Yahoo Tuning Groups Ultimate Backup tuning 7-limit vals, wedgies, EDO pairs etc.

Hi there.

#1. I thought that a proper 7-limit val for 573-EDO was <573, 908, 1331, 1609>. To my surprise, Graham's online scripts suggest <573, 908, 1330, 1608>, which means that they choose a stretched octave for this particular case while I was shrinking it. It also means that they had to approximate some other intervals first and then find the new degree for 7/1 while I would first approximate the three most "in-tune" intervals (2/1, 3/1, 7/1) as well as possible and then find the new degree for 5/1. Could someone explain what I'm doing wrong here?

#2. If temperaments as complex as mutt or misty can be useful and can have webpages about them, then what do you think of the 99&84 temperament? It also splits the octave into three periods and seems to be similarly complex as the other two. The same goes for comparing marvo with the other temperament that maps the 7/1 to (14, -26) rather than (-17, 46).

Petr

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "petrparizek2000" <petrparizek2000@...> wrote:
>
> Hi there.
>
> #1. I thought that a proper 7-limit val for 573-EDO was <573, 908, 1331, 1609>.

That is indeed the patent val, if that is what you mean.

To my surprise, Graham's online scripts suggest <573, 908, 1330, 1608>, which means that they choose a stretched octave for this particular case while I was shrinking it.

You don't need to worry about the octave unless you want to; you're comparing vals.

> It also means that they had to approximate some other intervals first and then find the new degree for 7/1 while I would first approximate the three most "in-tune" intervals (2/1, 3/1, 7/1) as well as possible and then find the new degree for 5/1. Could someone explain what I'm doing wrong here?

The second val, using your typical badness measure such as Tenney-Eulcidean which I presume Graham used, gets a lower badness figure than the first. Note that 7/6, 6/5 and 7/5 are among the intervals you want to approximate.

> #2. If temperaments as complex as mutt or misty can be useful and can have webpages about them, then what do you think of the 99&84 temperament? It also splits the octave into three periods and seems to be similarly complex as the other two.

It's your baby. What do you want to name it? It tempers out 6144/6125 and 250047/250000 in case that helps, so you could relate it to the rank three porwell temperament tempering out 6144/6125; both have the same optimal patent val of 381 so there's not much to distinguish the tunings.

>The same goes for comparing marvo with the other temperament that maps the 7/1 to (14, -26) rather than (-17, 46).

Rather than have me work out what this means, could you give a pair of patent vals again?

🔗genewardsmith <genewardsmith@...>

🔗Petr Parízek <petrparizek2000@...>

Gene wrote:

> That is indeed the patent val, if that is what you mean.

I'm not sure if I fully understand what patent vals are. Is it possible to
read an explanation somewhere on how to find them?

> You don't need to worry about the octave unless you want to; you're
> comparing vals.

Yes, but if I want to find a val for a particular EDO, don't I first have to
decide what the step size should be? That's what I was doing -- i.e. I realized
that 5/1 had the worst "mistuning" and therefore I focused on the other
primes and tried to make them as "in tune" as possible. And because I
eventually decided to use a slightly smaller step than exactly 1200/573
cents, the 5/1 then became closer to 1331 steps than 1330.

> The second val, using your typical badness measure such as
> Tenney-Eulcidean which I presume Graham used,
> gets a lower badness figure than the first. Note that 7/6, 6/5 and 7/5 are
> among the intervals you want to
> approximate.

I see, I suspect I should learn more about how to find the right vals then.

> It's your baby. What do you want to name it? It tempers out 6144/6125 and
> 250047/250000 in case that helps,
> so you could relate it to the rank three porwell temperament tempering out
> 6144/6125; both have the same
> optimal patent val of 381 so there's not much to distinguish the tunings.

Trying to make the name as short as possible while still somehow reminding me of what temperament it is, I'm currently thinking of nothing less "cryptic" than "nessafof" -- i.e.
#1. there are two <ne>utral <s>econds to a 6/5,
#2. a <s>emi-<a>ugmented <fo>urth, stacked <f>ive times, approximates 5/1. :-D (The fact that one semi-augmented fourth is 1 period + 1 generator is another story.)

> Rather than have me work out what this means, could you give a pair of
> patent vals again?

I think it's 65&72 but I mean the "non-pure-octave 65" since the pure-octave
65 gives me marvo.

Petr

🔗Carl Lumma <carl@...>

Hi Petr,

> #1. I thought that a proper 7-limit val for 573-EDO was
> <573, 908, 1331, 1609>. To my surprise, Graham's online
> scripts suggest <573, 908, 1330, 1608>, which means that
> they choose a stretched octave for this particular case
> while I was shrinking it. It also means that they had to
> approximate some other intervals first and then find the
> new degree for 7/1 while I would first approximate the
> three most "in-tune" intervals (2/1, 3/1, 7/1) as well as
> possible and then find the new degree for 5/1. Could
> someone explain what I'm doing wrong here?

Graham can tell you exactly what he does, but I believe
the idea is to find the val for which the sum of weighted
dyadic errors is lowest.

-Carl

🔗petrparizek2000 <petrparizek2000@...>

🔗Graham Breed <gbreed@...>

"Carl Lumma" <carl@...> wrote:

> Graham can tell you exactly what he does, but I believe
> the idea is to find the val for which the sum of weighted
> dyadic errors is lowest.

I don't know in what context there's only one result.
There are two here:

http://x31eq.com/cgi-bin/rt.cgi?ets=573&limit=7

They're sorted by TE error.

I don't know where 1331 comes into it. 5:1 is 1330.46
steps.

Graham

🔗Petr Parízek <petrparizek2000@...>

🔗Graham Breed <gbreed@...>

🔗Petr Parízek <petrparizek2000@...>

Graham wrote:

> 573 times the log to base 2 of 5, with normal floating
> point variables.

Okay, so both of us seem to do the same thing for the first part of the procedure.
And then, how do you decide that the other version should use 1608 steps for 7/1 if the pure-octave version uses 1609? Am I right in assuming that "something" suggests a slightly larger step size? And if so, what is that "something"?

Petr

🔗Graham Breed <gbreed@...>

Petr Parízek <petrparizek2000@...> wrote:
> Graham wrote:
>
> > 573 times the log to base 2 of 5, with normal floating
> > point variables.
>
> Okay, so both of us seem to do the same thing for the
> first part of the procedure.
> And then, how do you decide that the other version should
> use 1608 steps for 7/1 if the pure-octave version uses
> 1609? Am I right in assuming that "something" suggests a
> slightly larger step size? And if so, what is that
> "something"?

I don't know. What do you think I decided? The website
should be -- and was when I checked -- giving two results.
That's because there are two plausible mappings with a
reasonable error. They're sorted so that the one with the
lower error comes first.

The way the error's calculated is documented here:

http://x31eq.com/te.pdf

and also here:

http://xenharmonic.wikispaces.com/Tenney-Euclidean+temperament+measures

Graham

🔗Petr Parízek <petrparizek2000@...>

Graham wrote:

> I don't know. What do you think I decided? The website
> should be -- and was when I checked -- giving two results.
> That's because there are two plausible mappings with a
> reasonable error. They're sorted so that the one with the
> lower error comes first.

That's what I was asking - how do I know there are actually two reasonable mappings rather than one.

> The way the error's calculated is documented here:
>
> http://x31eq.com/te.pdf
>
> and also here:
>
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+temperament+measures

I see. Thanks, I'll skim through that -- maybe I'll find the answer to my question there.

Petr

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗Petr Parízek <petrparizek2000@...>

Gene wrote:

> http://xenharmonic.wikispaces.com/Patent+val

Thanks.
So if I'm understanding correctly, to get the desired step size, I would first find the patent val, then divide the rounded step counts by the corresponding log2(p), multiply these by the EDO cardinatliy and find the RMS value as the "Vals and tuning space" page describes.

But this doesn't tell me that, for example, 34-EDO or 65-EDO round the 7/1 to a different number of steps if the other primes are to be more "in tune" -- how do I reconcile that?

Petr

🔗Graham Breed <gbreed@...>

Petr Parízek <petrparizek2000@...> wrote:
> Graham wrote:
>
> > I don't know. What do you think I decided? The website
> > should be -- and was when I checked -- giving two
> > results. That's because there are two plausible
> > mappings with a reasonable error. They're sorted so
> > that the one with the lower error comes first.
>
> That's what I was asking - how do I know there are
> actually two reasonable mappings rather than one.

I have a function that produces all possible mappings (vals)
for a given division of the octave and error cutoff. It
isn't easy to explain how that works. It's in the file
parametric.py as "limitedMappings" and it's about 25 lines
of code. An explanation in English would probably take
longer and I'd make mistakes in it. (It's also 20 lines in
the somewhat less readable regular_terse.py.) All the
code's available at

http://x31eq.com/temper/regular.zip

For a given division of the octave, the TE error is
(almost) proportional to the relative TE error. The
relative TE error of an equal temperament is the standard
deviation of the weighted mapping vector. You can get rid
of a square root by calling it the variance of the weighted
mapping vector. If you multiply the variance by the number
of primes, you get a kind of sum instead of a kind of RMS.
It should get larger with each prime you add, so you can
predict what range of errors is accepable for that prime.

Graham

🔗Petr Parízek <petrparizek2000@...>

Hi again.

I've now stopped thinking about EDO approximations and come to an interesting idea, that if we accept that a 2D temperament may use a generator that exceeds the size of the period, then the wedgies can tell us the actual period/generator mapping. Unfortunately, so far I've been able to figure it out only in those cases where either 3/1 or 5/1 can be described with 0 periods and any number of generators. If both 3/1 and 5/1 need a number of periods, then I still don't know how to solve the task. Anyway, this procedure is usable in many cases and I hope I'm at least partialy on the right track.

For example, meantone and orwell allow both variants of generator mapping -- i.e. for describing 3/1 without any periods, meantone can use a 3/1-like generator itself and orwell splits it into 7 steps, while if you wish to describe 5/1 without any periods, then meantone splits it into 4 steps and orwell into 3 steps. The same can be applied to temperaments like hanson (3/1), magic (either 3/1 or 5/1), schismatic (3/1), superpyth (3/1), diaschismatic (3/1), wizard (5/1), wuerschmidt (5/1), mavila (3/1) and a few others.

For the general explanation of the procedure, let's assume the following symbols representing the period/generator mapping:
(PA, GA)
(PB, GB)
(PC, GC)
(PD, GD)
And let's assume <<W1, W2, W3>> to represent the 5-limit wedgie. Then:

If the temperament allows to map 3/1 without using periods, the following is true:
PA=GCD(W1,W2)
GA=0
PB=0
GB=W1/PA
PC=-W3/W1*PA
GC=W2/PA

If the temperament allows to map 5/1 without using periods, the following is true:
PA=GCD(W1,W2)
GA=0
PB=W3/W2*PA
GB=W1/PA
PC=0
GC=W2/PA

What I'm not sure about, however, is how I could find a similar "all-integer" mapping for temperaments like miracle where neither 3/1 nor 5/1 is expressed without involving periods.

But anyway, I think this could make wedgies even more understandable for people like me. I'm speaking about it primarily because now I'm thinking about wedgies regardless of which EDOs they come from. The fact that a wedgie fully characterizes a temperament is one thing and the fact that pairing up EDOs of "such and such" gives me a wedgie of "such and such" is another.

Comments and suggestions are appreciated.

Petr

🔗petrparizek2000 <petrparizek2000@...>

I wrote:

> If the temperament allows to map 3/1 without using periods, the following is
> true:
> PA=GCD(W1,W2)
> GA=0
> PB=0
> GB=W1/PA
> PC=-W3/W1*PA
> GC=W2/PA
>
> If the temperament allows to map 5/1 without using periods, the following is
> true:
> PA=GCD(W1,W2)
> GA=0
> PB=W3/W2*PA
> GB=W1/PA
> PC=0
> GC=W2/PA

I should also add that this can be easily extended to 7-limit wedgies as well (if we add W4, W5, W6 to the previously mentioned variables), in which case the number of periods for the 7/1 is either "-W5/W1*PA" IF 3/1 USES NO PERIODS or "-W6/W2*PA" IF 5/1 USES NO PERIODS. This explains the meaning of the number 12 in the 7-limit meantone wedgie of <<1, 4, 10, 4, 13, 12>>. For me that was something that I didn't understand for a long time. Actually, if there are 4 generators to a 5/1, then there are -3 periods (and 10 generators) to a 7/1, which is what it says (i.e. -12/4 = -3).

Petr