Tuning-Math Digests messages 9250 - 9274

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Message: 9250

Date: Sun, 18 Jan 2004 15:47:42

Subject: Re: Annotated Dave Keenan file

From: Carl Lumma

>Single chain:                               No. generators in
>                        Min        Min      interval
>Generator  No. tetrads  7-limit    7-limit  2  4  5  4  5  6
>(+-0.5c)   in 10 notes  RMS error  MA err.  3  5  6  7  7  7
>-------------------------------------------------------------
>125c           6        12.2c      17.9c   -4  3 -7 -2 -5  2
>tertiathirds

Why isn't this negri?

By the way, anybody know names for these...

!
 Two pentatonic chains of 7:4's rooted a 5:4 apart, tuned in 31-tet.
 10
!
 154.839 !.....4
 232.258 !.....6
 387.097 !....10
 464.516 !....12
 619.355 !....16
 696.774 !....18
 851.613 !....22
 967.742 !....25
 1083.871 !...28
 2/1 !........31
!
! Four 5-limit triads on 1-4-7, strictly proper.

!
 Two pentatonic chains of 3:2's rooted a 7:4 apart, tuned in 31-tet.
 10
!
 154.839 !......4
 193.548 !......5
 387.097 !.....10
 464.516 !.....12
 658.065 !.....17
 696.774 !.....18
 890.323 !.....23
 967.742 !.....25
 1161.290 !....30
 2/1 !.........31
!
! Four 5-limit triads on 1-4-7, not proper.

?

-Carl


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Message: 9251

Date: Sun, 18 Jan 2004 02:29:24

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:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> > > Dual to the 5-limit symmetrical lattice of intervals is a 5-
limit
> > > symmetrical lattice of vals whose first component is zero--which
> > > includes the generators in the period-generator of linear 
> > >temperaments.
> > 
> > Don't get it.
> 
> What's the hang-up? Do you understand the part about pairs of 
integers
> representing generators for 5-limit linear temperaments?

No, it would seem you need 4 integers to specify the mapping. Oh, 
you're throwing out the period. Seems dirty to do that.

> > What does that line mean? 
> 
> That's to be explored. We have, along a line, 
> 
> porcupine-->meantone-->tetracot-->amity
> 
> In terms of generator mapping, but not the period part of the map, 
and
> therefore not in terms of the generators in cents, we can transform
> one to the next continuously. That's the sort of thing I think is
> worth thiking about from a compositional point of view, for 
>starters.

What can you do with it?


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Message: 9254

Date: Sun, 18 Jan 2004 02:33:52

Subject: Re: summary -- are these right?

From: Paul Erlich

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

> >> 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?

> >> [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. 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.


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Message: 9255

Date: Sun, 18 Jan 2004 16:52:54

Subject: Re: TOP on the web

From: Carl Lumma

>Monzos: |...>
>Vals: <...|
>Linear temperament wedgies: <<...||
>Planar temperament wedgies: <<<...|||

Thank GOD you wrote this.  Where's monz?

>It makes sense to use <...| also for tuning maps, but I didn't above,
>as I'm not sure if doing so would sow confusion.

How could it sow confusion?  I didn't know there was a difference
between a map a some number of vals.

-Carl


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Message: 9256

Date: Sun, 18 Jan 2004 02:38:57

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> 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. But then it has nothing to do with odd-
limit. And is there a 2-axis too?

> >> 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.


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Message: 9259

Date: Sun, 18 Jan 2004 19:02:28

Subject: Re: Annotated Dave Keenan file

From: Carl Lumma

>> >Single chain:                               No. generators in
>> >                        Min        Min      interval
>> >Generator  No. tetrads  7-limit    7-limit  2  4  5  4  5  6
>> >(+-0.5c)   in 10 notes  RMS error  MA err.  3  5  6  7  7  7
>> >-------------------------------------------------------------
>> >125c           6        12.2c      17.9c   -4  3 -7 -2 -5  2
>> >tertiathirds
>> 
>> Why isn't this negri?
>
>Interesting question. This is the only 7-limit version of negri with
>a badness score which is much good, so using the "reasonable tuning"
>criterion perhaps it should be.
//
>However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much
>closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning.

The TOP tuning of what?

>What to do?

I don't get it.  Paul's temperament database doesn't list tertiathirds
so I don't know what comma(s) tertiathirds was based on.  If it's
previously been a 5-limit linear temperament I don't see how it could
have <-4 3] the same as negri.

-Carl


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Message: 9260

Date: Sun, 18 Jan 2004 02:51:03

Subject: Re: summary -- are these right?

From: Paul Erlich

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

> >> >> [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.

Right -- no logs, so no log(odd-limit) or log("ratio-of").

> 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.


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Message: 9263

Date: Sun, 18 Jan 2004 03:07:12

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 "Paul Erlich" <perlich@a...> 
wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> 
> > > 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,
> > 
> > That or something very similar to it, like perhaps
> > log(2*odd-limit - 1)
> > or
> > log(2*odd-limit + 1)
> > etc.
> > 
> > > and that it constitutes a norm
> > > on the triangular odd-limit lattice with log weighting?
> > 
> > Technically, it can't, because you don't have uniqueness, etc.
> 
> Here's something you might try. Take every consonance in a given odd
> limit, expressed as a monzo. Multiply this by the log of the odd 
limit
> of that consonance. In this way, get a collection of points, and 
take
> the convex hull. This gives a convex solid containing the origin, 
and
> therefore defines a metric in the usual way, where the usual way is 
to
> call this the ball of radius one, and then find the norm of a point 
by
> scaling the ball so that the point is on the boundry; the scale 
factor
> is the norm.

I'm interested in this approach.

Also (NB Carl), these alternatives to log(odd-limit) don't work:

> > log(2*odd-limit - 1)
> > or
> > log(2*odd-limit + 1)

since the length for 1-limit ratios must be log(1)=0 on the lattice. 
But I still wonder whether anything else might make sense here.


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Message: 9266

Date: Sun, 18 Jan 2004 03:18:43

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:
> 
> > 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.
> 
> Is the weighting by multiplying or dividing by the log of the odd
> limit? Presumably mutliplying will make more sense.

Divide. As in TOP, errors of more complex intervals are divided by 
larger numbers.

> Do we square and
> then multiply, since we will be taking square roots?

No, we want to apply the weighting directly to the errors, before 
deciding how overall error is calculated from the individual weighted 
errors.

> > > I think my idea of using
> > > the dual norm to my "geometric" norm makes more sense.
> > 
> > Why is that?
> 
> It's more or less reasonable to start with. We have
> 
> ||3/2|| = log2(3), ||9/8|| = 2log2(3), ||5/4|| = log2(5), 
> ||6/5|| = log2(5), ||7/6|| = ||7/5|| = ||7/4|| = log2(7),
> ||11/6|| = ||11/7|| = ||11/8|| = ||11/10|| = log2(11),
> ||9/5|| = sqrt(2log2(3)^2 + log2(5)^2)
> ||9/7|| = sqrt(2log2(3)^2 + log2(7)^2)
> ||11/9|| = sqrt(2log2(3)^2 + log2(11)^2)
> 
> which isn't too bad.

OK, but I think my proposal above will make even more sense than 
this. The two should agree for 7-odd-limit and below.


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Message: 9267

Date: Sun, 18 Jan 2004 00:11:23

Subject: Re: summary -- are these right?

From: Paul Erlich

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

You were talking about odd-limit thing:

Yahoo groups: /tuning-math/message/8662 *

When and where did you switch to a rectangular thing?


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Message: 9268

Date: Sun, 18 Jan 2004 03:23:43

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:
> 
> > No, it would seem you need 4 integers to specify the mapping. Oh, 
> > you're throwing out the period. Seems dirty to do that.
> 
> Dirty??

For one thing, the generator is not unique, and its multiplicity is 
proportional to periods per octave. For example, diaschismic can be 
understood as having, like meantone, a generator of a fourth, but 
its 'canonical' generator is sort of a minor second. Which do you 
use, and what's the rule to determine which?

> > > > What does that line mean? 
> > > 
> > > That's to be explored. We have, along a line, 
> > > 
> > > porcupine-->meantone-->tetracot-->amity
> > > 
> > > In terms of generator mapping, but not the period part of the 
map, 
> > and
> > > therefore not in terms of the generators in cents, we can 
transform
> > > one to the next continuously. That's the sort of thing I think 
is
> > > worth thiking about from a compositional point of view, for 
> > >starters.
> > 
> > What can you do with it?
> 
> I just said.

If you're not transforming continuously in terms of the generator in 
cents, what are you transforming continuously in terms of?


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Message: 9269

Date: Sun, 18 Jan 2004 00:46:15

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.

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


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Message: 9271

Date: Sun, 18 Jan 2004 03:37:49

Subject: Re: summary -- are these right?

From: Paul Erlich

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

A graph (as in graph-theory) but with lengths for each rung?

> >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 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.


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Message: 9273

Date: Sun, 18 Jan 2004 01:44:19

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

> > 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.
> > > 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?"?

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

> > 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.


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Message: 9274

Date: Sun, 18 Jan 2004 03:40:34

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 "Paul Erlich" <perlich@a...> 
wrote:
> 
> > I'm interested in this approach.
> 
> I just found out Maple's convex hull finder only works in two
> dimensions, which seems terribly lame. Maybe Mathematica or Matlab 
can
> do better?

Yes:

 CONVHULLN N-D Convex hull.
    K = CONVHULLN(X) returns the indices K of the points in X that 
    comprise the facets of the convex hull of X. X is an m-by-n array 
    representing m points in n-D space. If the convex hull has p 
facets
    then K is p-by-n. 
 
    [K,V] = CONVHULLN(X) also returns the volume of the convex hull
    in V.
 
    CONVHULLN is based on Qhull.
 
    See also CONVHULL, QHULL, DELAUNAYN, VORONOIN, TSEARCHN, DSEARCHN.

 QHULL  Copyright information for Qhull.
 
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