Tuning-Math messages 950 - 974

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Message: 950

Date: Mon, 03 Sep 2001 21:01:03

Subject: Re: Question for Gene

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Gene, could you please comment on this:
> 
> It looks to me like he's trying to do something which doesn't quite 
> work, which is to analyze a taxicab metric in Euclidean terms.

What if we forget about the Euclidean part -- can you shed any light on the "paradox" in pure taxicab terms?


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Message: 951

Date: Mon, 03 Sep 2001 21:02:23

Subject: Re: Distance measures cut to order

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> Rather than screwing around trying to transform taxicab metrics, I 
> would suggest starting off with a Euclidean metric which works the 
> way you want it to work.

Goodness no! I definitely want a taxicab metric.


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Message: 952

Date: Mon, 03 Sep 2001 21:06:36

Subject: Re: Black magic

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> I've already mentioned that if we take the N-et, and set r = N/ln(2), 
> and then calculate
> 
> r' = (r+G+1/8)/ln(r)
> 
> we get an adjusted tuning after setting N' = ln(2) r'. Here G 
> represents the nearest Gram point, which is round(g(r)), where
> 
> g(r) = r ln(r) - r - 1/8
> 
> and "round" rounds to the nearest integer. This strikes me as almost 
> black magic, it's so easy. Another piece of the same magic is this: 
> define a function 
> 
> tend(N) = 180 (g(r) - round(g(r))),
> 
> where again r = N/ln(2), and the "180" makes tend read out in degrees 
> from -180 to 180. Tend gives the tendency of an et, being positive 
> for ets with a sharp tendency, and negative for flat ets. We have for 
> example:
> 
> N  tend(N)
> 
>  7   -23
> 10     8
> 12    13
> 15    42
> 19   -40
> 22    22
> 27    75
> 31   -22
> 34    40
> 41   -11
> 46    15
> 53    -3
> 58    67
> 72   -55
> 99    54
> 
> When using these to create MOS of M steps out of N, it is better that 
> the tendencies of M and N agree. Thus 19, 31, and 41 are reasonable 
> fits to the flat 72, while 22, 46 (and 21, where we have tend(21) = 
> 14) are less apt, and 58 is downright awkward. On the other hand, 
> when adding two ets to get an et, then it is better if the tendencies 
> are opposite, where they tend to cancel. For instance both 22+31 and 
> 19+34 lead to the neutral 53, whereas adding the slightly sharp 12 to 
> the distinctly sharp 15 leads to the very sharp 27.
> 
> Both the meantone and the 72 systems tend towards flatness, and it 
> might be interesting to look to the sharp systems (such as the 15 out 
> of 27 system I mentioned) for something a little different. 22 out of 
> 46,

That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
out-of-15?  The diminished scale (8-out-of-12)?
> 
> I haven't had any zeta feedback--does any of this make sense?

Can you describe "tendency" a little more precisely for us ignoramuses?


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Message: 953

Date: Mon, 03 Sep 2001 21:37:31

Subject: Re: Symmetric 5-limit and 7-limit distance measures

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > If we take everything in a sphere of radius 1 around [0, 0] we get
> > 
> > 1 - 6/5 - 5/4 - 4/3 - 3/2 - 5/3 - (2),

> What happened to 8/5? This _should_ be the 5-limit Tonality Diamond.

What happened is that I forgot to put it in. Maybe I should have run 
a computer program. :)

> > More interesting is what happens when we 
> > center at [1/2, 1/2, -1/2]:
> > 
> > q7(v - [1/2, 1/2, -1/2]) <= 1/2
> > 
> > gives us
> > 
> > 1 - 15/14 - 5/4 - 10/7 - 3/2 - 12/7 - (2).
> 
> That's the hexany. You should familiarize yourself with CPS 
(Combination Product Set) scales . . . the hexany is the 2)4 
> (1,3,5,7) hexany, meaning it's the set of numbers you get when you 
take products of 2 numbers at a time out of {1,3,5,7}.

I've been thinking of it as an octahedron for the last quarter-
century, but by whatever name it is a fundamental doodad in the 7-
limit. Does anyone but me think of the hexany as a regular 
octahedron, and the 7-limit note-classes as forming the (unique) 
semiregular 3D honeycomb, with cells consisting of tetrahedra and 
octahedra, and vertex figure the cubeoctahedron? Or is the hexany a 
purely combinatorial idea, s your definition suggests?

 The 
> 3)6 (1,3,5,7,9,11) and 3)6 (1,3,7,9,11,15) are called Eikosanies, 
and are similarly symmetrical in the equilateral-triangular 
> lattice.  

Hmmm. In the first place, the hexany is not in an equilateral-
triangle lattice, but a 3D semiregular lattice, at least in my 
metric. In the second place, you have both 3 and 9 in the above list, 
so it can't be entirely symmetrical.

So is the 2)5 (1,3,5,7,9) dekany, when 9 gets its own axis -- Dave 
Keenan created a splendid rotating  dekany that 
> actually plays the notes as the dekany rotates in 4-dimensional 
space. 

3 and 9 have separate generators, you figure?


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Message: 954

Date: Mon, 03 Sep 2001 22:06:20

Subject: Re: Symmetric 5-limit and 7-limit distance measures

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > > If we take everything in a sphere of radius 1 around [0, 0] we get
> > > 
> > > 1 - 6/5 - 5/4 - 4/3 - 3/2 - 5/3 - (2),
> 
> > What happened to 8/5? This _should_ be the 5-limit Tonality Diamond.
> 
> What happened is that I forgot to put it in. Maybe I should have run 
> a computer program. :)
> 
> > > More interesting is what happens when we 
> > > center at [1/2, 1/2, -1/2]:
> > > 
> > > q7(v - [1/2, 1/2, -1/2]) <= 1/2
> > > 
> > > gives us
> > > 
> > > 1 - 15/14 - 5/4 - 10/7 - 3/2 - 12/7 - (2).
> > 
> > That's the hexany. You should familiarize yourself with CPS 
> (Combination Product Set) scales . . . the hexany is the 2)4 
> > (1,3,5,7) hexany, meaning it's the set of numbers you get when you 
> take products of 2 numbers at a time out of {1,3,5,7}.
> 
> I've been thinking of it as an octahedron for the last quarter-
> century, but by whatever name it is a fundamental doodad in the 7-
> limit. Does anyone but me think of the hexany as a regular 
> octahedron,

Everyone does. You should have been on the tuning list for the last year. Erv Wilson drew this octahedron in the 60's, I 
believe.

> and the 7-limit note-classes as forming the (unique) 
> semiregular 3D honeycomb, with cells consisting of tetrahedra and 
> octahedra,

I've been drawing those for at least 10 years now.  Didn't you see, for example, the lattices I just posted for Rami Vitale's 
scale (ASCII)  and Justin White's scale (.gif) on the tuning list?

> and vertex figure the cubeoctahedron?

I think George Olshevsky explained "vertex figure" to us. And the cuboctahedron is the shape of the 7-limit Tonality 
Diamond, etc,  etc. . . .

> Or is the hexany a 
> purely combinatorial idea, s your definition suggests?

Combinatorial _and_ geometrical.
> 
>  The 
> > 3)6 (1,3,5,7,9,11) and 3)6 (1,3,7,9,11,15) are called Eikosanies, 
> and are similarly symmetrical in the equilateral-triangular 
> > lattice.  
> 
> Hmmm. In the first place, the hexany is not in an equilateral-
> triangle lattice, but a 3D semiregular lattice, at least in my 
> metric.

What I  mean is that the lattice has a lot of equilateral triangles in it. Of course it has a lot of squares in it too, but I'm just 
trying to distinguish it, in layman's terms, from the Cartesian square lattice, which is common too (Euler and Fokker come 
to mind).

> In the second place, you have both 3 and 9 in the above list, 
> so it can't be entirely symmetrical.

9 gets its own axis,  and is _not_ assumed to equal 3*3.
> 
> So is the 2)5 (1,3,5,7,9) dekany, when 9 gets its own axis -- Dave 
> Keenan created a splendid rotating  dekany that 
> > actually plays the notes as the dekany rotates in 4-dimensional 
> space. 
> 
> 3 and 9 have separate generators, you figure?

Well, one can think of it either way, but the figure is more symmetrical when one puts 3 and 9 on different axes. The 
problem is that there are a couple of 9-limit consonances that don't show up as direct connections. But with the 
alternative, having only axes for prime numbers, one doesn't have direct connections for _any_ of the ratios of 9. One 
solution is to use a taxicab metric, along with "wormholes" (search the tuning list archives).


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Message: 955

Date: Mon, 03 Sep 2001 22:58:50

Subject: Re: Black magic

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > Both the meantone and the 72 systems tend towards flatness, and 
it 
> > might be interesting to look to the sharp systems (such as the 15 
out 
> > of 27 system I mentioned) for something a little different. 22 
out of 
> > 46,

> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> out-of-15?  The diminished scale (8-out-of-12)?

As you've probably figured out by now, there's lots I'm not familiar 
with--in fact, most of what I know is what I worked out for myself 20-
30 years ago, after reading Helmholtz and something called "Music, a 
Science and an Art" by a JI advocate whose name I can't recall. It 
all really started in grade school, where a music teacher came into 
class and told us about white keys and black keys, and how there were 
seven notes to the scale and twelve to the octave. When I 
asked "Why?", she said "That's just the way it is." This sort of 
answer never makes me happy.

However, let me guess:

Shrutar 2n mod 23, pattern 22222222223 * 2

Paul 10 out of 22, 2n mod 11, pattern 22223 * 2

(Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2)

Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5

Etc. Is that right?

> Can you describe "tendency" a little more precisely for us 
ignoramuses?

It measures the proportional distance to the nearest Gram point from 
the N-et. What it actually does is give a high reading if 3, 5, 7 
etc. pile up in one direction or another--tending to be sharp or 
flat, and a low reading if they are less consistent, with some sharp 
and some flat. In some sense all odd primes are considered, but 
things are heavily weighted in favor of the first few.


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Message: 956

Date: Mon, 03 Sep 2001 23:35:34

Subject: Re: Symmetric 5-limit and 7-limit distance measures

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I've been drawing those for at least 10 years now.  Didn't you see, 
for example, the lattices I just posted for Rami Vitale's 
> scale (ASCII)  and Justin White's scale (.gif) on the tuning list?

I did, and it looked suspiciously regular (and hence familiar) to me; 
that's why I suggested you could do a better job of drawing it.

> What I  mean is that the lattice has a lot of equilateral triangles 
in it. Of course it has a lot of squares in it too, but I'm just 
> trying to distinguish it, in layman's terms, from the Cartesian 
square lattice, which is common too (Euler and Fokker come 
> to mind).

Do people ever look at the reciprocal lattices--the hexagons in the 5-
limit, and in the 7-limit the bee honeycomb of rhombic dodecahedra, 
which is what you get by squashing the spheres in a regular close-
packing together. These are lattices of chords.

> Well, one can think of it either way, but the figure is more 
symmetrical when one puts 3 and 9 on different axes. 

I've never found an ideal solution to this either, though you can 
certainly make the 3 half as big as 5,7,9,11 and 13.


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Message: 957

Date: Tue, 04 Sep 2001 00:26:03

Subject: Scala

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> out-of-15?  The diminished scale (8-out-of-12)?

Now that I have a sound card, I want to creat midi files, and have 
just downloaded Scala. It seems rather formidable, but I noticed a 
lot of Paul Erlich .sla files in a list of files there.

Is there a FAQ or something to lead a person through this? Most of 
the FAQs seem to want to tell me how to do math!


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Message: 958

Date: Tue, 04 Sep 2001 02:05:49

Subject: Re: Distance measures cut to order

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:

> > Rather than screwing around trying to transform taxicab metrics, 
I 
> > would suggest starting off with a Euclidean metric which works 
the 
> > way you want it to work.

> Goodness no! I definitely want a taxicab metric.

Is this so your worms will have holes? I searched on "wormhole" and 
came up with nothing, but I wonder what taxicab metrics can do for 
you that a well-chosen Euclidean metric could not also do.


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Message: 959

Date: Tue, 04 Sep 2001 20:16:37

Subject: Re: Black magic

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > > Both the meantone and the 72 systems tend towards flatness, and 
> it 
> > > might be interesting to look to the sharp systems (such as the 
15 
> out 
> > > of 27 system I mentioned) for something a little different. 22 
> out of 
> > > 46,
> 
> > That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
> out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> > out-of-15?  The diminished scale (8-out-of-12)?
> 
> As you've probably figured out by now, there's lots I'm not 
familiar 
> with--in fact, most of what I know is what I worked out for myself 
20-
> 30 years ago, after reading Helmholtz and something called "Music, 
a 
> Science and an Art" by a JI advocate whose name I can't recall. It 
> all really started in grade school, where a music teacher came into 
> class and told us about white keys and black keys, and how there 
were 
> seven notes to the scale and twelve to the octave. When I 
> asked "Why?", she said "That's just the way it is." This sort of 
> answer never makes me happy.
> 
> However, let me guess:
> 
> Shrutar 2n mod 23, pattern 22222222223 * 2

Not quite . . . it's altered to get omnitetrachordality (the two 
instances of '3' are placed a 3:2 apart).
> 
> Paul 10 out of 22, 2n mod 11, pattern 22223 * 2

That's the symmetrical version . . . the omnitetrachodal version is 
altered, with the two '3's a 3:2 apart.
> 
> (Similarly, 12 out of 22, 2n mod 11, pattern 222221 * 2)

On my keyboard, I use an altered version, with the two '1's appearing 
a 3:2 apart (between E and F and between B and C).
> 
> Blackwood 10 out of 15, 2n mod 3, pattern 21 * 5

Right.
> 
> Etc. Is that right?
> 
> > Can you describe "tendency" a little more precisely for us 
> ignoramuses?
> 
> It measures the proportional distance to the nearest Gram point 
from 
> the N-et. What it actually does is give a high reading if 3, 5, 7 
> etc. pile up in one direction or another--tending to be sharp or 
> flat, and a low reading if they are less consistent, with some 
sharp 
> and some flat. In some sense all odd primes are considered, but 
> things are heavily weighted in favor of the first few.

Very, very interesting! I'm familiar with Riemann's zeta function 
from Schroeder's _Number Theory in Science and Communication_ but 
that's about it.


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Message: 960

Date: Tue, 04 Sep 2001 20:22:37

Subject: Re: Symmetric 5-limit and 7-limit distance measures

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I've been drawing those for at least 10 years now.  Didn't you 
see, 
> for example, the lattices I just posted for Rami Vitale's 
> > scale (ASCII)  and Justin White's scale (.gif) on the tuning list?
> 
> I did, and it looked suspiciously regular (and hence familiar) to 
me; 
> that's why I suggested you could do a better job of drawing it.
> 
> > What I  mean is that the lattice has a lot of equilateral 
triangles 
> in it. Of course it has a lot of squares in it too, but I'm just 
> > trying to distinguish it, in layman's terms, from the Cartesian 
> square lattice, which is common too (Euler and Fokker come 
> > to mind).
> 
> Do people ever look at the reciprocal lattices--the hexagons in the 
5-
> limit, and in the 7-limit the bee honeycomb of rhombic dodecahedra, 
> which is what you get by squashing the spheres in a regular close-
> packing together. These are lattices of chords.

So far, plotting notes has been more useful -- it encapsulates all 
the information in terms of the simplest elements musicians must deal 
with: individual tones.
> 
> > Well, one can think of it either way, but the figure is more 
> symmetrical when one puts 3 and 9 on different axes. 
> 
> I've never found an ideal solution to this either, though you can 
> certainly make the 3 half as big as 5,7,9,11 and 13.

Let's take a step back. If we don't assume octave equivalence, then 
the Tenney lattice, with a taxicab metric, is quite wonderful. 
There's an axis for each prime, and distance between rungs along each 
is log(p). Then the Tenney "harmonic distance" (HD) between two notes 
turns out to be log(n*d) where n/d is the ratio between the notes. 
That's quite wonderful -- the harmonic entropy of the simpler ratios 
n/d turns out to behave very much like log(n*d).

Now when we assume octave equivalence . . . and use 9-limit or 
higher . . . things get hairy.


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Message: 961

Date: Tue, 04 Sep 2001 20:23:34

Subject: Re: Scala

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > That's the Shrutar. Are you familiar with my 10-out-of-22 and 14-
> out-of-26 (or 14-out-of-38) systems? Blackwood's 10-
> > out-of-15?  The diminished scale (8-out-of-12)?
> 
> Now that I have a sound card, I want to creat midi files, and have 
> just downloaded Scala. It seems rather formidable, but I noticed a 
> lot of Paul Erlich .sla files in a list of files there.
> 
> Is there a FAQ or something to lead a person through this? Most of 
> the FAQs seem to want to tell me how to do math!

A FAQ for Scala? You can ask Manuel Op de Coul directly; he's on the 
Tuning List and very helpful.


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Message: 962

Date: Tue, 04 Sep 2001 20:30:19

Subject: Re: Distance measures cut to order

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > > Rather than screwing around trying to transform taxicab 
metrics, 
> I 
> > > would suggest starting off with a Euclidean metric which works 
> the 
> > > way you want it to work.
> 
> > Goodness no! I definitely want a taxicab metric.
> 
> Is this so your worms will have holes? I searched on "wormhole" and 
> came up with nothing,

Probably this discussion happened before the list moved off the Mills 
server.

> but I wonder what taxicab metrics can do for 
> you that a well-chosen Euclidean metric could not also do.

A lot of things. First, consider the Tenney lattice I just described. 
Second, see my post on this list where I conjectured that unison 
vectors with numbers of size S and difference between numerator and 
denominator D imply an amount of tempering proportional to D/(S^2). 
This depends on the Kees van Prooijen lattice with a taxicab metric. 
Lots of other reasons too, probably best enunciated by Paul Hahn back 
in the Mills days.


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Message: 963

Date: Tue, 04 Sep 2001 22:32:37

Subject: Re: Distance measures cut to order

From: Carl Lumma

>>Is this so your worms will have holes? I searched on "wormhole" and 
>>came up with nothing,
> 
>Probably this discussion happened before the list moved off the
>Mills server.

The discussion you're referring to happened while the list was
hosted on onelist, and those messages are in the Yahoo! archive.
The problem is that Yahoo!'s search only goes back arbitrarily
far... I tried this search three times, and got searches from
message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all
showing no matches.

Unexpectedly, when searching my personal archive to verify the
date of the thread, I did find a Mills post containing the
term "wormholes":

Date: Sat, 30 May 1998 11:43:43 -0400
From: Daniel Wolf <DJWOLF_MATERIAL@c...>
To: "INTERNET:tuning@e..." <tuning@e...>
Subject: TUNING digest 1431: Wilson's nines
Message-ID: <199805301143_MC2-3EA2-A973@c...>

Very often Erv Wilson will put his nines on a separate axis from
threes.  A look at his Xenharmonikon cover art will provide several
examples. (All back issues of XH are available from Frog Peak
Music).

One of the most interesting mappings done by Wilson is his mapping
onto a Penrose tiling, treating it as a two dimensional
representation of a 5 dimensional space.  When nines or fifteens
are treated as independent axes from threes and fives,
interesting 'wormholes' in the lattice start to appear, where
alternative representations of the same pitch class occur in
surprisingly different contexts.

Although I have not followed up on this work for some years, I was
getting interesting results in using the Penrose tilings as a
control mechanism over random walks over the lattice.

While treating nine as two steps on the three axis will certainly
be the more efficient means of avoiding redundancies, there can be
musically compelling grounds for preserving such redundancies or
ambiguities.

-Carl


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Message: 964

Date: Wed, 05 Sep 2001 03:15:54

Subject: Tenney's harmonic distance

From: genewardsmith@j...

I did a search on this, and found this old posting, to which I 
respond.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I don't think you should! I think n*d is a much better measure of 
> complexity than one you'd derive from your lattice formula. For 
> simple ratios, harmonic entropy is proportional to log(n*d). And log
> (n*d) is known as Tenney's Harmonic Distance, since it's the city 
> block distance in his lattice. Thus I think his (octave-specific) 
> lattice is much better than yours for depicting dissonance. 

Let's see what happens if we try to make this work in a Euclidean 
framework. In the 5-limit, we want the following distances from the 
origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6), 
d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic form 
is 

u^2 + v^2 + w^2 - 2uv - 2uw - 2vw, 

where u = ln(2)x, v = ln(3)y, w = ln(5)z.

The matrix for the corresponding bilinear form is

[ 1  -1  -1]
[-1   1  -1]
[-1  -1   1],

which is not positive definite, having eigenvalues of -1, 2, and 2. 
This is therefore a Lorentzian metric, like the geometry of space-
time, which does seem a little goofy--should the consonance of 30 
really be imaginary? You can pick eigenvalue coordinates, and 
collapse the -1 part belonging to [1 1 1] out of the picture and get 
something positive definite in two dimensions, but this collapses 30 
down to 1, which doesn't seem any better.

On the other hand if you stick with the obvious, namely u^2+v^2+w^2 
then you get 5/3 the same size as 15, which is what the taxicab 
metric gave you, and a measurement for consonance which seems 
generally in line with what I think you want:

d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), 
d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6),
d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10),
d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15).

Is there some reason not to use this?


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Message: 966

Date: Wed, 05 Sep 2001 20:21:25

Subject: Theorem Paul

From: genewardsmith@j...

Recall that a val was defined to be an element in the dual group to a 
note group; if in particular we take the p-limit note group N_p, with 
elements we think of concretely as row vectors with integer 
coordinate values, then vals are column vectors with integer values. 
We make the following definitions: consider the note group N_p where 
we take as usual the generating set to be the primes up to p, so that 
2 is represented by [1, 0, ..., 0], 3 by [0,1, 0, ..., 0] and so 
forth. If a val for notes in this basis has all of its coordinate 
values positive, and if the coordinate values  have no common 
divisor, we will say the val is *valid*. Another way to express this 
is to say v is valid if v(q) is positive for all primes q<=p and if
gcd({q_i})=1, where {q_i} is the set of primes up to p.

If {n_i} are a set of generators for the kernel of a valid val v (or 
in other words, if they generate the dual group to the group 
generated by v) then we call the set valid. We then have the 
following:

(Theorem Paul) Let n={n_i} be a valid set of generators associated to 
the val v in N_p, where p>3, and let m be n minus one generator. Let 
B be a block of v(2) notes in an octave, defined by n and octave 
equivalence. Let M be the group generated by m, and let u be a val 
which together with v generates the dual M` to M. Then there exists a 
nonnegative integer t such that if w is the val t*v + u we have a 
v(2) out of w(2) MOS which tempers out M from everything in the block 
B.

Proof: Since v is valid, there must be an odd prime q such that 
gcd(v(2), v(q))=1. Hence v(q) generates the cyclic group Z/v(2)Z of 
integers reduced modulo v(2). Consider the ratios v(q)/v(2) and 
w(q)/w(2). As t approaches infinity, w(q)/w(2) approaches v(q)/v(2).
Let us pick a positive t large enough so that all the coordinate 
values of w (ie its values w(q_i) on all the primes up to p) are 
positive, and w(2) is larger than v(2). If such a t is large enough, 
we also have
 
|v(q)/v(2) - w(q)/w(2)| < 1/v(2)^2, 

and v(q)/v(2) is a semiconvergent to w(q)/w(2) and hence v(q) is a 
generator for a v(2) out of w(2) MOS. Since w is in M`, all of the 
elements of M (and hence in particular of m) are sent to 0 by w--ie, 
they are tempered out. Therefore block B is tempered by w to the v(2) 
out of w(2) MOS obtained above, QED.

This is quite a weak condition for vals which are intended to 
represent ets, and hence a strong version of Theorem Paul, but 
nothing more seems to be needed. In practice we could confine our 
attentions to valid sets of generators arising from an h_n et for 
some integer n.


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Message: 967

Date: Wed, 05 Sep 2001 21:32:02

Subject: Re: Distance measures cut to order

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >>Is this so your worms will have holes? I searched on "wormhole" 
and 
> >>came up with nothing,
> > 
> >Probably this discussion happened before the list moved off the
> >Mills server.
> 
> The discussion you're referring to happened while the list was
> hosted on onelist, and those messages are in the Yahoo! archive.
> The problem is that Yahoo!'s search only goes back arbitrarily
> far... I tried this search three times, and got searches from
> message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all
> showing no matches.

Keep clicking "Next" . . . a whole slew of wormhole messages will be 
found. I'm surprised you and Gene both said you couldn't find them!


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Message: 968

Date: Wed, 05 Sep 2001 21:36:26

Subject: Re: Tenney's harmonic distance

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> I did a search on this, and found this old posting, to which I 
> respond.
> 
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I don't think you should! I think n*d is a much better measure of 
> > complexity than one you'd derive from your lattice formula. For 
> > simple ratios, harmonic entropy is proportional to log(n*d). And 
log
> > (n*d) is known as Tenney's Harmonic Distance, since it's the city 
> > block distance in his lattice. Thus I think his (octave-specific) 
> > lattice is much better than yours for depicting dissonance. 
> 
> Let's see what happens if we try to make this work in a Euclidean 
> framework. In the 5-limit, we want the following distances from the 
> origin: d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), d(3/2) = ln(6), 
> d(5/2) = ln(10) and d(5/3) = ln(15). The corresponding quadratic 
form 
> is 
> 
> u^2 + v^2 + w^2 - 2uv - 2uw - 2vw, 
> 
> where u = ln(2)x, v = ln(3)y, w = ln(5)z.
> 
> The matrix for the corresponding bilinear form is
> 
> [ 1  -1  -1]
> [-1   1  -1]
> [-1  -1   1],
> 
> which is not positive definite, having eigenvalues of -1, 2, and 2. 
> This is therefore a Lorentzian metric, like the geometry of space-
> time, which does seem a little goofy--should the consonance of 30 
> really be imaginary? You can pick eigenvalue coordinates, and 
> collapse the -1 part belonging to [1 1 1] out of the picture and 
get 
> something positive definite in two dimensions, but this collapses 
30 
> down to 1, which doesn't seem any better.
> 
> On the other hand if you stick with the obvious, namely u^2+v^2+w^2 
> then you get 5/3 the same size as 15, which is what the taxicab 
> metric gave you, and a measurement for consonance which seems 
> generally in line with what I think you want:
> 
> d(2) = ln(2), d(3) = ln(3), d(5) = ln(5), 
> d(3/2) = sqrt(ln(2)^2 + ln(3)^2) = d(6),
> d(5/2) = sqrt(ln(2)^2 + ln(5)^2) = d(10),
> d(5/3) = sqrt(ln(2)^2 + ln(5)^2) = d(15).
> 
> Is there some reason not to use this?

d(3/2) is between ln(3) and ln(4)! Not good.


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Message: 969

Date: Wed, 05 Sep 2001 21:40:58

Subject: Re: Theorem Paul

From: Paul Erlich

I'll have to come back to this later, but I don't like the idea (in 
the last line) of restricting our attention to MOSs that are within 
some ET. Also, did you make use of the hyperparallelepiped 
construction? If you don't, you don't necessarily get an MOS . . . 
Now do you have a quick way of determining the generator of the 
linear temperament, given n-1 commatic unison vectors?


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Message: 970

Date: Wed, 05 Sep 2001 22:00:43

Subject: Re: Tenney's harmonic distance

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> d(3/2) is between ln(3) and ln(4)! Not good.

You probably want to stick with your taxicab if you don't like this; 
but remember that coordinate transformations transform taxicab spaces 
to other taxicab spaces, you can't treat any of them like Euclidean 
spaces.


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Message: 971

Date: Wed, 05 Sep 2001 22:14:03

Subject: Re: Theorem Paul

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'll have to come back to this later, but I don't like the idea (in 
> the last line) of restricting our attention to MOSs that are within 
> some ET. 

I made that assumption and some others simply to make the proof easy, 
in the lazy way normal to mathematicians. There isn't any reason to 
restrict yourself to these in general, but I thought the point was to 
prove the theorem true. Of course, if what I have stated isn't what 
what you wanted proven (and I've been having problems with that, as 
you know) then the proof won't give you all you want.

Also, did you make use of the hyperparallelepiped 
> construction? 

You get a hyperparallepiped as the walls of the block defined by the 
generators {n_i} together with 2.


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Message: 972

Date: Wed, 05 Sep 2001 22:33:52

Subject: Re: Tenney's harmonic distance

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > d(3/2) is between ln(3) and ln(4)! Not good.
> 
> You probably want to stick with your taxicab if you don't like 
this; 
> but remember that coordinate transformations transform taxicab 
spaces 
> to other taxicab spaces, you can't treat any of them like Euclidean 
> spaces.

You better believe it! So, any comments on the questions I asked?


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Message: 973

Date: Wed, 05 Sep 2001 22:36:21

Subject: Re: Theorem Paul

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I'll have to come back to this later, but I don't like the idea 
(in 
> > the last line) of restricting our attention to MOSs that are 
within 
> > some ET. 
> 
> I made that assumption and some others simply to make the proof 
easy, 
> in the lazy way normal to mathematicians. There isn't any reason to 
> restrict yourself to these in general, but I thought the point was 
to 
> prove the theorem true. Of course, if what I have stated isn't what 
> what you wanted proven (and I've been having problems with that, as 
> you know) then the proof won't give you all you want.

I guess I'd ideally like my original proof to be used as an outline, 
with conditions tightened up where needed. But I'll try to take a 
close look at yours at a later point in time.
> 
> Also, did you make use of the hyperparallelepiped 
> > construction? 
> 
> You get a hyperparallepiped as the walls of the block defined by 
the 
> generators {n_i} together with 2.

So you did make use of this? I'll have to look more closely 
later . . .


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Message: 974

Date: Wed, 05 Sep 2001 22:43:56

Subject: Re: Distance measures cut to order

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > > Rather than screwing around trying to transform taxicab 
metrics, 
> I 
> > > would suggest starting off with a Euclidean metric which works 
> the 
> > > way you want it to work.
> 
> > Goodness no! I definitely want a taxicab metric.
> 
> Is this so your worms will have holes? I searched on "wormhole" and 
> came up with nothing

Gene, you have to keep clicking "Next" in the search dialog. It goes 
backwards . . . somewhere around message 4000, they start popping up.


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