxxx.xxx

Let z = s + i*t be a complex number with real part s and imaginary 
part t. If s>1 we can define a function

zeta(z) = sum_n n^(-z),

where n runs over the positive integers. This function analytically 
continues over the whole complex plane, giving a meromorphic function 
with a single simple pole at z=1. The sum is absolutely convergent 
for s>1, and we may pass in this case to a corresponding infinite 
product

zeta(z)= prod_p (1 - p^(-z))^(-1),

where the product is taken over all primes p. This is the key to the 
zeta function--it encodes information about the primes. If we take 
logarithms, we get

ln(zeta(z))= sum_p -ln(1-p^(-z)) = sum_p ln(1+p^z + p^(2z)+p^3z+...)

The real part of this is the log of the absolute value

Re(ln(zeta(z)) = ln(|zeta(z)|) = sum_p ln(|1-p^(-z)|).

If we fix s and let t vary, then each term of the above sum becomes a 
periodic function of t with period 2 pi / ln(p). If we look at the 
first-order approximation, we have 

ln(1+p^(-z)) ~ p^(-z) = p^(-s) (cos(ln(p) t) - i sin(ln(p) t)),

so that the real part of ln(zeta(z)) is approximately

sum_p p^(-s) cos(ln(p) t)

If we rescale by setting t = 2 pi n /ln(2), we get a sum weighted by 
factors of 1/p^s of cosine functions with periods of log_2(p). If n 
is a real number near a good et (e.g. n=12.018) then for the smallest 
primes, which have the highest weight, we will simultaneously be near 
a peak value of several of these cosine functions. The functions we 
are actually summing are also periodic but not exactly cosines; they 
in fact are an improvement, which take into account the powers of the 
prime p and so sharpen the peaks near their maximum, particularly for 
the smaller primes where we are especially concerned to accurately 
represent prime powers.

Since the infinite product converges when s>=1, we are justified in 
taking values of 
t=2 pi n /ln(2) which give relatively high maximums for the fixed 
value of s as representing good scale divisions. As s goes to 
infinity, this becomes increasingly a matter of finding good values 
for the fifth, which we can much more easily accomplish via a 
continued fraction. More interesting are the cases with smaller 
values of s, as these give more weight to the larger primes and prime 
powers.

It is a little harder to justify taking s into the critical strip 
between 0 and 1. However, the fact of analytic continuation and the 
Riemann-Siegel formula helps to make sense of this also, at least up 
to the critical line s=1/2; this is particularly the case if the 
Riemann hypothesis is true, in which case ln(zeta(z)) is analytic in 
the strip up to the critical line. We have a functional equation 
relating values of s to values of 1/2-s, so we don't want to push it 
past the critical line in any case. When we continue past s=1, we may 
follow a line of steepest ascent up to a high value of absolute value 
for zeta, and particularly when we get a good et which does well for 
a number of primes we might expect to reach a high maximum.

We may adjust the zeta function along the critical line by setting

Z(t) = zeta(1/2 + i t) pi^(-i t) Gamma(1/4 + i t/2)/|Gamma(1/4 + i 
t/2)|

this makes Z into a real function of the real variable t, whose 
absolute value is the same as that of zeta(1/2+i t). Here Gamma is 
the Gamma function, where Gamma(z) = (z-1)! for complex z.

We have an approximate formula for Z, 

Z(t) ~ 2 sum_{n <= L} cos(t ln(n) - theta(t))/sqrt(n)

where L = sqrt(t / 2 pi) and theta has an asymptotic expression

theta(t) ~~ (t/2) ln(t / 2 pi) - t/2 - pi/8 + 1/ (48 t) + 7/(5760 
t^3) + ...

which is extremely accurate for our purposes.

We see that if theta(t) is a multiple of pi, we have again a weighted 
sum of cosines. Hence we are justified at looking at maximums of Z(t) 
particularly at "Gram points", which are points where theta(t) is an 
integer multiple of pi and so cos(theta(t))=+-1, sin(theta(t))=0. If 
we divide theta by pi and set r = t / 2 pi, we get

f(r) = theta(t / 2 pi)/pi ~ r ln(r) - r - 1/8

and Gram points will be integer values of f(r). We can then use 
Newton's method to find the Gram point near a value of r where f(r) 
is close to an integer G. In this case, if we ignore the terms in 
inverse powers of r in the asymptotic formula we simply need iterate

r' = (G + r + 1/8) / ln(r)

If n is the number of steps in an n-et, then r = n/ln(2). For 
example, if n=12 then r=17.32... and f(r) = 31.927..., so we pick the 
Gram point where G=32. Applying Newton's method, we get an adjusted 
value r=17.337... which corresponds to n=12.017..., where octaves are 
flat by 1.764... cents. We may call such a tuning a Gram tuning, and 
it is interesting to consider what the flatness or sharpness of the 
Gram tuning for a given et is telling us, and whether they make 
practical sense.

I found an amazing applet on the web, which will graph Z(t) for 
anyone who wants to investigate this. (Don't you just love Java?) 
Unfortunately, it isn't set up for microtonalists, but if you 
multiply your et n by the magic number 2 pi / ln(2), you get a value 
of t which corresponds to that et and around which it is interesting 
to consider the graph. Maybe if we ask nicely he will put up a 
version for music theorists; he might be interested by the interest!

The url is:

Riemann Hypothesis in a Nutshell * [with cont.]  (Wayb.)

Message: 1463 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 21:27:10

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl

>/.../ What's the weakest such condition we can come up with? If
>at least one step size is smaller than the chromatic unison
>vector . . . will that work?


Well, in this non-CS example from way-back,

|4 -1|
|0  2| = 8
1/1  135/128  9/8  5/4  45/32  3/2  27/16  15/8

The smallest 2nd is 135:128, and the chromatic UV is 32:25.


>Whoops . . . I meant "at least one step size is larger than the
>chromatic unison vector" . . .


The largest 2nd is 9:8, so this condition is not met, though
still not 100% on why the 81:80 can't be chromatic (isn't this
just a matter of choice?).  IIRC, we agreed way back that all
steps being larger than all vectors would give CS, so this is
indeed weaker.

That's non-CS for you.  Not exactly clear on what property
Gene has found.

-Carl

Message: 1464 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 21:37:51

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., "Carl" <carl@l...> wrote:

>> /.../ What's the weakest such condition we can come up with? If
>> at least one step size is smaller than the chromatic unison
>> vector . . . will that work?
> 

> Well, in this non-CS example from way-back,
> 
> |4 -1|
> |0  2| = 8
> 1/1  135/128  9/8  5/4  45/32  3/2  27/16  15/8
> 
> The smallest 2nd is 135:128, and the chromatic UV is 32:25.
> 

>> Whoops . . . I meant "at least one step size is larger than the
>> chromatic unison vector" . . .
> 

> The largest 2nd is 9:8, so this condition is not met,


Cool . . . so my proposal looks good so far.


> though
> still not 100% on why the 81:80 can't be chromatic (isn't this
> just a matter of choice?).


You haven't tempered any unison vectors out yet, so at this point, 
there's really no difference between commatic and chromatic. If you 
called the 81:80 chromatic and 25:16 commatic, you'd have to temper 
out the 25:16 first before looking at the step sizes.

Message: 1465 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 21:34:48

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl

[I wrote...]

>That's non-CS for you.  Not exactly clear on what property
>Gene has found.


Sorry for the 3rd-person, Gene!

[Gene wrote...]

>The value for b is 77 cents flat from the 1200*(12/7) mark, well
>over the allowed amount; we would not have a MOS.


So it's MOS, then!

-Carl

Message: 1466 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 21:55:27

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl

>> >he largest 2nd is 9:8, so this condition is not met,
> 

> Cool . . . so my proposal looks good so far.


Well, to me the key thing about the Hypothesis is that L-s
is the chromatic vector.  So if this proposal gives MOS,
then the hypothesis is true.  But it seems there are scales
with more than two sizes of 2nd that meet the proposal
condition...


>> though still not 100% on why the 81:80 can't be chromatic (isn't
>> this just a matter of choice?).
> 

>You haven't tempered any unison vectors out yet, so at this point, 
>there's really no difference between commatic and chromatic. If you 
>called the 81:80 chromatic and 25:16 commatic, you'd have to temper 
>out the 25:16 first before looking at the step sizes.


Oh, right.  It blows my mind that this could work, or that anybody
could think of it.

-Carl

Message: 1467 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 22:39:57

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:


> Carl brought this sort of thing up a while back (actually, he 
showed 
> that some PBs are not CS, also using very large unison vectors). I 
> replied that there needs to be some notion of "good" PBs. What's 
the 
> weakest such condition we can come up with? If at least one step 
size 
> is smaller than the chromatic unison vector . . . will that work?


I don't know what the weakest condition is, and I haven't thought 
about the one you propose above. I'm not sure what the point of it is-
-a step of the 7-et is larger than 25/24, after all.

However, the point of what I was sketching out before was that it is 
certainly possible to come up with conditions which make sense and 
suffice to produce a MOS.

What's CS?

Message: 1468 - Contents - Hide Contents

Date: Mon, 27 Aug 2001 22:55:23

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl

--- In tuning-math@y..., genewardsmith@j... wrote:

> What's CS?


The property that every interval in a scale appears in only
one interval class.  For example, 3:2 appears only as a 5th
in the diatonic scale... but in 12-tET, the tritone appears
as both a 4th and a 5th, so the diatonic scale in 12-tET is
non-CS.

CS is our acronym for Constant Structures.  ...You don't
want to know.  :)

Actually, terms like MOS and CS come to us from Erv Wilson,
who admits he has trouble naming the things he thinks about,
and suggests we call them whatever we like.  But, names seem
to stick.

-Carl

Message: 1470 - Contents - Hide Contents

Date: Tue, 28 Aug 2001 16:52:16

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., BobWendell@t... wrote:


> In Just Intonation, these pitches form intervals with C that are 
not 
> equal. 


Your definition said every interval appears in only one class--
however, these are two different intervals. Naturally, they will be 
represented as different in many systems, and will be the same only 
in those systems that has their ratio in the kernel.

Message: 1471 - Contents - Hide Contents

Date: Tue, 28 Aug 2001 17:47:25

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

>>> >hat's CS?
>> 

>> The property that every interval in a scale appears in only
>> one interval class.  For example, 3:2 appears only as a 5th
>> in the diatonic scale... but in 12-tET, the tritone appears
>> as both a 4th and a 5th, so the diatonic scale in 12-tET is
>> non-CS.
>

>It seems to me that in a 12-et, a tritone would always be 6 steps. 
>Can you clarify?


Scale steps.  What recent threads have called "steps" are
actually 2nds.  Then, there's 3rds (major and minor), etc.

Here's what Rothenberg calls a "interval matrix" for the
diatonic scale,

           1sts 2nds 3rds 4ths 5ths 6ths 7ths 8ths
    ionian    0    2    4    5    7    9   11   12
    dorian    0    2    3    5    7    9   10   12
  phrygian    0    1    3    5    7    8   10   12
    lydian    0    2    4   [6]   7    9   11   12
mixolydian    0    2    4    5    7    9   10   12
   aeolian    0    2    3    5    7    8   10   12
   locrian    0    1    3    5   [6]   8   10   12

The tritone is what R. calls an "ambiguous interval".
CS is equivalent to no ambiguous intervals.

In meantone, the augmented 4th is smaller than the
diminished 5th, so the meantone diatonic is CS.

In Pythagorean tuning, the aug. 4th is larger than
the dim. 5th, so it is also CS.  But Rothenberg would
call this scale "improper", since its scale steps
overlap in interval space.

When certain other conditions are met, R. claims
proper scales make possible a compositional style in
which melodies may be transposed across the modes
of a scale without loosing their identity.

-Carl

Message: 1472 - Contents - Hide Contents

Date: Tue, 28 Aug 2001 17:53:55

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

>> >n Just Intonation, these pitches form intervals with C that are 
>> not equal. 
> 

>Your definition said every interval appears in only one class--
>however, these are two different intervals. Naturally, they will be 
>represented as different in many systems, and will be the same only 
>in those systems that has their ratio in the kernel.


Just for the record, that was Bob, not me.

"Interval class" is just a bad way to say "scale step".  "Every
interval" is just a bad way to say "every acoustic interval".
Does that help?  See also my previous message in this thread.

Gene, was it ever decided if a kernel is equivalent to a set
of unison vectors, as we use them?

-Carl

Message: 1473 - Contents - Hide Contents

Date: Tue, 28 Aug 2001 18:38:08

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 

>> Carl brought this sort of thing up a while back (actually, he 
> showed 

>> that some PBs are not CS, also using very large unison vectors). 
I 
>> replied that there needs to be some notion of "good" PBs. What's 
> the 

>> weakest such condition we can come up with? If at least one step 
> size 

>> is smaller than the chromatic unison vector . . . will that work?
> 

> I don't know what the weakest condition is, and I haven't thought 
> about the one you propose above. I'm not sure what the point of it 
is-
> -a step of the 7-et is larger than 25/24, after all.


You must have missed the message where I corrected this -- I 
meant "at least one step size is larger than the chromatic unison 
vector".

> 
> However, the point of what I was sketching out before was that it 
is 
> certainly possible to come up with conditions which make sense and 
> suffice to produce a MOS.
> 
> What's CS?


Every specific interval size is always subtended by the same number 
of steps. Seems to be synonymous with "good" PBs in the untempered JI 
case.

Message: 1474 - Contents - Hide Contents

Date: Tue, 28 Aug 2001 18:44:10

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., BobWendell@t... wrote:

> In 12-tET, six steps can be used "enharmonically" to represent 
either 
> an augmented fourth or a diminished fifth. If we think in the key 
of 
> C, this implies that the sixth step above C can be used as either 
an 
> F# or a Gb. 
> 
> In Just Intonation, these pitches form intervals with C that are 
not 
> equal. In 12-tET, the irrational approximation of both intervals 
(sq 
> root of 2) lies between F# and Gb. On the other hand, in Just 
> Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and Gb is 
> 36/25 (1.44)of C.


In Just Intonation there are actually several possible ratios for F#, 
as well as for Gb. So for these purposes, a well-defined tuning such 
as Pythagorean or meantone would have been better.

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