Tuning-Math Digests messages 9225 - 9249

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 10

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9200 - 9225 -



top of page bottom of page down


Message: 9225

Date: Sat, 17 Jan 2004 16:07:51

Subject: Re: summary -- are these right?

From: Carl Lumma

>> 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!

This was a different thing from our thread.

-Carl


top of page bottom of page up down


Message: 9227

Date: Sat, 17 Jan 2004 19:24:31

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.
>
>OK -- but without 'cheating', how can one do in the octave-equivalent 
>case what Tenney does in the octave-specific case?

My question exactly.

>> My point, if any, is that I think this will be impossible
>> with odd-limit < inf. on a triangular lattice.
>
>Well, that's exactly what this:
>
>lattice orientation *
>
>was attempting to address, at least for a prime limit of 5.

Hmm...

-Carl


top of page bottom of page up down


Message: 9229

Date: Sat, 17 Jan 2004 19:27:30

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.
>
>A 'lattice'-like thing, yes.

If we're going to be going over to the mathematical definition
of lattice, we should come up with a term that means "anything
with rungs".

>But then it has nothing to do with odd-limit. And is there a
>2-axis too?

What would happen either way?

>> >> 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...
>
>True, but if you use something different from what Tenney gives, 
>you'll be hard pressed to get all the consonant intervals within a 
>given range (say, 260-500 cents) in the correct order of consonance.

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?

On the other hand, given Gene's recent post, "we" might not
include him...

-Carl


top of page bottom of page up down


Message: 9231

Date: Sat, 17 Jan 2004 17:18:10

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >> >> 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).

I guess so.  Can you demonstrate how to get length log(9) out
of 9/5?

>> >> 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.

Geez, I'm so sorry, it's

[Paul Hahn] *

-Carl


top of page bottom of page up down


Message: 9232

Date: Sat, 17 Jan 2004 10:23:24

Subject: Re: TOP history

From: Graham Breed

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


top of page bottom of page up down


Message: 9233

Date: Sat, 17 Jan 2004 17:25:14

Subject: Re: summary -- are these right?

From: Carl Lumma

>> This was a different thing from our thread.
>
>You were talking about odd-limit thing:
>
>Yahoo groups: /tuning-math/message/8662 *
>
>When and where did you switch to a rectangular thing?

Let's start over.

I'm fishing for something we can use to weed down the
number of "lattices" we're interested in.  Am I correct
that you think log(odd-limit) is the best octave-equivalent
concordance heuristic, and that it constitutes a norm
on the triangular odd-limit lattice with log weighting?
Am I correct that you believe log(n*d) is the best
octave-specific concordance heuristic and that it
constitutes a norm on the Tenney lattice?

The two obvious variations are rectangular odd-limit
and triangular octave-specific.  What say you about those?

Finally, for each of these four lattice types, we can
inquire about what happens when we use no weighting,
("unit lengths").

-Carl


top of page bottom of page up down


Message: 9234

Date: Sat, 17 Jan 2004 03:33:00

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.
With unit lengths, it has length 1 on the 9 axis or length 2 on
the 3 axis.  You can allow this, or impose log weighting which
makes them the same length on a rectangular lattice.

-Carl


top of page bottom of page up down


Message: 9236

Date: Sat, 17 Jan 2004 19:59:46

Subject: Re: summary -- are these right?

From: Carl Lumma

>> If we're going to be going over to the mathematical definition
>> of lattice, we should come up with a term that means "anything
>> with rungs".
>
>A graph (as in graph-theory) but with lengths for each rung?

That could be a "directed graph" I think.  But all the flavors
of graph I'm aware of lack orientation, fixed dimensionality,
and so forth.  Maybe "space" would work here?

>> >But then it has nothing to do with odd-limit. And is there a
>> >2-axis too?
>> 
>> What would happen either way?
>
>If there is a 2-axis, a 9-axis in rectangular lattice seems 
>superfluous (it doesn't change anything in terms of the taxicab 
>distances you get, but adds an infinite number of copies of each 
>pitch), unless you have a reason for treating '9' as different 
>from '3*3' (and therefore '9/3' different than '3'), etc., such
>as a constraint to 768-equal partials.
>
>If there is no 2-axis, you get bad consonance evaluation, for the 
>usual reasons.

So rectangular must have 2, and triangular probably shouldn't...

>> 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?

-Carl


top of page bottom of page up down


Message: 9239

Date: Sun, 18 Jan 2004 04:20:41

Subject: Re: A new graph for Paul?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > For one thing, the generator is not unique, and its multiplicity 
is 
> > proportional to periods per octave.
> 
> Let's stick with octave periods for now.
> 
> > If you're not transforming continuously in terms of the generator 
in 
> > cents, what are you transforming continuously in terms of?
> 
> It might be something you can do by using 7-equal as a way to pass
> from one temperament to another. The lines in question are 7-et 
lines;
> if you take the generator val <0 a b| and wedge/cross-product it 
with 
> <1 11/7 16/7| you get the monzos for the various temperaments.

Let me get back to this after we're done talking about error 
functions and the metrics of their duals.


top of page bottom of page up down


Message: 9241

Date: Sun, 18 Jan 2004 02:14:00

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > Thanks for that. Sounds to come.
> 
> Thanks. I've listened to them. Definitely single pitches. Can you 
tell
> us the relative amplitides of all the partials.

sqrt(1), sqrt(2), . . . sqrt(6).

> And can we hear a
> sustained note around middle C.

Yahoo groups: /tuning_files/files/Erlich/dave1.wav *


> > > > 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.
> 
> OK. So is your question, "In what tonal systems other than
> diatonic/meantone is it going to be possible to have dissonant 
chords
> resolving to consonant chords?"?

Yes, in a sense, and with timbres that are primarily sine-wave but 
have spectral and envelopular aids to being individually heard out.

> The obvious answer would seem to be systems in which there are
> consonant chords, i.e which approximate (or are) JI at least 
>partially.

Right -- now is Top pelogic such a system?

> > > What's your point?
> > 
> > Did the above really not say anything to you?
> 
> Certainly not until you clarified the above. And it might still be a
> good idea for you to spell out the conclusion you intend.

That the phenomenon responsible for central (aka 'virtual') pitch 
allows x amount of Tenney-weighted error in the partials of a single 
pitch, and should the phenomenon be at least partially responsible 
also for the consonance of triads, temperaments with x amount of 
Tenney-weighted error stand a chance of exhibiting this triadic 
consonance, especially if roughness-inducing harmonic overtones are 
absent from each of the pitches.

So it's more than matching tuning and timbre to acheive 'sensory 
consonance' -- defined as a local minimum of roughness -- but it's 
less than the multiply-caused (central pitch, combinational tones, 
and low roughness) consonance that occurs with very-low-error 
temperaments or JI when using harmonic timbres.


top of page bottom of page up down


Message: 9245

Date: Sun, 18 Jan 2004 04:32:55

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:
> > >> If we're going to be going over to the mathematical definition
> > >> of lattice, we should come up with a term that means "anything
> > >> with rungs".
> > >
> > >A graph (as in graph-theory) but with lengths for each rung?
> > 
> > That could be a "directed graph" I think. 
> 
> A directed graph can have both ways or one way streets between the
> nodes. A multigraph allows for multiplicity in the connection, which
> is a little like a shorter length. Nodes connected by lines at 
various
> distances sounds most like a polytope.

?

Polytope -- from MathWorld *


top of page bottom of page up down


Message: 9246

Date: Sun, 18 Jan 2004 22:33:36

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> > And can we hear a
> > sustained note around middle C.
> 
> Yahoo groups: /tuning_files/files/Erlich/dave1.wav *

I didn't find a sustained note there.

> > OK. So is your question, "In what tonal systems other than
> > diatonic/meantone is it going to be possible to have dissonant 
> chords
> > resolving to consonant chords?"?
> 
> Yes, in a sense, and with timbres that are primarily sine-wave but 
> have spectral and envelopular aids to being individually heard out.
> 
> > The obvious answer would seem to be systems in which there are
> > consonant chords, i.e which approximate (or are) JI at least 
> >partially.
> 
> Right -- now is Top pelogic such a system?

Possibly. But if so, it's marginal. This is of course what I've been
saying for some time. I don't see that you've given me any reason to
change this position.

> > And it might still be a
> > good idea for you to spell out the conclusion you intend.
> 
> That the phenomenon responsible for central (aka 'virtual') pitch 
> allows x amount of Tenney-weighted error in the partials of a single 
> pitch, and should the phenomenon be at least partially responsible 
> also for the consonance of triads, temperaments with x amount of 
> Tenney-weighted error stand a chance of exhibiting this triadic 
> consonance, especially if roughness-inducing harmonic overtones are 
> absent from each of the pitches.
> 
> So it's more than matching tuning and timbre to acheive 'sensory 
> consonance' -- defined as a local minimum of roughness -- but it's 
> less than the multiply-caused (central pitch, combinational tones, 
> and low roughness) consonance that occurs with very-low-error 
> temperaments or JI when using harmonic timbres.

But remember that the disagreement on things like "beep" and "father"
is not whether they contain consonances but whether those consonances
have anything to do with their supposed 5-limit mappings.


top of page bottom of page up down


Message: 9248

Date: Sun, 18 Jan 2004 02:25:08

Subject: Re: Duals to ems optimization

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > 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.)
> 
> It weighs 3 more, but it still seems weird. The norm on vals is 
> 
> sqrt(-6x3x11-2x5x11-2x5x7-6x3x7-6x3x5-
2x7x11+21x3^2+5x5^2+5x7^2+5x11^2)
> 
> and the corresponding norm on pcs is
> 
> sqrt
(3e3^2+6e3e5+6e3e7+6e3e11+11e5^2+10e5e7+10e5e11+11e7^2+10e11e7+11e11^2
)
> 
> This gives us ||3|| = ||3/2|| = ||4/3|| = sqrt(3),

Perfect . . .
(1.7321)

> ||9/8||=2sqrt(3), 

Excellent . . .
(3.4641)

> ||5/4|| = ||7/4|| = ||11/8|| = sqrt(11).

OK . . .
(3.3166)

> What seems more dubious
> is ||11/6|| = ||7/6|| = ||6/5|| = 2sqrt(2).

(2.8284)
Doesn't seem unduly dubious, though, given that all the lengths are 
about equal here, except 3 is shorter. Pretty much in accordance with 
what we put in.

Now what if we apply 'odd-limit-weighting' to each of the intervals, 
including 9:3 which is treated as having an odd-limit of 9? Try 
using 'odd-limit' plus-or-minus 1 or 1/2 too.

> I think my idea of using
> the dual norm to my "geometric" norm makes more sense.

Why is that?


top of page bottom of page up down


Message: 9249

Date: Sun, 18 Jan 2004 05:19:25

Subject: Re: Question for Dave Keenan

From: Carl Lumma

> > And can we hear a
> > sustained note around middle C.
> 
> Yahoo groups: /tuning_files/files/Erlich/dave1.wav *

Wrong file, Paul?

-Carl


top of page bottom of page up

Previous Next

9000 9050 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 9600 9650 9700 9750 9800 9850 9900 9950

9200 - 9225 -

top of page