Tuning-Math Digests messages 8100 - 8124

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Message: 8100

Date: Tue, 11 Nov 2003 03:07:58

Subject: Re: Enharmonic diesis?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > > 
> > > i'm also trying to establish here a new standard usage
> > > of brackets.  since using commas to separate groups of 
> > > exponents in a monzo eliminates the need for contrasting
> > > brackets to indicate the presence or absence of prime-factor 2,
> > > i propose that we use angle-brackets for the prime-factors
> > > themselves, and square-brackets for the actual monzo of
> > > the exponents.
> > 
> > Well I suppose you could, but I don't think this is necessary,
> > and we could save the angle-brackets for something more 
> > important. There is obviously no need for selective use of
> > commas between the primes themselves. They provide no 
> > additional information there. 
> 
> 
> huh?  they would simply separate the groups of primes to
> show how the exponents are grouped.  i know that that's
> "a given", so i guess maybe you're right.

I assumed that's what you intended, and there's no harm in a bit of
redundancy. But I see this as _doubly_ redundant.

If you know the system with the commas in the monzos, then you know
that e.g. [1 1, 1] is a 2,3,5-monzo and not a 3,5,7-monzo, so adding
the words "2,3,5-monzo" is redundant (but still a good idea). And if
you're told it's a 2,3,5-monzo then you can ignore the commas and just
line up exponents with primes. So "2,3 5-monzo" is doubly redundant
and forces us to use something like the angle-brackets to hold it
together: <2,3 5>-monzo.

But while we intend the monzo [1, 2 3] to be quite different from the
monzo [1 2, 3], a 2,3,5-monzo is of course no different from a <2,3
5>-monzo.

Someone seeing the term "<2,3 5>-monzo" might wonder if there were
such things as <2 3,5>-monzos and <2 3 5>-monzos, as different categories.
 
> > I would simply call these 2,3,5-monzos. 
> 
> 
> simply using a comma between *every* prime, and no spaces.
> i suppose i like that.  (i don't sound too convinced, tho.)

I think mathematicians name things like that all the time. Don't they
Gene?

Don't forget to add the stuff about the selective comma(punctuation
sense) convention to your "monzo" dictionary entry when you get a chance.

> > Maybe a good use for angle-brackets would be for wedgies
> > since, as I understand it, these are in a _very_ different
> > domain from that of monzos, and the angle-brackets are
> > suggestive of wedges themselves.
> 
> great minds thinking alike!  ;-)
> 
> i had actually already thought of that too ... but since my
> understanding of wedgies is lagging far behind that of many
> of you others, i'll refrain from commenting further.

I'm afraid I don't understand them either. I couldn't even tell you
the difference between a wedgie and a val. :-) I read Gene's "val"
definition in your dictionary but I'm none the wiser. I don't see an
entry for "wedgie". I guess some time I'd like to see some carefully
explained examples of how vals and wedgies are used. I've never
managed to follow it on tuning-math. I fully expect that when I
eventually put in the effort to understand these things, I'll say "Is
that all they are? Well why didn't you _say_ so?". :-)

-- Dave Keenan


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Message: 8103

Date: Wed, 12 Nov 2003 00:03:19

Subject: Re: Vals?

From: Carl Lumma

>I've explained what a val is numerous times. I can't insist you pay
>attention to everything I say; these days you and George tend to lose
>me, after all, which is fair enough.

But you haven't explained how it works.

>If the standard 5-limit val
>for 12-equal is [12 19 28] or something, how does it come from
>round(n log2(p))?  Oh n is 12, eh?  So vals are uniquely identified
>by this n?
>
>So how does one find a standard val for an odd limit (the start of
>this thread, which perhaps George is still following)?  Where do you
>get your n?

-C.


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Message: 8105

Date: Wed, 12 Nov 2003 09:04:50

Subject: Re: Definition of microtemperament

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > Gene would like the limit set at 1 c, although I haven't read why.
> 
> "Micro" to me means small enough that the error hardly matters.

Exactly what I said in my definition. It's nice that we agree on
something. :-)

> > It's all pretty arbitrary, but I think we need to draw such a line
> > somewhere.
> 
> There's always my magnitude scale, with lines differing by a factor
> of two.

I can't find this by searching the archive. I tried all kinds of
things. Please explain or give a URL.

The term microtemperament has a long history of referring to
temperaments with errors less than half that of 1/4-comma meantone. So
the magnitude scale could go down by factors of two using the syntonic
comma as the basic unit. But it would be better to "carve nature at
its joints" if possible. That is, look at the minimax errors of large
numbers of the best temperaments and see where the gaps are, near to
these binary fractions of the comma. There seems to be one such gap
between about 2.8 c and 3.1 c for linear temperaments up to 15-limit.

Here's a definition from Feb 2000.
Yahoo groups: /tuning/message/8589 *

And here's the first use (Feb 1999) of the earlier term "wafso-just"
that "microtemperament" replaced.
Yahoo groups: /tuning/message/1012 *


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Message: 8107

Date: Wed, 12 Nov 2003 19:01:57

Subject: Re: Vals?

From: Carl Lumma

>Actually, if you need a shorter term than "prime-mapping", it seems
>like "mapping" would do. What other kinds of mappings do we use in
>tuning-math?

A "mapping", as it has been used, is sufficient to define a
linear temperament.  A val is not.  But choo got me as to the
exact relationship/difference between the two.

-Carl


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Message: 8108

Date: Wed, 12 Nov 2003 16:54:16

Subject: Re: Vals?

From: Graham Breed

Dave Keenan wrote:

> But it appears that little or no explanation would have been necessary
> if you had simply called them prime mappings.

I think I call them "equal mappings" to mean mappings of equal 
temperaments that don't imply you have an actual equal temperament. 
That is, I think this concept is the same as Gene's "Val".


                         Graham


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Message: 8113

Date: Wed, 12 Nov 2003 22:56:59

Subject: Re: Vals?

From: Carl Lumma

>Then a val is just a mapping-row,

What confuses the hell out of me is that Gene keeps using
the word "column" re. vals, but they don't give successive
approximations to the same prime, they give a single
approx. to various primes.

>I don't see how
>calling them vals adds anything to this. In fact I think it
>just obscures things.

By now it should be no surprise that I'm utterly confused
by your obsession over this word.  Considered a career in
postmodern critical theory?

At the very least, I'd hope understand what vals are good
for before trying to rename them.  Or maybe you understand
why the 11-limit has no standard val, and can explain it
to the rest of us.

-Carl


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Message: 8115

Date: Wed, 12 Nov 2003 23:06:50

Subject: Re: 7-limit optimal et vals

From: Carl Lumma

>And we have ED3 for the BP tunings.

Who's we?  I, for one, reject any and all EDx terminology
with the Iron Fist of Discountenance...

-Carl


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Message: 8118

Date: Wed, 12 Nov 2003 13:40:26

Subject: Re: Vals?

From: Carl Lumma

>> >I've explained what a val is numerous times. I can't insist you pay
>> >attention to everything I say; these days you and George tend to
>> >lose me, after all, which is fair enough.
>> 
>> But you haven't explained how it works.
>
>I have done just that many times.

Here's what you've given us so far...

>Consider the otonal chord of the n-odd-limit. This has (n+1)/2 octave 
>reduced elements, 1 < q[i] <= 2, where the q[i], i from 1 to (n+1)/2, 
>are arranged in increasing size. The n-odd-limit has pi(n) primes; we 
>may solve the (n+1)/2 linear equations for the val which sends q[1] 
>to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2. These linear 
>equations have a unique solution in the 3, 5, 7, 9, and 13 odd limits.
>For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard vals 
>in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.

>To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may 
>start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c]
>such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and 
>2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and
>a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the 
>val in question is uniquely determined to be [3, 5, 7], the standard 
>3-val for the 5-limit.

standard val-
>The vector consisting of round(n log2(p)) for primes p in ascending 
>order up to the chosen prime limit, considered as defining a val.

...It appears that in the case of the "standard 3-val for the 5-limit",
n=3.  Is that why you called it a 3-val?  Where did 3 come from?

Further, is the standard val supposed to be the val with the smallest
possible numbers that works?  Or, I don't get your criterion for
deciding you want "the val which sends q[1] to 1, q[2] to 2, up to
q[(n+1)/2]=2 to (n+1)/2".

Further, why are you sending 5/4 to 1 and 3/2 to 2 and 2/1 to 3,
instead of the reverse?  I thought "the q[i], i from 1 to (n+1)/2,
are arranged in increasing size".

-Carl


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Message: 8119

Date: Wed, 12 Nov 2003 23:18:45

Subject: Re: Vals?

From: Carl Lumma

>I'm just disappointed we got this far with "val"

...with only 1 -- two if we're lucky -- persons who know
how to use them for what they're capable of.

Gene, since you won't say what's desirable about being a
standard val, and you haven't said what the lack of a
standard 11-limit val means about the 11-limit, I can only
guess that the definition of standard val is an error,
since the 11-limit is one of the more useful ways to get
hexads.

-Carl


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Message: 8120

Date: Wed, 12 Nov 2003 21:41:37

Subject: Re: 7-limit optimal et vals

From: Paul Erlich

what is the optimality criterion?

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Here is a list of all of them which are not already standard vals, 
for
> n from 1 to 1 100. No torsion issues arise. In some cases other vals
> scored nearly as well.
> 
> <1 2 3 3]
> 
> <3 5 7 9]
> 
> <8 13 19 23]
> 
> <11 18 26 31]
> 
> <13 20 30 36]
> 
> <14 22 32 39]
> 
> <17 27 40 48]
> 
> <20 31 46 56]
> 
> <23 36 53 64]
> 
> <28 44 65 78]
> 
> <30 47 69 84]
> 
> <33 52 76 92]
> 
> <34 54 79 96]
> 
> <39 62 91 110]
> 
> <48 76 112 135]
> 
> <52 83 121 146]
> 
> <54 85 125 151]
> 
> <64 102 149 180]
> 
> <65 103 151 183]
> 
> <66 104 153 185]
> 
> <67 106 155 188]
> 
> <71 112 165 199]
> 
> <85 135 198 239]
> 
> <86 136 199 241]
> 
> <96 152 223 269]
> 
> <98 155 227 275]
> 
> <100 159 232 281]


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Message: 8121

Date: Wed, 12 Nov 2003 23:24:43

Subject: Re: 7-limit optimal et vals

From: Carl Lumma

>That's fine, but we still _have_ EDO and ED3 whether we want to use
>them or not. Or are you able to erase them from your memory? :-)

Actually, only a tiny fraction of the tiny fraction of theorists
who use these lists use this terminology.

>If so, sorry to remind you of them again, and don't ever look at the
>index to Monz's dictionary. At least keep away from the E's, OK. ;-)

There are ways of attacking a terminology.  Publishing papers with
similar but subtly different terminology, for example.  I don't think
this will be necessary, though, as the worthlessness of "EDO" should
be readily apparent to most onlookers.

-Carl


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