Second question about vals

Thanks all for the help. I'm starting to realize how deep this goes, really.

Another question: Let's say you have a 3-limit val,

[<1, 2], <0 -1]>

So here we have Pythagorean tuning. Here's meantone, a 5-limit expansion of this:

[<1, 2, 4], <0, -1, -4]>

Here's septimal meantone, a 7-limit expansion:

[<1, 2, 4, 7], <0, -1, -4, -10]>

Here's one of the 11-limit expansions:

[<1, 2, 4, 7, 11], <0, -1, -4, -10, -18]>

So, as you all know, the first few numbers of the val will stay the same each time.

My question, then is this: I have seen it proposed that using the zeta function can, in some sense, get us away from the concept of limits entirely then. So each of those vals then becomes a well-ordered infinite set, looking something like this:

[<1, 2, ..........], <0, -1, ..........]>

or, more precisely,

[<p_t(1), p_t(2), p_t(3), ...], <g_t(1), g_t(2), g_t(3), ...]>

Where p_t is a generating function for the period of this set, and g_t is a generating function for the generator of this set.

My question is, then:

What do these functions look like? They branch off at different points to yield different children of pythagorean tuning - how are the branches represented in these functions?

I assume whatever order there is that is psychoacoustically relevant would have to do with the distribution of the primes, so the Riemann hypothesis would be involved here. So then, could this generalize the concept of a val for w-limit spaces?

Mike wrote:

>Thanks all for the help. I'm starting to realize how deep this goes, really.
>
>Another question: Let's say you have a 3-limit val,
>
>[<1, 2], <0 -1]>

That looks like a pair of vals, but the notation is weird.
[<1 2], <0 -1]] would be more normal or (<1 2| <0 -1|) but
not what you wrote. Also it's much easier to read if you
put them on separate lines

<1 2|
<0 -1|

>So here we have Pythagorean tuning. Here's meantone, a 5-limit
>expansion of this:
>
>[<1, 2, 4], <0, -1, -4]>
>
>Here's septimal meantone, a 7-limit expansion:
>
>[<1, 2, 4, 7], <0, -1, -4, -10]>
>
>Here's one of the 11-limit expansions:
>
>[<1, 2, 4, 7, 11], <0, -1, -4, -10, -18]>
>
>So, as you all know, the first few numbers of the val will stay the
>same each time.
>
>My question, then is this: I have seen it proposed that using the zeta
>function can, in some sense, get us away from the concept of limits
>entirely then. So each of those vals then becomes a well-ordered
>infinite set, looking something like this:
>
>[<1, 2, ..........], <0, -1, ..........]>
>
>or, more precisely,
>
>[<p_t(1), p_t(2), p_t(3), ...], <g_t(1), g_t(2), g_t(3), ...]>
>
>Where p_t is a generating function for the period of this set, and g_t
>is a generating function for the generator of this set.
>
>My question is, then:
>
>What do these functions look like? They branch off at different points
>to yield different children of pythagorean tuning - how are the
>branches represented in these functions?
>
>I assume whatever order there is that is psychoacoustically relevant
>would have to do with the distribution of the primes, so the Riemann
>hypothesis would be involved here. So then, could this generalize the
>concept of a val for w-limit spaces?

Interesting question. Gene's applied the zeta function to *tuning*.
I'm not quite sure how, and by extension I'm very unsure how it might
apply to *mapping*.

Zeta function aside, presumably you're interested in branching
by the best extensions only. Gene's got this page
http://xenharmonic.wikispaces.com/Comma+sequences
which seems to be blank at the moment...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> Interesting question. Gene's applied the zeta function to *tuning*.
> I'm not quite sure how, and by extension I'm very unsure how it might
> apply to *mapping*.

Not in a way which strikes me as useful: taking the zeta tuning for two equal temperaments, you could use them to generate corresponding vals, and then stick them together and get temperaments, but the result wouldn't mean much beyond some point.

On Fri, Feb 4, 2011 at 10:53 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> > Interesting question. Gene's applied the zeta function to *tuning*.
> > I'm not quite sure how, and by extension I'm very unsure how it might
> > apply to *mapping*.
>
> Not in a way which strikes me as useful: taking the zeta tuning for two equal temperaments, you could use them to generate corresponding vals, and then stick them together and get temperaments, but the result wouldn't mean much beyond some point.

Why do you say that? It would be awesome if we could say that meantone
was f_g = sin(431!x^2+1/x - atan(x^x!)), or something. I also think it
might be useful for all of the stuff that Igs and I have been working
on in terms of assigning MOS's to entries in the scale tree, because
ideally we don't want to have to think in terms of prime-limit at all.

-Mike

On 5 February 2011 08:00, Mike Battaglia <battaglia01@gmail.com> wrote:

> Why do you say that? It would be awesome if we could say that meantone
> was f_g = sin(431!x^2+1/x - atan(x^x!)), or something. I also think it
> might be useful for all of the stuff that Igs and I have been working
> on in terms of assigning MOS's to entries in the scale tree, because
> ideally we don't want to have to think in terms of prime-limit at all.

It's possible to define a val without a prime limit. The simplest way
is to round each prime to the nearest whole number of steps -- which
gives Gene's patent val. Another way that correlates better with the
optimal mappings is to choose the best mapping of each prime in the
light of the mappings of the previous primes you've already chosen.
This assumes the small primes come first, because alternative mappings
of them are less likely to be valid.

You can even define objects like this in computer code. What you need
is a lazy list. In Scheme, you write a recursive function to generate
a list. In Python, you use a generator. In other object oriented
languages you can subclass the list type. You can then use these
objects to calculate the generator-period mappings and as long as you
only call a finite number of primes you can display them.

You won't always get the right mappings for subsets. If you want to
ignore 3 and map 9 to an odd number of steps that obviously won't
work. But if you're after a specific limit you should specify that
limit.

Measuring the errors and optimizing the tunings is where you have a
real problem. Usually these operations mean iterating over all the
primes. You need to find functions that are known to converge as the
prime limit approaches infinity. The Zeta tuning may be one such.
I'm dubious that anything like this has musical relevance because if
you know what primes you're working with it's better to get them in
tune and not worry about the rest.

Graham

On 5 February 2011 07:22, Carl Lumma <carl@lumma.org> wrote:
> Mike wrote:
>>
>>Another question: Let's say you have a 3-limit val,
>>
>>[<1, 2], <0 -1]>
>
> That looks like a pair of vals, but the notation is weird.
> [<1 2], <0 -1]] would be more normal or (<1 2| <0 -1|) but
> not what you wrote.  Also it's much easier to read if you
> put them on separate lines

It isn't a val. It's a way of writing a square mapping matrix in
bra/ket form. It contains two vals, the mappings for the period and
generator. They're placed in a ket as [period, generator>. If you
take a 3-limit vector (monzo) for, say, a perfect fourth, it's also a
ket: [2, -1>. You multiply it by the mapping to get

[<1, 2], <0, -1]> [2, -1>
= [<1, 2 | 2, -1>, <0, -1 | 2, -1>>

The bras and kets tell you which way to form the brackets. The result is

= [2-2, 0+1> = [0, 1>

This confirms that the fourth maps to a generator, which is [0, 1> as
a ket on the period/generator lattice.

Graham

On Sat, Feb 5, 2011 at 3:58 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> It's possible to define a val without a prime limit. The simplest way
> is to round each prime to the nearest whole number of steps -- which
> gives Gene's patent val.

Right, I thought that might be a good way to start doing this. So if
you do this, do different meantones correlate to different
combinations of meantone vals? Like, say, 7&12 will yield a different
19-limit meantone or whatever than 12&19, and that's one way of
distinguishing them?

> Another way that correlates better with the
> optimal mappings is to choose the best mapping of each prime in the
> light of the mappings of the previous primes you've already chosen.
> This assumes the small primes come first, because alternative mappings
> of them are less likely to be valid.

How would you work something like this out? So for instance, I've
always preferred to hear 17 as a really sharp meantone than as a
dicot. The former corresponds to 81/80 vanishing, out of tune though
it may be, and the latter corresponds to 25/24 vanishing, which is a
much badder interval to vanish.

> You can even define objects like this in computer code. What you need
> is a lazy list. In Scheme, you write a recursive function to generate
> a list. In Python, you use a generator. In other object oriented
> languages you can subclass the list type. You can then use these
> objects to calculate the generator-period mappings and as long as you
> only call a finite number of primes you can display them.

Right, that's the exact kind of thing I'm thinking about. Do functions
like these end up fitting into some well-known mathematical class of
functions? I assume there will be a round() in there somewhere, if
we're building linear temperaments off of patent vals. Or perhaps some
kind of continued fraction approximation is the way to go.

> You won't always get the right mappings for subsets. If you want to
> ignore 3 and map 9 to an odd number of steps that obviously won't
> work. But if you're after a specific limit you should specify that
> limit.

Why wouldn't that work?

> Measuring the errors and optimizing the tunings is where you have a
> real problem. Usually these operations mean iterating over all the
> primes. You need to find functions that are known to converge as the
> prime limit approaches infinity. The Zeta tuning may be one such.
> I'm dubious that anything like this has musical relevance because if
> you know what primes you're working with it's better to get them in
> tune and not worry about the rest.

I just like to understand things without worrying about the direct
application at the moment. Sometimes the dots connect later. In this
case, it came up as we tried to delve further into the scale tree. You
seem to have a lot of this mapped out (haha, pun (haha, pun (...))).
We're trying to figure out how to apply it to abstract entities like
3L1s. You've already figured out how to apply it to abstract entities
like 12-equal, but just taking the patent vals for 3 and 4 and
combining them isn't as sensible for something like 3L1s, since the
generator might be 5/4, 11/9, 6/5, or whatever.

In some sense temperaments are lumped together for 3L1s, and then get
less lumped together as you go further down the scale tree to 4L3s.
3L1s could have a generator of anything from 6/5 to 5/4, but these are
very obviously split off for 4L3s and 3L4s. And then 4L3s is
reasonably either dicot or magic (in the 5-limit), which then splits
off very obviously for 7L4s and 4L7s.

We first thought "oh, let's temper them together," so that since 3L1s
could have either a 6/5 or a 5/4 generator, it makes sense to say that
the generator is both and hence 3L1s gets the "dicot" name. But this
doesn't work for something like 1L2s, where, let's say, the generator
could be anything from 0 to 400 cents. So it could be 25/24, 16/15,
10/9, 9/8, 6/5, 5/4, etc. You can't temper them all together, because
6/5 and 5/4 differ by 25/24, so you can't temper 25/24 out and make it
also be the generator. Well, you can, but it's silly. The same problem
applies for a lot of 1Lxs scales, not just ones as pathological as
1L2s.

So we're trying to find ways to generalize the concept of a patent val
for 2d scales, which is a lot trickier since they tend to differ in
size. I thought that perhaps, in the generating function for some
w-bival <a b c d e ...| <a' b' c' d' e' ...|, there might be some
pattern there to generalize the concept of the patent val for any
limit and for a scale like 3L1s.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> How would you work something like this out? So for instance, I've
> always preferred to hear 17 as a really sharp meantone than as a
> dicot. The former corresponds to 81/80 vanishing, out of tune though
> it may be, and the latter corresponds to 25/24 vanishing, which is a
> much badder interval to vanish.

Why does it need to be either? 17edo does a decent job for the 2.3.11/7 subgroup, and you can consider, if you like, that 896/891 is being tempered out in this subgroup and you aren't getting either meantone or dicot.

On 2/5/2011 9:04 AM, Mike Battaglia wrote:
> On Sat, Feb 5, 2011 at 3:58 AM, Graham Breed<gbreed@gmail.com> wrote:
>>
>> It's possible to define a val without a prime limit. The simplest way
>> is to round each prime to the nearest whole number of steps -- which
>> gives Gene's patent val.
>
> Right, I thought that might be a good way to start doing this. So if
> you do this, do different meantones correlate to different
> combinations of meantone vals? Like, say, 7&12 will yield a different
> 19-limit meantone or whatever than 12&19, and that's one way of
> distinguishing them?

Yes, in 19-limit, 7&12 would be this temperament:

[<1, 2, 4, 2, 6, 2, 2, 3], <0, -1, -4, 2, -6, 4, 5, 3]]

and this is 12&19:

[<1, 2, 4, 7, 6, 2, 2, 3], <0, -1, -4, -10, -6, 4, 5, 3]]

5&7:

[<1, 2, 4, 2, 3, 5, 2, 3], <0, -1, -4, 2, 1, -3, 5, 3]]

19&31:

[<1, 2, 4, 7, -2, 10, 2, 3], <0, -1, -4, -10, 13, -15, 5, 3]]

But there are other 19-limit extensions of meantone that don't fit into that system, e.g.

[<1, 2, 4, 7, 11, 10, 2, 3], <0, -1, -4, -10, -18, -15, 5, 3]]

[<1, 2, 4, 7, 6, 10, 2, 3], <0, -1, -4, -10, -6, -15, 5, 3]]

[<1, 2, 4, 7, -2, 2, 2, 8], <0, -1, -4, -10, 13, 4, 5, -9]]

[<1, 2, 4, 7, 11, 15, 2, 3], <0, -1, -4, -10, -18, -27, 5, 3]]

Whether any of those are actually useful for any musical purposes is a harder question to answer, since there are so many of them when you get to 19-limit that it would be impractical to try them all in actual music. With very few exceptions, most of the familiar rank 2 temperaments can be defined with patent vals, and the few exceptions can be labeled with specific names (e.g. hedgehog, superpelog).

Graham wrote:

>>>Another question: Let's say you have a 3-limit val,
>>>
>>>[<1, 2], <0 -1]>
>>
>> That looks like a pair of vals, but the notation is weird.
>> [<1 2], <0 -1]] would be more normal or (<1 2| <0 -1|) but
>> not what you wrote. Also it's much easier to read if you
>> put them on separate lines
>
>It isn't a val. It's a way of writing a square mapping matrix in
>bra/ket form. It contains two vals, the mappings for the period and
>generator. They're placed in a ket as [period, generator>. If you
>take a 3-limit vector (monzo) for, say, a perfect fourth, it's also a
>ket: [2, -1>. You multiply it by the mapping to get
>
>[<1, 2], <0, -1]> [2, -1>
>= [<1, 2 | 2, -1>, <0, -1 | 2, -1>>

I see. Barely, through the commas. Which should never be used
to separate anything.

-Carl

On Sat, Feb 5, 2011 at 12:58 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> Why does it need to be either? 17edo does a decent job for the 2.3.11/7 subgroup, and you can consider, if you like, that 896/891 is being tempered out in this subgroup and you aren't getting either meantone or dicot.

I just mean that, if you I to pick one, that the supermajor triad
sounds like 4:5:6 to me more than the neutral triad, which sounds like
a neutral triad. But what kind of useful chords can you construct out
of 2.3.11/7?

-Mike

On Sat, Feb 5, 2011 at 1:37 PM, Herman Miller <hmiller@io.com> wrote:
>
> Whether any of those are actually useful for any musical purposes is a
> harder question to answer, since there are so many of them when you get
> to 19-limit that it would be impractical to try them all in actual
> music. With very few exceptions, most of the familiar rank 2
> temperaments can be defined with patent vals, and the few exceptions can
> be labeled with specific names (e.g. hedgehog, superpelog).

What is the mapping for the generator with superpelog? A really sharp
8/7, looks like?

-Mike

On 2/5/2011 9:13 PM, Mike Battaglia wrote:
> On Sat, Feb 5, 2011 at 1:37 PM, Herman Miller<hmiller@io.com> wrote:
>>
>> Whether any of those are actually useful for any musical purposes is a
>> harder question to answer, since there are so many of them when you get
>> to 19-limit that it would be impractical to try them all in actual
>> music. With very few exceptions, most of the familiar rank 2
>> temperaments can be defined with patent vals, and the few exceptions can
>> be labeled with specific names (e.g. hedgehog, superpelog).
>
> What is the mapping for the generator with superpelog? A really sharp
> 8/7, looks like?

[<1, 2, 1, 3], <0, -2, 6, -1]>
The generator is around 260 cents, closer to 7/6.

One possible generator tuning is 1202.95, 261.20. This is based on optimizing the superparticular intervals, which ends up sounding better than the 1208.92, 261.88 tuning from optimizing the primes.

On 5 February 2011 18:04, Mike Battaglia <battaglia01@gmail.com> wrote:

> Right, I thought that might be a good way to start doing this. So if
> you do this, do different meantones correlate to different
> combinations of meantone vals? Like, say, 7&12 will yield a different
> 19-limit meantone or whatever than 12&19, and that's one way of
> distinguishing them?

Oh, you wanted 19-limit? I did the 7-limit overnight. 7-equal is
inconsistent in the 7-limit. The patent val (nearest approximation of
primes) doesn't match the optimal TE mapping. The two vals are:

<7, 11, 16, 19] (optimal)
<7, 11, 16, 20] (patent)

Here are some meantones you get by combining the optimal mappings:

7&5 --> Sharptone
7&12 --> Meantone
5&12 --> Dominant
7&19 --> Meantone

With patent vals:

7&5 --> Dominant
7&12 --> Dominant
5&12 --> Dominant
7&19 --> Flattone

These are different results but I don't know how you decide which is
better. You can always get the usual meantone with 12&19, 19&31, and
so on.

>> Another way that correlates better with the
>> optimal mappings is to choose the best mapping of each prime in the
>> light of the mappings of the previous primes you've already chosen.
>> This assumes the small primes come first, because alternative mappings
>> of them are less likely to be valid.
>
> How would you work something like this out? So for instance, I've
> always preferred to hear 17 as a really sharp meantone than as a
> dicot. The former corresponds to 81/80 vanishing, out of tune though
> it may be, and the latter corresponds to 25/24 vanishing, which is a
> much badder interval to vanish.

You need to do some algebra. There's a quadratic formula that gives
you the TE error. You make the new prime mapping an unknown, and
solve for it. That gives a floating point result. So you round it
off both ways and check which is better. (You might even be able to
prove that rounding to the nearest integer always works.)

>> You can even define objects like this in computer code. What you need
>> is a lazy list. In Scheme, you write a recursive function to generate
>> a list. In Python, you use a generator. In other object oriented
>> languages you can subclass the list type. You can then use these
>> objects to calculate the generator-period mappings and as long as you
>> only call a finite number of primes you can display them.
>
> Right, that's the exact kind of thing I'm thinking about. Do functions
> like these end up fitting into some well-known mathematical class of
> functions? I assume there will be a round() in there somewhere, if
> we're building linear temperaments off of patent vals. Or perhaps some
> kind of continued fraction approximation is the way to go.

Patent vals use a round(), yes. That means the mapping of each prime
is independent of the other primes, but you're giving primes a
privileged status.

>> You won't always get the right mappings for subsets. If you want to
>> ignore 3 and map 9 to an odd number of steps that obviously won't
>> work. But if you're after a specific limit you should specify that
>> limit.
>
> Why wouldn't that work?

The mapping of 9 is always double the mapping of 3. You can't double
any integer to get an odd number.

> I just like to understand things without worrying about the direct
> application at the moment. Sometimes the dots connect later. In this
> case, it came up as we tried to delve further into the scale tree. You
> seem to have a lot of this mapped out (haha, pun (haha, pun (...))).
> We're trying to figure out how to apply it to abstract entities like
> 3L1s. You've already figured out how to apply it to abstract entities
> like 12-equal, but just taking the patent vals for 3 and 4 and
> combining them isn't as sensible for something like 3L1s, since the
> generator might be 5/4, 11/9, 6/5, or whatever.

You need to choose two representative equal temperaments, maybe 3&4 or
4&7. They'll give you the mapping of the generator.

> In some sense temperaments are lumped together for 3L1s, and then get
> less lumped together as you go further down the scale tree to 4L3s.
> 3L1s could have a generator of anything from 6/5 to 5/4, but these are
> very obviously split off for 4L3s and 3L4s. And then 4L3s is
> reasonably either dicot or magic (in the 5-limit), which then splits
> off very obviously for 7L4s and 4L7s.

That's it. You need to go down the tree.

> We first thought "oh, let's temper them together," so that since 3L1s
> could have either a 6/5 or a 5/4 generator, it makes sense to say that
> the generator is both and hence 3L1s gets the "dicot" name. But this
> doesn't work for something like 1L2s, where, let's say, the generator
> could be anything from 0 to 400 cents. So it could be 25/24, 16/15,
> 10/9, 9/8, 6/5, 5/4, etc. You can't temper them all together, because
> 6/5 and 5/4 differ by 25/24, so you can't temper 25/24 out and make it
> also be the generator. Well, you can, but it's silly. The same problem
> applies for a lot of 1Lxs scales, not just ones as pathological as
> 1L2s.

You can't give one meaningful temperament name to such a wide range of tunings.

> So we're trying to find ways to generalize the concept of a patent val
> for 2d scales, which is a lot trickier since they tend to differ in
> size. I thought that perhaps, in the generating function for some
> w-bival <a b c d e ...| <a' b' c' d' e' ...|, there might be some
> pattern there to generalize the concept of the patent val for any
> limit and for a scale like 3L1s.

You can generalize the concept of patent val easily by taking two of
them. What's the problem? You can also generalize the concept of
best mappings, which may give better results.

Graham

On Sat, Feb 5, 2011 at 10:14 PM, Herman Miller <hmiller@io.com> wrote:
>
> On 2/5/2011 9:13 PM, Mike Battaglia wrote:
> > On Sat, Feb 5, 2011 at 1:37 PM, Herman Miller<hmiller@io.com> wrote:
> >>
> >> Whether any of those are actually useful for any musical purposes is a
> >> harder question to answer, since there are so many of them when you get
> >> to 19-limit that it would be impractical to try them all in actual
> >> music. With very few exceptions, most of the familiar rank 2
> >> temperaments can be defined with patent vals, and the few exceptions can
> >> be labeled with specific names (e.g. hedgehog, superpelog).
> >
> > What is the mapping for the generator with superpelog? A really sharp
> > 8/7, looks like?
>
> [<1, 2, 1, 3], <0, -2, 6, -1]>
> The generator is around 260 cents, closer to 7/6.

Yeah, but 7 is mapped to -1 generators...? I'm a little confused.

-Mike

On 2/6/2011 8:28 AM, Mike Battaglia wrote:
> On Sat, Feb 5, 2011 at 10:14 PM, Herman Miller<hmiller@io.com> wrote:
>>
>> On 2/5/2011 9:13 PM, Mike Battaglia wrote:
>>> On Sat, Feb 5, 2011 at 1:37 PM, Herman Miller<hmiller@io.com> wrote:
>>>>
>>>> Whether any of those are actually useful for any musical purposes is a
>>>> harder question to answer, since there are so many of them when you get
>>>> to 19-limit that it would be impractical to try them all in actual
>>>> music. With very few exceptions, most of the familiar rank 2
>>>> temperaments can be defined with patent vals, and the few exceptions can
>>>> be labeled with specific names (e.g. hedgehog, superpelog).
>>>
>>> What is the mapping for the generator with superpelog? A really sharp
>>> 8/7, looks like?
>>
>> [<1, 2, 1, 3],<0, -2, 6, -1]>
>> The generator is around 260 cents, closer to 7/6.
>
> Yeah, but 7 is mapped to -1 generators...? I'm a little confused.
>
> -Mike

49/48 is a unison vector. So both 8/7 and 7/6 are mapped to the same tempered interval.