Yahoo Tuning Groups Ultimate Backup tuning "n EDO is good in the p-limit"

I'm having a discussion offlist with someone about vals and mapping. I'm
pretty comfortable with the process of tempering a scale by using an
appropriate val -- usually the patent val, but it doesn't have to be.

(Notation note: To prevent confusion between an EDO number and a prime, I'm
going to write, e.g., prime 19 as 19/1 in some cases where it wouldn't
normally be needed.)

A couple of questions came out of that discussion.

1. Do some EDOs map some primes better than others? (We were talking about
19/1 in 31 EDO, so I'll use that as my example.)

2. How can I find out which EDOs are better than others for a given prime?

I think the answer to #1 is "Yes, BUT that's only relevant in the context of
a given val and a given scale." Here's why.

-----

Using a val means using a particular substitute for a prime. The 19-limit
patent val for 31 EDO is < 31 49 72 87 107 115 127 132 |. That means it
takes 132 steps of 31 EDO to reach 19/1, which is 5097.51 cents.

That's not true, of course. Each step of 31 EDO is 38.70967742 cents, 132
steps of which is 5109.677 cents. But we're pretending that the 132-step
figure is right. That means we're substituting 2^(5109.677/1200), or
19.13397238, in place of 19/1.

Because you're substituting a too-big number for 19/1, any interval with
19/1 on top will be a little bit sharp (e.g., 19/16 is 309.7 cents instead
of the just 297.5 cents). Any interval with 19/1 on bottom will be a little
bit flat. (Of course, the errors from other primes may add to or cancel out
some of the sharpness or flatness that comes from 19/1.)

If the 19.13397238 is too big for you, you can change the tuning by changing
the val. Take the patent val:
< 31 49 72 87 107 115 127 132 |
...and subtract 1 from the 19/1 place...
< 31 49 72 87 107 115 127 131 |
...and now you're saying that 131 steps = 19/1, which means you're
substituting 18.71089225 in place of 19/1. That's pretty flat, but maybe
it's what you want.

Good so far?

Okay. Given all of that, it seems to me that a large error in 19/1 will make
some EDOs worse than others. The 12-EDO patent-val value for 19/1 is
19.02731384, which is considerably better than the 31-EDO patent-val value.
Furthermore, there's a limit to how much you can adjust 31 EDO, because the
farther up you go from 132 or the farther down you go from 131 steps, the
more you're distorting 19/1.

Thus 12 seems like a better EDO than 31 for 19/1, generally speaking.

By the same logic (glossing over the calculations), 31 is a better EDO than
12 for 5/1, generally speaking (5.0023 vs. 5.0397).

However, I glossed over something very important: "The errors from other
primes may add to or cancel out some of the sharpness or flatness that comes
from 19/1." So even an EDO that's "not very good" for a particular prime can
give good results as long as the errors cancel in the particular intervals
you want.

A great example of that is 9/7 in 11 EDO using the val < 11 17 26 30 |. (The
patent val would have 31 in the 7/1 place.)

3/1 is 2.918960211, so 9/1 is 8.520328716. Yuck. And 7/1 is 6.622026239. But
they're both so flat that when you divide one by the other, the error
cancels out, and you get an almost-perfect 9/7. Similarly, it has a really
good 29/24 because 29 and 3 have about the same (large) error.

Going back to 19/1 in 31 EDO: 19/18 has a high error (116 or 77 cents
depending on the val, when the just value is 94 cents), but 19/17 maps
really well with the patent val (within a cent of just). So saying "31 EDO
isn't great for 19" depends on whether 19/17 or 19/18 -- or both -- is in
your scale.

The lesson for me is that I shouldn't really care about whether "n EDO is
good in the p-limit" except in the context of a particular just scale that
I'm mapping to it. When I'm looking for a starting point, it doesn't hurt to
get relatively low errors for the primes I'm interested to start off with;
but nothing matters until I map the specific intervals in my scale.

That strikes me as odd, since I seem to hear "n EDO is good in the p-limit"
relatively often.

-----

So how do I find out which EDOs and vals map to given primes with relatively
little error?

My first approach would be to go to Graham's temperament finder. (This might
just be a question for Graham.)

Going here:
http://x31eq.com/temper/pregular.html
...and plugging in 19 for the limit gives me EDOs 27eg, 31, 26, 29g, 19egh,
12f, 9p, 15g, 14cf, 24p.

But I've already shown that 31 has a worse value of 19/1 than 12 does.
Moreover, Graham doesn't measure "error", he measures "badness", which I'm
struggling to understand. As a result, I don't really think this is what I'm
looking for.

I'm inclined to think that I should put all of the primes of all of my
intervals into the "limit" field, and that would give me a good idea of what
EDOs would work; however, the Temperament Finder can't know which primes I'm
coupling together in my individual intervals, so it can't tell me what the
error would be for my particular scale.

So I guess I'm not sure how to determine which EDOs you should use for any
given prime or set of primes. Right now I'm looking at what commas I want to
temper, finding an EDO of sufficient size to reduce the error to an
undetectable level, and loading the file into Scala.

It certainly works, and maybe it's all I need. If so, I'm happy to keep on
keeping on. :)

Thanks in advance for any commentary,
Jake

🔗Mike Battaglia <battaglia01@...>

On Mon, Aug 15, 2011 at 12:45 AM, Jake Freivald <jdfreivald@...> wrote:
>
> Using a val means using a particular substitute for a prime. The 19-limit patent val for 31 EDO is < 31 49 72 87 107 115 127 132 |. That means it takes 132 steps of 31 EDO to reach 19/1, which is 5097.51 cents.
>
> That's not true, of course. Each step of 31 EDO is 38.70967742 cents, 132 steps of which is 5109.677 cents. But we're pretending that the 132-step figure is right. That means we're substituting 2^(5109.677/1200), or 19.13397238, in place of 19/1.
>
> Because you're substituting a too-big number for 19/1, any interval with 19/1 on top will be a little bit sharp (e.g., 19/16 is 309.7 cents instead of the just 297.5 cents). Any interval with 19/1 on bottom will be a little bit flat. (Of course, the errors from other primes may add to or cancel out some of the sharpness or flatness that comes from 19/1.)
>
> If the 19.13397238 is too big for you, you can change the tuning by changing the val. Take the patent val:
> < 31 49 72 87 107 115 127 132 |
> ...and subtract 1 from the 19/1 place...
> < 31 49 72 87 107 115 127 131 |
> ...and now you're saying that 131 steps = 19/1, which means you're substituting 18.71089225 in place of 19/1. That's pretty flat, but maybe it's what you want.
>
> Good so far?

Very good! Exactly right.

> Okay. Given all of that, it seems to me that a large error in 19/1 will make some EDOs worse than others. The 12-EDO patent-val value for 19/1 is 19.02731384, which is considerably better than the 31-EDO patent-val value. Furthermore, there's a limit to how much you can adjust 31 EDO, because the farther up you go from 132 or the farther down you go from 131 steps, the more you're distorting 19/1.

You can also stretch or compress the octave slightly. TOP and TOP-RMS
(now called TE-Optimal I believe on Graham's temperament finder) are
ways to spread the error evenly out over all primes, even if it means
stretching the octave.

> Thus 12 seems like a better EDO than 31 for 19/1, generally speaking.

Yep.

> By the same logic (glossing over the calculations), 31 is a better EDO than 12 for 5/1, generally speaking (5.0023 vs. 5.0397).

Right.

> However, I glossed over something very important: "The errors from other primes may add to or cancel out some of the sharpness or flatness that comes from 19/1." So even an EDO that's "not very good" for a particular prime can give good results as long as the errors cancel in the particular intervals you want.

Sure, and also you find that flatter 3/1's and 5/1's make a flatter
7/1 sound better than a sharper one, which leads to situations where
the "best val" is not the same as the "patent val." This can be
formalized by saying that it's often the case that the patent val
isn't the one with the lowest TE error (or TOP error, etc). The val
with the lowest error usually also corresponds to which one sounds the
best, and it also usually corresponds to the above (though not always,
such as in 22-equal, where you have a slightly flat 5/4 and a slightly
sharp 3/2).

For example, the best val for 23-equal is not the patent val - the
best val is "23d," and the patent val is "23p":

http://x31eq.com/cgi-bin/rt.cgi?ets=23&limit=7

The one on top is best. So in this case, the val that gives the lowest
error in some sense over "all" intervals in the 7-limit isn't the one
in which you use the best direct approximation to 7/4. (Assuming the
theory works, which to my ears it does.)

> Going back to 19/1 in 31 EDO: 19/18 has a high error (116 or 77 cents depending on the val, when the just value is 94 cents), but 19/17 maps really well with the patent val (within a cent of just). So saying "31 EDO isn't great for 19" depends on whether 19/17 or 19/18 -- or both -- is in your scale.

I haven't worked this out but if you're saying that one val gives you
a better 19/18 and the other gives you a better 19/17, then this is
saying that 31-EDO is -inconsistent- in the 19-limit. Paul Erlich
really hates this. I tend to be more of the "just pick a val and close
your eyes and think no more" type, although in practice with this
22-equal guitar sometimes I adjust my mapping for 13 on the fly.

> The lesson for me is that I shouldn't really care about whether "n EDO is good in the p-limit" except in the context of a particular just scale that I'm mapping to it.

I guess there are two main realizations to make here:
1) When people say "n-EDO" at all, you have to realize that the phrase
"n-EDO" is code for some val <n a b c d e ...|. For example, <19 30
44| (the patent val for 19-equal) is great in the 5-limit, but <19 31
45| isn't so great. Whether or not you're talking about the patent val
is usually left open - actually when I talk about 17-equal I'm usually
talking about the 17c val - or maybe it's assumed that you're willing
to switch the mapping around on the fly like I do with 22-equal in the
13-limit. But an EDO isn't good in any limit at all until you put a
val on it.

2) Let's say that you're in the 5-limit and you want to add 19. So now
you'd be in the 2.3.5.19-limit, or the 2.3.5.19 subgroup, or the
2.3.5.19 subgroup of the 19-limit, depending on how pedantic you want
to be. If you know that you only want to use some interval in the
19-limit, like 19/17, then you'd be in the 2.3.5.19/17 subgroup. A val
might do a lot better for the 2.3.5.19/17 subgroup than it does in the
full 2.3.5.19 subgroup. For example, 11-equal is great in the
2.5/3.7.9.11 subgroup; not so great in the full 11-limit.

> Going here:
> http://x31eq.com/temper/pregular.html
> ...and plugging in 19 for the limit gives me EDOs 27eg, 31, 26, 29g, 19egh, 12f, 9p, 15g, 14cf, 24p.
>
> But I've already shown that 31 has a worse value of 19/1 than 12 does. Moreover, Graham doesn't measure "error", he measures "badness", which I'm struggling to understand. As a result, I don't really think this is what I'm looking for.

That's because the primes are weighted; 19 is assumed to be less
important than something like 3 is. The 19-limit includes a lot more
primes than just 19, and 31-equal handles most of them really well,
with 19 being more or less the worst.

> I'm inclined to think that I should put all of the primes of all of my intervals into the "limit" field, and that would give me a good idea of what EDOs would work; however, the Temperament Finder can't know which primes I'm coupling together in my individual intervals, so it can't tell me what the error would be for my particular scale.

Try this: http://x31eq.com/cgi-bin/pregular.cgi?limit=2.3.5.7.11.13.19%2F17&error=5.0

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

Mike, Gene, and Keenan, thanks for the replies.

Mike said:
> Sure, and also you find that flatter 3/1's and 5/1's make a
> flatter 7/1 sound better than a sharper one, which leads to
> situations where the "best val" is not the same as the "patent
> val."

Interesting. I knew that sometimes you wanted a different val, but
hadn't heard this reason. I had assumed it was because you'd want to,
say, slightly sharpen 7/1 to fix one interval, and slightly flatten
5/1 to compensate for the sharpening in another interval. And thanks
for the example of 23p -- examples help a lot. I'll go study that when
I get the chance.

> I haven't worked this out but if you're saying that one val gives
> you a better 19/18 and the other gives you a better 19/17,
> then this is saying that 31-EDO is -inconsistent- in the 19-limit.

That seems weird to me. It seems to me that everything would be
"inconsistent" by that definition, because the amount of error in a
given interval will always depend on the accuracy of the other primes
used. 19/18 in 31 EDO is bad because 19 is bad and 3 is good; 19/17 in
31 EDO is good because 19 is bad and 17 is bad in the same way. I
wouldn't say "11 EDO is inconsistent in the 3-limit", because 11 EDO
is terrible in the 3-limit -- it just happens to be terrible in the
same way that 29 is, which makes 29/24 work out well.

I guess, based on what we say later, that I could alternately say that
11 EDO is great in the 2.9/7.29/3 subgroup. (Though I confess that I
tend not to think of these low EDOs as temperaments as much as just
EDOs.)

> I tend to be more of the "just pick a val and close your eyes and
> think no more" type,

I can see that having its appeal... :)

> 1) When people say "n-EDO" at all, you have to realize that the
> phrase "n-EDO" is code for some val <n a b c d e ...|.

Yep, got that.

> 2) Let's say that you're in the 5-limit and you want to add 19. So
> now you'd be in the 2.3.5.19-limit, or the 2.3.5.19 subgroup, or
> the 2.3.5.19 subgroup of the 19-limit, depending on how
> pedantic you want to be. If you know that you only want to use
> some interval in the 19-limit, like 19/17, then you'd be in the
> 2.3.5.19/17 subgroup. A val might do a lot better for the
> 2.3.5.19/17 subgroup than it does in the full 2.3.5.19 subgroup.

That makes sense, especially with Gene's and Keenan's follow-up comments.

That also answers a different question I had once, which is why you'd
use fractions in subgroups or tonality diamonds. Although it could be
arbitrary, or artistic preference, or whatever, you might also be
building a scale that you want to use on a physical instrument, like a
22 EDO guitar. Because of the errors in 22 EDO, you might choose to
explore 2.5.19/13 because you know it'll be playable on your guitar.
(Not that that's a *good* example, and frankly I'm not seeing many
good examples, but it's an example.)

> That's because the primes are weighted;

Great, makes sense. Plugging in the subgroup, with fractions as
needed, should clear things up, then.

Thanks for the help.

Regards,
Jake

On 8/15/11, Mike Battaglia <battaglia01@...> wrote:
> On Mon, Aug 15, 2011 at 2:59 AM, Keenan Pepper <keenanpepper@...>
> wrote:
>>
>> But I could see subgroups being confusing for some people, for the
>> following reason. You might say "Oh, I don't want all possible 7-limit
>> intervals. The only three I care about are 7/6, 6/5, and 7/5, as well as
>> octaves." But in fact there is no subgroup containing these three
>> intervals other than the full 7 limit.
>
> 7/6 * 6/5 = 7/5, so this is actually the 2.5/3.7/3 subgroup...
>
> -Mike
>
>
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🔗Michael <djtrancendance@...>

Just a general observation...

Before I got in "trouble" with Jake for using a non-patent val mapping for 31EDO to map an interval nearing 19/18 in my scale...the problem, as I understand it...being that soon as a use a non-patent val, certain "regular temperament" musical consistencies IE "a neutral 3rd plus a neutral 3rd = a fifth" can become untrue as errors add up in such additions...and often you end up on EDO step too far or short during such "interval additions".

31EDO, on the other end, seems to have a good habit...that is, of "badly mapping" higher limit intervals to lower limit ones that still retain a not-unsimilar feel.

>"I had assumed it was because you'd want to, say, slightly sharpen 7/1 to fix one interval, and slightly flatten 5/1 to compensate for the sharpening in another interval."

Usually when I have two notes in a scale that form a flat interval relative to each other, I simply move the lower one down by a small amount and a higher one up (IE each one down or up by no more than 7 cents)...and then check to see if any of the notes, relative to those two notes, form a "no longer good" interval.

The thing/problem with using EDOs (at least ones which aren't very high, such as 31EDO) is that you often don't have the option of only moving by 7 cents...you end up moving by something over 20 cents...so your chance of messing up another interval is much higher.
Then again, since 31EDO is good at so many different limits (just about everything but 13 limit)...chances are even if an interval becomes inconsistent with the type of interval addition usually possible in musical theory (see above), it will end up becoming something else that is still a pretty good sounding (if a bit unexpected) interval.

So I often end up using 31EDO anyhow, thinking even if a few fractions in a JI scale I've made come out terrible (IE 13/9, which has a huge error in 31EDO)...virtually everything else will work very well (it even turns out the octave inverse of 13/9 can map to 7/5, making 13/9 "map" to 10/7...which, IMVHO, really isn't a bad thing as I often prefer 10/7 to 13/9 in the first place)...

>"That seems weird to me. It seems to me that everything would be

"inconsistent" by that definition, because the amount of error in a

given interval will always depend on the accuracy of the other primes

used. 19/18 in 31 EDO is bad because 19 is bad and 3 is good;"

Meaning to say that 19 has a large error of how it's approximated in 31EDO and 18 (which has a limit of 3) resolves around the error for 3-limit in 31EDO, which is small. So a distorted numerator over a not similarly distorted denominator means a fraction with high error, correct?
And it turns out, even if you use the "closer" approximation for 19/18 in 31TET...it's still about 20 cents off and quite a lousy approximation.

🔗Mike Battaglia <battaglia01@...>

On Aug 15, 2011, at 10:57 AM, Jake Freivald <jdfreivald@...> wrote:

> I haven't worked this out but if you're saying that one val gives
> you a better 19/18 and the other gives you a better 19/17,
> then this is saying that 31-EDO is -inconsistent- in the 19-limit.

When we talk about consistency in the n-limit, we generally refer to all
dyads with both numerator and denominator only up to n (assuming octave
equivalence). So 5-limit consistency would deal with 3/2, 5/4, and 6/5 -
despite that 6 is greater than 5, we're assuming octave equivalence, so this
reduces to 3/5 (or 5/3). However, 5-limit consistency would NOT include
something like 9/8 or 25/16, because the largest odd factor in these dyads
is greater than 5. This is sort of an alternate definition of the word
"limit" that's sometimes called "odd-limit" for clarity, whereas the usual
definition is often called "prime-limit." If the patent val for the EDO also
gives you the best match for each individual dyad in some n-(odd)-limit,
then it's consistent in the n-(odd)-limit.

This means it becomes meaningful to talk about things like the
9-(odd)-limit: while 16-equal is consistent in the 3-odd-limit, it's not
consistent in the 9-odd-limit, because the best approximation to 9/8 is 225
cents, whereas the best mapping for 3/2 leads to a 150 cent approximation.
This can be solved by using two approximations to 9/8 - the "prime" version
(best direct mapping) and the "composite" version (best indirect mapping).
You'd call this the 2.3.5.9' subgroup, where 9' is read "9 prime." Which 9/8
you'd use in any context depends on context and on your musical judgement.

So you could always explore the 2.3.5.5/3' subgroup, or the 2.3.5.19.19/17
or the 2.3.5.17.19.19/17' subgroup if you want.

As a last note, keep in mind that the 9-odd-limit actually exists as part of
the 7-prime-limit, although nothing's to stop you from talking about the
5-prime-limit, 9-odd-limit, or the 2.7.9.11-prime-limit, 11 odd limit, etc.

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗Keenan Pepper <keenanpepper@...>

🔗Mike Battaglia <battaglia01@...>

🔗Jake Freivald <jdfreivald@...>

I'm going to say a lot of things here in the hopes that someone who knows
better will correct me when I'm wrong.

> Before I got in "trouble" with Jake for using a non-patent val
> mapping for 31EDO to map an interval nearing 19/18 in my scale...

I'm too new to this stuff to get you in trouble! :)

I don't have a problem with using a non-patent val. Using a non-patent val
is still using a regular temperament.

I have a "problem" with arbitary rounding vs. using a val for your scale.

"Problem" is in quotes because you shouldn't let math get in the way of
creativity and making music. We should recognize, though, that regular
temperaments provide a framework for creativity, which has a great deal of
value that you give up if you don't use them.

> the problem, as I understand it...being that soon as a use a
> non-patent val, certain "regular temperament" musical consistencies
> IE "a neutral 3rd plus a neutral 3rd = a fifth" can become untrue
> as errors add up in such additions...

Well, no. When you use a non-patent val, you're using a different regular
temperament. It may or may not be fine, depending on what you want, but it's
regular.

That's different from rounding. When you round, you're *not* using a regular
temperament.

If you're not using a regular temperament, then things like "31 EDO tempers
out 243/242" no longer have a basis in fact. If you can't say "31 EDO
tempers out 243/242", then you can't guarantee that (11/9)^2 = 3/2. Maybe
sometimes it will, maybe sometimes it won't. It depends on what you rounded
to what.

If you *are* using a regular temperament that tempers out 243/242, then I
can guarantee you that anytime you take two steps of 11/9, you'll get a 3/2.

To say the exact same thing with different words: If every interval in your
scale is mapped using the same val, and that val tempers out 243/242 in 31
EDO, then I can guarantee you (etc.). (This happens to work with the patent
val in 31 EDO, but I don't care if you're using the patent val or not, the
statement is still true.)

In other words, rounding doesn't *increase errors*, it *eliminates
regularity and predictability*.

Mapping doesn't eliminate errors, it makes them regular and predictable.
Because the errors are regular and predictable, you can guarantee specific
errors will always be there, such as 243/242 = 1/1.

Guaranteeing that those errors will always be there is what leads to the
note relationships in things like Meantone and Porcupine.

Back to what you were saying...

> and often you end up on EDO step too far or short during such
> "interval additions".

You started by saying, "as soon as I use a non-patent val". Let's correct
that to "as soon as I start to round instead of using a val". Then, yes, you
might end up an EDO step too far or too short during these interval
additions.

> 31EDO, on the other end, seems to have a good habit...that is, of
> "badly mapping" higher limit intervals to lower limit ones that
> still retain a not-unsimilar feel.

Well, except you're not mapping. You're rounding. Which means you're playing
willy-nilly with the primes. The more you do it, the less your choices have
anything to do with any particular JI intervals.

> Usually when I have two notes in a scale that form a flat interval
> relative to each other, I simply move the lower one down by a small
> amount and a higher one up (IE each one down or up by no more than 7
> cents)... [snip]

There's nothing inherently wrong with what you're doing here, but you're not
applying a regular temperament.

> The thing/problem with using EDOs (at least ones which aren't very
> high, such as 31EDO) is that you often don't have the option of only
> moving by 7 cents...you end up moving by something over 20 cents...

Right. Lower EDOs have higher errors, which isn't surprising given their
step sizes.

The issue we're discussing is more easily noticed in higher EDOs, because
there the step sizes are small, and the error sizes can approach or go
beyond the step sizes.

It's okay that you like using 31 EDO, but sentences like this confuse me:

> (it even turns out the octave inverse of 13/9 can map to 7/5, making
> 13/9 "map" to 10/7...

How does one JI interval "map" to another JI interval? If you mean that the
13/9 interval maps to the same 31 EDO step as the 10/7 interval does, then I
think you mean that 16 steps of 31 EDO (which is a very good approximation
for 10/7) = 13/9.

If that's what you want, you can round, in which case I can tell you
literally nothing about what your scale tempers.

Or you could use this val:
< 31 49 72 87 107 114 |

...instead of the patent val, which is this:
< 31 49 72 87 107 115 |

...and get a regular temperament, with all its benefits. For instance, I can
tell you that this val tempers out 225/224, 385/384, and 364/363, which
makes it a Marvel temperament called "Deecee". It also tempers out 65/64 and
91/90.

If you use this val to map all of the intervals consistently across your
entire scale, I can guarantee you those relationships, just like I can
guarantee you that 5 fifths = an octave reduced Major Third in 12-tET.

Or you can round. Your call.

> Meaning to say that 19 has a large error of how it's approximated in
> 31EDO and 18 (which has a limit of 3) resolves around the error for
> 3-limit in 31EDO, which is small. So a distorted numerator over a not
> similarly distorted denominator means a fraction with high error, correct?

Yes, correct.

> And it turns out, even if you use the "closer" approximation for 19/18 in
> 31TET...it's still about 20 cents off and quite a lousy approximation.

True. But that makes sense: The only time you'd be tempted to round to a
step other than what the patent val tells you is if neither the patent val
nor the alternate val are great approximations.

Regards,
Jake

🔗Michael <djtrancendance@...>

Jake>"I have a "problem" with arbitary rounding vs. using a val for your scale."
However...if I have it right, what I did was the equivalent of using something equivalent to the patent val MINUS one change...I moved the number of steps to represent 19/1 down by one.
The counter question is how does arbitrary rounding differ from the effect of changing a val... That is, unless I came across a special case where I changed the 19-limit val and it only effects one ratio because there's only one 19-limit ratio in the whole scale I made (and it would have knocked things off miserably if there was, say, a 19/16 and more along with the 19/18). In which case there would be an (often gross) inconsistency with the amount of error in 19/18 vs the other 19-limit ratios, thus making it "irregular"...correct?

>You started by saying, "as soon as I use a non-patent val". Let's correct that to "as soon as I start to round instead of using a val". Then, yes, you might end up an EDO step too far or too short during these interval additions.

This, again, would seem to say that the problem you have is ultimately my using the prime 19 mapped in more than one way within a single scale...
For the record, what makes a patent val different than a non-patent val, since now it seems obvious both types of vals are consistently mapped (that one obeys musical interval addition rules IE 11/9 + 11/9 (logarithmically added) = 3/2...and one doesn't)?

>"Well, except you're not mapping. You're rounding. Which means you're playing willy-nilly with the primes. The more you do it, the less your choices have anything to do with any particular JI intervals."

So (if I have it right), my supposed scale's 13/9 may slip into a 10/7 in one case (IE the case when I rounded), but my scale may have a 13/10 that's almost perfect (virtually no error)...so there would be no easy way to tell how accurate any 13-limit chords, for example, would be.

>"Mapping doesn't eliminate errors, it makes them regular and predictable."
IE, using a mapping including a step for 7/1, everything 7-limit, be it 8/7, 7/6,7/5...will always have the same error, which would be the error of the prime 7/1 from the EDO's nearest value for 7/1.

>"If you use this val to map all of the intervals consistently across your entire scale..."
Right, so everything with an odd-limit of 13 gets mapped with the same error?

>"The only time you'd be tempted to round to a step other than what the patent val tells you is if neither the patent val nor the alternate val are great approximations."
Exactly... This goes back to our (unfinished) off-list discussion about finding out what the valuable commas are in a scale (vs. non-valuable commas) to match those commas to EDOs that suit them well.

🔗Mike Battaglia <battaglia01@...>

On Aug 15, 2011, at 3:05 PM, Jake Freivald <jdfreivald@...> wrote:

I'm going to say a lot of things here in the hopes that someone who knows
better will correct me when I'm wrong.

> Before I got in "trouble" with Jake for using a non-patent val
> mapping for 31EDO to map an interval nearing 19/18 in my scale...

I'm too new to this stuff to get you in trouble! :)

I don't have a problem with using a non-patent val. Using a non-patent val
is still using a regular temperament.

I have a "problem" with arbitary rounding vs. using a val for your scale.

There can be plenty of value in using more than one mapping for an equal
temperament. For instance, 25-edo has utility both as a Blackwood
temperament (using the 720 cent mapping for 3/2), and as a mavila
temperament (using the 674 cent mapping for 3/2). The first would be the 25p
patent val, the other would be 25b. There's no reason you can't have both,
and treat it as a 2.3.3b.5 temperament. This is the sort of thing that has
been explored more in the compositional community than on this list, where
folks will intuitively treat a temperament as having both a normal fifth and
a "pelog fifth" or something, but there's no reason you can't do it.

-Mike

🔗Jake Freivald <jdfreivald@...>

This might be a re-post, in which case I apologize. I thought this went out
last night. I've slightly re-written it anyway, which might be a good thing.
Michael said:
> if I have it right, what I did was the equivalent of using something
> equivalent to the patent val MINUS one change...I moved the number
> of steps to represent 19/1 down by one.

Yes.

> The counter question is how does arbitrary rounding differ from the
> effect of changing a val... That is, unless I came across a special
> case where I changed the 19-limit val and it only effects one ratio
> because there's only one 19-limit ratio in the whole scale I made
> (and it would have knocked things off miserably if there was, say, a
> 19/16 and more along with the 19/18). In which case there would be
> an (often gross) inconsistency with the amount of error in 19/18 vs
> the other 19-limit ratios, thus making it "irregular"...correct?

Bingo.

> the problem you have is ultimately my using the prime 19 mapped in more
> than one way within a single scale...

Yes.

> For the record, what makes a patent val different than a non-patent val,
> since now it seems obvious both types of vals are consistently mapped
> (that one obeys musical interval addition rules IE 11/9 + 11/9
> (logarithmically added) = 3/2...and one doesn't)?

Someone else can give you a more technical definition, but I'll give you how
I've been calculating it in my spreadsheet, using 31 EDO as an example. I'll
overexplain, as usual, to make sure we're on the same sheet of music, not
because I want to insult your intelligence.

The val contains the number of steps it takes to get to a given prime
number, in prime number order:
< [2/1] [3/1] [5/1] [7/1] [etc.] |

By definition, for any EDO, the number of steps to 2/1 is the EDO division:
31 for 31 EDO. The 2-limit patent val is < 31 |.

What't the number of steps to 3/1? The step size for 31 EDO is 38.70967742
cents. 3/1 is 1901.96 in cents.

1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.

This is an EDO, though -- I can't take .13383752 steps. So I round. This is
clearly closer to 49 steps, so that's the "obvious" or "patent" choice.

The 3-limit patent val is < 31 49 |.

Do the same thing up through 17, and you get an 17-limit patent val of
< 31 49 72 87 107 115 127 |.

How about 19? Same math: 19/1 = 5097.51 cents, 5097.51 / 38.70967742
cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is
< 31 49 72 87 107 115 127 132 |.

If I understand it correctly, the patent val will always have the least
error on a per-prime basis for a given EDO. That doesn't always make it the
best for mapping particular intervals to it, as the subgroup discussion that
Mike B. presented shows. Plus, your ear may tell you that you really want
some interval to be a little sharp, even though that has greater error than
if you went flat.

So much for the patent val. What about vals that aren't the patent val?

Any 19-limit val other than this one is still a valid val, it's just not the
"patent" val. Remember that "patent" here means something like "obvious" --
just because a choice is obvious doesn't mean it's best. So there's an
unlimited number of vals you could use here, each one with its own
approximation for each prime in it. And as long as you're using the same
approximations for the primes in all of the intervals of a given scale, you
have a regular temperament.
If you use a non-patent val, then some of the relationships that you found
with the patent val no longer hold; however, new relationships take their
place. So, for instance -- and this is a random example -- if you use the
patent val in 31 EDO, you temper out 105/104, while if you use < 31 49 72 87
107 114 | -- where the value for 13/1 is one less than in the patent val --
we don't temper 105/104, but we do temper 91/90, which the patent val
didn't.

Some relationships remain the same, of course. 81/80 is a 5-limit comma, so
changing the 13/1 place doesn't affect it. 81/80 is tempered out using
either val in 31 EDO.

> So (if I have it right), my supposed scale's 13/9 may slip into a 10/7
> in one case (IE the case when I rounded), but my scale may have a 13/10
> that's almost perfect (virtually no error)...so there would be no easy
> way to tell how accurate any 13-limit chords, for example, would be.

I still don't know what you mean by "13/9 may slip into a 10/7".

Are you saying the 13/9 approximation sounds like a 10/7 to you? Because
that's probably true, since 16 steps of 31 EDO is just 2 cents away from a
pure 10/7.

Are you saying you're using the 13/9 approximation as you would a 10/7? That
makes sense, too, for the same reason, but then why call it a 13/9?

You're not dealing with literal JI ratios, you're dealing with EDO steps. It
seems to me that the most you could say is that "The closest thing I have to
a 13/9 is the same thing I use for a 10/7."

There *is* a way to tell how good any given 13-limit chords would be, as it
turns out: Go to Graham's temperament finder and type in the subgroup for
the chord(s) you want, and it'll tell you which EDOs and vals provide which
badness measures.

> IE, using a mapping including a step for 7/1, everything 7-limit, be it
> 8/7, 7/6,7/5...will always have the same error, which would be the error
> of the prime 7/1 from the EDO's nearest value for 7/1.

Yes.

> > If you use this val to map all of the intervals consistently across
> > your entire scale...
>
> Right, so everything with an odd-limit of 13 gets mapped with the same
error?

Prime limit, not odd-limit. Vals have nothing to do with odd-limit.

> This goes back to our (unfinished) off-list discussion about finding out
> what the valuable commas are in a scale (vs. non-valuable commas) to
> match those commas to EDOs that suit them well.

Yes. I felt I had to answer the prime-mapping-to-EDO question before we went
back to that, and I didn't know the answer.

For me, the most interesting (and challenging) aspect of learning what
little I have in the past year has been understanding why tempering out a
particular comma helps, or what character it adds to a temperament. I get
meantone -- flatten the fifths, sweeten the thirds -- but for most other
things I look on a scale-by-scale basis. When I use 11/9, I'm probably going
to want to temper 363/362 so that 2 neutral thirds = a perfect fifth.
Eventually I think I'd need to survey some temperament families to see how,
say, Marvel differs from Kleismic differs from Porcupine, in terms of the
intervals that might arise in their scales.

Regards.
Jake

🔗Graham Breed <gbreed@...>

Jake Freivald <jdfreivald@...> wrote:

> There *is* a way to tell how good any given 13-limit
> chords would be, as it turns out: Go to Graham's
> temperament finder and type in the subgroup for the
> chord(s) you want, and it'll tell you which EDOs and vals
> provide which badness measures.

There is? I thought I hid the badness. It will give equal
temperaments in order of badness.

> > IE, using a mapping including a step for 7/1,
> > everything 7-limit, be it 8/7, 7/6,7/5...will always
> > have the same error, which would be the error of the
> > prime 7/1 from the EDO's nearest value for 7/1.
>
> Yes.

Where did this come from? Somebody must be assuming that
the 5-limit is perfectly tuned. But also neglecting that
the 5-limit is a subset of the 7-limit.

> > > If you use this val to map all of the intervals
> > > consistently across your entire scale...
> >
> > Right, so everything with an odd-limit of 13 gets
> > mapped with the same
> error?
>
> Prime limit, not odd-limit. Vals have nothing to do with
> odd-limit.

You can certainly get odd-limit errors from vals (equal
temperament mappings). First you calculate the error in
each prime. Then you weight the primes according to how
often they come up, so 3 is weighted double in the
9-limit. (Really, you have to consider partials, not
primes, because 15 has to be considered independently
when you get to it.) Then the worst error of the equal
temperament is max-min of these errors, and there are
functions to give you RMS errors as well.

Graham

🔗Jake Freivald <jdfreivald@...>

Thanks for the response, Graham.

> > There *is* a way to tell how good any given 13-limit
> > chords would be, as it turns out: Go to Graham's
> > temperament finder and type in the subgroup for the
> > chord(s) you want, and it'll tell you which EDOs and vals
> > provide which badness measures.
>
> There is? I thought I hid the badness. It will give equal
> temperaments in order of badness.

Sorry, you're right -- it doesn't give the actual badness measure. My point
was that it doesn't rank in order of error, but in order of badness.

> > > IE, using a mapping including a step for 7/1,
> > > everything 7-limit, be it 8/7, 7/6,7/5...will always
> > > have the same error, which would be the error of the
> > > prime 7/1 from the EDO's nearest value for 7/1.
> >
> > Yes.
>
> Where did this come from? Somebody must be assuming that
> the 5-limit is perfectly tuned. But also neglecting that
> the 5-limit is a subset of the 7-limit.

Sorry, that's a good catch. We had already gone through the fact that the 5
will have an error, which means the error for each interval will be
different. What I was agreeing to was that each interval would have some
error contributed from the error in 7/1

> > Prime limit, not odd-limit. Vals have nothing to do with
> > odd-limit.
>
> You can certainly get odd-limit errors from vals (equal
> temperament mappings). First you calculate the error in
> each prime. Then you weight the primes according to how
> often they come up, so 3 is weighted double in the
> 9-limit. (Really, you have to consider partials, not
> primes, because 15 has to be considered independently
> when you get to it.) Then the worst error of the equal
> temperament is max-min of these errors, and there are
> functions to give you RMS errors as well.

I spoke too strongly. Sure, just like you can get the errors of any dyad,
you can also get the errors of any multiples of any primes and thereby
calculate odd-limit errors. We were talking about using vals to do mappings
instead of rounding, though, and in that context the odd-limit doesn't seem
relevant. If I'm still missing something, I'd be interested to hear why.

Thanks,
Jake

🔗Graham Breed <gbreed@...>

🔗Jake Freivald <jdfreivald@...>

🔗Jason Conklin <jason.conklin@...>

On Tue, Aug 16, 2011 at 12:51, Jake Freivald <jdfreivald@...> wrote:

> **
> ...
>
> Someone else can give you a more technical definition, but I'll give you
> how I've been calculating it in my spreadsheet, using 31 EDO as an example.
> I'll overexplain, as usual, to make sure we're on the same sheet of music,
> not because I want to insult your intelligence.
>

I just wanted to say thanks for taking the time to do so. I've been
struggling for the past few days with the basic concept of vals -- I know
how important it is to discussion here, but reading and rereading the
definition on the xenwiki, trying think about it in different ways, I just
wasn't making a lot of sense out it. This concrete "procedural" explanation
kinda cracked the nut and pointed me in what I hope is a more fruitful
direction for understanding other concepts. There's still a lot of basics I
don't get, but I feel like I'm on my way now... not as much head-bashing.

Thanks,
Jason

> The val contains the number of steps it takes to get to a given prime
> number, in prime number order:
> < [2/1] [3/1] [5/1] [7/1] [etc.] |
>
> By definition, for any EDO, the number of steps to 2/1 is the EDO division:
> 31 for 31 EDO. The 2-limit patent val is < 31 |.
>
> What't the number of steps to 3/1? The step size for 31 EDO is 38.70967742
> cents. 3/1 is 1901.96 in cents.
>
> 1901.96 cents / 38.70967742 cents/step = 49.13383752 steps.
>
> This is an EDO, though -- I can't take .13383752 steps. So I round. This is
> clearly closer to 49 steps, so that's the "obvious" or "patent" choice.
>
> The 3-limit patent val is < 31 49 |.
>
> Do the same thing up through 17, and you get an 17-limit patent val of
> < 31 49 72 87 107 115 127 |.
>
> How about 19? Same math: 19/1 = 5097.51 cents, 5097.51 / 38.70967742
> cents/step = 131.6857529 steps. Round to get 132. The 19-limit patent val is
> < 31 49 72 87 107 115 127 132 |.
>
> If I understand it correctly, the patent val will always have the least
> error on a per-prime basis for a given EDO. That doesn't always make it the
> best for mapping particular intervals to it, as the subgroup discussion that
> Mike B. presented shows. Plus, your ear may tell you that you really want
> some interval to be a little sharp, even though that has greater error than
> if you went flat.
>
> So much for the patent val. What about vals that aren't the patent val?
>
> [etc.]
>

🔗Mike Battaglia <battaglia01@...>

On Mon, Aug 15, 2011 at 5:02 PM, Jake Freivald <jdfreivald@...> wrote:
>
> > The first would be the 25p patent val, the other would be 25b.
> > There's no reason you can't have both, and treat it as a 2.3.3b.5
> > temperament.
> I don't understand. You would use both the 25p and the 25b to map the same intervals in the same scale? How do you map a 3/2 to two different values? And how does that affect the commas you would temper out if I were only using one of the vals?

You'd have two different 3/2's; you'd have the "blackwood 3/2" and the
"pelogic 3/2." Which you'd use depends on musical context and personal
taste. You could say that 135b/128 vanishes, but that 256/243 vanishes
- not 135/128 or 256/243b.

I used the "b" wart above just to signify that we're using a different
mapping for 3. I could also have said 2.3.3'.5 or 2.3.3*.5; doesn't
matter. Either way, if we're in the 2.3.3*.5 limit or the 2.3.3b.5 or
2.3.3*'!#!@.5 limit, our val would be <25 40 39 58|, where we're
understanding that the first column represents prime 2, the second
column represents prime 3, the third column represents a different 3,
and the last column represents prime 5.

The larger point that I'm making is that we should strive to turn
practice into theory whenever possible. Lots of people like to think
about 25-equal as having two sizes of 3/2, for example, and the math
provides for you to work it out by mapping the two 3s to different
places in a val.

-Mike