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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: 8101

Date: Tue, 11 Nov 2003 03:08:07

Subject: A Riemann zeta peek at 75 equal

From: Gene Ward Smith

I've put a jpg of Z(x) for values of x near 75 in the Photos
directory. Here Z is the Riemann zeta function along the critical
line, twisted in the usual way to get a real function of a real
variable, and then rescaled by the factor ln(2)/2 pi in the argument,
so as to make it correspond to divisions of the octave.

The interest of 75 is that there is a zero of the Reimann zeta
function near 75, and two reasonably large peaks, rather than the
usual one, near to it. The highest peak is the flat octaves peak,
which for a range of values around the peak corresponds to the
19-limit val

[75, 119, 174, 211, 260, 278, 307, 319]

which has 7-limit comma basis

[225/224, 1728/1715, 15625/15309]

The other peak is the sharp octaves peak; it has 19-limit val

[75, 119, 174, 210, 259, 277, 306, 318]

and comma basis

[686/675, 875/864, 5120/5103]

The standard val, in case anyone cares, would be

[75, 119, 174, 211, 259, 278, 307, 319]

The three are already distinct in the 11-limit.


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Message: 8102

Date: Tue, 11 Nov 2003 03:21:23

Subject: Re: Enharmonic diesis?

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> > 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 see no point in it myself; one is hardly likely to confuse a wedgie 
with an interval, though telling an 11-limit linear wedgie from an 11-
limit planar wedgie is another matter. There are also vals to 
consider; it's too hard writing these as column vectors, which is 
really the clearest way to do it.


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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 * [with cont.] 

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


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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: 8109

Date: Wed, 12 Nov 2003 17:17:34

Subject: Re: Definition of microtemperament

From: George D. Secor

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...>
> wrote:
> > I would have said "would always be less than about 3 cents" 
or "... 
> > less than 3.5 cents" in order to include Miracle.  Or don't you 
> > consider that a microtemperament, and if not, then what should we 
> > call it?
> 
> I've always considered miracle to be a microtemperament at the 7-
limit
> (2.4 c) but not at the 9 or 11 limits (3.3 c).

I don't follow this.  The error of 4:5 in Miracle (with minimax 
generator) is ~3.323c.

Anyway, what should I call a temperament in which all of the 
consonances are within 3.33 cents and all of the 7-limit consonances 
are within half of that amount, which I believe sounds like "JI 
brought to life" (i.e., with the "stagnation" or "deadness" 
eliminated with a small amount of tempering)?  I have a particular 13-
limit temperament in mind, and I could send you a recording of a live 
1975 performance on the Scalatron in that temperament, so you could 
judge whether it sounds like JI or not.  Or I could make an mp3 file 
from the recording available, if anyone else wants to hear it.

> I originally said "less than half the 5-limit error of 1/4-comma
> meantone", i.e. less than 2.7 c.

I think you're comparing apples and oranges here.  The max error of 9-
limit 1/4-comma meantone is twice that of 5-limit meantone, simply 
because 8:9 will have twice the error of 2:3.  If you're going to use 
anything on the order of half the error of meantone as your cutoff, 
then you should also extend this to half the error of 8:9 in meantone 
for a 9 limit.  Otherwise, when you evaluate 9-limit (or higher) 
temperaments against your standard, you are really using a 1/4-the-
error-of meantone standard.

The beating harmonics in a tempered 8:9 are much more difficult to 
hear than for 2:3, hence that interval is more difficult to play in 
tune with flexible-pitch instruments, hence the actual error for that 
interval in a live performance is likely to be greater.

> I let it creep up already so a couple of temperaments with 2.8 c
> errors could scrape in, and I went up to 3 for this definition just
> because it seemed silly to be as precise as 2.8 c, so I definitely
> wouldn't want it to creep _past_ 3 cents.
> 
> Gene would like the limit set at 1 c, although I haven't read why.
> However I believe this definition caters for that, by allowing the 
ear
> to arbitrate, and mentionaing the context dependence. In some 
contexts
> a temperament with an error between 1 and 3 cents may not be a
> microtemperament.
> 
> All I'm saying with the 3 cent thing is that there is no context in
> which an error _greater_ than 3 cents would be considered a
> microtemperament, ear or no ear.

Then I guess that (without knowing exactly how much 3 cents has been 
exceeded) you'll have to listen to my recording and then decide.  But 
whatever you decide, I think that you would definitely agree that the 
accuracy is a magnitude or two better than meantone (and even 
noticeably better than 72-ET).

> It's all pretty arbitrary, but I think we need to draw such a line
> somewhere.

Yes.

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> ...
> 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 * [with cont.] 

If you want to draw a boundary at ~2.8 cents, then draw it there, not 
at 3.0 cents, because you're inviting others to want the boundary to 
creep upward.

BTW, what is the next (larger-error) gap?  I am inclined to think 
that a factor of two is too large for establishing magnitudes of 
error.

--George


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Message: 8110

Date: Wed, 12 Nov 2003 20:37:42

Subject: Re: Vals?

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> > <0 1 4 10| should be familiar from the meantone temperament. 
> 
> This looks like one row of the 7-limit prime-mapping for the 
meantone
> linear temperament using a fifth as the generator, in particular the
> row giving the mapping to fifth generators. Isn't it somewhat
> incomplete without the other row that gives the mapping to octave
> generators (periods)?

The octaves would be another val.

> Why do we want to give the same name to something which in one case 
is
> the complete mapping for an ET (a 1D temperament), and in the other
> case only a part of the mapping for an LT (a 2D temperament)?

Is it just barely possible I might know something about mathematics? 
To me this is a bit like asking why we would care about a single 
comma, when we need more than one of them for a Fokker block, BTW.

> But assuming that there's a good reason, I'd simply call them
> "prime-mappings" or "1D-prime mappings".

Call them what you like, but clearly your names are clumsier.

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

They aren't prime mappings per se; that's just a basis. I *did* 
explain they were homomorphic mappings, and give examples with column 
vectors, etc etc.

> So is a val, as applied to tuning theory, simply a prime-mapping, 
or a
> 1D-prime-mapping?

More or less.


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Message: 8111

Date: Wed, 12 Nov 2003 20:39:13

Subject: Re: Vals?

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >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.


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Message: 8112

Date: Wed, 12 Nov 2003 20:48:59

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> > 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.

Maximum error in cents

Magnitude 0: 0.25-0.5
Magnitude 1: 0.5-1.0
Magnitude 2: 1.0-2.0
Magnitude 3: 2.0-4.0

-log2(4 * error) is the formula.

Miracle is a third magnitude temperament by this.

> The term microtemperament has a long history of referring to
> temperaments with errors less than half that of 1/4-comma meantone.

I didn't know that. I do recall wafso-just.


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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: 8114

Date: Wed, 12 Nov 2003 21:04:51

Subject: Re: Linear Temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:

> So, can a mapping (of primes in terms of generators) be derived from
> two temperaments (say 12 & 19) without using commas? or wedgies? 
How 
> would that be done? Thanks

A method I often use is to reduce the matrix whose rows are the two 
vals in quesition to integer Hermite normal form; this means reducing 
the front square part, the rest of the matrix following along. The 
reason I so often use it is that in Maple, if the two vals are 
written in terms of lists, v1, and v2, and I make a lists of lists 
[v1, v2], then ihermite([v1, v2]) does this for me immediately.

Hermite Normal Form -- from MathWorld * [with cont.] 


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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: 8116

Date: Wed, 12 Nov 2003 21:08:06

Subject: Re: Gaps between Zeta function zeros and ets

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:

> Are these in order of the largest gap to the smallest for the first
> 10000 zeros, or some other ordering. How do you derive 954, for 
> example. Thanx

954 is the largest in the sense that the size of the gap, times
log2(954), is the largest up through 1089. If I'd cut things off at 
900, 311 would have been the largest, which is kind of cool. I'm 
going to check and see what happens if I integrate Z(x) between the 
two zeros of the gap.


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Message: 8117

Date: Wed, 12 Nov 2003 21:10:51

Subject: Re: Definition of microtemperament

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> 
wrote:

I have a particular 13-
> limit temperament in mind...

Which is?


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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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Message: 8122

Date: Wed, 12 Nov 2003 21:52:34

Subject: Re: Vals?

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

I've given way, way way more than that. I can't force anyone to read 
it.

> ...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?

A division of the octave into three parts, or in other words, a 
mapping of 2 to 3.

> 
> Further, is the standard val supposed to be the val with the 
smallest
> possible numbers that works? 

It's simply what you get by rounding log2(p) to the nearest integer.

 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".

Eh? clearly 5/4 < 3/2 < 2.


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Message: 8123

Date: Wed, 12 Nov 2003 21:53:47

Subject: Re: 7-limit optimal et vals

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> what is the optimality criterion?

Minimax error in the 7-limit.


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Message: 8124

Date: Wed, 12 Nov 2003 00:43:12

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:

> so how about "mapping" instead of "val" with the implication 
> (preferably stated along with n) that we are talking about ET.

I don't see the point. What about optimal et?


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