Tuning-Math Digests messages 9105 - 9129

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Message: 9105

Date: Mon, 12 Jan 2004 19:03:58

Subject: Re: summary -- are these right?

From: Carl Lumma

>> TOP is a single-comma technique last I heard. 
>
>Where have you been? It applies to any number of commas.

So where are the TOP 7-limit linear temperaments?

>> >...did Gene or Graham say there's a version of TOP equivalent
>> >to weighted rms?  And Paul, have you looked at the non-weighted
>> >Tenney lattice?
>
>I don't recall saying it, but you could do something along those 
>lines if you wished.

RMS lines, or unweighted lines?

-Carl


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Message: 9106

Date: Mon, 12 Jan 2004 19:42:23

Subject: Re: Temperament agreement

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 don't like these two-curve boundaries when it's clear one 
simple 
> > curve could do. I personally could do without 78732/78125 and 
> > 20000/19683, but not without 531441/524288.
> 
> Is this subjective, or can you quantify it?

Actually, it's absurd. I misremembered the four curves that I drew, 
which was easy since no one has referred to them yet. What I really 
meant (:)) was that probably, all three of these should be in, or all 
three should be out.


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Message: 9107

Date: Mon, 12 Jan 2004 19:21:47

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >Where have you been? It applies to any number of commas.
>> 
>> So where are the TOP 7-limit linear temperaments?
>
>Yahoo groups: /tuning-math/message/8504 *

Oh dear.  Does anybody besides Gene understand this yet?
Last I heard Paul was trying to use heron's formula to get
around straightness.

-Carl


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Message: 9108

Date: Mon, 12 Jan 2004 19:43:50

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:
> 
> > You should hang it on your refrigerator. Once you do, you may be 
> able 
> > to understand this: for the kernel of a temperament, it will be a 
> > list of linearly independent commas that don't lead to torsion; 
for 
> a 
> > temperament, it will be a list of linearly independent intervals 
> that 
> > generate the whole temperament.
> 
> Do you mean "vals" and not "intervals"?

No, I mean intervals -- for example, for meantone temperament, the 
list could consist of the meantone fifth and octave.


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Message: 9109

Date: Mon, 12 Jan 2004 20:29:53

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> I've put up a TOP web page. It needs to have, at least, a 
discussion 
> of equal and linear temperaments and Tenney complexity and badness 
> added to it, but it should be valuable as a starter. Here it is:
> 
> /root/tentop.htm *
> 
> Paul, could you tell me what you want attributed to you?

My eyes glazed over. I'll have to look at this again some other time.

I came up with the idea of tempering uniformly by length, and then 
with the observation that this minimizes maximum weighted error over 
all intervals. Strictly codimension one. I also came up with a way to 
minimize maximum weighted error over all intervals, for dimension one 
(ETs).


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Message: 9111

Date: Mon, 12 Jan 2004 21:32:19

Subject: Re: TOP on the web

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 came up with the idea of tempering uniformly by length, and 
then 
> > with the observation that this minimizes maximum weighted error 
> over 
> > all intervals. Strictly codimension one. 
> 
> OK.
> 
> I also came up with a way to 
> > minimize maximum weighted error over all intervals, for dimension 
> one 
> > (ETs).
> 
> I think several people were doing ets; I know I was.

Yes, you and Graham both, but from what I could tell, you were both 
using more complicated methods. Though mine was fairly obvious and 
I'm sure either of you could have come up with it anyway.


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Message: 9112

Date: Mon, 12 Jan 2004 13:57:42

Subject: Re: summary -- are these right?

From: Carl Lumma

>> and is there
>> a definition of "basis" somewhere?
>
>You should hang it on your refrigerator. Once you do, you may be able 
>to understand this: for the kernel of a temperament, it will be a 
>list of linearly independent commas that don't lead to torsion; for a 
>temperament, it will be a list of linearly independent intervals that 
>generate the whole temperament.

?  I can't hang it on my refrigerator if I don't have it! 

>> Standard val -> canonical val
>> 
>> ...the standard val is just the best approximation of each
>> identity in the ET, right?  Are there any other contenders
>> for canonical val?
>
>Yes.
>
>(I'm in a hurry, my apologies)

Paul, please take your time.  At your convenience, I'd love to have
your full comments on my message.

And did you see the posts where I compare zeta, gram, and TOP-et
tunings?

Thanks,

-Carl


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Message: 9113

Date: Mon, 12 Jan 2004 22:07:40

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> and is there
> >> a definition of "basis" somewhere?
> >
> >You should hang it on your refrigerator. Once you do, you may be 
able 
> >to understand this: for the kernel of a temperament, it will be a 
> >list of linearly independent commas that don't lead to torsion; 
for a 
> >temperament, it will be a list of linearly independent intervals 
that 
> >generate the whole temperament.
> 
> ?  I can't hang it on my refrigerator if I don't have it! 

Vector Space Basis -- from MathWorld *

> >> Standard val -> canonical val
> >> 
> >> ...the standard val is just the best approximation of each
> >> identity in the ET, right?  Are there any other contenders
> >> for canonical val?
> >
> >Yes.
> >
> >(I'm in a hurry, my apologies)
> 
> Paul, please take your time.  At your convenience, I'd love to have
> your full comments on my message.
> 
> And did you see the posts where I compare zeta, gram, and TOP-et
> tunings?

Yup . . .


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Message: 9114

Date: Mon, 12 Jan 2004 23:10:14

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> I've put up a TOP web page. It needs to have, at least, a 
discussion 
> of equal and linear temperaments and Tenney complexity and badness 
> added to it, but it should be valuable as a starter. Here it is:
> 
> /root/tentop.htm *

"Because of the transcendence and linear independence of the logs of 
odd primes, the coordinates of TOP(S) can never be the same as any of 
the coordinates of JIP. TOP(S) therefore retunes every rational 
number by some amount, and this includes octaves; hence the 
alternative acronym of Tempered Octaves, Please for TOP."

Something must be wrong with either the premise or the inference, 
since, for example, Top Beep has pure octaves.


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Message: 9116

Date: Tue, 13 Jan 2004 01:20:25

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > "Because of the transcendence and linear independence of the logs 
of 
> > odd primes, the coordinates of TOP(S) can never be the same as 
any of 
> > the coordinates of JIP. TOP(S) therefore retunes every rational 
> > number by some amount, and this includes octaves; hence the 
> > alternative acronym of Tempered Octaves, Please for TOP."
> > 
> > Something must be wrong with either the premise or the inference, 
> > since, for example, Top Beep has pure octaves.
> 
> I'm getting this for TOP(27/25):
> 
> [1214.176 1879.486 2819.230]
> 
> What are you getting?

See Yahoo groups: /tuning/message/51193 * -- the same 3 
and 5, but not the same 2. I guess the tuning of 2 actually has a 
range of possibilities while maintaining the same maximum weighted 
error for the tuning as a whole.

Hopefully you've seen my formula for TOP tempering of a single 
comma . . .


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Message: 9117

Date: Tue, 13 Jan 2004 02:06:31

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> > is there a definition of "basis" somewhere?
> //
> >Vector Space Basis -- from MathWorld *
> 
> Ah, good.  That's what I thought.
> 
> >> >You should hang it on your refrigerator. Once you do, you may be
> >> >able to understand this: for the kernel of a temperament, it 
will
> >> >be a list of linearly independent commas that don't lead to
> >> >torsion;
> 
> This is the only sense I've ever noticed it used around here, and
> it's what I meant by "TM reduction -> canonical basis".
> 
> >> >for a temperament, it will be a list of linearly independent
> >> >intervals that generate the whole temperament.
> 
> Generate the pitches in the temperament.  One also needs the map.

If it's a regular tuning, rather than a regular temperament, one 
doesn't need a map.

> >> And did you see the posts where I compare zeta, gram, and TOP-et
> >> tunings?
> >
> >Yup . . .
> 
> I've been wondering about working backwards from the technique
> to TOP for codimension > 1 temperaments.  How would it apply to
> a pair of vals?

A pair of vals -> dimension = 2. How would what apply?

> Which commas is it tempering in the single-val
> case?

Nothing new to TOP here.

> etc.
> 
> -Carl

Not sure what these questions mean, but working forwards from my 
technique for ETs to dimension>1 seems possible, a linearly-
constrained minimax problem, good for linear programming . . .


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Message: 9122

Date: Tue, 13 Jan 2004 03:07:40

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> TOP is a single-comma technique last I heard. 
> >
> >Where have you been? It applies to any number of commas.
> 
> So where are the TOP 7-limit linear temperaments?

Yahoo groups: /tuning-math/message/8504 *


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Message: 9123

Date: Tue, 13 Jan 2004 03:53:54

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> >Where have you been? It applies to any number of commas.
> >> 
> >> So where are the TOP 7-limit linear temperaments?
> >
> >Yahoo groups: /tuning-math/message/8504 *
> 
> Oh dear.  Does anybody besides Gene understand this yet?
> Last I heard Paul was trying to use heron's formula to get
> around straightness.

Right, but that was pre-TOP. Now I can at least understand how to do 
these as a linear programming exercise, if no other way . . .


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Message: 9124

Date: Tue, 13 Jan 2004 06:15:15

Subject: The Exotemperaments in the dual representation

From: Paul Erlich

Yahoo groups: /tuning_files/files/Erlich/dualxoom.gif *


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