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Message: 9450 - Contents - Hide Contents
Date: Tue, 27 Jan 2004 19:17:54
Subject: Re: rank complexity explanation updated
From: Carl Lumma
>>>> >00.0 : 2 4 5 7 9 11 12
>>>> 200.0 : 2 3 5 7 9 10 12
>>>> 400.0 : 1 3 5 7 8 10 12
>>>> 500.0 : 2 4 6 7 9 11 12
>>>> 700.0 : 2 4 5 7 9 10 12
>>>> 900.0 : 2 3 5 7 8 10 12
>>>> 1100.0: 1 3 5 6 8 10 12
//
>>> One can obtain "my" interval vector from "your" interval matrix
>>> by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
>>> You subsequently obtain (2,5,4,3,6,1)
>>
>> Sorry, but how does tallying numbers in the above matrix lead to
>> (2,5,4,3,6,1)?
>
>Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
Aha! Are these known as "interval vectors" in the trade? And have
they ever been applied to tunings other than 12-equal?
>Jon Wild and I have been discussing this, and he has compiled
>interval vector lists for sets and their subsets all the way up
>to "C{31,15}-reduced" (Gene hates that I use the C{m,n} notation,
>he has proposed a better way, to name these sets, somewhere in
>the archives, I'll have to hunt for it)
What is this notation? Also, much appreciated if find Gene's
suggestion.
-Carl
Message: 9451 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 22:06:25
Subject: Re: 114 7-limit temperaments
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
> wrote:
>> This comparison of different outputs for the same temperament shows
> up
>> the need to correctly normalise the new weighted error and
> complexity
>> figures so they actually have units we can relate to. i.e. cents for
>> the error and gens per interval for the complexity.
>>
>> This should be simple to do.
>>
>> I think the correct normalisation of a weighted norm is the one
> where,
>> if every individual value happened to be X then the, the norm would
>> also be X, irrespective of the weights.
>>
>> e.g. if the individual errors are E1, E2, ... En,
>
> You realize that there are an infinite number of errors in the TOP
> case.
No I didn't. How do you mean? Obviously you don't do an infinite
number of calculations. But even so, the normalisation factor may
still converge.
If it's a minimax type of thing then all we need is to divide the
current result by the maximum weight of any interval. Let me guess:
this is lg2(2) = 1 so there would be no change?
Is the weighting the same for the complexity? Minimax where gens per
interval is divided by lg2(product_complexity(interval))?
Message: 9452 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 18:06:30
Subject: Re: rank complexity explanation updated
From: Carl Lumma
>>>>> >he interval matrix often, if not typically, has all
>>>>> unisons/octaves along the diagonal. This one is merely
>>>>> a reshuffling so that the diagonal becomes a right vertical.
>>>>
>>>> No, the interval matrix is always written as above (the values
>>>> after the colons, at least).
>>>
>>> Where do you get *always*??
>>
>> I have never seen it printed this way. Where do you get
>> "often, if not typically"?
>>
>> -Carl
>
Oh, maybe that is how R. printed them. I'll look when I get home.
-Carl
Message: 9453 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 22:54:34
Subject: Re: 114 7-limit temperaments
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> Can one of you easily plot these 7-limit temperaments on an error
>>> vs. complexity graph (log log or whatever seemed best with 5-
limit)
>>> so we can all think about what our subjective badness contours
might
>>> look like.
>>
>> I could do that, but as this was done with a log-flat badness
cutoff,
>> there will be a huge gaping hole in the graph. That's why I'm
trying
>> to figure out the whole deal for myself, but no one's helping.
>
> Since I usually like the way you figure things out, I'll do whatever
> I can to help, which may not be much. If there are any particular
> msg. #s associated with this, I'll reread them. I didn't follow
your
> orthogonalization posts at all. :(
>
> Part of the problem is these contours represent musical values, so
> they're ultimately a matter of opinion.
Not the problem here, as Dave simply wanted a graph, for which the
best dataset would involve a single error cutoff and a single
complexity cutoff, both generous enough to satisfy everyone. Too big
a dataset to post to this list, but I hope to be doing this
eventually, starting with wedgies.
Message: 9454 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 22:56:01
Subject: Re: rank complexity explanation updated
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> I'm afraid I don't know what an "external" interval is. Here's
>>> the interval matrix of the diatonic scale in 12-equal, as given
>>> by Scala...
>>>
>>> 100.0 : 2 4 5 7 9 11 12
>>> 200.0 : 2 3 5 7 9 10 12
>>> 400.0 : 1 3 5 7 8 10 12
>>> 500.0 : 2 4 6 7 9 11 12
>>> 700.0 : 2 4 5 7 9 10 12
>>> 900.0 : 2 3 5 7 8 10 12
>>> 1100.0: 1 3 5 6 8 10 12
>>
>> Q: Shouldn't the first row be 000.0?
>
> It is quite safe to ignore the values before the colon, as they
> are merely an artifact of scala's output.
What kind of artifact are they, if not an error?
Message: 9455 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 18:51:30
Subject: Re: Graef article on rationalization of scales
From: Carl Lumma
>> >ince you practically single-handedly launched the 'popular scales
>> are good PBs' program, I find it highly unusual that you are now
>> asking me what it is.
>
>Now I understand you better. Yet, 'popular scales' will often have a
>large number of plausible derivations from a PB,
You mean to a PB, I think?
>in terms of its shape, its position in the lattice, and the unison
>vectors involved, so going from the scale to the PB still seems like
>a step backwards, a step from greater generality to lesser generality.
True, but we're not necessarily interested in a particular one.
If we are, it seems Gene has or is close to having tools to transform
between them, define a canonical one, etc. But if your 'the basis
doesn't matter' reasoning applied to blocks, we might say that all
alternate PB versions ought to have the same goodness. If we could
then further show that historical scales correspond to gooder blocks
than random scales...
-Carl
Message: 9456 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 22:59:30
Subject: Re: 114 7-limit temperaments
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
>> I could do that, but as this was done with a log-flat badness
cutoff,
>> there will be a huge gaping hole in the graph. That's why I'm
trying
>> to figure out the whole deal for myself, but no one's helping.
>
> Try asking for some specific form of help.
I'm afraid I've done my best. Perhaps the way I'm seeing this stuff
bears no resemblance to the way anyone else is, but I do think it's
important that I understand it all myself, in at least one way.
Message: 9457 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 19:00:39
Subject: Re: 114 7-limit temperaments
From: Carl Lumma
>> >anuel, at least, has
>> always insisted that simpler ratios need to be tuned more accurately,
>> and harmonic entropy and all the other discordance functions I've
>> seen show that the increase in discordance for a given amount of
>> mistuning is greatest for the simplest intervals.
>
>But surely it's obvious that beat rates go up as something like error
>_times_ complexity, not down as error _divided_by_ complexity, even
>though beat amplitudes do go down.
Without evaluating this, I'll claim that beat rates are not a good
indicator of discordance.
-Carl
Message: 9458 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 22:57:42
Subject: Re: Graef article on rationalization of scales
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> This raises an interesting question. What is our approved method
>>> for finding Fokker blocks for an arbitrary irrational scale?
>>> Such a method would surely make Graf's look silly.
>>
>> All such methods are silly,
>
> Perhaps you mean Graf's idea of people wanting "just" versions of
> arbitrary scales is silly. That's for sure.
Yup.
>> but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
>> blocks that result from the min-"odd-limit" criterion. But the
>> whole idea of rationalizing a tempered scale is completely
>> backwards and misses the point in a big way.
>
> I think you missed my point.
Would you, then, clarify your point, perhaps with examples?
Message: 9459 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 23:01:15
Subject: Re: rank complexity explanation updated
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
>>>> I'm afraid I don't know what an "external" interval is. Here's
>>>> the interval matrix of the diatonic scale in 12-equal, as given
>>>> by Scala...
>>>>
>>>> 100.0 : 2 4 5 7 9 11 12
>>>> 200.0 : 2 3 5 7 9 10 12
>>>> 400.0 : 1 3 5 7 8 10 12
>>>> 500.0 : 2 4 6 7 9 11 12
>>>> 700.0 : 2 4 5 7 9 10 12
>>>> 900.0 : 2 3 5 7 8 10 12
>>>> 1100.0: 1 3 5 6 8 10 12
>>>
>>> Q: Shouldn't the first row be 000.0?
>>
>> It is quite safe to ignore the values before the colon, as they
>> are merely an artifact of scala's output.
>>
>>> Another point,
>>>
>>> One can obtain "my" interval vector from "your" interval matrix
>>> by tallying all the intervals from 1 to 6 and ignoring 7 to 12.
>>> You subsequently obtain (2,5,4,3,6,1)
>>
>> Sorry, but how does tallying numbers in the above matrix lead to
>> (2,5,4,3,6,1)?
>> -Carl
>
> Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
I see 2 6's ;)
>(Also works
> for 7-11, in reverse from 6:, so the full vector is
> (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
>> - Paul
This time there *really* should be 2 6's.
Message: 9460 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 23:03:33
Subject: Re: rank complexity explanation updated
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>
>> 100.0 : 2 4 5 7 9 11 12
>> 200.0 : 2 3 5 7 9 10 12
>> 400.0 : 1 3 5 7 8 10 12
>> 500.0 : 2 4 6 7 9 11 12
>> 700.0 : 2 4 5 7 9 10 12
>> 900.0 : 2 3 5 7 8 10 12
>> 1100.0: 1 3 5 6 8 10 12
>
> Why is the first row 100.0 and not 000.0? I also see I had the wrong
> definition of interval matrix; maybe this one would have a more
> interesting characteristic polynomial. I'd like something that told
me
> something about the scale!
The interval matrix often, if not typically, has all unisons/octaves
along the diagonal. This one is merely a reshuffling so that the
diagonal becomes a right vertical.
Message: 9461 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 23:19:38
Subject: Re: 114 7-limit temperaments
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
>> wrote:
>>> This comparison of different outputs for the same temperament
shows
>> up
>>> the need to correctly normalise the new weighted error and
>> complexity
>>> figures so they actually have units we can relate to. i.e.
cents for
>>> the error and gens per interval for the complexity.
>>>
>>> This should be simple to do.
>>>
>>> I think the correct normalisation of a weighted norm is the one
>> where,
>>> if every individual value happened to be X then the, the norm
would
>>> also be X, irrespective of the weights.
>>>
>>> e.g. if the individual errors are E1, E2, ... En,
>>
>> You realize that there are an infinite number of errors in the
TOP
>> case.
>
> No I didn't. How do you mean?
It's the minimax over *all* intervals.
> Obviously you don't do an infinite
> number of calculations.
No, you only need to set the primes, and the rest falls out correctly.
>But even so, the normalisation factor may
> still converge.
Yes, it may.
> If it's a minimax type of thing then all we need is to divide the
> current result by the maximum weight of any interval.
This doesn't seem right -- what's your reasoning?
> Let me guess:
> this is lg2(2) = 1 so there would be no change?
There is no need to use base 2 -- the result is the same regardless
of which base you use in the logarithms. The error is measured using
logarithms, but so is the complexity = log(n*d), so error divided by
complexity, which is what you're minimizing the maximum of, is
insensitive to choice of base.
> Is the weighting the same for the complexity? Minimax where gens per
> interval is divided by lg2(product_complexity(interval))?
No particular generator basis is assumed in the TOP complexity
calculations. Instead, it's a direct measure of how much the
tempering simplifies the lattice, and reduces (Gene seems to
imply/agree) to the number of notes per acoustic whatever in the case
of equal (1-dimensional) temperaments.
Message: 9462 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 19:28:24
Subject: Re: Graef article on rationalization of scales
From: Carl Lumma
>>> >ow I understand you better. Yet, 'popular scales' will often have
>>> a large number of plausible derivations from a PB,
>>
>> You mean to a PB, I think?
>
>A gets derived *from* B, it doesn't get derived *to* B, right?
Sure. I guess what threw me is that you expect multiple PBs,
and "from a PB" sounds very singular.
-Carl
Message: 9465 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 23:41:23
Subject: Re: rank complexity explanation updated
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
>> wrote:
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>>
>>>>>>> I'm afraid I don't know what an "external" interval is.
>> Here's
>>>>>>> the interval matrix of the diatonic scale in 12-equal,
as
>> given
>>>>>>> by Scala...
>>>>>>>
>>>>>>> 100.0 : 2 4 5 7 9 11 12
>>>>>>> 200.0 : 2 3 5 7 9 10 12
>>>>>>> 400.0 : 1 3 5 7 8 10 12
>>>>>>> 500.0 : 2 4 6 7 9 11 12
>>>>>>> 700.0 : 2 4 5 7 9 10 12
>>>>>>> 900.0 : 2 3 5 7 8 10 12
>>>>>>> 1100.0: 1 3 5 6 8 10 12
>>>>>>
>>>>>> Q: Shouldn't the first row be 000.0?
>>>>>
>>>>> It is quite safe to ignore the values before the colon, as
> they
>>>>> are merely an artifact of scala's output.
>>>>>
>>>>>> Another point,
>>>>>>
>>>>>> One can obtain "my" interval vector from "your" interval
> matrix
>>>>>> by tallying all the intervals from 1 to 6 and ignoring 7
to
> 12.
>>>>>> You subsequently obtain (2,5,4,3,6,1)
>>>>>
>>>>> Sorry, but how does tallying numbers in the above matrix
lead
> to
>>>>> (2,5,4,3,6,1)?
>>>>> -Carl
>>>>
>>>> Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
>>>
>>> I see 2 6's ;)
>>
>> Good eye!
>>>
>>>> (Also works
>>>> for 7-11, in reverse from 6:, so the full vector is
>>>> (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same
as '0')
>>>>> - Paul
>>>
>>> This time there *really* should be 2 6's.
>>
>> However, Since this is the same tritone (F-B and B-F) John Rahn
>> and Allen Forte both divide this by 2. 2/2 =1
>
> Oops! In this vector there should be 2 6's. Your right, I'm left!
Looks like we're fully agreed, then.
Message: 9466 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 23:40:47
Subject: Re: rank complexity explanation updated
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>
>>>>>> I'm afraid I don't know what an "external" interval is.
> Here's
>>>>>> the interval matrix of the diatonic scale in 12-equal, as
> given
>>>>>> by Scala...
>>>>>>
>>>>>> 100.0 : 2 4 5 7 9 11 12
>>>>>> 200.0 : 2 3 5 7 9 10 12
>>>>>> 400.0 : 1 3 5 7 8 10 12
>>>>>> 500.0 : 2 4 6 7 9 11 12
>>>>>> 700.0 : 2 4 5 7 9 10 12
>>>>>> 900.0 : 2 3 5 7 8 10 12
>>>>>> 1100.0: 1 3 5 6 8 10 12
>>>>>
>>>>> Q: Shouldn't the first row be 000.0?
>>>>
>>>> It is quite safe to ignore the values before the colon, as
they
>>>> are merely an artifact of scala's output.
>>>>
>>>>> Another point,
>>>>>
>>>>> One can obtain "my" interval vector from "your" interval
matrix
>>>>> by tallying all the intervals from 1 to 6 and ignoring 7 to
12.
>>>>> You subsequently obtain (2,5,4,3,6,1)
>>>>
>>>> Sorry, but how does tallying numbers in the above matrix lead
to
>>>> (2,5,4,3,6,1)?
>>>> -Carl
>>>
>>> Easy! There are 2 1's, 5 2's 4 3's 3 4's 6 5's and 1 6.
>>
>> I see 2 6's ;)
>
> Good eye!
>>
>>> (Also works
>>> for 7-11, in reverse from 6:, so the full vector is
>>> (2,5,4,3,6,1,6,3,4,5,2, and 7 if you include '12') (same as '0')
>>>> - Paul
>>
>> This time there *really* should be 2 6's.
>
> However, Since this is the same tritone (F-B and B-F) John Rahn
> and Allen Forte both divide this by 2. 2/2 =1
I didn't know they ever used the "full vector", but if they did, I
would disagree. Only in the tally of interval *classes*, which go
from 1 to 6, should you divide the tritone count by 2. That's why I
wrote a smiley-face in the first case above -- I was winking to you
because in that case, we both know the tritone count should be
divided by 2.
Message: 9467 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 02:40:13
Subject: Re: 114 7-limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> I could do that, but as this was done with a log-flat badness cutoff,
> there will be a huge gaping hole in the graph. That's why I'm trying
> to figure out the whole deal for myself, but no one's helping.
Try asking for some specific form of help.
Message: 9469 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 17:08:50
Subject: Re: rank complexity explanation updated
From: Carl Lumma
>>>>>> >ttp://lumma.org/tuning/rank-complexity.txt
>>>>>
>>>>>> Hi Paul,
>>>>
>>>> I'm afraid I don't know what an "external" interval is. Here's
>>>> the interval matrix of the diatonic scale in 12-equal, as given
>>>> by Scala...
>>>>
>>>> 100.0 : 2 4 5 7 9 11 12
>>>> 200.0 : 2 3 5 7 9 10 12
>>>> 400.0 : 1 3 5 7 8 10 12
>>>> 500.0 : 2 4 6 7 9 11 12
>>>> 700.0 : 2 4 5 7 9 10 12
>>>> 900.0 : 2 3 5 7 8 10 12
>>>> 1100.0: 1 3 5 6 8 10 12
>>>
>>> Q: Shouldn't the first row be 000.0?
>>
>> It is quite safe to ignore the values before the colon, as they
>> are merely an artifact of scala's output.
>
>What kind of artifact are they, if not an error?
They indicate the interval between 1/1 and the degree of the
original 'native' mode, on which the mode shown on the particular
line is based.
In this case they are artifactal only because they are given in
different units than is the interval matrix itself.
This happened because I wanted to give the interval matrix in
'steps of 12-tET' units. Unfortunately (and one of my biggest
desired features) Scala does not offer 'degrees of n-ET' units.
So what you see above is actually the RANK ORDER MATRIX of the
diatonic scale in 12-tET. Because the scale covers every degree
of that tuning the two are the same.
-Carl
Message: 9470 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 01:15:56
Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)
From: Paul Erlich
Since Herman has expressed his preferences as regards badness
functions, and his interest in "#82", I thought I'd cull the list of
114 by applying a more stringent cutoff of 1.355*comp + error <
10.71. This is an arbitrary choice among the linear functions of
complexity and error that could be chosen; it's chosen so that
Miracle, Blackwood, and Diaschismic make it in, but unfortunately
Waage does not. A slightly higher cutoff would take us outside Gene's
search range, but would probably still add temperaments of interest
to Herman. Anyway, here's the resulting list top-41 list:
Number 1 Meantone
[1, 4, 10, 4, 13, 12] [[1, 2, 4, 7], [0, -1, -4, -10]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3370.548328]
TOP generators [1201.698520, 504.1341314]
bad: 6.5251 comp: 3.562072 err: 1.698521
Number 2 Magic
[5, 1, 12, -10, 5, 25] [[1, 0, 2, -1], [0, 5, 1, 12]]
TOP tuning [1201.276744, 1903.978592, 2783.349206, 3368.271877]
TOP generators [1201.276744, 380.7957184]
bad: 7.0687 comp: 4.274486 err: 1.276744
Number 3 Pajara
[2, -4, -4, -11, -12, 2] [[2, 3, 5, 6], [0, 1, -2, -2]]
TOP tuning [1196.893422, 1901.906680, 2779.100462, 3377.547174]
TOP generators [598.4467109, 106.5665459]
bad: 7.1567 comp: 2.988993 err: 3.106578
Number 4 Semisixths
[7, 9, 13, -2, 1, 5] [[1, -1, -1, -2], [0, 7, 9, 13]]
TOP tuning [1198.389531, 1903.732520, 2790.053107, 3364.304748]
TOP generators [1198.389531, 443.1602931]
bad: 7.8851 comp: 4.630693 err: 1.610469
Number 5 Dominant Seventh
[1, 4, -2, 4, -6, -16] [[1, 2, 4, 2], [0, -1, -4, 2]]
TOP tuning [1195.228951, 1894.576888, 2797.391744, 3382.219933]
TOP generators [1195.228951, 495.8810151]
bad: 8.0970 comp: 2.454561 err: 4.771049
Number 6 Injera
[2, 8, 8, 8, 7, -4] [[2, 3, 4, 5], [0, 1, 4, 4]]
TOP tuning [1201.777814, 1896.276546, 2777.994928, 3378.883835]
TOP generators [600.8889070, 93.60982493]
bad: 8.2512 comp: 3.445412 err: 3.582707
Number 7 Kleismic
[6, 5, 3, -6, -12, -7] [[1, 0, 1, 2], [0, 6, 5, 3]]
TOP tuning [1203.187308, 1907.006766, 2792.359613, 3359.878000]
TOP generators [1203.187309, 317.8344609]
bad: 8.3168 comp: 3.785579 err: 3.187309
Number 8 Hemifourths
[2, 8, 1, 8, -4, -20] [[1, 2, 4, 3], [0, -2, -8, -1]]
TOP tuning [1203.668842, 1902.376967, 2794.832500, 3358.526166]
TOP generators [1203.668841, 252.4803582]
bad: 8.3374 comp: 3.445412 err: 3.66884
Number 9 Negri
[4, -3, 2, -14, -8, 13] [[1, 2, 2, 3], [0, -4, 3, -2]]
TOP tuning [1203.187308, 1907.006766, 2780.900506, 3359.878000]
TOP generators [1203.187309, 124.8419629]
bad: 8.3420 comp: 3.804173 err: 3.187309
Number 10 Tripletone
[3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
TOP tuning [1197.060039, 1902.640406, 2793.140092, 3377.079420]
TOP generators [399.0200131, 92.45965769]
bad: 8.4214 comp: 4.045351 err: 2.939961
Number 11 Schismic
[1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
TOP generators [1200.760624, 498.1193303]
bad: 8.5260 comp: 5.618543 err: .912904
Number 12 Superpythagorean
[1, 9, -2, 12, -6, -30] [[1, 2, 6, 2], [0, -1, -9, 2]]
TOP tuning [1197.596121, 1905.765059, 2780.732078, 3374.046608]
TOP generators [1197.596121, 489.4271829]
bad: 8.6400 comp: 4.602303 err: 2.403879
Number 13 Orwell
[7, -3, 8, -21, -7, 27] [[1, 0, 3, 1], [0, 7, -3, 8]]
TOP tuning [1199.532657, 1900.455530, 2784.117029, 3371.481834]
TOP generators [1199.532657, 271.4936472]
bad: 8.6780 comp: 5.706260 err: .946061
Number 14 Augmented
[3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
TOP generators [399.9922103, 107.3111730]
bad: 8.7811 comp: 2.147741 err: 5.870879
Number 15 Porcupine
[3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
TOP generators [1196.905960, 162.3176609]
bad: 8.9144 comp: 4.295482 err: 3.094040
Number 16
[6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
TOP generators [598.4467109, 162.3159606]
bad: 8.9422 comp: 4.306766 err: 3.106578
Number 17 Supermajor seconds
[3, 12, -1, 12, -10, -36] [[1, 1, 0, 3], [0, 3, 12, -1]]
TOP tuning [1201.698521, 1899.262909, 2790.257556, 3372.574099]
TOP generators [1201.698520, 232.5214630]
bad: 9.1819 comp: 5.522763 err: 1.698521
Number 18 Flattone
[1, 4, -9, 4, -17, -32] [[1, 2, 4, -1], [0, -1, -4, 9]]
TOP tuning [1202.536420, 1897.934872, 2781.593812, 3361.705278]
TOP generators [1202.536419, 507.1379663]
bad: 9.1883 comp: 4.909123 err: 2.536420
Number 19 Diminished
[4, 4, 4, -3, -5, -2] [[4, 6, 9, 11], [0, 1, 1, 1]]
TOP tuning [1194.128460, 1892.648830, 2788.245174, 3385.309404]
TOP generators [298.5321149, 101.4561401]
bad: 9.2912 comp: 2.523719 err: 5.871540
Number 20
[6, 10, 3, 2, -12, -21] [[1, 2, 3, 3], [0, -6, -10, -3]]
TOP tuning [1202.659696, 1907.471368, 2778.232381, 3359.055076]
TOP generators [1202.659696, 82.97467050]
bad: 9.3161 comp: 4.306766 err: 3.480440
Number 21
[0, 0, 12, 0, 19, 28] [[12, 19, 28, 34], [0, 0, 0, -1]]
TOP tuning [1197.674070, 1896.317278, 2794.572829, 3368.825906]
TOP generators [99.80617249, 24.58395811]
bad: 9.3774 comp: 4.295482 err: 3.557008
Number 22
[3, -7, -8, -18, -21, 1] [[1, 3, -1, -1], [0, -3, 7, 8]]
TOP tuning [1202.900537, 1897.357759, 2790.235118, 3360.683070]
TOP generators [1202.900537, 570.4479508]
bad: 9.5280 comp: 4.891080 err: 2.900537
Number 23
[3, 12, 11, 12, 9, -8] [[1, 3, 8, 8], [0, -3, -12, -11]]
TOP tuning [1202.624742, 1900.726787, 2792.408176, 3361.457323]
TOP generators [1202.624742, 569.0491468]
bad: 9.6275 comp: 5.168119 err: 2.624742
Number 24 Nonkleismic
[10, 9, 7, -9, -17, -9] [[1, -1, 0, 1], [0, 10, 9, 7]]
TOP tuning [1198.828458, 1900.098151, 2789.033948, 3368.077085]
TOP generators [1198.828458, 309.8926610]
bad: 9.7206 comp: 6.309298 err: 1.171542
Number 25 Miracle
[6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
TOP generators [1200.631014, 116.7206423]
bad: 9.8358 comp: 6.793166 err: .631014
Number 26 Beatles
[2, -9, -4, -19, -12, 16] [[1, 1, 5, 4], [0, 2, -9, -4]]
TOP tuning [1197.104145, 1906.544822, 2793.037680, 3369.535226]
TOP generators [1197.104145, 354.7203384]
bad: 9.8915 comp: 5.162806 err: 2.895855
Number 27 -- formerly Number 82
[6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
TOP tuning [1203.400986, 1896.025764, 2777.627538, 3379.328030]
TOP generators [601.7004928, 230.8749260]
bad: 10.0002 comp: 4.619353 err: 3.740932
Number 28
[3, -5, -6, -15, -18, 0] [[1, 3, 0, 0], [0, -3, 5, 6]]
TOP tuning [1195.486066, 1908.381352, 2796.794743, 3356.153692]
TOP generators [1195.486066, 559.3589487]
bad: 10.0368 comp: 4.075900 err: 4.513934
Number 29
[8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
TOP tuning [1198.553882, 1907.135354, 2778.724633, 3378.001574]
TOP generators [599.2769413, 272.3123381]
bad: 10.1077 comp: 5.047438 err: 3.268439
Number 30 Blackwood
[0, 5, 0, 8, 0, -14] [[5, 8, 12, 14], [0, 0, -1, 0]]
TOP tuning [1195.893464, 1913.429542, 2786.313713, 3348.501698]
TOP generators [239.1786927, 83.83059859]
bad: 10.1851 comp: 2.173813 err: 7.239629
Number 31 Quartaminorthirds
[9, 5, -3, -13, -30, -21] [[1, 1, 2, 3], [0, 9, 5, -3]]
TOP tuning [1199.792743, 1900.291122, 2788.751252, 3365.878770]
TOP generators [1199.792743, 77.83315314]
bad: 10.1855 comp: 6.742251 err: 1.049791
Number 32
[8, 1, 18, -17, 6, 39] [[1, -1, 2, -3], [0, 8, 1, 18]]
TOP tuning [1201.135544, 1899.537544, 2789.855225, 3373.107814]
TOP generators [1201.135545, 387.5841360]
bad: 10.2131 comp: 6.411729 err: 1.525246
Number 33
[6, 0, 15, -14, 7, 35] [[3, 5, 7, 9], [0, -2, 0, -5]]
TOP tuning [1197.060039, 1902.856975, 2793.140092, 3360.572393]
TOP generators [399.0200131, 46.12154491]
bad: 10.2154 comp: 5.369353 err: 2.939961
Number 34
[0, 12, 12, 19, 19, -6] [[12, 19, 28, 34], [0, 0, -1, -1]]
TOP tuning [1198.015473, 1896.857833, 2778.846497, 3377.854234]
TOP generators [99.83462277, 16.52294019]
bad: 10.2188 comp: 5.168119 err: 3.215955
Number 35
[5, 8, 2, 1, -11, -18] [[1, 2, 3, 3], [0, -5, -8, -2]]
TOP tuning [1194.335372, 1892.976778, 2789.895770, 3384.728528]
TOP generators [1194.335372, 99.13879319]
bad: 10.3332 comp: 3.445412 err: 5.664628
Number 36
[6, 0, 3, -14, -12, 7] [[3, 4, 7, 8], [0, 2, 0, 1]]
TOP tuning [1199.400031, 1910.341746, 2798.600074, 3353.970936]
TOP generators [399.8000105, 155.5708520]
bad: 10.4461 comp: 3.804173 err: 5.291448
Number 37
[1, -8, -2, -15, -6, 18] [[1, 2, -1, 2], [0, -1, 8, 2]]
TOP tuning [1195.155395, 1894.070902, 2774.763716, 3382.790568]
TOP generators [1195.155395, 496.2398890]
bad: 10.4972 comp: 4.075900 err: 4.974313
Number 38 Superkleismic
[9, 10, -3, -5, -30, -35] [[1, 4, 5, 2], [0, -9, -10, 3]]
TOP tuning [1201.371917, 1904.129438, 2783.128219, 3369.863245]
TOP generators [1201.371918, 322.3731369]
bad: 10.5077 comp: 6.742251 err: 1.371918
Number 39
[9, 0, 9, -21, -11, 21] [[9, 14, 21, 25], [0, 1, 0, 1]]
TOP tuning [1197.060039, 1897.499011, 2793.140092, 3360.572393]
TOP generators [133.0066710, 35.40561749]
bad: 10.6719 comp: 5.706260 err: 2.939961
Number 40
[6, 0, 0, -14, -17, 0] [[6, 10, 14, 17], [0, -1, 0, 0]]
TOP tuning [1194.473353, 1901.955001, 2787.104490, 3384.341166]
TOP generators [199.0788921, 88.83392059]
bad: 10.7036 comp: 3.820609 err: 5.526647
Number 41 Diaschismic
[2, -4, -16, -11, -31, -26] [[2, 3, 5, 7], [0, 1, -2, -8]]
TOP tuning [1198.732403, 1901.885616, 2789.256983, 3365.267311]
TOP generators [599.3662015, 103.7870123]
bad: 10.7079 comp: 6.966993 err: 1.267597
Message: 9471 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 17:11:34
Subject: Re: Graef article on rationalization of scales
From: Carl Lumma
>>>> >his raises an interesting question. What is our approved method
>>>> for finding Fokker blocks for an arbitrary irrational scale?
>>>> Such a method would surely make Graf's look silly.
>>>
>>> All such methods are silly,
>>
>> Perhaps you mean Graf's idea of people wanting "just" versions of
>> arbitrary scales is silly. That's for sure.
>
>Yup.
>
>>> but I prefer the hexagonal (rhombic dodecahedral, etc.) or Kees
>>> blocks that result from the min-"odd-limit" criterion. But the
>>> whole idea of rationalizing a tempered scale is completely
>>> backwards and misses the point in a big way.
>>
>> I think you missed my point.
>
>Would you, then, clarify your point, perhaps with examples?
That the diatonic scale is a 'good' PB seems like the best example.
Since you practically single-handedly launched the 'popular scales
are good PBs' program, I find it highly unusual that you are now
asking me what it is.
-Carl
Message: 9472 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 02:38:49
Subject: Re: rank complexity explanation updated
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 100.0 : 2 4 5 7 9 11 12
> 200.0 : 2 3 5 7 9 10 12
> 400.0 : 1 3 5 7 8 10 12
> 500.0 : 2 4 6 7 9 11 12
> 700.0 : 2 4 5 7 9 10 12
> 900.0 : 2 3 5 7 8 10 12
> 1100.0: 1 3 5 6 8 10 12
Why is the first row 100.0 and not 000.0? I also see I had the wrong
definition of interval matrix; maybe this one would have a more
interesting characteristic polynomial. I'd like something that told me
something about the scale!
Message: 9473 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 17:20:20
Subject: Re: rank complexity explanation updated
From: Carl Lumma
>>> >00.0 : 2 4 5 7 9 11 12
>>> 200.0 : 2 3 5 7 9 10 12
>>> 400.0 : 1 3 5 7 8 10 12
>>> 500.0 : 2 4 6 7 9 11 12
>>> 700.0 : 2 4 5 7 9 10 12
>>> 900.0 : 2 3 5 7 8 10 12
>>> 1100.0: 1 3 5 6 8 10 12
Gene wrote...
>> Why is the first row 100.0 and not 000.0?
Hopefully you've seen the bit about ignoring these numbers
by now.
>> I also see I had the wrong definition of interval matrix;
That's pretty incredible considering that you've got the
Rothenberg papers on the matter.
Paul wrote...
>The interval matrix often, if not typically, has all unisons/octaves
>along the diagonal. This one is merely a reshuffling so that the
>diagonal becomes a right vertical.
No, the interval matrix is always written as above (the values
after the colons, at least). You're thinking maybe of the
tonality diamond.
-Carl
Message: 9474 - Contents - Hide Contents
Date: Wed, 28 Jan 2004 07:28:32
Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
> Number 15 Porcupine
>
> [3, 5, -6, 1, -18, -28] [[1, 2, 3, 2], [0, -3, -5, 6]]
> TOP tuning [1196.905961, 1906.858938, 2779.129576, 3367.717888]
> TOP generators [1196.905960, 162.3176609]
> bad: 8.9144 comp: 4.295482 err: 3.094040
>
> Number 16
>
> [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> TOP tuning [1196.893422, 1906.838962, 2779.100462, 3377.547174]
> TOP generators [598.4467109, 162.3159606]
> bad: 8.9422 comp: 4.306766 err: 3.106578
These two are related, and very close in terms of error and
complexity. Bi porcupine makes me think of porcupine sex, and I'm not
sure I want to, but some sort of porky name seems apt.
> Number 25 Miracle
>
> [6, -7, -2, -25, -20, 15] [[1, 1, 3, 3], [0, 6, -7, -2]]
> TOP tuning [1200.631014, 1900.954868, 2784.848544, 3368.451756]
> TOP generators [1200.631014, 116.7206423]
> bad: 9.8358 comp: 6.793166 err: .631014
This seems absurdly far down the list. I think mine was better.
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