Yahoo Tuning Groups Ultimate Backup tuning-math Who came up with the term "Harmonic Lattice Diagrams"?

Hi, I'm Alex from Brasil.

we have a research group here, and we want to have a translation of
this term in portuguese. But the word in portuguese for lattice is
hardly ever used, and it refers to a grid of wood that people put in
thei windows... Math books in portuguese do not have a word in
potuguese for the mathematical diagrams that in english are refered
with the apropiation of the word "lattice", they give more a phrase
with the basic description of the concept, but no "term" like "lattice".

But then, I'm interested in knowing who came up with the idea of
applying the mathematical concept of "Lattice" into the J.I. theory...
and also, as I'm not well aquainted with math, if the use of "lattice"
in J.I. theory is a natural and literal use, or if it was another
apropriation, meaning that "harmonic lattice diagrams" are quite
diferent from the mathematical "lattice" concepts.

So, Who is credited for developing the "Harmonic Lattice Diagram", are
you in the group anyway, can you clear it all up for me?

Thanks

Alex

🔗Herman Miller <hmiller@IO.COM>

mentalosmosis wrote:
> Hi, I'm Alex from Brasil.
> > we have a research group here, and we want to have a translation of
> this term in portuguese. But the word in portuguese for lattice is
> hardly ever used, and it refers to a grid of wood that people put in
> thei windows... Math books in portuguese do not have a word in
> potuguese for the mathematical diagrams that in english are refered
> with the apropiation of the word "lattice", they give more a phrase
> with the basic description of the concept, but no "term" like "lattice".
> > But then, I'm interested in knowing who came up with the idea of
> applying the mathematical concept of "Lattice" into the J.I. theory...
> and also, as I'm not well aquainted with math, if the use of "lattice"
> in J.I. theory is a natural and literal use, or if it was another
> apropriation, meaning that "harmonic lattice diagrams" are quite
> diferent from the mathematical "lattice" concepts.
> > So, Who is credited for developing the "Harmonic Lattice Diagram", are
> you in the group anyway, can you clear it all up for me?
> > Thanks
> > Alex

I don't know who first started using the word "lattice" to refer to the tuning diagrams, but the particular meaning of the word "lattice" that is used is from mineralogy. (Mathematicians use the word "lattice" with a different meaning.) So you might borrow a similar term from Portuguese as used in mineralogy to describe crystal structures.

http://pt.wikipedia.org/wiki/Estrutura_cristalina

It appears that what English-speaking mineralogists describe as "Bravais lattices" are "redes de Bravais" in Portuguese. An early diagram of the sort we would refer to as a tonal lattice was called a "Tonnetz" by Hugo Riemann. Since German "Netz" and Portuguese "rede" are both translated with the primary meaning of "net" in English, it seems reasonable to borrow the German word (which predates our usage of "lattice" for these diagrams).

🔗PORRES <mentalosmosis@yahoo.com>

Thanks a lot, I always thought it was connected with the Math, and I still find it preety close in meaning, haven`t quite got the difference yet, what is it?

Yes, "Rede" is "Net" in English! Good Job.

So, Tonnetz = Tonal Net = Rede Tonal...

It sure makes more sense to link to this reference than a translation for the "lattice" as used for the mathematical concept...

I have been actually using the Term "Diagramas Reticulares", which is "Smal net Diagramas", I like it...

I can see at: http://tonalsoft.com/enc/l/lattice.aspx a lot about the development of diagrams for tonal space, but I`m still after who brought up the usage of "Harmonic Lattice Diagram", which is commonly used nowadays.

Thanks A Lot.

Cheers
Alex

Herman Miller <hmiller@IO.COM> wrote: mentalosmosis wrote:
> Hi, I'm Alex from Brasil.
>
> we have a research group here, and we want to have a translation of
> this term in portuguese. But the word in portuguese for lattice is
> hardly ever used, and it refers to a grid of wood that people put in
> thei windows... Math books in portuguese do not have a word in
> potuguese for the mathematical diagrams that in english are refered
> with the apropiation of the word "lattice", they give more a phrase
> with the basic description of the concept, but no "term" like "lattice".
>
> But then, I'm interested in knowing who came up with the idea of
> applying the mathematical concept of "Lattice" into the J.I. theory...
> and also, as I'm not well aquainted with math, if the use of "lattice"
> in J.I. theory is a natural and literal use, or if it was another
> apropriation, meaning that "harmonic lattice diagrams" are quite
> diferent from the mathematical "lattice" concepts.
>
> So, Who is credited for developing the "Harmonic Lattice Diagram", are
> you in the group anyway, can you clear it all up for me?
>
> Thanks
>
> Alex

I don't know who first started using the word "lattice" to refer to the
tuning diagrams, but the particular meaning of the word "lattice" that
is used is from mineralogy. (Mathematicians use the word "lattice" with
a different meaning.) So you might borrow a similar term from Portuguese
as used in mineralogy to describe crystal structures.

http://pt.wikipedia.org/wiki/Estrutura_cristalina

It appears that what English-speaking mineralogists describe as "Bravais
lattices" are "redes de Bravais" in Portuguese. An early diagram of the
sort we would refer to as a tonal lattice was called a "Tonnetz" by Hugo
Riemann. Since German "Netz" and Portuguese "rede" are both translated
with the primary meaning of "net" in English, it seems reasonable to
borrow the German word (which predates our usage of "lattice" for these
diagrams).

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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> I don't know who first started using the word "lattice" to refer to the
> tuning diagrams, but the particular meaning of the word "lattice" that
> is used is from mineralogy. (Mathematicians use the word "lattice" with
> a different meaning.)

Mathematicians use "lattice" in two different senses, one of which
does correspond precisely to note classes. Minerology does not
correspond so precisely, and I think it is a bad idea to use terms
from minerology when trying to apply mathematics unless the terms
clearly fit better, which they do not. When I use "lattice", at any
rate, it *always* means in the sense a mathematician would give it. It
makes more sense simply on objective merits, and it also makes more
sense, I think, to use standard mathematical vocabulary for
mathematics unless there is some reason not to.

One difference is that to a mathemaician a lattice is a group, and
that, it seems to me, is something we want, since note-classes form a
group. Minerology is not concerned with groups aside from symmetry
groups, and is not as good place to go for terminology it seems to me.
I think Paul Erlich has led people in a bad direction here.

On the other hand, aside from the special case of the 7-limit, chords
do not have a group structure. But crystals don't help us here either,
because the problem arises in dimensions higher than three. We can
easily enough express the resulting structures as the union of two
mathematical lattices.

🔗Carl Lumma <ekin@lumma.org>

Hi Alex,

In English, "lattice" is also the wooden thing in windows.
More comments below...

[Herman wrote...]
> (Mathematicians use the word "lattice" with a different
> meaning.)

But a "point lattice" is more like it...

http://mathworld.wolfram.com/PointLattice.html

...also called a "grid" or "mesh", according to this.

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "mentalosmosis"
<mentalosmosis@...> wrote:

> we have a research group here, and we want to have a translation of
> this term in portuguese. But the word in portuguese for lattice is
> hardly ever used, and it refers to a grid of wood that people put in
> thei windows... Math books in portuguese do not have a word in
> potuguese for the mathematical diagrams that in english are refered
> with the apropiation of the word "lattice", they give more a phrase
> with the basic description of the concept, but no "term" like "lattice".

I doubt very, very much that Portuguese does not have a word for
mathematical lattice in both senses of the word. Most likely, it has
two instead of the one that English has, like French does. But it
could, for instance, be "rêde", which I think is "net".

I wonder if there is some good on-line discussion group, akin to
sci.math, with Portuguese being the language? If so, you could ask
there. If not, email someone in the math department of a Portuguese
speaking university. Or I could try to find Paulo Ney De Souza who was
a Berkeley grad student when I was there, I suppose. Lots of people
must know the answer.

> But then, I'm interested in knowing who came up with the idea of
> applying the mathematical concept of "Lattice" into the J.I. theory...

The germ of the idea came from Euler, and was further developed by
Hugo Riemann and Shohe Tanaka. I don't know of anyone before me who
actually thought of it as a mathematical lattice and made use of the
theory of such things.

🔗Graham Breed <gbreed@gmail.com>

PORRES wrote:
> Thanks a lot, I always thought it was connected with the Math, and I still find it preety close in meaning, haven`t quite got the difference yet, what is it? > > Yes, "Rede" is "Net" in English! Good Job.
> > So, Tonnetz = Tonal Net = Rede Tonal...
> > It sure makes more sense to link to this reference than a translation for the "lattice" as used for the mathematical concept... > > I have been actually using the Term "Diagramas Reticulares", which is "Smal net Diagramas", I like it...
> > I can see at: http://tonalsoft.com/enc/l/lattice.aspx a lot about the development of diagrams for tonal space, but I`m still after who brought up the usage of "Harmonic Lattice Diagram", which is commonly used nowadays.

Fokker used the term "rï¿½seau harmonique" which sounds like "harmonic lattice" to me and is at least in a Romance language. My dictionary translates "rï¿½seau" as "network" and it sounds related to "rede".

The reference is from the Huygens-Fokker site. I don't have the URL to hand, but here's how it declares itself:

A.D. Fokker: Les mathï¿½matiques et la musique. Trois confï¿½rences

Archives du Musï¿½e Teyler vol. 10, Martinus Nijhoff, Den Haag, 1947.

Graham

🔗PORRES <mentalosmosis@yahoo.com>

Hi Carl, thanks. Now, checking the links from math world and wikipedia... I`m assuming "lattice" is a synonym for grid, and that in the math use it is restricted to the bi-dimentional space, and that for the mineralogy you have a 3-dimentional space (5-limit and 7-limit respectively for exemple), that is what I realized as the main difference.

But then, I haven`t seen any 4-dimentional diagram with the word "lattice" apart from the "harmonic lattices" from the tuning folks... I see that only in the higher than 7 limit diagrams from the tuning theory they are called as lattices, because of the apropriation of the term by Gene, correct?

Thanks follks...

Graham Breed <gbreed@gmail.com> wrote: PORRES wrote:
> Thanks a lot, I always thought it was connected with the Math, and I still find it preety close in meaning, haven`t quite got the difference yet, what is it?
>
> Yes, "Rede" is "Net" in English! Good Job.
>
> So, Tonnetz = Tonal Net = Rede Tonal...
>
> It sure makes more sense to link to this reference than a translation for the "lattice" as used for the mathematical concept...
>
> I have been actually using the Term "Diagramas Reticulares", which is "Smal net Diagramas", I like it...
>
> I can see at: http://tonalsoft.com/enc/l/lattice.aspx a lot about the development of diagrams for tonal space, but I`m still after who brought up the usage of "Harmonic Lattice Diagram", which is commonly used nowadays.

Fokker used the term "rï¿½seau harmonique" which sounds like "harmonic
lattice" to me and is at least in a Romance language. My dictionary
translates "rï¿½seau" as "network" and it sounds related to "rede".

The reference is from the Huygens-Fokker site. I don't have the URL to
hand, but here's how it declares itself:

A.D. Fokker: Les mathï¿½matiques et la musique. Trois confï¿½rences

Archives du Musï¿½e Teyler vol. 10, Martinus Nijhoff, Den Haag, 1947.

Graham

Fokker used the term "rï¿½seau harmonique" which sounds like "harmonic
lattice" to me and is at least in a Romance language. My dictionary
translates "rï¿½seau" as "network" and it sounds related to "rede".

The reference is from the Huygens-Fokker site. I don't have the URL to
hand, but here's how it declares itself:

A.D. Fokker: Les mathï¿½matiques et la musique. Trois confï¿½rences

Archives du Musï¿½e Teyler vol. 10, Martinus Nijhoff, Den Haag, 1947.

Graham

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🔗PORRES <mentalosmosis@yahoo.com>

Gene Ward Smith Tells me:

> I don't know of anyone before me who
> actually thought of it as a mathematical
> lattice and made use of the theory of
> such things.

Now, that`s what I`m really interested here... I first inquired about who came up with the "harmonic lattice diagram" because I was considering it as an apropiation of the term "lattice" into the tuning theory... and I wanted to know who proclaimed the appropiation.

So Gene, can you tell me more about how you did it? Like if you`d published a paper on it? And why and how it became the standard concept between tuning experts?

Cheers
Thanks

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🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

>Hi Carl, thanks. Now, checking the links from math world and wikipedia...
>I'm assuming "lattice" is a synonym for grid, and that in the math use it
>is restricted to the bi-dimentional space, and that for the mineralogy you
>have a 3-dimentional space (5-limit and 7-limit respectively for exemple),
>that is what I realized as the main difference.
>
>But then, I haven`t seen any 4-dimentional diagram with the word
>"lattice" apart from the "harmonic lattices" from the tuning folks... I
>see that only in the higher than 7 limit diagrams from the tuning theory
>they are called as lattices, because of the apropriation of the term by
>Gene, correct?

Hi Alex- Here are general definitions of lattice and grid in
English

http://www.dict.org/bin/Dict?Form=Dict2&Database=*&Query=lattice
http://www.dict.org/bin/Dict?Form=Dict2&Database=*&Query=grid

I don't see that in the math use it is restricted to 2 dimensions.

In tuning theory, the most common lattices are 2-dimensional,
5-limit lattices, but lattices of 3, 4, 5, and higher dimensions
are sometimes seen (as in the projections of Erv Wilson, for
example).

Gene is apparently first to use things like wedge products and
LLL reduction in tuning theory, but Fokker used determinants to
find the size of periodicity blocks (I believe).

-Carl

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, PORRES <mentalosmosis@...> wrote:
>
> Gene Ward Smith Tells me:
>
> > I don't know of anyone before me who
> > actually thought of it as a mathematical
> > lattice and made use of the theory of
> > such things.
>
> Now, that`s what I`m really interested here... I first inquired
about who came up with the "harmonic lattice diagram" because I was
considering it as an apropiation of the term "lattice" into the tuning
theory... and I wanted to know who proclaimed the appropiation.

Apparently the one who did this was Fokker, not me.

> So Gene, can you tell me more about how you did it? Like if you`d
published a paper on it? And why and how it became the standard
concept between tuning experts?

I tried to publish this paper in the mid-eighties:

/tuning-math/files/DJVU/GWS1983.djvu

but Balzano turned it down for being incomprehensibly mathematical.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Graham Breed <gbreed@gmail.com>

Carl Lumma wrote:
>>The germ of the idea came from Euler, and was further developed by
>>Hugo Riemann and Shohe Tanaka. I don't know of anyone before me who
>>actually thought of it as a mathematical lattice and made use of the
>>theory of such things.
> > > Who does Fokker cite? The one paper of his I found online (didn't
> have citations).

From the French paper,

* Leonhard Euler, Tentamen novae theoviae musicae, Petropoli, 1739, Opera Omnia, series III, volume I, p. 252.
* Giuseppe Tartini, Trattato di Musica, Padova, 1754.
* Mart. J. Lï¿½rsen, Grondslagen van de muziektheorie, Den Haag, 1944.
* Mart. J. Lï¿½rsen, Modi antiche, musiche nuove, Haarlem, 1947.
* Ernst Krenek, ï¿½ber neue Musik, Wien, 1937.
* R.H.M. Bosanquet, The theory of the division of the octave, and the practical treatment of the musical systems thus obtained, Proc. Roy. Soc. London XXIII, 390, 1875.

Graham

🔗Carl Lumma <ekin@lumma.org>

>>>The germ of the idea came from Euler, and was further developed by
>>>Hugo Riemann and Shohe Tanaka. I don't know of anyone before me who
>>>actually thought of it as a mathematical lattice and made use of the
>>>theory of such things.
>>
>> Who does Fokker cite? The one paper of his I found online (didn't
>> have citations).
>
> From the French paper,
>
>* Leonhard Euler, Tentamen novae theoviae musicae, Petropoli, 1739,
>Opera Omnia, series III, volume I, p. 252.
>* Giuseppe Tartini, Trattato di Musica, Padova, 1754.
>* Mart. J. Lürsen, Grondslagen van de muziektheorie, Den Haag, 1944.
>* Mart. J. Lürsen, Modi antiche, musiche nuove, Haarlem, 1947.
>* Ernst Krenek, Über neue Musik, Wien, 1937.
>* R.H.M. Bosanquet, The theory of the division of the octave, and the
>practical treatment of the musical systems thus obtained, Proc. Roy.
>Soc. London XXIII, 390, 1875.

I'm not aware of any latticework in Bosanquet, but he did
investigate mappings to chains of fifths out to 612.

-Carl

🔗PORRES <mentalosmosis@yahoo.com>

Wow, so, as I first said, I was using the term "Diagramas reticulares", which is the same as reticulado, but I found a page both in english and portugues about point lattices:

http://www.gregosetroianos.mat.br/latticeEnglish.asp

http://www.gregosetroianos.mat.br/lattice.asp

And as I said you can see that Point Lattices is refered as "Integer Coordinate Points", which is more a description of the concept than a term.

Anyway, "RetÃculo" is like a "smal net", so there you go again the word "net" comes along one more time...

But I sse as a main carachteristic of this diagrams the idea of having angle vectors representing the limit dimentions. In other words: a multi-dimentional diagram, and I don`t know if multi-dimentional diagrams have a clearer term than "net" for it in portuguese. Only saying "net" doesnt mention about the structure based on angle vectors... but anyway, I`ll reach my portuguese speaking fellows to work on it, I think I`m done on the subject, thanks a lot for clearing it all up to me... it`s been a great help

cheers everybody!!!

Alexandre Torres Porres
Post Graduate Student in Composition at UNICAMP (State University of Campinas/Sao Paulo - Brasil)

Gene Ward Smith <genewardsmith@coolgoose.com> wrote: --- In tuning-math@yahoogroups.com, "mentalosmosis"
<mentalosmosis@...> wrote:

According to Paulo de Souza, the word "reticulado" in Portuguese, like
the word "lattice" in English, is used in two distinct mathematical
senses, meaning both a kind of partial order and a kind of discrete
group. Hence, the word you want is "reticulado".

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🔗monz <monz@tonalsoft.com>

Hi Alexandre,

Many years ago, i used an internet translator to make
translations of my webpage about my lattice diagrams.

Some French and German speakers helped me to correct
those, but i never had an opportunity to fix those in
Portuguese, Spanish, or Italian.

Here's the original:

English
http://tonalsoft.com/monzo/lattices/lattices.htm

(with links to French and German versions)

Here are the internet translations ... all probably quite bad:

Portuguese
http://tonalsoft.com/monzo/lattices/lat-port.htm

Spanish
http://tonalsoft.com/monzo/lattices/lat-span.htm

Italian
http://tonalsoft.com/monzo/lattices/lat-ital.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗PORRES <mentalosmosis@yahoo.com>

Hi Monzo, I remember I tried to translate it for you once a few years ago... so, this seems reasonably fine, I have to decide on the math issues first, then I can give you a feedback on the portuguese translation...

cheers

alex

monz <monz@tonalsoft.com> wrote: Hi Alexandre,

Many years ago, i used an internet translator to make
translations of my webpage about my lattice diagrams.

Some French and German speakers helped me to correct
those, but i never had an opportunity to fix those in
Portuguese, Spanish, or Italian.

Here's the original:

English
http://tonalsoft.com/monzo/lattices/lattices.htm

(with links to French and German versions)

Here are the internet translations ... all probably quite bad:

Portuguese
http://tonalsoft.com/monzo/lattices/lat-port.htm

Spanish
http://tonalsoft.com/monzo/lattices/lat-span.htm

Italian
http://tonalsoft.com/monzo/lattices/lat-ital.htm

-monz
http://tonalsoft.com
Tonescape microtonal music software

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🔗monz <monz@tonalsoft.com>

🔗mentalosmosis <mentalosmosis@yahoo.com>

Gene, I cant open this .djvu file, what do i need to do?
cheers

I tried to publish this paper in the mid-eighties:

/tuning-math/files/DJVU/GWS1983.djvu

but Balzano turned it down for being incomprehensibly mathematical.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, PORRES <mentalosmosis@> wrote:
> >
> > Gene Ward Smith Tells me:
> >
> > > I don't know of anyone before me who
> > > actually thought of it as a mathematical
> > > lattice and made use of the theory of
> > > such things.
> >
> > Now, that`s what I`m really interested here... I first inquired
> about who came up with the "harmonic lattice diagram" because I was
> considering it as an apropiation of the term "lattice" into the tuning
> theory... and I wanted to know who proclaimed the appropiation.
>
> Apparently the one who did this was Fokker, not me.
>
> > So Gene, can you tell me more about how you did it? Like if you`d
> published a paper on it? And why and how it became the standard
> concept between tuning experts?
>
> I tried to publish this paper in the mid-eighties:
>
> /tuning-math/files/DJVU/GWS1983.djvu
>
> but Balzano turned it down for being incomprehensibly mathematical.
>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗hstraub64 <hstraub64@telesonique.net>

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Many years ago, i used an internet translator to make
> translations of my webpage about my lattice diagrams.
>
> Some French and German speakers helped me to correct
> those, but i never had an opportunity to fix those in
> Portuguese, Spanish, or Italian.
>
> Here's the original:
>
> English
> http://tonalsoft.com/monzo/lattices/lattices.htm
>
> (with links to French and German versions)
>
>
> Here are the internet translations ... all probably quite bad:
>
> Portuguese
> http://tonalsoft.com/monzo/lattices/lat-port.htm
>
> Spanish
> http://tonalsoft.com/monzo/lattices/lat-span.htm
>
> Italian
> http://tonalsoft.com/monzo/lattices/lat-ital.htm
>

I just looked at the italian version. Looks a little strange, indeed -
it seems that some of the word endings were cut off.
I might try to correct it , but my italian is not so good...
--
Hans Straub

🔗monz <monz@tonalsoft.com>

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Italian
> > http://tonalsoft.com/monzo/lattices/lat-ital.htm
> >
>
> I just looked at the italian version. Looks a little strange,
> indeed - it seems that some of the word endings were cut off.
> I might try to correct it , but my italian is not so good...

Thanks, Hans. I would appreciate any improvement anyone can
make to these.

Actually i haven't touched this page for many many years.
I should try running it thru an internet translator again,
to see if i get better results now. I'll try to get to that
on Tuesday.

Leonardo Perretti had done some really excellent nice
translation jobs for me a few years ago:

http://tonalsoft.com/monzo/aron/toscanello/aron_toscanello.htm

http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.htm

(hmm ... on this page i made a lattice with a linear plot
of 2/7-comma meantone, exactly as i've been describing the
last few days on the main tuning list ... this is nice
because it shows the whole tuning unfolding as Zarlino
describes it.)

... i wonder if Leonardo is around and would take a stab at this?

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗yahya_melb <yahya@melbpc.org.au>

Monz,

Indeed, even the title "Harmonic Lattice Diagrams" is translated
into Spanish as "Harmonic Diagrams of the Sieve", which I'm sure is
pretty wide of the mark ... Unfortunately, as with Hans' Italian, my
Spanish, particularly w.r.t. musical and mathematical terminology,
isn't good enough to vet the translation thoroughly. You really do
need someone fluent *and* familiar with the subject-matter *in both
tongues* to make a decent fist of translation. If you wish, I could
go thru the page correcting any obvious grammatical problems.

FWIW, I use the ImTranslator free web translation tool for a first
cut. I've found it superior to all other free tools I've tried or
used in the last few years.

Regards,
Yahya

--- In tuning-math@yahoogroups.com, "hstraub64" wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Many years ago, i used an internet translator to make
> > translations of my webpage about my lattice diagrams.
> >
> > Some French and German speakers helped me to correct
> > those, but i never had an opportunity to fix those in
> > Portuguese, Spanish, or Italian.
> >
> > Here's the original:
> >
> > English
> > http://tonalsoft.com/monzo/lattices/lattices.htm
> >
> > (with links to French and German versions)
> >
> >
> > Here are the internet translations ... all probably quite bad:
> >
> > Portuguese
> > http://tonalsoft.com/monzo/lattices/lat-port.htm
> >
> > Spanish
> > http://tonalsoft.com/monzo/lattices/lat-span.htm
> >
> > Italian
> > http://tonalsoft.com/monzo/lattices/lat-ital.htm
> >
>
> I just looked at the italian version. Looks a little strange,
> indeed -
> it seems that some of the word endings were cut off.
> I might try to correct it , but my italian is not so good...
> --
> Hans Straub