Tuning-Math Digests messages 2575 - 2599

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Message: 2575

Date: Wed, 19 Dec 2001 00:48:51

Subject: Re: Badness with gentle rolloff

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > > Yes. http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip - Ok *
> > 
> > Got it!
> > 
> > >> I'd love to see consistency plotted against steps*rms.
> > > 
> > > This has been added at your request.
> > 
> > Thanks!  I wish I could say I knew what I was looking at.
> > Cut-off badness on x and "is consistent" on y?  I was thinking
> > steps on x and real-number (unrounded) Hahn consistency on y
> > (it looks like you're using boolean consistency).
> 
> If you're talking about the maximum real-number consistency _level_ 
> obeyed by an ET, this will be equal to 1/(max_error*1200*steps) 
> wherever it's greater than 1.5. So for the good ETs, you'll just be 
> plotting maximum error vs. rms error.

Oops -- that's true if you plot maximum real-number consistency level 
against 1/(steps*rms), which is what I'm guessing you really had in 
mind anyway. . . ?


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Message: 2576

Date: Wed, 19 Dec 2001 00:50:04

Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff)

From: genewardsmith

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

> > You or someone might try graphing log steps vs badness for both of 
> these, or log steps vs number less than a cut-off value.

> Gene, I'm disappointed. Dave has produced tons of graphs, in case you 
> didn't notice. 

Did he do the graphs I suggested? They might explain flatness better than some other sort of graph.

Dave wants to understand how you use Diophantine 
> approximation theory here.

Well, in case you haven't noticed, I did explain it. What is left to do?


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Message: 2577

Date: Wed, 19 Dec 2001 01:12:36

Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff)

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > You or someone might try graphing log steps vs badness for both of 
> these, or log steps vs number less than a cut-off value.
> 
> Gene, I'm disappointed. Dave has produced tons of graphs, in case 
you 
> didn't notice. Dave wants to understand how you use Diophantine 
> approximation theory here.

Not quite true. While a gentle introduction to Diophantine 
approximation theory would be welcome, I would also like to see it 
tested empirically.

Paul, your criticism of my graphs was that they didn't go far enough. 
I only went to 612-tET. Going much further than 2000-tET is probably 
not practical for my spreadsheet approach, but you have plotted 
1/(steps^(4/3)*cents) to 10^7-tET or some such. Could you maybe do the 
same for 1/(steps*cents)?

However, plotting stuff and eyeballing it is one thing, but we should 
really be able to _calculate_ the flatness of a given badness measure 
out to large number ETs. 

Gene seems to be saying that for any given badness cutoff, the number 
less than that should be about the same in every decade (1 to 9-tET, 
10 to 99-tET, 100 to 999-tET, etc). Could you check that with Matlab 
Paul, for both steps^(4/3)*cents and steps*cents? For various cutoffs?


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Message: 2578

Date: Wed, 19 Dec 2001 01:45:31

Subject: Re: formula for meantone implications?

From: joemonz

--- In tuning-math@y..., "unidala" <JGill99@i...> wrote:
Yahoo groups: /tuning-math/message/1955 *

> Joe,
> 
> The more that I stared at my "correction" in message #1926,
> the more I realized that I (erroneously) added something (2^)
> to the solution for Z as a function of X. With the understanding
> that the dependent variables Y and Z (as well as the independent
> variable X), being in the *exponents" of your (nicely done)
> diagram's algebraic identity [which was (3^(X/3))*(5^(X/6))
> ~= (3^Y)*(5^Z)] represent quantaties in the "logarithmic domain"
> (and not the "linear domain"), the correction below should work
> well for the such determinations of Y and Z as functions of X as
> (it would seem) you have requested.
>
> > Z=(LN of(5^(X/2)-(3*(Z+(dY/dX)*X)+ X)/(LN of 3)))/(3*(LN of 5)))


There's one parenthesis too many at the end.
Can you go over this again and remove it?  Thanks.
I'm running your formulas thru Excel to see what the produce.



-monz


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Message: 2579

Date: Wed, 19 Dec 2001 01:55:14

Subject: Re: Badness with gentle rolloff

From: dkeenanuqnetau

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > Yes. http://uq.net.au/~zzdkeena/Music/7LimitETBadness.xls.zip - Ok *
> 
> Got it!
> 
> >> I'd love to see consistency plotted against steps*rms.
> > 
> > This has been added at your request.
> 
> Thanks!  I wish I could say I knew what I was looking at.
> Cut-off badness on x and "is consistent" on y?  I was thinking
> steps on x and real-number (unrounded) Hahn consistency on y
> (it looks like you're using boolean consistency).

Yes. steps*rms on x and Boolean consistency on y.

Tell me how to calculate real-number Hahn consistency.

Do you mean you want to see both Hahn consistency and steps*cents 
badness (or do you want 1/(steps*cents) goodness? plotted against 
steps?


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Message: 2581

Date: Thu, 20 Dec 2001 03:00:15

Subject: Re: Two conditions on temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> It seems to me there are two conditions it would make sense to put 
on something intended as an odd-limit temperament:
> 
> (1) Weak condition--no element of the tonality diamond is allowed 
to be a unison

Fine. Then we won't need a lower bound on g anymore, will we?
> 
> (2) Strong condition--all elements of the tonality diamond are 
distinct

That would make me cry, as it excludes twintone in the 7-limit.


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Message: 2582

Date: Thu, 20 Dec 2001 13:56 +0

Subject: Re: Meantone & co

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9vsgv8+d425@xxxxxxx.xxx>
Gene wrote:

> Graham is more likely to include things I wouldn't than vice-versa--his 
> methods can include more than one version of the same system, so that 
> you have for instance meantone, and another meantone with a seemingly 
> useless doubling of the generator steps. There are at least three 
> version of kleismic on his list--one the usual, one with doubled 
> generator steps, and one with a half-ocatave period.

Yes.  That's an interesting, uh, feature of my algorithm for going from 
ETs to linear temperaments.

A normal 5-limit kleismic can be generated from 19- and 34-equal:

>>> h19 = temper.PrimeET(19, temper.primes[:2])
>>> h34 = temper.PrimeET(34, temper.primes[:2])
>>> (h19&h34).mapping
[(1, 0), (0, 6), (1, 5)]

The defining wedge product is

>>> (h19^h34).complement().flatten()
(-6, -5, 6)

One example where it goes wrong is 15- and 23-equal

>>> h23 = temper.PrimeET(23, temper.primes[:2])
>>> h15 = temper.PrimeET(15, temper.primes[:2])
>>> (h23&h15).mapping
[(1, 0), (0, 12), (1, 10)]

the wedge product is

>>> (h15^h23).complement().flatten()
(12, 10, -12)

and so simplifies to be equivalent to the above.  In this case, it's 
obvious from looking at the generator mapping that something's wrong, 
because it has a common factor of 2.

The other strange example is 4&34:

>>> h4 = temper.PrimeET(4, temper.primes[:2])
>>> (h4&h34).mapping
[(2, 0), (6, -6), (7, -5)]

for the same reason

>>> (h4^h34).complement().flatten()
(-12, -10, 12)

Here, though, there's nothing obviously wrong with the resulting 
temperament.


So are these behaviours a problem?  They follow from the generator and 
period being chosen ahead of the mapping.  That's probably correct, 
because this is supposed to be an extension of the Scale Tree(s).  They 
won't usually get to the top of the list, because they have a higher 
complexity than the equivalents.  But it does mean temperaments will be 
underrated if they appear in this way.  I'm hoping that any set of ETs 
containing a pair with torsion will also contain a pair without torsion.  
Also that it'll be far less common in the higher limits.  If not, we can 
always get to those temperaments by using wedge products.  With the 
current implementation, at least, I expect that would slow things down.


                      Graham


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Message: 2583

Date: Thu, 20 Dec 2001 03:02:56

Subject: Re: Two conditions on temperaments

From: genewardsmith

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

> That would make me cry, as it excludes twintone in the 7-limit.

That occurred to me also; I think the strong condition is too strong, but I see no problem with the weak condition.


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Message: 2584

Date: Thu, 20 Dec 2001 19:11:55

Subject: Re: Badness with gentle rolloff

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >Tell me how to calculate real-number Hahn consistency.
> 
> According to Paul, it's 1/(max_error*1200*steps)

No, that's only correct if it's greater than 1.5.


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Message: 2585

Date: Thu, 20 Dec 2001 03:05:53

Subject: Re: Meantone & co

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are four more:
> 
> 81/80
> 
> Map:
> 
> [ 0  1]
> [-1  2]
> [-4  4]
> 
> Generators: a = 20.9931/50; b = 1
> 
> badness: 108
> rms: 4.22
> g: 2.944
> errors: [-5.79, -1.65, 4.14]
> 
> Nothing left to say about this one. :)
> 
> 
> 2048/2025
> 
> Map:
> 
> [ 0  2]
> [-1  4]
> [ 2  3]
> 
> Generators: 14.0123/34 (~4/3); b = 1/2
> 
> badness: 211
> rms: 2.613
> g: 4.32
> errors: [3.49, 2.79, -.70]
> 
> A good way to take advantage of the 34-ets excellent 5-limit 
harmonies
> is two gothish 17-et chains of fifths a sqrt(2) apart.

This is my interpretation of the Indian scales.

> 
> 
> 78732/78125 = 2^2 3^9 5^-7
> 
> Map:
> 
> [ 0  1]
> [ 7 -1]
> [ 9 -1]
> 
> Generators: 23.9947/65 (~9/7); b = 1
> 
> badness: 346
> rms: 1.157
> g: 6.68
> errors: [-1.1, 0.5, 1.6]

Would you call this one a "unique facet of 65-tET"? Is this the kind 
of thing that Graham's searching pairs of ETs is likely to miss?

> 
> 
> 393216/390625 = 2^17 3 5^-8
> 
> Map:
> 
> [ 0  1]
> [ 8 -1]
> [ 1  2]
> 
> Generators: a = 31.9951/99 (~5/4); b = 1
> Works with 31,34,65,99,164
> 
> badness: 251
> rms: 1.072
> g: 6.16
> error: [.602, 1.506, .904]

Magic. Rather than continuing, Gene, could you start over going from 
lowest badness to highest badness?


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Message: 2586

Date: Thu, 20 Dec 2001 03:07:47

Subject: Re: 55-tET

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > > It makes sense, but I don't think it defines a unique interval.
> > 
> > Meaning you can always find smaller and smaller examples? Even if 
you 
> > disallow "potential torsion"? What if you fix all the commas 
except 
> > one, and just have to find the smallest candidate for the 
remaining 
> > comma. Isn't that choice unique?
> 
> That should do it, though it seems a little arbitary. If you don't 
>fix all but one, I can prove easily enough you get arbitarily small 
>commas, so this may be your best shot. For the 55-et, or anything 
>else where there is a clear set of all-but-one keepers in the comma 
>department, it would make sense.

Well? Did I find it? 3^15*5^10.


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Message: 2587

Date: Thu, 20 Dec 2001 03:16:11

Subject: Re: Meantone & co

From: genewardsmith

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

> Magic.

Not Magic; another 5/4 system.

 Rather than continuing, Gene, could you start over going from 
> lowest badness to highest badness?

This way puts similar systems together; why do it the other way?


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Message: 2588

Date: Thu, 20 Dec 2001 03:17:35

Subject: Re: supernatural superparticulars?

From: paulerlich

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> 
> Yahoo groups: /tuning/message/31663 *
> 
> 
> > >Curiously, J Gill 
> > 
> > No -- but what seems to be the case very often, is that when one 
> > comes up with such a scale in the form of a periodicity block, 
one 
> > has quite a few arbitrary choices to make as to which version of 
a 
> > particular scale degree one wants (the different versions 
differing 
> > by a unison vector), and then _one such set_ of arbitrary choices 
> > does lead to a scale with superparticular step sizes. 
> 
> Hi Paul...
> 
> Well, that's pretty *mysterious* isn't it? Why does that happen 
that 
> the superparticular step sizes result?

Usually they don't "result", but very often you can arbitrarily 
choose to use them.

> Is it just the way the system 
> is set up.

Well, superparticulars are not favored _by design_, if that's what 
you mean. You should ask Kraig Grady about superparticular step sizes 
too.

> Spooky stuff! (If we can't believe in "magic primes" 
> that surely is something a little weird... yes?)

One of the themes on the tuning-math list (busier than ever) is 
superparticulars . . . for example, we found that the graph of 
ET "goodness" has "waves" in it, and the most prominent visible wave 
by far for 7-limit rises and falls every 1664 ETs . . . and the 
superparticular ratio that Graham made famous in his Blackjack 
progression, 2401:2400, fits 1663.9 times in an octave.


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Message: 2589

Date: Thu, 20 Dec 2001 03:29:21

Subject: Re: Meantone & co

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Magic.
> 
> Not Magic; another 5/4 system.
> 
>  Rather than continuing, Gene, could you start over going from 
> > lowest badness to highest badness?
> 
> This way puts similar systems together; why do it the other way?

To get a sense of whether I can accept your kind of "flatness" in 
this context.


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Message: 2590

Date: Thu, 20 Dec 2001 03:30:40

Subject: Re: Meantone & co

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Magic.
> 
> Not Magic; another 5/4 system.

Is this better or worse than Magic in the 5-limit?


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Message: 2591

Date: Thu, 20 Dec 2001 03:27:42

Subject: The Magic of Superparticulars

From: paulerlich

Basically, the magic of superparticulars is that they're the smallest 
possible intervals for a given complexity (or distance on the 
lattice). Which is kind of obvious if you think about it. If the two 
numbers in a ratio differ by 1, how can the interval be any smaller 
without increasing the two numbers in the ratio?

Things fizzle out, though, once you get past the point where there 
are no superparticulars left. Think 5-limit; you can look at the page

S235 *

and in the second column from the right, you'll see a bunch of 
ratios. Certain rows have the first few columns in parentheses, but 
those that don't correspond to the SMALLEST unison vectors for some 
given DISTANCE LIMIT in the lattice, or ODD-LIMIT, or INTEGER-LIMIT 
if you prefer. , you can see, are all superparticular, until you 
reach the last two, your old friends 25:24 and 81:80. Then there are 
no more superparticulars possible in the 5-limit.

Now look at 

S2357 *

and again, all the non-parenthesized entries are superparticular, 
until you reach the last two, 2401:2400 and 4375:4374. The latter is 
the last superparticular possible in the 7-limit.


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Message: 2592

Date: Thu, 20 Dec 2001 03:33:18

Subject: Re: Two conditions on temperaments

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > That would make me cry, as it excludes twintone in the 7-limit.
> 
> That occurred to me also; I think the strong condition is too 
strong, but I see no problem with the weak condition.

I agree. but we'd like to know when the strong condition applies to a 
temperament. This is shown as "unique" in Graham Breed's output, as in 
all elements of the diamond are represented by a unique interval in 
the temperament.


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Message: 2593

Date: Thu, 20 Dec 2001 03:40:38

Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff)

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > This is all sounding a little contrived. How low is low enough? 
How 
> > far from 1-tET is far enough? If we have to go out beyond what is 
> > musically relevant, then what's the point?
> 
> Why not just exlude the inconsistent ets? The point is to get rid of 
the crap parade at the very beginning, and this looks like a use of 
consistency I could go for.
> 

I understand this to be equivalent to putting a sharp cutoff at 600 on 
steps*cents. The usual objection to sharp cutoffs applies, namely 
people don't usually apply sharp cutoffs when making decisions about 
the usefulness of tunings.


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Message: 2594

Date: Thu, 20 Dec 2001 03:42:50

Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff)

From: dkeenanuqnetau

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I understand this to be equivalent to putting a sharp cutoff at 600 
on 
> steps*cents. The usual objection to sharp cutoffs applies, namely 
> people don't usually apply sharp cutoffs when making decisions about 
> the usefulness of tunings.

Oops! Only when cents is max-absolute error, not rms.


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Message: 2595

Date: Thu, 20 Dec 2001 03:46:04

Subject: Re: Flat 7 limit ET badness? (was: Badness with gentle rolloff)

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > I understand this to be equivalent to putting a sharp cutoff at 
600 
> on 
> > steps*cents. The usual objection to sharp cutoffs applies, namely 
> > people don't usually apply sharp cutoffs when making decisions 
about 
> > the usefulness of tunings.
> 
> Oops! Only when cents is max-absolute error, not rms.

Maybe it should be minimax. Maybe we should give a _range_ of optimal 
generators, rather than just one, when the same minimax is achieved 
for all within the range. Maybe we should also give the points at 
which the minimax is doubled. This would give an idea of the 
sensitivity of the tuning.


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Message: 2596

Date: Thu, 20 Dec 2001 03:50:55

Subject: Re: Meantone & co

From: genewardsmith

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

> > This way puts similar systems together; why do it the other way?
> 
> To get a sense of whether I can accept your kind of "flatness" in 
> this context.

I don't get it--how will mixing the apples with the oranges do that?


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Message: 2597

Date: Thu, 20 Dec 2001 03:58:04

Subject: Re: Meantone & co

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > > This way puts similar systems together; why do it the other way?
> > 
> > To get a sense of whether I can accept your kind of "flatness" in 
> > this context.
> 
> I don't get it--how will mixing the apples with the oranges do that?

I just want to get the sense that some fixed value of badness is 
going to remove unimportant ones while retaining all the important 
ones. For example, I'd definitely set tbe badness cutoff high enough 
so that meantone, augmented, and diminished, even if they had errors 
as large as in 12-tET, make it in.


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Message: 2599

Date: Thu, 20 Dec 2001 04:44:35

Subject: [tuning] Re: great explanation [periodicity block]

From: paulerlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> <<Hmm? Those would be the stepsizes for the 12-tone scale in this
> series, not the whole series, correct?>>
> 
> Right, following the expansion or contraction rules will walk you up
> and back in the series. So the stepsizes for the 22-tone scale in 
this
> series would be 81/80 128/125 25/24, and the 2D UVs would be 
3125/3072
> and 2048/2025.

There's probably one more, if you take a look at the lattice.

Why don't you make a quick-and-dirty periodicity-revealing lattice 
for the 12- and 22-tone scales that result in the "Tribonacci" system?


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