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Message: 10476 - Contents - Hide Contents

Date: Tue, 02 Mar 2004 23:55:11

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

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

>>>>> I don't know where sqrt would be coming from.  I thought 
> everything

>>>>> would have to have whole number lengths.
>>>> 

>>>> I'm using Euclidean distances.
>>> 

>>> I wish you had said that.
>> 

>> If I'm talking about symmetric lattices, that would be assumed.
> 

> You didn't assume it the other week, when it seemed (even upon 
> clarification) that you referred to *two* possible metrics in the 
> symmetric lattice -- the euclidean one, and the 'taxicab' one. 


Those are two different lattices, and my point is that if someone says
"symmetric lattice" without qualification they presumably mean the
stuff you guys already knew about when I got here.


> That's a bogus assumption. We live in a nearly euclidean universe so 
> people have no choice in the matter when it comes to diagrams. If I 
> draw a Tenney lattice, which assumes a taxicab metric, how am I 
> supposed to avoid drawing it in a way that you'd interpret as 
> euclidean??


You tell us it's a Tenney lattice. This is really the only possible
method.

Message: 10477 - Contents - Hide Contents

Date: Tue, 02 Mar 2004 23:57:41

Subject: Re: Hanzos

From: Gene Ward Smith

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


>> What I don't understand is why anyone would want to. You can also
>> throw away the 7, and keep the 2,3 and 5. Would anyone propose
doing
>> that? It strikes me as an absurd proceedure. Where
octave-equivalent
>> vectors are useful is in octave-equivalent contexts.
> 

> We weren't talking about temperaments, we were talking about finding
> chord sequences in JI; clearly an occasion for octave equivalence.


You seem to be mixing up to completely different threads, unless there
was a connection I was supposed to have noticed and didn't.

Message: 10478 - Contents - Hide Contents

Date: Tue, 02 Mar 2004 16:26:19

Subject: Re: Hanzos

From: Carl Lumma

>>> >hat I don't understand is why anyone would want to. You can also
>>> throw away the 7, and keep the 2,3 and 5. Would anyone propose
>>> doing that? It strikes me as an absurd proceedure. Where
>>> octave-equivalent vectors are useful is in octave-equivalent
>>> contexts.
>> 

>> We weren't talking about temperaments, we were talking about finding
>> chord sequences in JI; clearly an occasion for octave equivalence.
>

>You seem to be mixing up to completely different threads, unless there
>was a connection I was supposed to have noticed and didn't.


The only reason I ever brought up Hahn was in the stepwise thread.
You created the Hanzos thread, you tell me what it's about.

-Carl

Message: 10479 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 05:45:56

Subject: Re: Stepwise scales

From: Gene Ward Smith

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


> If we look at the intervals this gives us, the three smallest are
> 2401/2400, 225/224, and 1029/1024, making it a natural candidate for
> miracle tempering.


Hemiwuerschmidt is also a natural. If we take, not the smallest scale
steps, but the closest approximatations to 7-limit consonances, we get
as the five smallest commas, in order, 2401/2400, 6144/6125,
3136/3125, 225/224, 1029/1024. Taking the first three together gives
us hemiwuerschmidt.

In his Ancient Worlds record, Michael Harrison uses an (unspecified)
7-limit scale of 24 notes. It occured to me that if you wanted a
7-limit microtemperament for 24 notes, and if miracle was not quite
accurate enough for you, your best bet would probably be
Hemiweurschmidt[24]. Hemiwuerschmidt[25] is the DE, not 24, but I
don't think that matters much.

Here's the 55-note scale, reduced to 51 notes in hemiwuerschmidt:

[-32, -30, -28, -27, -25, -22, -21, -19, -18, -17, -16, -14, -13, -12,
-11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 25, 27, 28, 30, 32]

Message: 10480 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 17:37:12

Subject: Re: Hanzos

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:

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

>>>>>> I don't know where sqrt would be coming from.  I thought 
>> everything

>>>>>> would have to have whole number lengths.
>>>>> 

>>>>> I'm using Euclidean distances.
>>>> 

>>>> I wish you had said that.
>>> 

>>> If I'm talking about symmetric lattices, that would be assumed.
>> 

>> You didn't assume it the other week, when it seemed (even upon 
>> clarification) that you referred to *two* possible metrics in the 
>> symmetric lattice -- the euclidean one, and the 'taxicab' one. 
> 

> Those are two different lattices,


But they both can only be drawn with the usual tetrahedra and 
octahedra, right? So they *look* like the same lattice, and the 
choice of metric has to be specified *verbally*.


> and my point is that if someone says
> "symmetric lattice" without qualification they presumably mean the
> stuff you guys already knew about when I got here.


Which pretty much used a 'taxicab' metric (as per Paul Hahn's 
algorithm) when a metric was used at all.


>> That's a bogus assumption. We live in a nearly euclidean universe 
so 
>> people have no choice in the matter when it comes to diagrams. If 
I 
>> draw a Tenney lattice, which assumes a taxicab metric, how am I 
>> supposed to avoid drawing it in a way that you'd interpret as 
>> euclidean??
> 

> You tell us it's a Tenney lattice. This is really the only possible
> method.


So you see why your assumption was bogus, yes?

Message: 10482 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:00:13

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


>>> You didn't assume it the other week, when it seemed (even upon 
>>> clarification) that you referred to *two* possible metrics in the 
>>> symmetric lattice -- the euclidean one, and the 'taxicab' one. 
>> 

>> Those are two different lattices,
> 

> But they both can only be drawn with the usual tetrahedra and 
> octahedra, right? So they *look* like the same lattice, and the 
> choice of metric has to be specified *verbally*.



Strictly speaking, only one of them can be drawn at all.


>> and my point is that if someone says
>> "symmetric lattice" without qualification they presumably mean the
>> stuff you guys already knew about when I got here.
> 

> Which pretty much used a 'taxicab' metric (as per Paul Hahn's 
> algorithm) when a metric was used at all.


Are you saying you actually were clueless about this until I arrived?
Did you not know a hexany was a literal octahedron, and tetrads
literal tetrahedra?


> So you see why your assumption was bogus, yes?


Most of the world means "Euclidean" when they say "lattice" at all,
and don't bother to say so; and assumes it when they speak of
octahedra or other geometrical figures. If you want to do things
differently, it is up to you to specify but please don't claim I am
being ambiguous. I'm not; I'm doing things how pretty well everyone
does them.

I'd be interested to know what people meant by "lattice" when I
arrived, though, and whether we have been miscommunicating from the start.

Message: 10483 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:01:11

Subject: Re: Canonical generators for 7-limit planar temperaments

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

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

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>> > 

>>>> ***[-1/13*a-4/13*b-6/13*c, 14/13*b+3/26*c+7/26*a, c]
> 

> With      a                      b                      c ?


3^a 5^b 7^c defining the note class.

Message: 10484 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:06:44

Subject: Re: Hanzos

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 didn't assume it the other week, when it seemed (even 
upon 
>>>> clarification) that you referred to *two* possible metrics in 
the 
>>>> symmetric lattice -- the euclidean one, and the 'taxicab' 
one. 
>>> 

>>> Those are two different lattices,
>> 

>> But they both can only be drawn with the usual tetrahedra and 
>> octahedra, right? So they *look* like the same lattice, and the 
>> choice of metric has to be specified *verbally*.
> 
> 

> Strictly speaking, only one of them can be drawn at all.


So you can't draw a graph?


>>> and my point is that if someone says
>>> "symmetric lattice" without qualification they presumably mean 
the
>>> stuff you guys already knew about when I got here.
>> 

>> Which pretty much used a 'taxicab' metric (as per Paul Hahn's 
>> algorithm) when a metric was used at all.
> 

> Are you saying you actually were clueless about this until I 
arrived?
> Did you not know a hexany was a literal octahedron, and tetrads
> literal tetrahedra?


That's the way we were drawing them, so I don't know what you could 
mean by 'clueless'.


> I'd be interested to know what people meant by "lattice" when I
> arrived, though, and whether we have been miscommunicating from the 
>start.


In any case, hopefully it's clear to you that Paul Hahn's algorithm 
(at least if we stick to 7-limit and don't mess with 9-limit) gives 
the taxicab distance in the symmetric oct-tet lattice. Since it 
measures the number of 'consonances' needed to get from one note to 
another, it's certainly a meaningful measure, while the euclidean 
distance doesn't seem to signify anything musically meaningful.

Message: 10485 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:25:30

Subject: Re: Hanzos

From: Gene Ward Smith

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


> Most of the world means "Euclidean" when they say "lattice" at all,
> and don't bother to say so...


Here's the Wikipedia article:

Lattice - Wikipedia, the free encyclopedia * [with cont.]  (Wayb.)(group)

The definition is assuming a Euclidean metric, since R^n without
qualification means R^n under a Euclidean norm. It then goes on to
reference usage in materials science (assumed Euclidean) and
computational physics, where you are generally thinking Euclidean, but
where the lattice is really a kind of graph, and so may be close to
what people are thinking here. Then it finishes talking about Lie
groups and Haar measure, pretty much off on a tangent.

Here is World of Mathematics:

http://mathworld.wolfram.com/PointLattice.html * [with cont.] 

"Formally, a lattice is a discrete subgroup of Euclidean space"

This entry is explicitly ruling out other possibilities! I prefer to
say something like it is a discrete subgroup of a finite dimensional
normed real vector space with compact quotient, but norms other than
L2 norms come up seldom; you can read long and sophisticated books
stuffed full of lattices which never bother to mention them.

Message: 10486 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:31:31

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


>> Strictly speaking, only one of them can be drawn at all.
> 

> So you can't draw a graph?


A figure in Tenney space is not a mere graph; it has far more structure.


>> Are you saying you actually were clueless about this until I 
> arrived?

>> Did you not know a hexany was a literal octahedron, and tetrads
>> literal tetrahedra?
> 

> That's the way we were drawing them, so I don't know what you could 
> mean by 'clueless'.


Did you or did you not know you could put 7-limit note classes into a
symmetric lattice pattern in three dimensional Euclidean space?


> In any case, hopefully it's clear to you that Paul Hahn's algorithm 
> (at least if we stick to 7-limit and don't mess with 9-limit) gives 
> the taxicab distance in the symmetric oct-tet lattice. Since it 
> measures the number of 'consonances' needed to get from one note to 
> another, it's certainly a meaningful measure, while the euclidean 
> distance doesn't seem to signify anything musically meaningful.


I agree it is meaningful and quite interesting; I don't agree
Euclidean is meaningless. You've been sleeping through my postings if
you think that; Euclidean draws finer distinctions than Hahn's measure
does, and those distinctions do relate to things musical.

Message: 10487 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:40:17

Subject: Re: Hanzos

From: Paul Erlich

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 

>>> Strictly speaking, only one of them can be drawn at all.
>> 

>> So you can't draw a graph?
> 

> A figure in Tenney space is not a mere graph; it has far more 
structure.


OK . . . but can't you think of it as existing in euclidean space and 
define a new kind of distance in which one is restricted to traveling 
along the edges?


>>> Are you saying you actually were clueless about this until I 
>> arrived?

>>> Did you not know a hexany was a literal octahedron, and tetrads
>>> literal tetrahedra?
>> 

>> That's the way we were drawing them, so I don't know what you 
could 
>> mean by 'clueless'.
> 

> Did you or did you not know you could put 7-limit note classes into 
a
> symmetric lattice pattern in three dimensional Euclidean space?


Since we were actually doing so, it seems absurd to question whether 
we knew that we could do so. We simply rarely or never measured 
euclidean distances in this structure, since they don't carry a 
straightforward musical meaning.


>> In any case, hopefully it's clear to you that Paul Hahn's 
algorithm 
>> (at least if we stick to 7-limit and don't mess with 9-limit) 
gives 
>> the taxicab distance in the symmetric oct-tet lattice. Since it 
>> measures the number of 'consonances' needed to get from one note 
to 
>> another, it's certainly a meaningful measure, while the euclidean 
>> distance doesn't seem to signify anything musically meaningful.
> 

> I agree it is meaningful and quite interesting; I don't agree
> Euclidean is meaningless. You've been sleeping through my postings 
if
> you think that;


I don't appreciate that comment.


> Euclidean draws finer distinctions than Hahn's measure
> does,


Clearly -- but are they meaningful distinctions?


> and those distinctions do relate to things musical.


Namely . . . ?

Message: 10488 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 18:59:40

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
>> 

>>>> Strictly speaking, only one of them can be drawn at all.
>>> 

>>> So you can't draw a graph?
>> 

>> A figure in Tenney space is not a mere graph; it has far more 
> structure.
> 

> OK . . . but can't you think of it as existing in euclidean space and 
> define a new kind of distance in which one is restricted to traveling 
> along the edges?


Sure, and that may be a good approach. You are now using a Euclidean
lattice, and defining a corridor-travel distance function on top of
it. It is certainly familiar to us that as-the-crow-flies and as the
taxicab drives are both valid and can exist simultaneously.


>> Did you or did you not know you could put 7-limit note classes into 
> a

>> symmetric lattice pattern in three dimensional Euclidean space?
> 

> Since we were actually doing so, it seems absurd to question whether 
> we knew that we could do so. We simply rarely or never measured 
> euclidean distances in this structure, since they don't carry a 
> straightforward musical meaning.


It never occurred to you to do orthogonal transformations? In any
case, you should be aware that there *are* differences between
intervals like 15/14 and 21/20 as compared to 25/24, 36/35 or 49/48,
and that these again are different from 50/49. They all take two
7-limit consonaces to get to, which is why they all turn up in the
stepwise problem I've recently been posting on, and Hahn's measure
captures that. They behave differently in terms of how the chord
relationships work, and the finer distinctions of the Euclidean metric
capture that.


> I don't appreciate that comment.


Nevertheless, it seems you've been missing a lot.


>> Euclidean draws finer distinctions than Hahn's measure
>> does,
> 

> Clearly -- but are they meaningful distinctions?


See above. Yes, of course they are meaningful. If you want to try to
linearly transform 50/49 to 49/48, be my guest, but don't be surprised
when it works very differently than a linear transformation sending
15/14 to 21/20.

Message: 10489 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 19:14:43

Subject: Re: Hanzos

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:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" 
<perlich@a...> 
>> wrote:
>>> 

>>>>> Strictly speaking, only one of them can be drawn at all.
>>>> 

>>>> So you can't draw a graph?
>>> 

>>> A figure in Tenney space is not a mere graph; it has far more 
>> structure.
>> 

>> OK . . . but can't you think of it as existing in euclidean space 
and 
>> define a new kind of distance in which one is restricted to 
traveling 
>> along the edges?
> 

> Sure, and that may be a good approach. You are now using a Euclidean
> lattice, and defining a corridor-travel distance function on top of
> it. It is certainly familiar to us that as-the-crow-flies and as the
> taxicab drives are both valid and can exist simultaneously.
> 

>>> Did you or did you not know you could put 7-limit note classes 
into 
>> a

>>> symmetric lattice pattern in three dimensional Euclidean space?
>> 

>> Since we were actually doing so, it seems absurd to question 
whether 
>> we knew that we could do so. We simply rarely or never measured 
>> euclidean distances in this structure, since they don't carry a 
>> straightforward musical meaning.
> 

> It never occurred to you to do orthogonal transformations?


We've transformed between the 48 elements of the symmetry group of 
the 7-limit lattice. Are "orthogonal transformations" the ones that 
involve a 90-degree rotation, or something else?


> In any
> case, you should be aware that there *are* differences between
> intervals like 15/14 and 21/20 as compared to 25/24, 36/35 or 49/48,
> and that these again are different from 50/49.


We could refer to these as "meta", "ortho", and "para", respectively.


> They all take two
> 7-limit consonaces to get to, which is why they all turn up in the
> stepwise problem I've recently been posting on, and Hahn's measure
> captures that. They behave differently in terms of how the chord
> relationships work, and the finer distinctions of the Euclidean 
metric
> capture that.


But there's nothing *special* about the euclidean metric here. The 
chord relationships work differently, but what is it about them that 
implies the specific values that the euclidean metric gives?


>> I don't appreciate that comment.
> 

> Nevertheless, it seems you've been missing a lot.


I don't think so.


>>> Euclidean draws finer distinctions than Hahn's measure
>>> does,
>> 

>> Clearly -- but are they meaningful distinctions?
> 

> See above. Yes, of course they are meaningful. If you want to try to
> linearly transform 50/49 to 49/48, be my guest, but don't be 
surprised
> when it works very differently than a linear transformation sending
> 15/14 to 21/20.


You misunderstood my question, and in fact I find this response 
highly condescending.

Message: 10490 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 19:16:43

Subject: Re: Hanzos

From: Paul Erlich

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


> It never occurred to you to do orthogonal transformations? In any
> case, you should be aware that there *are* differences between
> intervals like 15/14 and 21/20 as compared to 25/24, 36/35 or 49/48,
> and that these again are different from 50/49.


I meant to say "ortho", "meta", and "para", respectively.

Message: 10492 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 20:14:13

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


>> It never occurred to you to do orthogonal transformations?
> 

> We've transformed between the 48 elements of the symmetry group of 
> the 7-limit lattice. Are "orthogonal transformations" the ones that 
> involve a 90-degree rotation, or something else?


I meant isometries--Euclidean distance preserving mappings--fixing the
origin, and sending the lattice to itself. This defines the
automorphism group aut(L) of the lattice L. Below is a definition of
othogonal transformation.

Orthogonal Transformation -- from MathWorld * [with cont.] 



>> In any
>> case, you should be aware that there *are* differences between
>> intervals like 15/14 and 21/20 as compared to 25/24, 36/35 or 49/48,
>> and that these again are different from 50/49.
> 

> We could refer to these as "meta", "ortho", and "para", respectively.


I called them shell 2, shell 3 and shell 4 before, which is lattice
terminology.


>> They all take two
>> 7-limit consonaces to get to, which is why they all turn up in the
>> stepwise problem I've recently been posting on, and Hahn's measure
>> captures that. They behave differently in terms of how the chord
>> relationships work, and the finer distinctions of the Euclidean 
> metric
>> capture that.
> 

> But there's nothing *special* about the euclidean metric here. The 
> chord relationships work differently, but what is it about them that 
> implies the specific values that the euclidean metric gives?


The Euclidean metric gives the most refined distinctions. The Hahn
metric replaces spheres with rhombic dodecahedra. This means to figure
out what it means in general, and not just between lattice points,
you'd need to work that out, which sounds like a bit of a pain but is
probably worth doing; but it also means you end up conflating a lot of
things the Euclidean metric classifies as distinct. If you want to
really understand things, Euclidean is often the way to go. 

Here's an example: from a Euclidean point of view, 81/80 and 1029/1024
are symmetrically located with respect to 1, and this is interesting;
we can immediately conclude that 81/80 planar and 1029/1024 planar can
be transformed between. From a Hahn point of view, they are still
symmetrically located, and this is less interesting, because we would
need further analysis to tell us that we can, in fact, transform 81/80
planar to 1029/1024 planar.


>> See above. Yes, of course they are meaningful. If you want to try to
>> linearly transform 50/49 to 49/48, be my guest, but don't be 
> surprised

>> when it works very differently than a linear transformation sending
>> 15/14 to 21/20.
> 

> You misunderstood my question, and in fact I find this response 
> highly condescending.


Touchy touchy. Why should I be the only one on this thread to be
condescended to, if that really is what the above involves? Is it in
fact so obvious that 49/48<-->50/49 is a very different propostion
than 15/14<-->21/20, and if so, why? It's tough if I am going to be
dumped on half the time for obscurity, and the other half for
condescendingly explaining the blindingly obvious.

Message: 10493 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 20:41:56

Subject: Re: Hanzos

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 never occurred to you to do orthogonal transformations?
>> 

>> We've transformed between the 48 elements of the symmetry group 
of 
>> the 7-limit lattice. Are "orthogonal transformations" the ones 
that 
>> involve a 90-degree rotation, or something else?
> 

> I meant isometries--Euclidean distance preserving mappings--fixing 
the
> origin, and sending the lattice to itself.


So are there 48 of these for the 7-limit symmetrical (oct-tet) 
lattice? Did I forget to count mirror inversions?


>>> They all take two
>>> 7-limit consonaces to get to, which is why they all turn up in 
the
>>> stepwise problem I've recently been posting on, and Hahn's 
measure
>>> captures that. They behave differently in terms of how the chord
>>> relationships work, and the finer distinctions of the Euclidean 
>> metric
>>> capture that.
>> 

>> But there's nothing *special* about the euclidean metric here. 
The 
>> chord relationships work differently, but what is it about them 
that 
>> implies the specific values that the euclidean metric gives?
> 

> The Euclidean metric gives the most refined distinctions. The Hahn
> metric replaces spheres with rhombic dodecahedra. This means to 
figure
> out what it means in general, and not just between lattice points,
> you'd need to work that out, which sounds like a bit of a pain but 
is
> probably worth doing; but it also means you end up conflating a lot 
of
> things the Euclidean metric classifies as distinct. If you want to
> really understand things, Euclidean is often the way to go. 


It's perfectly clear that Hahn's metric 'conflates' things. But 
Euclidean seems arbitrary among an infinite class of possibilities 
here.


> Here's an example: from a Euclidean point of view, 81/80 and 
1029/1024
> are symmetrically located with respect to 1, and this is 
interesting;
> we can immediately conclude that 81/80 planar and 1029/1024 planar 
can
> be transformed between. From a Hahn point of view, they are still
> symmetrically located, and this is less interesting, because we 
would
> need further analysis to tell us that we can, in fact, transform 
81/80
> planar to 1029/1024 planar.


Hahn's metric isn't intended to tell us anything about configuration, 
so I don't see this as a flaw.


>>> See above. Yes, of course they are meaningful. If you want to 
try to
>>> linearly transform 50/49 to 49/48, be my guest, but don't be 
>> surprised

>>> when it works very differently than a linear transformation 
sending
>>> 15/14 to 21/20.
>> 

>> You misunderstood my question, and in fact I find this response 
>> highly condescending.
> 

> Touchy touchy. Why should I be the only one on this thread to be
> condescended to, if that really is what the above involves? Is it in
> fact so obvious that 49/48<-->50/49 is a very different propostion
> than 15/14<-->21/20, and if so, why?


It's immediately evident from looking at the lattice.


> It's tough if I am going to be
> dumped on half the time for obscurity, and the other half for
> condescendingly explaining the blindingly obvious.


It's possible to explain the blindingly obvious without being 
condescending. Don't take it personally, but do try to improve.

Message: 10494 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 20:57:53

Subject: Re: Hanzos

From: Paul Erlich

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


> The Hahn
> metric replaces spheres with rhombic dodecahedra. This means to 
figure
> out what it means in general, and not just between lattice points,
> you'd need to work that out, 


Why would anyone ever use it on non-lattice points?

Message: 10495 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 21:07:08

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


> So are there 48 of these for the 7-limit symmetrical (oct-tet) 
> lattice? Did I forget to count mirror inversions?


No; there are 48. In fcc coordinates, there are 8 sign changes and six
permutation, and 8*6 = 48.



> It's perfectly clear that Hahn's metric 'conflates' things. But 
> Euclidean seems arbitrary among an infinite class of possibilities 
> here.


Hardly arbitrary if you accept the above 48 transformations as
interesting, since they are orthogonal. The Euclidean metric falls out
immediately in terms of the invariants of the group, actually.


>> Here's an example: from a Euclidean point of view, 81/80 and 
> 1029/1024

>> are symmetrically located with respect to 1, and this is 
> interesting;

>> we can immediately conclude that 81/80 planar and 1029/1024 planar 
> can

>> be transformed between. From a Hahn point of view, they are still
>> symmetrically located, and this is less interesting, because we 
> would

>> need further analysis to tell us that we can, in fact, transform 
> 81/80

>> planar to 1029/1024 planar.
> 

> Hahn's metric isn't intended to tell us anything about configuration, 
> so I don't see this as a flaw.


I'm pointing out that Euclidean gives us more information.


>> Is it in
>> fact so obvious that 49/48<-->50/49 is a very different propostion
>> than 15/14<-->21/20, and if so, why?
> 

> It's immediately evident from looking at the lattice.


It's immediately obvious from looking at the *Euclidean* lattice.
Aren't you proving my point for me?


>> It's tough if I am going to be
>> dumped on half the time for obscurity, and the other half for
>> condescendingly explaining the blindingly obvious.
> 

> It's possible to explain the blindingly obvious without being 
> condescending. Don't take it personally, but do try to improve.


First I need to learn what is blindingly obvious and what isn't. It
seems I am often wrong in those assessments; in other words, to me
"blindingly obvious" is not itself blindingly obvious. And if I am
supposed to stop being condescending, will other people undertake to
do so also?

Message: 10496 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 21:19:54

Subject: Re: Hanzos

From: Paul Erlich

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 

>> So are there 48 of these for the 7-limit symmetrical (oct-tet) 
>> lattice? Did I forget to count mirror inversions?
> 

> No; there are 48. In fcc coordinates, there are 8 sign changes and 
six
> permutation, and 8*6 = 48.
> 
> 

>> It's perfectly clear that Hahn's metric 'conflates' things. But 
>> Euclidean seems arbitrary among an infinite class of 
possibilities 
>> here.
> 

> Hardly arbitrary if you accept the above 48 transformations as
> interesting, since they are orthogonal.


I don't get it. How does this list of transformations depend on your 
metric?


> The Euclidean metric falls out
> immediately in terms of the invariants of the group, actually.


Can you explain this?


>>> Here's an example: from a Euclidean point of view, 81/80 and 
>> 1029/1024

>>> are symmetrically located with respect to 1, and this is 
>> interesting;

>>> we can immediately conclude that 81/80 planar and 1029/1024 
planar 
>> can

>>> be transformed between. From a Hahn point of view, they are 
still
>>> symmetrically located, and this is less interesting, because we 
>> would

>>> need further analysis to tell us that we can, in fact, 
transform 
>> 81/80

>>> planar to 1029/1024 planar.
>> 

>> Hahn's metric isn't intended to tell us anything about 
configuration, 
>> so I don't see this as a flaw.
> 

> I'm pointing out that Euclidean gives us more information.


But all kinds of other shapes, besides a sphere, would also imply 
these same distinct shells.


>>> Is it in
>>> fact so obvious that 49/48<-->50/49 is a very different 
propostion
>>> than 15/14<-->21/20, and if so, why?
>> 

>> It's immediately evident from looking at the lattice.
> 

> It's immediately obvious from looking at the *Euclidean* lattice.
> Aren't you proving my point for me?


Absolutely not. For instance, for 50:49 or any other "para" dyad with 
Hahn distance 2, there's only one note consonant with both pitches in 
the dyad. So it's straightforward to take the "lattice" (or whatever 
you want to call it) and transform 50:49 to any dyad which has this 
property (such as 9:8, 25:18, 25:16, etc.) while it would take some 
severe gymnastics to transform it to one that doesn't have this 
property, 


>>> It's tough if I am going to be
>>> dumped on half the time for obscurity, and the other half for
>>> condescendingly explaining the blindingly obvious.
>> 

>> It's possible to explain the blindingly obvious without being 
>> condescending. Don't take it personally, but do try to improve.
> 

> First I need to learn what is blindingly obvious and what isn't.


No you don't -- just explain it without being condescending, and even 
if people already know it, they won't be offended.


> And if I am
> supposed to stop being condescending, will other people undertake to
> do so also?


I'm doing my best, and still improving, I think.

Message: 10497 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 21:20:06

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


> Why would anyone ever use it on non-lattice points?


Constructing scales, for starters. In any case, whether you use it or
not it has to be defineable or you don't have a lattice.

Message: 10498 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 21:23:47

Subject: Re: Hanzos

From: Paul Erlich

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

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 

>> Why would anyone ever use it on non-lattice points?
> 

> Constructing scales, for starters.


Which scales involve non-lattice points? Your "shells" only led you 
to choose certain lattice points, and the non-lattice points were 
irrelevant, as far as I could tell . . .


> In any case, whether you use it or
> not it has to be defineable or you don't have a lattice.


I'm happy with whatever it is you do have, I think.

Message: 10499 - Contents - Hide Contents

Date: Wed, 03 Mar 2004 21:44:45

Subject: Re: Hanzos

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:


>> Hardly arbitrary if you accept the above 48 transformations as
>> interesting, since they are orthogonal.
> 

> I don't get it. How does this list of transformations depend on your 
> metric?


It defines a metric.


>> The Euclidean metric falls out
>> immediately in terms of the invariants of the group, actually.
> 

> Can you explain this?


I checked Wikipedia, and it directed me to the Molien series article,
which doesn't exist. Maybe I should write one, which could serve as an
explanation.


>> I'm pointing out that Euclidean gives us more information.
> 

> But all kinds of other shapes, besides a sphere, would also imply 
> these same distinct shells.


They imply distict shells, but not the *same* distinct shells.


>>>> Is it in
>>>> fact so obvious that 49/48<-->50/49 is a very different 
> propostion

>>>> than 15/14<-->21/20, and if so, why?
>>> 

>>> It's immediately evident from looking at the lattice.
>> 

>> It's immediately obvious from looking at the *Euclidean* lattice.
>> Aren't you proving my point for me?
> 

> Absolutely not. For instance, for 50:49 or any other "para" dyad with 
> Hahn distance 2, there's only one note consonant with both pitches in 
> the dyad. So it's straightforward to take the "lattice" (or whatever 
> you want to call it) and transform 50:49 to any dyad which has this 
> property (such as 9:8, 25:18, 25:16, etc.) while it would take some 
> severe gymnastics to transform it to one that doesn't have this 
> property, 


Which is clear from the Euclidean lattice, which says that 9/8, 25/18,
25/16, 50/49 are all located at a distance of 2 from the unison, and
(this is harder; you can use the invariant stuff I mentioned as one
method) that they are all in the same geometric relationship to the
lattice. It is hardly clear using Hahn distance, which says all of the
above are at a distance of 2, but also says 10/9, 16/15, 25/24, 36/35
etc for which the Euclidean distance is sqrt(3) are at a distance of
2, and even 15/14, 21/20 etc for which the Euclidean distance is
sqrt(2) are at a distance of 2. The Hahn distance is making far less
refined distictions, and is not providing the help in sorting these
questions out that the Euclidean distance immediately gives.


>>>> It's tough if I am going to be
>>>> dumped on half the time for obscurity, and the other half for
>>>> condescendingly explaining the blindingly obvious.
>>> 

>>> It's possible to explain the blindingly obvious without being 
>>> condescending. Don't take it personally, but do try to improve.
>> 

>> First I need to learn what is blindingly obvious and what isn't.
> 

> No you don't -- just explain it without being condescending, and even 
> if people already know it, they won't be offended.


Your claim is that I was being condescending. On what basis are you
making it? What, exactly, was condescending about my remark?

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