Yahoo Tuning Groups Ultimate Backup tuning What am I doing wrong?

This seems so simple, but my spreadsheet is giving me weird results.
I'll show stuff that works first, and then show what seems to be
broken.

I have a monzo: take | -1 1> for 3/2, 701.96 cents. In the spreadsheet
I actually have | -1 1 0 0 0 0> because I want to go up to the
13-limit. (Crazy for a novice, right? But Canton is 13-limit, and
that's what I'm working in right now.)

I have the 3-limit 12-EDO val: <12 19 |. In the spreadsheet it's
actually <12 19 28 34 42 44 |, but having the higher numbers there
doesn't matter when the monzo has zeroes for those primes.

I generate the homomorphism ("do the mapping"?): (-12)+(19)=7 steps of
12 EDO, or 700 cents. Good. Works for 4/3 = | 2 -1>, 5/4 = | -2 0 1>,
7/6 = | -1 -1 0 1>, and others, using the higher-limit vals.

Then I start with some commas. The limma = | 8 -5> works (90 cents
mapped to 1 12-EDO step), as does the apotome (114 cents to 1 12-EDO
step), and the Pythagorean (| -19 12> = 23 cents mapped to 0 steps).

So far, so good.

But then I get the Porcupine comma, 250/243 or | 1 -5 3>, which is
49.2 cents. That maps to 1 step of 12 EDO: (1*12)+(-5*19)+(3*28) =
12-95+84=1.

The Magic comma, 3125 / 3072 or | -10 -1 5>, is only 29.6 cents, but
it maps to 1 step of 12 EDO: (-10*12)+(-1*19)+5*28)=-120+-19+140=1.

Mercator's comma, | -84 53> is only 3.6 cents, and it gets mapped to
-1 step of 12 EDO. It sure seems like that should be zero. And it gets
mapped to -7 steps of 31 EDO (val <31 49 72 87 107 |, and yes, I know
that's more places than I need, but it's irrelevant for what I'm
doing). That just can't be right.

Here's another: Use the monzo for 256/243 = 90.2 cents, or | 8 -5>,
and I see a mapping of <V|M> = 3 steps of 31 EDO, or 116.1 cents. But
two steps is actually closer, at 77.4 cents, than three steps is:
-12.8 cents vs. 25.9 cents.

The math doesn't seem complicated, and I even feel like I have an
intuitive grasp for why the mapping should work -- but it seems wrong
to me that 49- and 30-cent commas would map to 1 step of 12 EDO
instead of 0, or why a mapping would prefer a larger difference to a
smaller one.

In short, I think I must be doing something wrong. Any thoughts?

Thanks,
Jake

🔗Mike Battaglia <battaglia01@...>

On Tue, May 10, 2011 at 11:22 AM, Jake Freivald <jdfreivald@...> wrote:
>
> But then I get the Porcupine comma, 250/243 or | 1 -5 3>, which is
> 49.2 cents. That maps to 1 step of 12 EDO: (1*12)+(-5*19)+(3*28) =
> 12-95+84=1.

This is correct. In meantone, 250/243 gets equated with 25/24, since
they differ by 81/80. In 12-tet, 25/24 is one step out of 12. So this
is absolutely right.

> The Magic comma, 3125 / 3072 or | -10 -1 5>, is only 29.6 cents, but
> it maps to 1 step of 12 EDO: (-10*12)+(-1*19)+5*28)=-120+-19+140=1.

This is also right.

> Mercator's comma, | -84 53> is only 3.6 cents, and it gets mapped to
> -1 step of 12 EDO. It sure seems like that should be zero. And it gets
> mapped to -7 steps of 31 EDO (val <31 49 72 87 107 |, and yes, I know
> that's more places than I need, but it's irrelevant for what I'm
> doing). That just can't be right.

Mercator's comma isn't tempered out in 12-equal or 31-equal.
Mercator's comma means that 53 fifths is equated with some octaves.
This isn't the case in 12-equal or 31-edo.

> Here's another: Use the monzo for 256/243 = 90.2 cents, or | 8 -5>,
> and I see a mapping of <V|M> = 3 steps of 31 EDO, or 116.1 cents. But
> two steps is actually closer, at 77.4 cents, than three steps is:
> -12.8 cents vs. 25.9 cents.

Welcome to the wild world of temperaments.

> The math doesn't seem complicated, and I even feel like I have an
> intuitive grasp for why the mapping should work -- but it seems wrong
> to me that 49- and 30-cent commas would map to 1 step of 12 EDO
> instead of 0, or why a mapping would prefer a larger difference to a
> smaller one.
>
> In short, I think I must be doing something wrong. Any thoughts?

Map the primes, and the rest will follow. If you for some reason
wanted to use a closer representation of 256/243 than the one the
tempered primes give you, you could have a 2.3.(256/243)' temperament,
where the ' is read "256/243 prime." This ends up being more commonly
the case when you have something like a 2.3.5.9' temperament, with a
"prime" 9'/8 and a "composite" 9/8. 30-equal is a good example of such
a temperament. I can't see why you'd want that for something like
256/243 though.

Hey, you picked all of this up fast! Do you have lots of prior
education in group theory or something like that?

-Mike

🔗Jake Freivald <jdfreivald@...>

>> But then I get the Porcupine comma, 250/243 or | 1 -5 3>, which is
>> 49.2 cents. That maps to 1 step of 12 EDO: (1*12)+(-5*19)+(3*28) =
>> 12-95+84=1.
>
> This is correct. In meantone, 250/243 gets equated with 25/24, since
> they differ by 81/80. In 12-tet, 25/24 is one step out of 12. So this
> is absolutely right.

*Gulp.*

So I'm in this weird situation:

| 1 -5 3> gives 250/243 or 49.17 cents, mapping to 1 step of 12 EDO = 100 cents.
| 2 -10 6> (adding another porcupine comma) gives 62500/59049 or 98.33
cents, mapping to 2 steps = 200 cents.
| 3 -15 9> gives 3503957 / 3217789 or 147.5 cents, mapping to 3 steps
= 300 cents.
| 4 -20 12> gives 9853013 / 8794965 or 196.66 cents, mapping to 4
steps = 400 cents.
| 5 -25 15> gives 7718434 / 6696695 or 245.83 cents, mapping to 5
steps = 500 cents.

In other words, each step of the porcupine comma maps to a 12-TET tone
that's twice its actual size, _whether or note there's another 12-TET
tone that's closer._ The *tempered note value* isn't necessarily the
note from the EDO scale that's closest to the *actual note value*.

That means my intuition for why the mapping works is somewhat off --
even if it's partly right, there's a subtlety I'm missing.

I think this just took me one step closer to understanding why I don't
understand this list most of the time. :)

I think I just learned why, when tempering to an EDO, you're not just
looking for an EDO that has notes close to the notes you're tempering;
you have to use EDOs that temper out the same commas, so that just
intervals will map to the closest step in the scale. (I've known the
EDOs are supposed to temper the same commas, but not *why*.) So, for
example, here's the same process with the Porcupine comma in 1200 EDO
(val < 1200 1902 2786 |)

| 1 -5 3> gives 49.17 cents, mapping to 48 step of 1200 EDO = 48 cents.
| 2 -10 6> gives 98.33 cents, mapping to 96 steps = 96 cents.
| 3 -15 9> gives 147.5 cents, mapping to 144 steps = 144 cents.
| 4 -20 12> gives 196.66 cents, mapping to 192 steps = 192 cents.
| 5 -25 15> gives 245.83 cents, mapping to 240 steps = 240 cents.

I can see that 1200 EDO doesn't support Porcupine -- does that mean
the same thing as "doesn't temper the Porcupine comma?" -- because if
I were just rounding to the closest step I'd round 49.17 cents to 49
instead of 48 cents! And I'm five cents flat by the time I get to 5
steps of the Porcupine comma -- in an EDO that should be able to
always get me within 1 cent of what I'm shooting for.

I have a vague sense that I'm onto something, but it's tough to put
into words. At the very least, I think I understand a process
differently than I used to: I thought you'd temper a comma out of a
scale, and then, if you wanted it in an EDO, you'd choose the right
EDO to put it into. But that's not really how it works. Instead, you
take your JI scale as it stands, pick the EDO that tempers the commas
that you want (and perhaps meets other criteria), and then do the
mapping of the JI scale into the EDO.

I hate to say this, but that actually seems pretty straightforward to me now.

Thanks,
Jake

🔗Graham Breed <gbreed@...>

On 10 May 2011 18:54, Jake Freivald <jdfreivald@...> wrote:

> I can see that 1200 EDO doesn't support Porcupine -- does that mean
> the same thing as "doesn't temper the Porcupine comma?" -- because if
> I were just rounding to the closest step I'd round 49.17 cents to 49
> instead of 48 cents! And I'm five cents flat by the time I get to 5
> steps of the Porcupine comma -- in an EDO that should be able to
> always get me within 1 cent of what I'm shooting for.

Let's assert the positives: supporting Porcupine is the same thing as
tempering out the Porcupine comma. Remember the "out". If the
Porcupine comma ends up larger than it should be, it's still tempered,
but it isn't tempered out.

When you choose a regular temperament, you give an inherent error to
each prime interval (mapping of a prime number). Those errors
accumulate when you look at smaller intervals. After a point, you'll
always something with a higher error than the step size you're dealing
with.

> I have a vague sense that I'm onto something, but it's tough to put
> into words. At the very least, I think I understand a process
> differently than I used to: I thought you'd temper a comma out of a
> scale, and then, if you wanted it in an EDO, you'd choose the right
> EDO to put it into. But that's not really how it works. Instead, you
> take your JI scale as it stands, pick the EDO that tempers the commas
> that you want (and perhaps meets other criteria), and then do the
> mapping of the JI scale into the EDO.

Yes, that's it, for tempering *out* the commas.

> I hate to say this, but that actually seems pretty straightforward to me now.

You hate that? You'd rather it stayed impenetrable?

Graham

🔗Jake Freivald <jdfreivald@...>

> Remember the "out". If the Porcupine comma ends up larger
> than it should be, it's still tempered, but it isn't tempered out.

Good caveat, thanks.

> When you choose a regular temperament, you give an inherent error to
> each prime interval (mapping of a prime number).

That part I've understood for a while: When we say "364/363 vanishes"
or "13/11 + 14/11 = 3/2" we're really replacing 2, 3, 7, 11, and/or 13
with other numbers that are close to them.

> Those errors accumulate when you look at smaller intervals. After
> a point, you'll always something with a higher error than the step
> size you're dealing with.

I guess that makes sense. I noticed it with the commas because the
commas are small, so I reached that point earlier than if I had used a
larger step size.

>> you take your JI scale as it stands, pick the EDO that tempers
>> the commas that you want (and perhaps meets other criteria),
>> and then do the mapping of the JI scale into the EDO.
>
> Yes, that's it, for tempering *out* the commas.

Super. One of the things we'd want, then, as a community, is a way to
see which commas a given EDO tempers out. It looks like those things
are inconsistently documented in the wiki -- is there a way to
determine that for ourselves, or a resource that contains that
information in an easy-to-use way?

It seems a little weird to think about an EDO, which is created by
stacking irrational nth roots of 2 on top of each other, tempering out
commas, which are rational.

>> I hate to say this, but that actually seems pretty straightforward to me now.
>
> You hate that? You'd rather it stayed impenetrable?

I was being facetious, of course, but it makes me wonder if I should
document it in a way that I think I would have understood a month ago.

Thanks,
Jake

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> In other words, each step of the porcupine comma maps to a 12-TET tone
> that's twice its actual size, _whether or note there's another 12-TET
> tone that's closer._ The *tempered note value* isn't necessarily the
> note from the EDO scale that's closest to the *actual note value*.

Correct. In a temperament, we are mapping prime numbers to numbers that are actually a bit smaller or larger than the primes themselves. What this means is that error is multiplied in intervals that are composite numbers. Since the porcupine comma is (2*5*5*5)/(3*3*3*3*3), and in 12-TET those "3"'s are actually 2.996614153753363's, and those "5"'s are actually 5.039684199579493's (the 2 is still a 2, though), the more porcupine commas you add together, the more the error gets multiplied. So in a temperament, we are not mapping every interval to its nearest approximation, we are only mapping the primes to their nearest approximation, and everything else is produced by multiplying these tempered primes together. Although we can also map primes to intervals that *aren't* their nearest approximation, too; see below.

First of all, if you're mapping the Porcupine comma to a non-zero value, you're not tempering it out. 1200 can support Porcupine, but it requires that you set 49.17 cents equal to 0 cents. You also have to use a non-optimal mapping of the primes in 1200 to get porcupine temperament out of it, but there's nothing stopping you from doing that. I'm not sure what the porcupine val in 1200 would be, exactly, but I'm sure there's a way to figure it out.

Right, and the commas will vary with your JI scale. It helps to lay the scale out as a Fokker periodicity block so that you can see the lattice connecting all the tones. If you haven't read it yet, I strongly recommend reading Paul Erlich's "Middle Path" paper, which can be found here:

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

In actuality, you can make any interval into a comma and temper it out, but the simpler the interval, the smaller and less accurate the temperament will become. A temperament's accuracy is more or less proportional to the size of its comma, whereas its complexity is proportional to the size of the numerator and denominator of the comma. For example, 16/15, the Father "comma", is large (111.73128526977776 cents) but has a small numerator and denominator; this leads to a simple temperament with a large amount of error. The schismatic comma of 32805/32768, OTOH, is small (1.9537207879341596 cents), but the numerator and denominator are both large; this leads to a complex temperament with a small amount of error. There's also interaction between the size of the comma and the size of the numbers in the numerator and denominator, such that a large comma with large numerator and denominator will be more accurate than a comma of roughly the same size but with a small numerator and denominator, because in complex temperaments the error is spread out over a larger chain of intervals. Likewise, if the complexity is more or less the same between two temperaments but the size of the comma is greater in one, that one will be less accurate. Meantone and Pelogic temperaments are good examples, because they have the same complexity (more or less), but the Pelogic comma of 135/128 is 92.17871646099708 cents, more than four times the size of the Syntonic comma of 81/80 (21.50628959671478 cents). This means Pelogic temperaments will be less accurate but have around the same complexity as Meantones.

Hope this helps!

-Igs

🔗Graham Breed <gbreed@...>

Jake Freivald <jdfreivald@...> wrote:

> That part I've understood for a while: When we say
> "364/363 vanishes" or "13/11 + 14/11 = 3/2" we're really
> replacing 2, 3, 7, 11, and/or 13 with other numbers that
> are close to them.

That's it!

> >> you take your JI scale as it stands, pick the EDO that
> >> tempers the commas that you want (and perhaps meets
> >> other criteria), and then do the mapping of the JI
> >> scale into the EDO.
> >
> > Yes, that's it, for tempering *out* the commas.
>
> Super. One of the things we'd want, then, as a community,
> is a way to see which commas a given EDO tempers out. It
> looks like those things are inconsistently documented in
> the wiki -- is there a way to determine that for
> ourselves, or a resource that contains that information
> in an easy-to-use way?

There are two ways to get that:

1) Produce along list of commas you might be interested,
and filter them by each equal temperament you want to show.

2) Find a basis for the null space or kernel of the lattice
defined by the equal temperaments. Reduce it to give
simple ratios (weighted LLL works well). This is fairly
easy in Pari, either with the GP or Sage front-ends, but
there's nothing online to do those calculations. I think I
know how to do it in a more general language (it involves
two Hermite reductions) but I haven't done it so it still
isn't online.

If you want to work through (1), you can put your favorite
commas in here, and note the equal temperaments that come
out:

http://x31eq.com/temper/uv.html

> It seems a little weird to think about an EDO, which is
> created by stacking irrational nth roots of 2 on top of
> each other, tempering out commas, which are rational.

Commas may be rational, or may be ratio-space vectors.
In algebraic terms, the equal temperament (val) space is
dual to the unison vector (tempered out comma) space. When
no unison vectors are tempered out, you have just
intonation.

> I was being facetious, of course, but it makes me wonder
> if I should document it in a way that I think I would
> have understood a month ago.

Yes, please!

Graham

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> Super. One of the things we'd want, then, as a community, is a way to
> see which commas a given EDO tempers out. It looks like those things
> are inconsistently documented in the wiki -- is there a way to
> determine that for ourselves, or a resource that contains that
> information in an easy-to-use way?

That's because every EDO can be said to temper out an infinite number of commas, especially if we allow the "limit" to be arbitrarily large. Also if we let error be arbitrarily large (which we shouldn't, but mathematically there is nothing stopping us), we can say any EDO tempers out *any* comma. The results will be psychoacoustically absurd, but still mathematically sound. There doesn't seem to be any easy, systematic, and rigourous way to place finite boundaries on the commas tempered out by each ET, or even to pick out which temperaments are the most psychoacoustically valid. However, given a comma (or commas), it *is* possible to find EDOs that temper it out reasonably well. However, I don't think anyone's put together a comprehensive list of all the EDOs of reasonable size that are reasonably psychoacoustically-compatible with all the known temperaments. Most seem to focus on EDOs that have the lowest "badness".

> It seems a little weird to think about an EDO, which is created by
> stacking irrational nth roots of 2 on top of each other, tempering out
> commas, which are rational.

That's because EDOs can be "created" in a variety of ways. Stacking irrational nth roots of 2 is one way, but stacking tempered rationals and wrapping to fit within an octave until an equal scale is reached is another way. An equal scale may not always be reached, of course.

-Igs

🔗Jake Freivald <jdfreivald@...>

Thanks as always, Igs.

>> I can see that 1200 EDO doesn't support Porcupine
[snip]

> First of all, if you're mapping the Porcupine comma to a non-zero value,
> you're not tempering it out.

Right, I wasn't thinking. I was actually talking about approximating
the Porcupine comma as a step size; "supporting Porcupine" would mean
supporting Porcupine temperament, which tempers that comma out.

> Right, and the commas will vary with your JI scale. It helps to
> lay the scale out as a Fokker periodicity block so that you can
> see the lattice connecting all the tones. If you haven't read it yet,
> I strongly recommend reading Paul Erlich's "Middle Path" paper,
> which can be found here:
>
> http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

I started reading that, but I don't think I knew enough to absorb what
it was trying to tell me. That was true of the wiki, too, though, and
re-reading an entry on the wiki started this discussion. :) I'll try
again. I have to try Fokker blocks again as well.

> the simpler the interval, the smaller and less accurate the temperament will become.

This whole discussion was interesting.

It's pretty easy to see why a large comma would lead to high error,
and I'm not sure why people would temper out larger intervals. (Not
knocking the idea, of course, just saying I haven't played with it
enough to see how it would be useful musically.)

It sounds like you're saying that complexity goes up when you have to
multiply more (distorted) primes together: (2*2*2)/(3*5) is "less
complicated" than (3*3*3*3*3*3*3*3*5)/(2*2*2*...*2). Is that what
"complexity" means, or is it a reasonable proxy for it?

It looks like the more you multiply the distorted primes together to
get the comma, the smaller the error in the distorted primes can be.
In a complex comma, the small amount of distortion in the primes gets
multiplied over and over again, which increases the error enough to
literally make the comma go away; meanwhile, in a simpler ratio like
7/6 or 5/4, the small amount of distortion isn't multiplied that much,
so the distorted interval isn't that different from just.

It sounds like the "best" commas to temper out, if you want something
closer to just intonation, are high in complexity and low in
magnitude. Small-looking ratios might be simpler, but large ones might
be sweeter.

Thanks,
Jake

🔗Jake Freivald <jdfreivald@...>

Graham wrote:

> 1) Produce along list of commas you might be interested,
> and filter them by each equal temperament you want to show.
[snip]

> http://x31eq.com/temper/uv.html

Thanks for this, both the link and the programming work that went into
it. (Thanks to Carl, too, for the same link.) I've seen this before,
but didn't understand why I'd need to use it for EDOs.

Based on Gene's and Igs' comments, it sounds like there's no magic
bullet for determining which EDOs are returned by your temperament
finder. May I assume you're using some sort of measure and you set
some sort of cutoff on the measure to say the EDO is good for this
temperament?

> 2) Find a basis for the null space or kernel of the lattice

I didn't understand this, so I'm not doing it anytime soon. :) One
thing at a time.

>> I was being facetious, of course, but it makes me wonder
>> if I should document it in a way that I think I would
>> have understood a month ago.
>
> Yes, please!

I probably shouldn't have opened my mouth -- I'm not sure if I know
how to get here from where I was! :)

Thanks,
Jake

🔗Jake Freivald <jdfreivald@...>

🔗Graham Breed <gbreed@...>

"cityoftheasleep" <igliashon@...> wrote:

> That's because every EDO can be said to temper out an
> infinite number of commas, especially if we allow the
> "limit" to be arbitrarily large. Also if we let error be
> arbitrarily large (which we shouldn't, but mathematically
> there is nothing stopping us), we can say any EDO tempers
> out *any* comma. The results will be psychoacoustically
> absurd, but still mathematically sound. There doesn't
> seem to be any easy, systematic, and rigourous way to
> place finite boundaries on the commas tempered out by
> each ET, or even to pick out which temperaments are the
> most psychoacoustically valid. However, given a comma
> (or commas), it *is* possible to find EDOs that temper it
> out reasonably well. However, I don't think anyone's put
> together a comprehensive list of all the EDOs of
> reasonable size that are reasonably
> psychoacoustically-compatible with all the known
> temperaments. Most seem to focus on EDOs that have the
> lowest "badness".

Yes, you shouldn't let the error be arbitrarily large. So
don't. There is something mathematical stopping you: the
list of commas becomes infinite.

There's an easy and systematic way to place a finite
boundary on the commas defined by each ET: draw lines of
maximum error and maximum complexity. Whether this is
rigorous depends on what you mean by that. Whether it's
psychoacoustically correct depends on a better knowledge of
psychoacoustics than I have. Producing lists of commas
according to these cutoffs can be done. So far, nobody has
stepped up to do it.

Yes, we tend to focus on badness. Where we know about
temperaments, it's often because they came up on lists of
low badness. Some temperaments are known about -- and
named -- purely because they sit on those lists. Where
there are outliers, I don't know if they're significant.
They tend to be higher-limit extensions of low badness
temperaments. I can list them if you want to investigate.

Low badness ETs tend to lead to low badness temperaments of
a higher rank. For Cangwu badness, this relationship is
mathematically rigorous. For a close approximation of TE
error and complexity, there's also a mathematically
rigorous way of choosing the equal temperaments:

http://x31eq.com/complete.pdf

A list of all equal temperaments supporting a regular
temperament will also get equally large, for rank 2 and
beyond. A size threshold isn't enough. You need an error
threshold as well. Or badness. Badness works.

Graham

🔗Carl Lumma <carl@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> Right, I wasn't thinking. I was actually talking about approximating
> the Porcupine comma as a step size; "supporting Porcupine" would mean
> supporting Porcupine temperament, which tempers that comma out.

Right. It's hard to keep it all straight at first, believe me I know!

> I started reading that, but I don't think I knew enough to absorb what
> it was trying to tell me. That was true of the wiki, too, though, and
> re-reading an entry on the wiki started this discussion. :) I'll try
> again. I have to try Fokker blocks again as well.

It took me about 4 years to fully understand it :->

> It's pretty easy to see why a large comma would lead to high error,
> and I'm not sure why people would temper out larger intervals. (Not
> knocking the idea, of course, just saying I haven't played with it
> enough to see how it would be useful musically.)

Mostly because of the radical simplicity and for extreme "puns". Tempering out 25/24, for example, equates 5/4 and 6/5, and also equates 25/16 (as well as 36/25) with 3/2. This is "dicot" temperament, and is one way of looking at 7-EDO (and also 10-EDO), a temperament in which the main "pun" is that major chords and minor chords are the same thing. Tempering out 16/15 leads to 5/4 and 4/3 being equated, and this is "father" temperament, and is one way of looking at 8-EDO (you can also do it in 13 and 21 if you use a non-optimal val). Having one interval that functions as both 4/3 and 5/4 is really wild and xenharmonic. Then there's 27/25, the "bug" comma, which equates two 6/5's (36/25) with 4/3 (36/27). 9-EDO can be looked at this way, as can 14-EDO. Very simple and very xenharmonic.

> It sounds like you're saying that complexity goes up when you have to
> multiply more (distorted) primes together: (2*2*2)/(3*5) is "less
> complicated" than (3*3*3*3*3*3*3*3*5)/(2*2*2*...*2). Is that what
> "complexity" means, or is it a reasonable proxy for it?

Pretty much, yes.

> It looks like the more you multiply the distorted primes together to
> get the comma, the smaller the error in the distorted primes can be.
> In a complex comma, the small amount of distortion in the primes gets
> multiplied over and over again, which increases the error enough to
> literally make the comma go away; meanwhile, in a simpler ratio like
> 7/6 or 5/4, the small amount of distortion isn't multiplied that much,
> so the distorted interval isn't that different from just.

Precisely. Another implication of this is that the "acceptable tuning range" for the generator of a very complex temperament will be much narrower than for a simpler temperament. You have a range of about +/- 4 cents (692-700 cents) to tune meantone, but to tune schismatic you have only maybe +/- 1 or 2 cents (700-703 cents).

> It sounds like the "best" commas to temper out, if you want something
> closer to just intonation, are high in complexity and low in
> magnitude. Small-looking ratios might be simpler, but large ones might
> be sweeter.

Generally-speaking, yes. But at the same time, the closer you are to JI, the less advantage you actually get from the tempering. The whole point of tempering is to do as little damage as possible to perceived consonance while reducing the complexity of the system by as *much* as possible. So the "best" comma would be one that is low in complexity *and* low in magnitude, and it turns out that in the 5-limit, that is unambiguously 81/80. Srutal/Pajara, Porcupine, Magic, Hanson, Negri, Augmented, and Diminished are the main runners-up (maybe Blackwood, too) in terms of the balance between simplicity and accuracy. Schismatic, Wuerschmidt, Amity, Tetracot, Orson, and Sensi are probably the main runners-up if we're willing to let simplicity slide in favor of extreme accuracy.

As far as EDOs that support these temperaments, Srutal, Porcupine, and Orson are tuned well in 22-EDO; Magic, Hanson, Meantone, Sensi, and Negri are all tuned well in 19-EDO; Augmented and Diminished are both close to optimal in 12-EDO, and Blackwood is done pretty ideally in 15-EDO (which half-asses Porcupine also). Schismatic doesn't really show up at all until 29-EDO (which also supports Porcupine and Negri), and really only comes into its own at 41-EDO (which also supports Tetracot and Magic, but is more useful for 7 or 11-limit temperaments). Amity doesn't really have a good tuning in anything I've looked at from 41 down...maybe 39 but the 5-limit is pretty f'ed up. Wuerschmidt works pretty well in 31 and 34; 31 also supports Meantone and Orson, while 34 also supports Srutal, Hanson, and Tetracot.

In the 7-limit...I can't really go there right now, actually. The 7-limit is a lot more complicated. Suffice to say that for extreme accuracy in the 5-limit, 34, 31, 19, and 22 (followed by 29 and 12) are the main contenders, with 19 supporting the largest number of 5-limit temperaments.

Hope that helps!

-Igs

🔗Jake Freivald <jdfreivald@...>

Lots of good things here, Igs.

With respect to puns: I'm mostly noodling with Cantonpenta right now.
I like its sound, and I figure if the whole Western world can devote
most of its energy to 12 EDO then I should be able to at least put
*some* semi-consistent energy into a single scale. :) The major pun in
Cantonpenta is 13/11 + 14/11 = 3/2. Thing is, once the scale has been
generated, I don't think about the fact that it's a pun: I just have
nice triads with more roots. Maybe I need to start playing with more
radically xenharmonic scales so I think in terms of puns more often.

By the way, this:

> the closer you are to JI, the less advantage you actually get from the tempering. The
> whole point of tempering is to do as little damage as possible to perceived consonance
> while reducing the complexity of the system by as *much* as possible. So the
> "best" comma would be one that is low in complexity *and* low in magnitude

...Is a superb point, and helps me balance the direction my head was headed in.

The list of temperaments was great, too. Sometimes taxonomic issues
only make sense when you see them in the right context.

Thanks,
Jake

On 5/10/11, cityoftheasleep <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
>> Right, I wasn't thinking. I was actually talking about approximating
>> the Porcupine comma as a step size; "supporting Porcupine" would mean
>> supporting Porcupine temperament, which tempers that comma out.
>
> Right. It's hard to keep it all straight at first, believe me I know!
>
>> I started reading that, but I don't think I knew enough to absorb what
>> it was trying to tell me. That was true of the wiki, too, though, and
>> re-reading an entry on the wiki started this discussion. :) I'll try
>> again. I have to try Fokker blocks again as well.
>
> It took me about 4 years to fully understand it :->
>
>> It's pretty easy to see why a large comma would lead to high error,
>> and I'm not sure why people would temper out larger intervals. (Not
>> knocking the idea, of course, just saying I haven't played with it
>> enough to see how it would be useful musically.)
>
> Mostly because of the radical simplicity and for extreme "puns". Tempering
> out 25/24, for example, equates 5/4 and 6/5, and also equates 25/16 (as well
> as 36/25) with 3/2. This is "dicot" temperament, and is one way of looking
> at 7-EDO (and also 10-EDO), a temperament in which the main "pun" is that
> major chords and minor chords are the same thing. Tempering out 16/15 leads
> to 5/4 and 4/3 being equated, and this is "father" temperament, and is one
> way of looking at 8-EDO (you can also do it in 13 and 21 if you use a
> non-optimal val). Having one interval that functions as both 4/3 and 5/4 is
> really wild and xenharmonic. Then there's 27/25, the "bug" comma, which
> equates two 6/5's (36/25) with 4/3 (36/27). 9-EDO can be looked at this
> way, as can 14-EDO. Very simple and very xenharmonic.
>
>> It sounds like you're saying that complexity goes up when you have to
>> multiply more (distorted) primes together: (2*2*2)/(3*5) is "less
>> complicated" than (3*3*3*3*3*3*3*3*5)/(2*2*2*...*2). Is that what
>> "complexity" means, or is it a reasonable proxy for it?
>
> Pretty much, yes.
>
>> It looks like the more you multiply the distorted primes together to
>> get the comma, the smaller the error in the distorted primes can be.
>> In a complex comma, the small amount of distortion in the primes gets
>> multiplied over and over again, which increases the error enough to
>> literally make the comma go away; meanwhile, in a simpler ratio like
>> 7/6 or 5/4, the small amount of distortion isn't multiplied that much,
>> so the distorted interval isn't that different from just.
>
> Precisely. Another implication of this is that the "acceptable tuning
> range" for the generator of a very complex temperament will be much narrower
> than for a simpler temperament. You have a range of about +/- 4 cents
> (692-700 cents) to tune meantone, but to tune schismatic you have only maybe
> +/- 1 or 2 cents (700-703 cents).
>
>> It sounds like the "best" commas to temper out, if you want something
>> closer to just intonation, are high in complexity and low in
>> magnitude. Small-looking ratios might be simpler, but large ones might
>> be sweeter.
>
> Generally-speaking, yes. But at the same time, the closer you are to JI,
> the less advantage you actually get from the tempering. The whole point of
> tempering is to do as little damage as possible to perceived consonance
> while reducing the complexity of the system by as *much* as possible. So
> the "best" comma would be one that is low in complexity *and* low in
> magnitude, and it turns out that in the 5-limit, that is unambiguously
> 81/80. Srutal/Pajara, Porcupine, Magic, Hanson, Negri, Augmented, and
> Diminished are the main runners-up (maybe Blackwood, too) in terms of the
> balance between simplicity and accuracy. Schismatic, Wuerschmidt, Amity,
> Tetracot, Orson, and Sensi are probably the main runners-up if we're willing
> to let simplicity slide in favor of extreme accuracy.
>
> As far as EDOs that support these temperaments, Srutal, Porcupine, and Orson
> are tuned well in 22-EDO; Magic, Hanson, Meantone, Sensi, and Negri are all
> tuned well in 19-EDO; Augmented and Diminished are both close to optimal in
> 12-EDO, and Blackwood is done pretty ideally in 15-EDO (which half-asses
> Porcupine also). Schismatic doesn't really show up at all until 29-EDO
> (which also supports Porcupine and Negri), and really only comes into its
> own at 41-EDO (which also supports Tetracot and Magic, but is more useful
> for 7 or 11-limit temperaments). Amity doesn't really have a good tuning in
> anything I've looked at from 41 down...maybe 39 but the 5-limit is pretty
> f'ed up. Wuerschmidt works pretty well in 31 and 34; 31 also supports
> Meantone and Orson, while 34 also supports Srutal, Hanson, and Tetracot.
>
> In the 7-limit...I can't really go there right now, actually. The 7-limit
> is a lot more complicated. Suffice to say that for extreme accuracy in the
> 5-limit, 34, 31, 19, and 22 (followed by 29 and 12) are the main contenders,
> with 19 supporting the largest number of 5-limit temperaments.
>
> Hope that helps!
>
> -Igs
>
>
>
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