Tuning-Math Digests messages 9200 - 9224

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Message: 9200

Date: Sat, 17 Jan 2004 14:26:28

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
>> >
>> >Doesn't 9 have length 2?
>> 
>> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
>
>Is this "lattice" one of those goofy things people insist on calling
>lattices even though they are not?

I don't know.

>What in the world do you mean?

It's a rectangular Thing with an axis for each odd number.

>9 is always 3^2, so it necessarily is twice as far from the origin as
>3, and in the same direction.

Right.

>If 9 and 11 are the same length, 3 is half that length.

There are two different routes to 9 in this Thing, with two
different lengths.  Maybe you can tell me what nice properties
such a setup violates.

Only if you impose log-weighting is the above true.

-Carl


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Message: 9201

Date: Sat, 17 Jan 2004 14:44:41

Subject: Re: Dual space example

From: Carl Lumma

>I had thought of this business of subspaces and the JIP as a way of
>formulating what we were already doing, but didn't realize it led to
>anything new. I wish I'd seen the possible link to Paul's heuristic,
>but another interesting aspect I didn't consider is this bounded
>relative error business. Here's a basic example of what I mean.
>
>As I've pointed out from time to time, the norm which gives the
>equilateral triangular lattice for 5-limit octave classes is
>
>||3^a 5^b|| = sqrt(a^2 + ab + b^2)
>
>where the class is represented by 3^a 5^b.

I can believe that.

>The norm on the dual space is then
>
>||(x3, x5)|| = sqrt(x3^2 - x3 x5 + x5^2)
>
>where x2 is the tuning of 3 (in log2, cents or whatever your favorite
>log is terms) and x5 is the tuning of 5.

I'll take your word on that.

>The nearest point to [log(3), log(5)] on a subspace corresponding
>to a temperament is the 5-limit rms tuning. If we look on the line
>x5=4x3 - 4, for instance, we get the Woolhouse tuning.

Yes!!

Is there anything fundamentally keeping the 2s out of this?

-Carl


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Message: 9203

Date: Sat, 17 Jan 2004 18:23:49

Subject: Re: summary -- are these right?

From: Carl Lumma

>> Can you demonstrate how to get length log(9) out
>> of 9/5?
>
>9/5 is a ratio of 9.

I meant on the lattice.

>> [Paul Hahn] *
>
>OK, which part were we talking about?

You were looking for Paul Hahn's algorithm, which is
like the 2nd or 3rd message in there.  It isn't that
long in any case.

-Carl


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Message: 9204

Date: Sat, 17 Jan 2004 21:08:33

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >A graph (as in graph-theory) but with lengths for each rung?
>> 
>> That could be a "directed graph" I think.
>
>Directed means each rung has a specific beginning point and ending 
>point.

Whoops, got my mix crossed up.  I meant "network".

>> But all the flavors
>> of graph I'm aware of lack orientation, fixed dimensionality,
>> and so forth.  Maybe "space" would work here?
>
>A space has an infinite number of points between any two points.

Crap.

>> >> So summing up, can we say that we're happy with our
>> >> octave-specific concordance heuristic and associated
>> >> lattice/metric, and that we have an octave-equivalent
>> >> concordance heuristic but *no* associated lattice/metric?
>> >
>> >I'd prefer not to say 'concordance heuristic', but yes.
>> 
>> What would you say?
>
>concordance function?

Ok.

-Carl


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Message: 9207

Date: Sat, 17 Jan 2004 14:58:05

Subject: Re: summary -- are these right?

From: Carl Lumma

>> There are two different routes to 9 in this Thing, with two
>> different lengths.  Maybe you can tell me what nice properties
>> such a setup violates.
>
>Uniqueness and linearity.

And does this interfere with having...

() a lattice
() a metric
() a norm

?

With log weighting we restore linearity, I'm guessing.  Then
do we have...

() a lattice
() a metric
() a norm

?

-Carl


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Message: 9208

Date: Sat, 17 Jan 2004 18:32:36

Subject: Re: summary -- are these right?

From: Carl Lumma

>> The two obvious variations are rectangular odd-limit
>
>How can odd-limit be rectangular? Makes no sense to me.

One can certainly have a rectangular lattice with a 9-axis.

>> and triangular octave-specific.
>
>Then the metric is not log(n*d) anymore.

We actually haven't specified how to find the lengths of
rungs like 9:5...

-Carl


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Message: 9209

Date: Sat, 17 Jan 2004 23:13:14

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> By "unweighted" I probably mean a norm without coefficents for
> >> >> an interval's coordinates.
> >> //
> >> >> The norm on Tenney space...
> >> >> 
> >> >> || |u2 u3 u5 ... up> || =
> >> >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
> >> >> 
> >> >> The 'coefficients on the intervals coordinates' here are
> >> >> log2(2), log2(3) etc.
> >> > 
> >> > So 'unweighted', 9 has a length of 2 but 11 has a length of
> >> > 1 . . . :(
> >> 
> >> Unless you use odd-limit.
> >
> >Please elaborate on how that's 'unweighted' in your view.
> 
> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> I'm not claiming anything necessarily good about this, note.
> I am asking for comments about it however.

OK . . . this, I suppose, is Paul Hahn's lattice or something?

> >> >> The taxicab distance on this lattice is log(odd-limit).
> >> > 
> >> > No it isn't -- try 9:5 for example.
> >> 
> >> This is what you were claiming in 1999.
> >> 
> >> ""
> >> But the basic insight is that a triangular lattice, with
> >> Tenney-like lengths, a city-block metric, and odd axes or
> >> wormholes, agrees with the odd limit perfectly, and so is
> >> the best octave-invariant lattice representation (with
> >> associated metric) for anyone as Partchian as me.
> >> ""
> >
> >Right -- you need those odd axes, which screws up uniqueness,
> >and thus most of how we've been approaching temperament.
> 
> But does the metric agree with log(odd-limit) or not?
> For 9:5, log(oddlimit) is log(9).  If you run it through
> the "norm" you get... 2log(3) + log(5).

No, because 9 has its own axis.

> Not the same,
> it seems.  However if you followed the
> lumma.org/stuff/latice1999.txt link,

The page cannot be found.

> apparently Paul Hahn
> did present a metric that agrees with log(odd-limit).

Can you re-present it?

> 
> >> >> I was thinking stuff like ||9|| = ||3|| = 1
> >> >> and thus ||3+3|| < ||3|| + ||3|| but that's ok.  It seems
> >> >> bad though, since the 3s are pointed in the same direction.
> >> > 
> >> > What lattice/metric was this about?
> >> 
> >> Unweighted odd-limit taxicab.
> >
> >In which 9 has its own axis . . . so the following:
> >
> >> By "3+3" I meant adding two 3
> >> vectors.  The equation is 1 < 2.
> >
> >does not apply.
> 
> Sure it does.  As you say, the 9 appears in two places.  If
> the metric comes out the same either way,

It doesn't -- see above.

> I don't see how this
> fact would "screws up uniqueness, and thus most of how we've
> been approaching temperament."

Even if the 'metric came out the same either way', every note would 
appear in an infinite number of places in the lattice, since for 
every integer n the ratio p/q can also be expressed as
p/q*3^(-2n)*9^n
.


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Message: 9210

Date: Sat, 17 Jan 2004 21:48:58

Subject: Re: A new graph for Paul?

From: Carl Lumma

>> <1 -3| 125/128
>
>135/128

Gene and all,

What if, instead of issuing a correction post like this,
we were to post a full corrected version and delete the
original from the archives?  Posterity may thank us...

Just a thought.

-Carl


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Message: 9211

Date: Sat, 17 Jan 2004 23:20:16

Subject: Re: TOP history

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Paul Erlich wrote:
> 
> > It's exactly what I've been pleading to you guys to help me 
figure 
> > out last year and probably even earlier, except without octave-
> > equivalence. The idea was to temper out commas uniformly over 
their 
> > length in the lattice, to see what error function this was 
optimal 
> > with respect to, and to then apply this same error function to 
> > optimize temperaments with more than one comma. The posts asking 
> > about this can be found in the archives here.
> 
> Was it?  Oh.  Well, I found this thread:
> 
> Yahoo groups: /tuning-math/message/2857 *
> 
> The new thing is the concept of weighted minimax.
> 
> 
>                   Graham

Right, because I didn't know that weighted minimax is exactly what 
results from 'applying' the octave-specific 'heuristic' (meaning 
tempering such that it's exactly correct) until I actually thought 
about it. Being sick (though mentally impaired by fever) and away 
from the list for a while seemed to help revive independent thought.


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Message: 9212

Date: Sat, 17 Jan 2004 15:29:52

Subject: Re: summary -- are these right?

From: Carl Lumma

>> On a unit-length odd-limit lattice both 9 and 11 have length 1.
>> I'm not claiming anything necessarily good about this, note.
>> I am asking for comments about it however.
>
>OK . . . this, I suppose, is Paul Hahn's lattice or something?

For his diameter measure I think it is.

>> >> >> The taxicab distance on this lattice is log(odd-limit).
>> >> > 
>> >> > No it isn't -- try 9:5 for example.
>> >> 
>> >> This is what you were claiming in 1999.
>> >> 
>> >> ""
>> >> But the basic insight is that a triangular lattice, with
>> >> Tenney-like lengths, a city-block metric, and odd axes or
>> >> wormholes, agrees with the odd limit perfectly, and so is
>> >> the best octave-invariant lattice representation (with
>> >> associated metric) for anyone as Partchian as me.
>> >> ""
>> >
>> >Right -- you need those odd axes, which screws up uniqueness,
>> >and thus most of how we've been approaching temperament.
>> 
>> But does the metric agree with log(odd-limit) or not?
>> For 9:5, log(oddlimit) is log(9).  If you run it through
>> the "norm" you get... 2log(3) + log(5).
>
>No, because 9 has its own axis.

It's still different than log(odd-limit), and in fact
log(5) + log(9) = 2log(3) + log(5).

>> Not the same,
>> it seems.  However if you followed the
>> lumma.org/stuff/latice1999.txt link,
>
>The page cannot be found.

Typo here; "lattice".

>> apparently Paul Hahn
>> did present a metric that agrees with log(odd-limit).
>
>Can you re-present it?

See the above url.

>> I don't see how this
>> fact would "screws up uniqueness, and thus most of how we've
>> been approaching temperament."
>
>Even if the 'metric came out the same either way', every note would 
>appear in an infinite number of places in the lattice, since for 
>every integer n the ratio p/q can also be expressed as
>p/q*3^(-2n)*9^n
>.

Hmm.

-Carl


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Message: 9213

Date: Sat, 17 Jan 2004 18:40:42

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >> Can you demonstrate how to get length log(9) out
>> >> of 9/5?
>> >
>> >9/5 is a ratio of 9.
>> 
>> I meant on the lattice.
>
>Yes, that's how this 'lattice' is defined, isn't it?

I was asking for any way it could be defined to make it
equal odd-limit, but this seems like cheating because
you require odd-limit infinity, and thus you're never
taking any multi-stop routes.

>> >> [Paul Hahn] *
>> >
>> >OK, which part were we talking about?
>> 
>> You were looking for Paul Hahn's algorithm, which is
>> like the 2nd or 3rd message in there.  It isn't that
>> long in any case.
>
>OK -- that's the algorithm when each consonance in a given
>odd-limit is given a rung of length 1.

Right.

>So going back to the above, if the given 
>odd-limit is less than 9, 9/5 will have to be constructed out of 3 
>and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs
>get involved there.

Right.  It's easy.  But it doesn't correspond to the "ratio-of"
the ratio.  My point, if any, is that I think this will be impossible
with odd-limit < inf. on a triangular lattice.

-Carl


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Message: 9214

Date: Sat, 17 Jan 2004 23:35:38

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> > >
> > >Doesn't 9 have length 2?
> > 
> > 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
> 
> Is this "lattice" one of those goofy things people insist on calling
> lattices even though they are not? What in the world do you mean?
> 
> 9 is always 3^2, so it necessarily is twice as far from the origin 
as
> 3, and in the same direction. If 9 and 11 are the same length, 3 is
> half that length.

What if you're dealing with a world where all sine wave components 
are constrained to be in 768-equal? And '9' is not the same as '3^2'?


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Message: 9215

Date: Sat, 17 Jan 2004 00:06:43

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
> >> By "unweighted" I probably mean a norm without coefficents for
> >> an interval's coordinates.
> //
> >> The norm on Tenney space...
> >> 
> >> || |u2 u3 u5 ... up> || =
> >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
> >> 
> >> The 'coefficients on the intervals coordinates' here are
> >> log2(2), log2(3) etc.
> > 
> > So 'unweighted', 9 has a length of 2 but 11 has a length of
> > 1 . . . :(
> 
> Unless you use odd-limit.

Please elaborate on how that's 'unweighted' in your view.

> >> >> This ruins the correspondence with taxicab distance on
> >> >> the odd-limit lattice given by Paul's/Tenney's norm,
> >> >
> >> >Huh? Which odd-limit lattice and which norm?
> >> 
> >> It's the same norm on a triangular lattice with a dimension
> >> for each odd number.
> > 
> > That's not a desirable norm.
> > 
> >> The taxicab distance on this lattice is log(odd-limit).
> > 
> > No it isn't -- try 9:5 for example.
> 
> This is what you were claiming in 1999.
> 
> ""
> But the basic insight is that a triangular lattice, with
> Tenney-like lengths, a city-block metric, and odd axes or
> wormholes, agrees with the odd limit perfectly, and so is
> the best octave-invariant lattice representation (with
> associated metric) for anyone as Partchian as me.
> ""

Right -- you need those odd axes, which screws up uniqueness, and 
thus most of how we've been approaching temperament. Kees apparently 
saw the issue the same way I did, and I'm still puzzling over some 
issues with his framework, but no one here seemed able to help (last 
year).

> >> It's also the same distance as on the Tenney lattice,
> >> except perhaps for the action of 2s in the latter (I
> >> forget the reasoning there).
> > 
> > Try building up the reasoning from scratch.
> 
> Here's what I was trying to remember...

citation?

> """
> The reason omitting the 2-axis forces one to make the lattice
> triangular is that typically many more powers of two will be
> needed to bring a product of prime factors into close position
> than to bring a ratio of prime factors into close position. So
> the latter should be represented by a shorter distance than the
> former. Simply ignoring distances along the 2-axis and sticking
> with a rectangular (or Monzo) lattice is throwing away
> information.
> //
> ... a weight of log(axis) should be applied to all axes, and
> if a 2-axis is included, a rectangular lattice is OK. If a
> 2-axis is not included, a triangular lattice is better.
> //
> ... in an octave-specific rectangular (or parallelogram)
> lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
> an octave-specific sense, 7:1 and 5:1 really are simpler than
> 7:5; the former are more consonant.  7:4 and 5:4 are each three
> rungs in the rectangular lattice, but they still come out a little
> simpler than 7:5 since the rungs along the 2-axis are so short.
> If you can buy that 35:1 is as simple as 7:5, then the octave-
> specific lattice really should be rectangular, not triangular.
> 35:1 is really difficult to compare with 7:5 -- it's much less
> rough but also much harder to tune . . .
> """
> 
> >> I was thinking stuff like ||9|| = ||3|| = 1
> >> and thus ||3+3|| < ||3|| + ||3|| but that's ok.  It seems
> >> bad though, since the 3s are pointed in the same direction.
> > 
> > What lattice/metric was this about?
> 
> Unweighted odd-limit taxicab.

In which 9 has its own axis . . . so the following:

> By "3+3" I meant adding two 3
> vectors.  The equation is 1 < 2.

does not apply.


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Message: 9216

Date: Sat, 17 Jan 2004 23:32:27

Subject: Re: A new graph for Paul?

From: Carl Lumma

>Not a bad plan; I do wonder how it works for people who read
>via email.

I just got your correction to the message in question by
e-mail.  It would help to have a standard header for the
subject line.  Maybe...

Correction01: [old subject]

...to be followed by Correction02 and so forth.

-Carl


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Message: 9217

Date: Sat, 17 Jan 2004 23:36:49

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >> On a unit-length odd-limit lattice both 9 and 11 have length 
1.
> >> >
> >> >Doesn't 9 have length 2?
> >> 
> >> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
> >
> >Is this "lattice" one of those goofy things people insist on 
calling
> >lattices even though they are not?
> 
> I don't know.
> 
> >What in the world do you mean?
> 
> It's a rectangular Thing with an axis for each odd number.

It's actually triangular, and is what Erv Wilson uses to map out his 
CPSs.


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Message: 9218

Date: Sat, 17 Jan 2004 00:11:54

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > > What does "yes" mean here?
> > 
> > the sound holds together as a single pitch.
> 
> My guess is that it will be experienced as a single pitch, but one
> that cannot be accurately determined. The pitch will be fuzzy or 
vague
> in a similar way to that of a harmonic note of very short duration.

Thanks for that. Sounds to come.

> > > > If I take any inharmonic timbre with one loud partial and 
some 
> > quiet, 
> > > > unimportant ones (very many fall into this category), and use 
a 
> > > > tuning system where
> > > > 
> > > > 2:1 off by < 10.4 cents
> > > > 3:1 off by < 16.5 cents
> > > > 4:1 off by < 20.8 cents
> > > > 5:1 off by < 24.1 cents
> > > > 6:1 off by < 26.9 cents
> > > > 
> > > > and play a piece with full triadic harmony, doesn't it follow 
> > that 
> > > > the harmony should 'hold together' the way 5-limit triads 
should?
> > > 
> > > I don't know. What has the single loud partial got to do with 
it? Is
> > > this partial one of those mentioned above? 
> > 
> > No, it essentially determines the pitch of the timbre.
> 
> So the waveform is essentially sinusoidal? Why not use sinusoidal
> waves for this thought experiment?

They're not especially musical -- you'll have an easier time hearing 
chords as sets of separate notes when the timbre is not a pure sine 
wave.

> > > We know that with quiet sine waves nothing special happens with 
any
> > > dyad except a unison, and that loud sine waves work like 
harmonic
> > > timbres presumably due to harmonics
> > 
> > and combinational tones . . .
> 
> Good point.
> 
> > > being generated in the
> > > nonlinearities of the ear-brain system.
> > 
> > quiet harmonic timbres don't generate combinational tones, so 
they 
> > won't "work like" loud sine waves.
> > 
> > > Don't we?
> > 
> > That also ignores virtual pitch. A set of quiet sine waves can 
evoke 
> > a single pitch which does not agree with any combinational 
tone . . . 
> > at certain intervals, the pitch evoked will be least ambiguous, 
which 
> > is certainly 'something special happening' . . .
> 
> How many sine waves in an approximate harmonic series do you need 
for
> this to be experienced? And what arrangements work? I was only
> speaking of dyads.

The effect has been observed with dyads, but most of the experiments 
concern three (or more) sine waves, since they evoke the phenomenon 
more readily.

> > The fact is that, when using inharmonic timbres of the sort I 
> > described, Western music seems to retain all it meaning: certain 
> > (dissonant) chords resolving to other (consonant) chords, etc., 
all 
> > sounds quite logical. My sense (and the opinion expressed in 
> > Parncutt's book, for example) is that *harmony* is in fact very 
> > closely related to the virtual pitch phenomenon. We already know, 
> > from our listening tests on the harmonic entropy list, that the 
> > sensory dissonance of a chord isn't a function of the sensory 
> > dissonances of its constituent dyads. Furthermore, you seem to be 
> > defining "something special" in a local sense as a function of 
> > interval size, but in real music you don't get to evaluate each 
> > sonority by detuning various intervals various amounts, which 
> > this "specialness" would seem to require for its detection.
> > 
> > The question I'm asking is, with what other tonal systems, 
besides 
> > the Western one, is this going to be possible in.
> 
> If by "Western tonal systems", you mean any based on approximating
> small whole number ratios of frequency,

No, I meant diatonic/meantone.

> What's your point?

Did the above really not say anything to you?


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Message: 9219

Date: Sat, 17 Jan 2004 15:38:20

Subject: Re: summary -- are these right?

From: Carl Lumma

>> It's a rectangular Thing with an axis for each odd number.
>
>It's actually triangular, and is what Erv Wilson uses to map out his 
>CPSs.

The Thing I was referring to here was most certainly rectangular.

-Carl


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Message: 9221

Date: Sat, 17 Jan 2004 23:44:40

Subject: Re: Duals to ems optimization

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> In the 11-limit and beyond, of course, things become more 
complicated
> because we will want to introduce ratios of odd numbers which are 
not
> necessarily primes. If we take ratios of odd numbers up to 11 for 
our
> set of consonances, we get
> 
> sqrt(20x3^2+5x5^2-2x7x11-6x3x5+5x7^2+5x11^2-6x3x7-2x5x7-2x5x11-
6x3x11)
> 
> as our norm on vals, and correspondingly,
> 
> sqrt
(18e3^2+36e3e5+36e3e7+36e3e11+62e5^2+58e5e7+58e5e11+62e7^2+58e11e7+62e
11^2)
> 
> as our norm on octave classes. This norm is not altogether
> satisfactory; for instance it gives a length of sqrt(44) to 5/3 and
> 6/5, and a length of sqrt(62) to 5/4. This suggests to me that there
> is something a little dubious in theory about using unweighted rms
> optimization, at least in the 11 limit and beyond.

Gene, note that I've always counted 3/1 and 9/3, etc., separately in 
these optimizations. If you use that "weighting", do things look less 
dubious? (The weight is proportional to the number of ways the 
interval class can be represented by a ratio of odd numbers within 
the limit.)


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Message: 9223

Date: Sat, 17 Jan 2004 23:48:48

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> >> I'm not claiming anything necessarily good about this, note.
> >> I am asking for comments about it however.
> >
> >OK . . . this, I suppose, is Paul Hahn's lattice or something?
> 
> For his diameter measure I think it is.
> 
> >> >> >> The taxicab distance on this lattice is log(odd-limit).
> >> >> > 
> >> >> > No it isn't -- try 9:5 for example.
> >> >> 
> >> >> This is what you were claiming in 1999.
> >> >> 
> >> >> ""
> >> >> But the basic insight is that a triangular lattice, with
> >> >> Tenney-like lengths, a city-block metric, and odd axes or
> >> >> wormholes, agrees with the odd limit perfectly, and so is
> >> >> the best octave-invariant lattice representation (with
> >> >> associated metric) for anyone as Partchian as me.
> >> >> ""
> >> >
> >> >Right -- you need those odd axes, which screws up uniqueness,
> >> >and thus most of how we've been approaching temperament.
> >> 
> >> But does the metric agree with log(odd-limit) or not?
> >> For 9:5, log(oddlimit) is log(9).  If you run it through
> >> the "norm" you get... 2log(3) + log(5).
> >
> >No, because 9 has its own axis.
> 
> It's still different than log(odd-limit), and in fact
> log(5) + log(9) = 2log(3) + log(5).

You're forgetting that 5:3 has its own rung in this lattice, with 
length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly, 5:3 
is a ratio of 5).

> 
> >> Not the same,
> >> it seems.  However if you followed the
> >> lumma.org/stuff/latice1999.txt link,
> >
> >The page cannot be found.
> 
> Typo here; "lattice".

The page cannot be found.

> >> apparently Paul Hahn
> >> did present a metric that agrees with log(odd-limit).
> >
> >Can you re-present it?
> 
> See the above url.

The page still cannot be found.


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Message: 9224

Date: Sat, 17 Jan 2004 23:49:32

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> It's a rectangular Thing with an axis for each odd number.
> >
> >It's actually triangular, and is what Erv Wilson uses to map out 
his 
> >CPSs.
> 
> The Thing I was referring to here was most certainly rectangular.
> 
> -Carl

Well then it's no Thing that I've ever thought about or talked about 
or heard of before!


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