Tuning-Math Digests messages 4900 - 4924

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Message: 4900

Date: Mon, 27 May 2002 12:59:08

Subject: Re: A 7-limit best list

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> [[1, 0, 4, 1], [0, 1, -1, 1]]   [1200, 2025]   1.354 154.263

This one slipped by me--ignore it.


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Message: 4901

Date: Tue, 28 May 2002 10:07:31

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> This is a first pass at a 7-limit best list. The first entry is the 
mapping matrix, the second period and generator, the third 
(unweighted) rms generator steps to consonances, the fourth 
(unweighted) rms error. Badness is not listed to save space, but is 
less than 300; generator steps are less than 40, and rms error less 
than 50 cents. The ordering is by badness, lowest to highest.
> 

For future reference, the blank lines between temperaments were a 
nuisance re importing to Excel. And I think we should have 
at least one decimal place for the rms-optimum generators, since 
we're sometimes stacking 10 or more of them. Everything else about the 
format was just fine, thanks.

Gene, I'm puzzled. How come we didn't see the 5-limit versions of 
these four in any of your earlier 5-limit lists.

> [[1, 9, 9, 8], [0, 10, 9, 7]]   [1200, -890]   6.377   3.32016
> [[1, 1, 5, 4], [0, 2, -9, -4]]   [1200, 356]  6.8678 6.2453
> [[1, 0, -12, 6], [0, 1, 9, -2]]   [1200, 1910]   6.831 6.410
> [[2, 1, 3, 4], [0, 4, 3, 3]]   [600, 326]   4.899 10.132

In lowest terms the generators are 310, 356, 490 and 274 cents 
respectively. These are the only three of possible interest that I 
don't have names for, except I tentatively call the 490 cent one 
superpythagorean (since its generator can be considered to be a 710 
cent fifth). 

For your next pass, you can limit the complexity (preferably 
odd-limit-weighted) to 17, and the rms error to 30 cents, but let your 
badness go up to about 500. Some of those that barely made it onto 
your list are reasonably high on mine.


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Message: 4902

Date: Tue, 28 May 2002 11:24:51

Subject: Re: A 7-limit best list

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Gene, I'm puzzled. How come we didn't see the 5-limit versions of 
> these four in any of your earlier 5-limit lists.
> 
> > [[1, 9, 9, 8], [0, 10, 9, 7]]   [1200, -890]   6.377   3.32016
> > [[1, 1, 5, 4], [0, 2, -9, -4]]   [1200, 356]  6.8678 6.2453
> > [[1, 0, -12, 6], [0, 1, 9, -2]]   [1200, 1910]   6.831 6.410
> > [[2, 1, 3, 4], [0, 4, 3, 3]]   [600, 326]   4.899 10.132

That's because as 5-limit systems they are all pretty bad. On the other hand, here is something I should have included on my 7s list:

[19, 19, 57, 79, -37, -14]   [[19, 0, 14, -37], [0, 1, 1, 3]]

badness   160.2710884   g   33.81074780   rms   .1401992317

generators   [63.15789474, 1901.874626]


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Message: 4905

Date: Tue, 28 May 2002 22:06 +0

Subject: Re: definitions of period, equivalence, etc. (was: Re: graham's line

From: graham@xxxxxxxxxx.xx.xx

genewardsmith wrote:

> That seems to be saying the tone group is a circle group, R/1200R if we 
> use cents. This means that the group is not ordered and its image under 
> the log map does not embed into a field, both of which don't help you. 
> On the plus side, it is a topological group with an invariant metric, 
> which gives us a notion of closeness.

Um, right.

> I'd say from a mathematician's point of view, having made things harder 
> in this way, we would want a payoff of some kind.

It doesn't matter if there's a payoff or not, so long as we can do it.

> > But there you're giving an octave-specific definition of a 
> > temperament you say won't work in octave-equivalent space.
> 
> I give octave-specific definitions of everything; you are the one 
> saying it might be better not to.

You're being totally obtuse here.  I don't even know how to answer that.  
I asked for an example that wouldn't work in octave-equivalent terms, and 
you give me an octave-specific one.  How much less relevant could that be?

>   Obviously an octave equivalent 
> > system can't do octave-specific things, but it works fine on its own 
> > terms.
> 
> It works when it works? It seems to me it works because you can lift it 
> to octave-specific in cases of practical interest. Why bother to do the 
> heavy lifting? What's the payoff?

How many times to I have to repeat that neither I nor anybody else who has 
expressed an opinion cares about this?  It doesn't matter.  Nobody thinks 
it would be an improvement.  You can talk in octave-equivalent terms and 
add the octaves to do the calculations.  All I'm saying is that the octave 
equivalent algebra+whatever works fine in it's own terms and you do *not* 
have to add in the octaves.


                         Graham


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Message: 4907

Date: Wed, 29 May 2002 13:05:58

Subject: scale collection

From: Carl Lumma

Gene, you had also given Qm(2) previously (e=15, c=4).

What's this:

10-tone scale, e=24 c=4, in 72-tET
(0 5 14 19 28 33 42 49 58 63 72)

?

-Carl


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Message: 4908

Date: Wed, 29 May 2002 13:20:46

Subject: Re: scale collection

From: Carl Lumma

Dave, this is one of yours:

!
  Single-chain MOS of 7:4's in 41-tet.
  11
!
  204.878 !......7
  234.146 !......8
  439.024 !......15
  468.293 !......16
  673.171 !......23
  702.439 !......24
  907.317 !......31
  936.585 !......32
  1141.463 !.....39
  1170.732 !.....40
  1200.000 !.....41
!

I don't see any generators near 7:4 in the catalog.  What is this?

-Carl


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Message: 4910

Date: Wed, 29 May 2002 00:28:02

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Gene, I'm puzzled. How come we didn't see the 5-limit versions of 
> > these four in any of your earlier 5-limit lists.
> > 
> > > [[1, 9, 9, 8], [0, 10, 9, 7]]   [1200, -890]   6.377   3.32016
> > > [[1, 1, 5, 4], [0, 2, -9, -4]]   [1200, 356]  6.8678 6.2453
> > > [[1, 0, -12, 6], [0, 1, 9, -2]]   [1200, 1910]   6.831 6.410
> > > [[2, 1, 3, 4], [0, 4, 3, 3]]   [600, 326]   4.899 10.132
> 
> That's because as 5-limit systems they are all pretty bad. 

Sure they wouldn't have made the final list, but they are way better 
than many others on the combined list I made of all those you 
generated back then.

> On the 
> other hand, here is something I should have included on my 7s list:
> > [19, 19, 57, 79, -37, -14]   [[19, 0, 14, -37], [0, 1, 1, 3]]
> 
> badness   160.2710884   g   33.81074780   rms   .1401992317
> 
> generators   [63.15789474, 1901.874626]

You did include it.

However your list does not include the following (from Graham's 
catalog):

twin meantone (double diatonic) [[2 . . .] [0 1 4 4]]
shrutar (double diaschismic)    [[2 . . .] [0 2 -4 7]]
porcupine                       [[1 . . .] [0 3 5 -6]]
diminished                      [[4 . . .] [0 1 1 4]]
diaschismic (15-limit variant)  [[2 . . .] [0 1 -2 -8]]
diaschismic (56-ET variant)     [[2 . . .] [0 1 -2 9]]

It seems that your current badness is too lenient on complexity and 
too hard on error.

Can you explain why you used complexity^3 * error for 5-limit and are 
now using complexity^2 * error for 7-limit?


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Message: 4911

Date: Wed, 29 May 2002 01:53:22

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> > Gene, I'm puzzled. How come we didn't see the 5-limit versions of 
> > these four in any of your earlier 5-limit lists.
> > 
> > > [[1, 9, 9, 8], [0, 10, 9, 7]]   [1200, -890]   6.377   3.32016
> > > [[1, 1, 5, 4], [0, 2, -9, -4]]   [1200, 356]  6.8678 6.2453
> > > [[1, 0, -12, 6], [0, 1, 9, -2]]   [1200, 1910]   6.831 6.410
> > > [[2, 1, 3, 4], [0, 4, 3, 3]]   [600, 326]   4.899 10.132
> 
> That's because as 5-limit systems they are all pretty bad.

Having found their 5-limit rms optima, I can see that a cutoff on your 
badness (= complexity^3 * error) would explain the non-appearance of 
all of them except the superpythagorean one (the 3rd one above). It's 
non-appearance can only be explained by such a cutoff if your badness 
uses unweighted complexity. Using weighted complexity it has a badness 
of 1202 which is better than 

1220703125/1207959552   semiminorsixths     badness 1352
48828125/47775744       quintaminorthirds   badness 1902

Maybe these two were only generated when you were using unweighted 
complexity and maybe you always had your badness limit set below 1202 
once you started using weighted complexity.

In my own badness ranking, superpythagorean (where 19683/20480 
vanishes) comes 23rd (after septathirds and parakleismic). 

Gene, I afraid we might have missed other 5-limit temperaments that 
were too bad by your measure, but not mine. Could you please rerun 
your 5-limit temperament generator with an rms error cutoff of 35 
cents, weighted rms complexity cutoff of 10 generators, and badness = 
weighted_complexity^3*error cutoff of 1900.


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Message: 4912

Date: Thu, 30 May 2002 00:01:55

Subject: Re: scale collection

From: Carl Lumma

>It's definitely not one of mine.  What made you think it was?

I have it in my records as being yours.  It might be my mistake; I
don't remember a source.

 >A mistake? It would take quite a stretch of my imagination to consider 
 >the generator of this scale as a 4:7. It has a 32 cent error. If it 
 >has a 7-limit mapping at all, it must be [2, -6, 1] in which case the 
 >generator can be considered to be (the inversion of) half the pelogic 
 >fifth. That temperament would be better represented in 37-ET, but 11 
 >notes would no longer be MOS there. Even with the optimum generator 
 >(259.64 cents) it has a max abs error of 28.5 cents.

I'll delete it.

-Carl


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Message: 4913

Date: Thu, 30 May 2002 12:39:04

Subject: Re: A 7-limit best list

From: genewardsmith

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> i'm not seeing the injera temperament here (generators should be 600, 
> 694);
that concerns me . . .

The feedback I'm getting from both you and Dave is that my badness
cutoff was set too low. If we go all the way out to ennealimmal with a
higher cutoff, we'll have a lot of temperaments. Perhaps we need to do
either one of Dave's rolloffs or two different lists.


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Message: 4915

Date: Thu, 30 May 2002 02:02:04

Subject: Re: scale collection

From: dkeenanuqnetau

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
> Dave, this is one of yours:
> 
> !
>   Single-chain MOS of 7:4's in 41-tet.
>   11
> !
>   204.878 !......7
>   234.146 !......8
>   439.024 !......15
>   468.293 !......16
>   673.171 !......23
>   702.439 !......24
>   907.317 !......31
>   936.585 !......32
>   1141.463 !.....39
>   1170.732 !.....40
>   1200.000 !.....41
> !

It's definitely not one of mine. What made you think it was?

> I don't see any generators near 7:4 in the catalog.

It might appear with a 7:8 generator in the catalog. But it doesn't 
and shouldn't.

> What is this?

A mistake? It would take quite a stretch of my imagination to consider 
the generator of this scale as a 4:7. It has a 32 cent error. If it 
has a 7-limit mapping at all, it must be [2, -6, 1] in which case the 
generator can be considered to be (the inversion of) half the pelogic 
fifth. That temperament would be better represented in 37-ET, but 11 
notes would no longer be MOS there. Even with the optimum generator 
(259.64 cents) it has a max abs error of 28.5 cents.


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Message: 4916

Date: Thu, 30 May 2002 15:18 +0

Subject: neutral third scale

From: graham@xxxxxxxxxx.xx.xx

Mark Gould wrote:

> Thoughts please on the following scale
> (in cents)
> 0 150 300 500 650 850 1000 (0
> 
> Interesting properties under transposition, it has. Can anyone name it, 
> or
> provide any other info

That's neutral Phrygian in Manuel's list.  Another mode of it is Mohajira, 
an uncommon Arabic scale.  I've got a page about these kind of things at 
<7+3 scales *>.


                    Graham


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Message: 4921

Date: Thu, 30 May 2002 22:01:11

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> > 
> > > i'm not seeing the injera temperament here (generators should be 
> 600, 
> > > 694); that concerns me . . .
> > 
> > The feedback I'm getting from both you and Dave is that my badness 
> >cutoff was set too low. If we go all the way out to ennealimmal 
with 
> >a higher cutoff, we'll have a lot of temperaments. Perhaps we need 
> >to do either one of Dave's rolloffs or two different lists.
> 
> if you're happy doing one of dave's rolloffs, then by all means (and 
> for 5-limit too!) . . . but i don't agree with dave that 
temperaments 
> like Shrutar necessarily have to show up in the 7-limit list . . .

They don't have to show up in the final list, but the fact that they 
are not showing up in Gene's wide-open list leads to fears that we are 
missing other possibly interesting ones that we haven't seen before.


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Message: 4922

Date: Thu, 30 May 2002 22:06:48

Subject: Re: A 7-limit best list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> 
> > i'm not seeing the injera temperament here (generators should be 
600, 
> > 694); that concerns me . . .
> 
> The feedback I'm getting from both you and Dave is that my badness 
cutoff was set too low. If we go all the way out to ennealimmal with a 
higher cutoff, we'll have a lot of temperaments. Perhaps we need to do 
either one of Dave's rolloffs or two different lists.

You haven't addressed my question: Why complexity^2 * error now, when 
you used complexity^3 * error for 5-limit?

I think if you change that, and put cutoffs at 17 gens for weighted 
complexity and 25 or 30 cents for error, the list shouldn't be too 
big. My earlier suggestion of 2000 or so badness cutoff is only if you 
insist on continuing to use complexity^2 * error.


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Message: 4924

Date: Thu, 30 May 2002 23:49 +0

Subject: Re: neutral third scale

From: graham@xxxxxxxxxx.xx.xx

emotionaljourney22 wrote:

> i think mark would also be interested to know (if he doesn't already) 
> about the academic paper considering this in a class of "anti-
> diatonic" scales. anyone have the reference handy?

The only reference I know of for "anti-diatonic" scales is the page on my 
website that I've already given.  I assume you're thinking of 
Carey&Clampitt's "Self-Similar Pitch Structures, Their Duals, Rhythmic 
Analogues" that appeared in Perspectives of New Music, probably the issue 
before Mark's own paper.  The neutral third scales are only described in a 
footnote, as the mean of two different diatonic scales.  In the body of 
the text a matrix is given which is a dual to the diatonic one, but not in 
a particularly important way (it comes down to them both being a 7 note 
MOS).  So it's not really a good place to find out about neutral third 
scales.

I got the paper online.  I think it's still available somewhere, maybe the 
files area for one of these groups.


                      Graham


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