Tuning-Math Digests messages 9275 - 9299

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Message: 9275

Date: Sun, 18 Jan 2004 01:49:24

Subject: Re: A new graph for Paul?

From: Paul Erlich

--- 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.

> The 3 axis and the 5 axis for intervals is 60 degrees apart; for a
> graph of the lattice of generators, |0 1> and |1 0> should be 120
> degrees apart. There are interesting lines to draw on such a graph;
> the |-3 -5> of porcupine, |1 4> of meantone and |5 13> of amity lie
> along a line, for instance.

What does that line mean? The dual graph I recently produced 
puts 'linear temperaments' that share an ET on straight lines . . .

> Each generator can be graphed twice, by
> graphing +-|1 4>, etc. This would give us additional lines; the line
> between |-1 -4> and |-3 -5> includes pelogic at |1 -3>.

Meaning?


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Message: 9276

Date: Sun, 18 Jan 2004 01:50:38

Subject: Re: summary -- are these right?

From: Paul Erlich

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

9/5 is a ratio of 9.

> >> >> 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] *

OK, which part were we talking about?


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Message: 9277

Date: Sun, 18 Jan 2004 04:08:56

Subject: Re: summary -- are these right?

From: Paul Erlich

--- 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.

Directed means each rung has a specific beginning point and ending 
point.

> 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.

> >> 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?


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Message: 9278

Date: Sun, 18 Jan 2004 01:53:30

Subject: Re: summary -- are these right?

From: Paul Erlich

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

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.

> 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?

Yes.

> The two obvious variations are rectangular odd-limit

How can odd-limit be rectangular? Makes no sense to me.

> and triangular octave-specific.

Then the metric is not log(n*d) anymore.



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Message: 9283

Date: Mon, 19 Jan 2004 12:08:03

Subject: Harmonic Entropy (was: Re: Question for Dave Keenan)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > It hasn't been done yet, but I promise to do it post haste, if 
you 
> > first give me a sincere effort to improve the computational 
> > efficiency of the calculation (starting, of course, with the 
dyadic 
> > case). I'd prefer to discuss this on the harmonic entropy list, 
for 
> > the convenience of those who wish to follow its development.
> 
> It's your definition; I've thought about trying to compute it, but 
it 
> looks nasty.

The idea is to find an easier way to express it or compute it, but 
the direct computation is  pretty easy for dyads. Let's use means 
instead of mediants, which will allow us to use voronoi cells in the 
generalization to higher dimensions. The harmonic entropy of a dyad i 
is simply

-SUM (p(j,i) log(p(j,i)))
  j

where j runs in order of cents size over all lowest-term ratios where 
n*d < 10000 or 65536 or some large number, and p(j,i) is proportional 
to

(cents(j+1)-cents(j-1))*exp( -(cents(j)-cents(i))^2 / (2s^2) )

and the constant of proportionality is such that

-SUM (p(j,i)) = 1
  j

or, if you prefer, you could define p(j,i) as 1/(s*sqrt(2*pi)) times 
the integral from
(cents(j-1) + cents(j))/2
to
(cents(j+1) + cents(j))/2
of
exp( -(cents(t)-cents(i))^2 / (2s^2) ) dt

> Are you suggesting trying to find something similar, but 
> easier to compute?

Yes -- similar or even equal.


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Message: 9287

Date: Mon, 19 Jan 2004 21:48:40

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> 
> > After all, Gene and others would have us 
> > believe that meantone dominant seventh chords were "experienced as" 
> > 4:5:6:7 chords, even though the 6:7 interval would typically be tuned 
> > far closer to 5:6.
> 
> Would harmonic entropy suggest something different?

I assume you mean here the _minimisation_ of harmonic entropy.

There's also the posssibility that the dominant seventh chord
functions best when its harmonic entropy (or maybe only the HE of one
of its dyads) is locally _maximised_.

George Secor alluded to this recently (on the tuning list I think).


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Message: 9288

Date: Mon, 19 Jan 2004 08:37:10

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > After all, Gene and others would have us 
> > believe that meantone dominant seventh chords were "experienced 
as" 
> > 4:5:6:7 chords, even though the 6:7 interval would typically be 
tuned 
> > far closer to 5:6.
> 
> Would harmonic entropy suggest something different?

It might, depending on the value of 's' or hearing resolution assumed 
(this is essentially the only free parameter in harmonic entropy, 
which subsumes considerations of timbre, register, etc.). The 
meantone dominant seventh could land outside the low-entropy region 
surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the 
axes corresponding to these harmonics lie at 60-degree angles).


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Message: 9289

Date: Mon, 19 Jan 2004 22:06:43

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > e.g. with TOP Beep, you were asking us to believe that the 260 c
> > generator could be "experienced as" an approximate 5:6 even though it
> > is only 7 cents away from 6:7, and 55 cents away from 5:6.
> 
> A feeble objection. 
...
> In other words, 7-limit beep equates 7/6 and 6/5 anyway.

Mad if it didn't. However I don't see that that affects anything I
said about 5-limit TOP Beep.

Paul's objection was better, namely that if you provide a large enough
context of other pitches in an approximate harmonic-series segment,
then maybe even a 55 cent error can be pulled into line, as it were.

But even if that were true, there is a very significant difference
between a normal 5-limit temperament where all the 5-limit harmonies
work (including bare dyads and utonalities), and one in which only
otonal tetrads and larger otonalities "work".


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Message: 9292

Date: Mon, 19 Jan 2004 22:12:50

Subject: Re: Annotated Dave Keenan file

From: Dave Keenan

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

> Tertiathirds is a name for the 7-limit temperament with TM comma base
> {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
> [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
> whether it should be called "negri", and I suppose it should be; it's
> pretty closely tied to 2/19 as a generator either way.

I assumed that tertiathirds and negri were alternative names for the
same thing, in my own 7-limit temperament spreadsheet from long ago.
One being a systematic name, the other a common name, like sodium
chloride and salt.


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Message: 9295

Date: Mon, 19 Jan 2004 15:50:21

Subject: Re: Question for Dave Keenan

From: Carl Lumma

>There's also the posssibility that the dominant seventh chord
>functions best when its harmonic entropy (or maybe only the HE of one
>of its dyads) is locally _maximised_.

What would this look like?  The dominant seventh chord is defined
as a local minimum of entropy.

-Carl


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Message: 9296

Date: Mon, 19 Jan 2004 08:49:41

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > It might, depending on the value of 's' or hearing resolution 
assumed 
> > (this is essentially the only free parameter in harmonic entropy, 
> > which subsumes considerations of timbre, register, etc.). The 
> > meantone dominant seventh could land outside the low-entropy 
region 
> > surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where 
the 
> > axes corresponding to these harmonics lie at 60-degree angles).
> 
> Can you calculate this for various s?

It hasn't been done yet, but I promise to do it post haste, if you 
first give me a sincere effort to improve the computational 
efficiency of the calculation (starting, of course, with the dyadic 
case). I'd prefer to discuss this on the harmonic entropy list, for 
the convenience of those who wish to follow its development.


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Message: 9297

Date: Mon, 19 Jan 2004 15:51:39

Subject: Re: Annotated Dave Keenan file

From: Carl Lumma

>> Tertiathirds is a name for the 7-limit temperament with TM comma base
>> {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
>> [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
>> whether it should be called "negri", and I suppose it should be; it's
>> pretty closely tied to 2/19 as a generator either way.
>
>I assumed that tertiathirds and negri were alternative names for the
>same thing, in my own 7-limit temperament spreadsheet from long ago.
>One being a systematic name, the other a common name, like sodium
>chloride and salt.

Works for me.

-Carl


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Message: 9299

Date: Mon, 19 Jan 2004 08:50:39

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > It might, depending on the value of 's' or hearing resolution 
assumed 
> > (this is essentially the only free parameter in harmonic entropy, 
> > which subsumes considerations of timbre, register, etc.). The 
> > meantone dominant seventh could land outside the low-entropy 
region 
> > surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where 
the 
> > axes corresponding to these harmonics lie at 60-degree angles).
> 
> If we analyze 1--5/4--3/2--9/5 as a chord of septimal meantone, 
using
> the approximations native to that, we get that 9/5 is a 9-limit
> consonance and 5/4--9/5 is equivalent to 10/7 by 126/125, so this
> would be a 9-limit magic chord for septimal meantone.

Yup.


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