Tuning-Math Digests messages 5875 - 5899

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Message: 5875

Date: Tue, 07 Jan 2003 22:51:44

Subject: Re: Temperament notation

From: Carl Lumma

> > >What, exactly, do you mean by "septimal notation"?
> > 
> > A notation with 7 nominals!
> 
> What I would prefer to call a heptatonic notation.

Of course; much better.

>>And in harmonics 6-12, the aug 3rd and dim 4th don't function
>>differently?
//
>It is interesting to contemplate that if we used a notation
>with 12 nominals for 12-ET that we would be unable to observe
>a distinction on the printed page.

Exactly.

>>Msg. #s 35809 and 41680.
> 
>Thanks!  I'll have to take a look.

Promising technology for microtonalists, to be sure.  For
infinite flexibility we loose velocity and aftertouch, so
we'd be stuck to organ-type patches if we wanted them to
sound good.

We also loose tactile feedback, which would probably make
sight reading impossible.  Notation could be replaced on
these instruments by a 'follow the lights' approach, in
which the whole key can light up, if you like!  Graham,
are you listening?

Certainly exciting that technology exists to bring the holy
grail of an infinitely configurable, extremely portable
keyboard within reach of the consumer (indeed: cheap!).

In the year between msg. 35809 and 41680, it went from
trade-show demo to at least two companies providing OEM
kits.  Assuming there's a flexible inferface in there
somewhere, a microtonal keyboard software project (perhaps
two projectors would be needed for a full keyboard) might
not be too difficult...

-Carl


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Message: 5878

Date: Tue, 07 Jan 2003 04:02:38

Subject: Re: 31, 112 and 11-limit Meantone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> The rms (and least-fourth power) value is 7/26-comma, and the
minimax value is 1/4-comma, which I would call a small difference.

OK.

With regard to sagittal notation, 112-ET fails the following test.

(b) The best fifth (approx 2:3) in the ET must be the same as the fifth
calculated by applying the temperament's mapping to the best
approximation of the generator (and period) in that ET.

The best fifth in 112-ET is 66 steps. The best approximation of the
meantone fifth generator in 112-ET is 65 steps.

I agree with George ("At last!", he says) that sagittal notation of
open meantone chains of up to 30 notes should be based on 31-ET notation.

31-ET passes Gene's earlier test, of being the highest denominator of
a convergent for both RMS and max-absolute generators.


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Message: 5882

Date: Tue, 7 Jan 2003 10:24:49

Subject: Re: thanks manuel

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Sure, I've enjoyed it too.

> so one might either be interested in the *average*
>complexity of the intervals formed by the note in question from all
>the other notes in the scale, or, in special cases, the complexity of
>the interval formed by the note from the tonic (1/1).

There's an idea, it could be added to the output of
"show/attribute intervals" which I didn't show you.

Manuel


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Message: 5885

Date: Wed, 8 Jan 2003 12:28:29

Subject: Re: thanks manuel

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:
>i think it would be good to have a graphical scale analysis tool.

Do you mean with a colour representation of attribute values,
like in your gif picture?
There are some other graphical analysis things in the to-do list 
already, don't think I'll get to it soon.

Manuel


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Message: 5889

Date: Wed, 08 Jan 2003 00:31:52

Subject: Re: Poptimal generators

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
> <d.keenan@u...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus 
> > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith 
> > > <genewardsmith@j...>" <genewardsmith@j...> wrote:
> > > > "Poptimal" is short for "p-optimal". The p here is a real variable
> > > > p>=2, which is what analysts normally use when discussing these 
> > > Holder
> > > > type normed linear spaces.
> > > > 
> > > > A pair of generators [1/n, x] for a linear temperament is 
> > > >*poptimal* if there is some p, 2 <= p <= infinity,
> > > 
> > > why not go all the way to 1? MAD, or p=1, error certainly seems 
> > most 
> > > appropriate for dissonance curves such as vos's or secor's -- which 
> > > are in fact even pointier at the local minima (resembling exp
> > > (|error|)) . . .
> > 
> > Possibly because no one in the history of this endeavour has ever 
> > before now suggested that mean-absolute error corresponds in any way 
> > to the human perception of these things.
> 
> no one in history? you've gotta be kidding me. sum or mean of absolute 
> errors is a quite common error criterion.

By "this endeavour" I meant specifically the mathematical modelling of
perceptual optimality of generators for musical temperaments, not
mathematic or statistics in general. I also only said "possibly". I'm
happy to be corrected.

> >> "A "poptimal" generator can lay claim to being absolutely and ideally
> >> perfect as a generator for a given temperament ..."
> > 
> > When we're talking about human perception, as we are, it should be 
> > obvious that nothing can be absolutely and ideally perfect for 
> > everyone. Even a single person might prefer slightly different 
> > generators for different purposes. To validate such a claim 
> > of "perfection" you would at least need to produce statistics on the 
> > opinions of many listeners.
> 
> clearly dave missed the clever mockery hidden in gene's statement.

Sorry. I must have missed some previous discussion that would have
made it clear that irony was intended. A smiley or winky after it
wouldn't have gone astray. I just read it and thought, hey this is the
sort of talk that gets the non-math folk pissed at us math folk.
Thanks Gene, for being kind enough to ignore my patronising pedantry.


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Message: 5890

Date: Wed, 08 Jan 2003 01:15:21

Subject: Re: Nonoctave scales and linear temperaments

From: Carl Lumma

>>Perhaps I'm not seeing it, but I don't think we need to change
>>our concept of limit.
> 
>we certainly would, and could use "integer limit" as gene
>suggests, or use product limit (tenney).

Maybe so, but I don't see why.  I'm suggesting we think only of
the map, and let it do the walking.  We get to pick what goes
in the map.  Picking 2, 3, 5, 7 and calling it "7-limit" seems
fine to me.

>>If you weight the error right, you shouldn't have to weight
>>the complexity.
>
>I'll return to this after I've had my breakfast coffee. :)

Sorry, all I should have said is, * is communitive.  So it's
really...

Sum ( raw-error(i) * graham-complexity(i) * weighting-factor(i) )

I'll wager a coke this eliminates the need for an averaging
function over the intervals of the limit.  If so, it would
approximate traditional badness, and this could be checked.

-Carl


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Message: 5894

Date: Wed, 08 Jan 2003 01:41:08

Subject: Re: 8-limit meantone

From: Carl Lumma

>If we take the ratio, we get .580551, which still doesn't
>quite do it for us.

What ratio is that?  Blackwood's r?

-Carl


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Message: 5896

Date: Thu, 9 Jan 2003 13:29:40

Subject: Re: thanks manuel

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>to start with, it would be good enough to simply have all the
>consonant intervals (say within a given odd limit) show up as
>diagonal lines -- the rest of the chart can be all white or all black
>for now . . . the point is you could visually tweak the scale with an
>eye toward approximating this consonance here and that consonance
>there . . . donīt know of a better way to achieve this goal than an
>applet like this!

Ok I understand. It probably won't be much work to expand the triad 
player to do this.

Manuel


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