Yahoo Tuning Groups Ultimate Backup tuning-math Re: [tuning-math] Re: Paul's lattice math and my diagrams

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 26, 2001 2:58 PM
> Subject: [tuning-math] Re: Paul's lattice math and my diagrams
>
>
> But alas, you are not getting the infinite strips I refer to above.

And exactly how am I supposed to portray "infinite strips" on a
computer screen, other than leaving the "infinite" part to the
reader's imagination?!

> > The 5-limit periodicity blocks are bounded by 2 unison-vectors,
> > one of which is tempered out (the 81:80 syntonic comma) and
> > one of which isn't -- and that one is the one which appears
> > at the end of each meantone chain.
>
> Right -- but since the 81:80 _is_ tempered out, your lattices should
> be proceeding infinitely in the direction of the 81:80.

Well... this is the part of your post that I was least sure about.
My lattices obviously proceed infinitely in the direction of
the interval that's not tempered out. The syntonic comma is
the interval that sets the boundaries on the *other* two sides.
But *isn't* that how meantones work? I seem to be missing
something in this...

> > >
> > > Now let's go back to "any vector in the lattice". This vector,
> > > added to itself over and over, will land one back at a pitch
> > > in the same equivalence class as the pitch one started with,
> > > after N iterations (and more often if the vector represents
> > > a generic interval whose cardinality is not relatively prime
> > > with N). In general, the vector will have a length that is
> > > some fraction M/N of the width of one strip/layer/hyperlayer,
> > > measured in the direction of this vector (NOT in the direction
> > > of the chromatic unison vector). M must be an integer, since
> > > after N iterations, you're guaranteed to be in a point in the
> > > same equivalence class as where you started, hence you must be
> > > an exact integer M strips/layers/hyperlayers away. As a
> > > special example, the generator has length 1/N of the width
> > > of one strip/layer/hyperlayer, measured in the direction of
> > > the generator.
> >
> >
> > This is precisely what was in my mind when I came up with
> > these meantone lattices.
>
> Really? I don't see the strips, and I don't see how the generator
> could be said to have any property resembling this in your lattices.

I only show part of one periodicity-block. To do it properly,
I should have a nice big grid representing the infinite lattice,
then simply draw the unison-vectors as boundaries to the various
tiled periodicity-blocks. Then you'd see the strips, each one
at the same angle as the meantone chain itself, and each one as
wide as a syntonic comma. The actual lateral width of each strip
would vary as the meantone chain's axis angle varies in relation
to the fixed vector of the syntonic comma.

> > > Anyhow, each occurence of the vector will cross either
> > > floor(M/N) or ceiling(M/N) boundaries between
> > > strips/layers/hyperlayers. Now, each time one crosses
> > > one of these boundaries in a given direction, one shifts
> > > by a chromatic unison vector. Hence each specific occurence
> > > of the generic interval in question will be shifted by
> > > either floor(M/N) or ceiling(M/N) chromatic unison vectors.
> > > Thus there will be only two specific sizes of the interval
> > > in question, and their difference will be exactly 1 of the
> > > chromatic unison vector. And since the vectors in the chain
> > > are equally spaced and the boundaries are equally spaced,
> > > the pattern of these two sizes will be an MOS pattern.
> >
> > Isn't this exactly how my pseudo-code works? (posted here:
> > </tuning-math/messages/2069?expand=1>).
>
> Monz, I don't see anything in your pseudocode that would give you any
> of this -- have you actually managed to produce MOSs with it?

Um... well... I never actually checked that anything I my code
produced was MOS or had any other scalar property. I simply
compared my periodicity-block coordinates (and those of all
their contents) with the ones you, Gene, and Fokker produced
using the same unison-vectors and kept working on the code
until it produced the same results.

-monz

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🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, December 26, 2001 2:58 PM
> > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> >
> >
> > But alas, you are not getting the infinite strips I refer to
above.
>
>
> And exactly how am I supposed to portray "infinite strips" on a
> computer screen, other than leaving the "infinite" part to the
> reader's imagination?!

Well, at least have two or three repetition of the pitches along that
direction, to *suggest* the infinitude . . . currently, you have only
one instance of most of the pitches, and two for a few others.

> > > The 5-limit periodicity blocks are bounded by 2 unison-vectors,
> > > one of which is tempered out (the 81:80 syntonic comma) and
> > > one of which isn't -- and that one is the one which appears
> > > at the end of each meantone chain.
> >
> > Right -- but since the 81:80 _is_ tempered out, your lattices
should
> > be proceeding infinitely in the direction of the 81:80.
>
>
> Well... this is the part of your post that I was least sure about.
> My lattices obviously proceed infinitely in the direction of
> the interval that's not tempered out.

If you're latticing a periodicity block, they shouldn't -- you should
hit a "wolf" at some point in that direction.

> The syntonic comma is
> the interval that sets the boundaries on the *other* two sides.
> But *isn't* that how meantones work?

No -- there are no boundaries in the direction of 81:80. These
*other* two sides that you refer to -- they meet one another when you
roll the lattice into a cylinder.

> > > > Now let's go back to "any vector in the lattice". This vector,
> > > > added to itself over and over, will land one back at a pitch
> > > > in the same equivalence class as the pitch one started with,
> > > > after N iterations (and more often if the vector represents
> > > > a generic interval whose cardinality is not relatively prime
> > > > with N). In general, the vector will have a length that is
> > > > some fraction M/N of the width of one strip/layer/hyperlayer,
> > > > measured in the direction of this vector (NOT in the direction
> > > > of the chromatic unison vector). M must be an integer, since
> > > > after N iterations, you're guaranteed to be in a point in the
> > > > same equivalence class as where you started, hence you must be
> > > > an exact integer M strips/layers/hyperlayers away. As a
> > > > special example, the generator has length 1/N of the width
> > > > of one strip/layer/hyperlayer, measured in the direction of
> > > > the generator.
> > >
> > >
> > > This is precisely what was in my mind when I came up with
> > > these meantone lattices.
> >
> > Really? I don't see the strips, and I don't see how the generator
> > could be said to have any property resembling this in your
lattices.
>
>
> I only show part of one periodicity-block. To do it properly,
> I should have a nice big grid representing the infinite lattice,
> then simply draw the unison-vectors as boundaries to the various
> tiled periodicity-blocks. Then you'd see the strips, each one
> at the same angle as the meantone chain itself, and each one as
> wide as a syntonic comma.

That's not correct. The strips I'm referring to would be as wide as
the other comma -- the one that's not tempered out.

> > > > Anyhow, each occurence of the vector will cross either
> > > > floor(M/N) or ceiling(M/N) boundaries between
> > > > strips/layers/hyperlayers. Now, each time one crosses
> > > > one of these boundaries in a given direction, one shifts
> > > > by a chromatic unison vector. Hence each specific occurence
> > > > of the generic interval in question will be shifted by
> > > > either floor(M/N) or ceiling(M/N) chromatic unison vectors.
> > > > Thus there will be only two specific sizes of the interval
> > > > in question, and their difference will be exactly 1 of the
> > > > chromatic unison vector. And since the vectors in the chain
> > > > are equally spaced and the boundaries are equally spaced,
> > > > the pattern of these two sizes will be an MOS pattern.
> > >
> > > Isn't this exactly how my pseudo-code works? (posted here:
> > > </tuning-math/messages/2069?
expand=1>).
> >
> > Monz, I don't see anything in your pseudocode that would give you
any
> > of this -- have you actually managed to produce MOSs with it?
>
>
> Um... well... I never actually checked that anything I my code
> produced was MOS or had any other scalar property. I simply
> compared my periodicity-block coordinates (and those of all
> their contents) with the ones you, Gene, and Fokker produced
> using the same unison-vectors and kept working on the code
> until it produced the same results.

Right, but you quoted a paragraph of mine and then asked "Isn't this
exactly how my pseudo-code works?" -- so what was it in that
paragraph that you think your pseudocode does? That paragraph was
meant to demonstrate the truth of my hypothesis about the
relationship between PBs and MOS, which appears to be original to me
and not something Fokker ever mentioned.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 26, 2001 3:26 PM
> Subject: [tuning-math] Re: Paul's lattice math and my diagrams
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Wednesday, December 26, 2001 2:58 PM
> > > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> > >
> > >
> > > But alas, you are not getting the infinite strips I refer to
> above.
> >
> >
> > And exactly how am I supposed to portray "infinite strips" on a
> > computer screen, other than leaving the "infinite" part to the
> > reader's imagination?!
>
> Well, at least have two or three repetition of the pitches along that
> direction, to *suggest* the infinitude . . . currently, you have only
> one instance of most of the pitches, and two for a few others.
>
> > > > The 5-limit periodicity blocks are bounded by 2 unison-vectors,
> > > > one of which is tempered out (the 81:80 syntonic comma) and
> > > > one of which isn't -- and that one is the one which appears
> > > > at the end of each meantone chain.
> > >
> > > Right -- but since the 81:80 _is_ tempered out, your lattices
> should
> > > be proceeding infinitely in the direction of the 81:80.
> >
> >
> > Well... this is the part of your post that I was least sure about.
> > My lattices obviously proceed infinitely in the direction of
> > the interval that's not tempered out.
>
> If you're latticing a periodicity block, they shouldn't -- you should
> hit a "wolf" at some point in that direction.

Right, of course... they continue infinitely in the direction
of the meantone chain if you don't close the chain somewhere.
I *am* interested in closing it so that I get a periodicity-block.

I'm almost finished with a lattice for the UV: (4 -1),(19 9)
periodicity-block which contains a 55-tone 1/6-comma chain.
I'll post it as soon as it's done.

> > The syntonic comma is
> > the interval that sets the boundaries on the *other* two sides.
> > But *isn't* that how meantones work?
>
> No -- there are no boundaries in the direction of 81:80. These
> *other* two sides that you refer to -- they meet one another when you
> roll the lattice into a cylinder.
>
> > > > > Now let's go back to "any vector in the lattice". This vector,
> > > > > added to itself over and over, will land one back at a pitch
> > > > > in the same equivalence class as the pitch one started with,
> > > > > after N iterations (and more often if the vector represents
> > > > > a generic interval whose cardinality is not relatively prime
> > > > > with N). In general, the vector will have a length that is
> > > > > some fraction M/N of the width of one strip/layer/hyperlayer,
> > > > > measured in the direction of this vector (NOT in the direction
> > > > > of the chromatic unison vector). M must be an integer, since
> > > > > after N iterations, you're guaranteed to be in a point in the
> > > > > same equivalence class as where you started, hence you must be
> > > > > an exact integer M strips/layers/hyperlayers away. As a
> > > > > special example, the generator has length 1/N of the width
> > > > > of one strip/layer/hyperlayer, measured in the direction of
> > > > > the generator.
> > > >
> > > >
> > > > This is precisely what was in my mind when I came up with
> > > > these meantone lattices.
> > >
> > > Really? I don't see the strips, and I don't see how the generator
> > > could be said to have any property resembling this in your
> > > lattices.
> >
> >
> > I only show part of one periodicity-block. To do it properly,
> > I should have a nice big grid representing the infinite lattice,
> > then simply draw the unison-vectors as boundaries to the various
> > tiled periodicity-blocks. Then you'd see the strips, each one
> > at the same angle as the meantone chain itself, and each one as
> > wide as a syntonic comma.
>
> That's not correct. The strips I'm referring to would be as wide as
> the other comma -- the one that's not tempered out.

That's how I understood it when I first read your post.
So you mean that on your ideal lattice you'd have long
(or I probably should say wide) strips of cylinders, right?

> > > > > Anyhow, each occurence of the vector will cross either
> > > > > floor(M/N) or ceiling(M/N) boundaries between
> > > > > strips/layers/hyperlayers. Now, each time one crosses
> > > > > one of these boundaries in a given direction, one shifts
> > > > > by a chromatic unison vector. Hence each specific occurence
> > > > > of the generic interval in question will be shifted by
> > > > > either floor(M/N) or ceiling(M/N) chromatic unison vectors.
> > > > > Thus there will be only two specific sizes of the interval
> > > > > in question, and their difference will be exactly 1 of the
> > > > > chromatic unison vector. And since the vectors in the chain
> > > > > are equally spaced and the boundaries are equally spaced,
> > > > > the pattern of these two sizes will be an MOS pattern.
> > > >
> > > > Isn't this exactly how my pseudo-code works? posted here:
/tuning-math/messages/2069?expand=1
> > >
> > > Monz, I don't see anything in your pseudocode that would give
> > > you any of this -- have you actually managed to produce MOSs
> > > with it?
> >
> >
> > Um... well... I never actually checked that anything I my code
> > produced was MOS or had any other scalar property. I simply
> > compared my periodicity-block coordinates (and those of all
> > their contents) with the ones you, Gene, and Fokker produced
> > using the same unison-vectors and kept working on the code
> > until it produced the same results.
>
> Right, but you quoted a paragraph of mine and then asked "Isn't this
> exactly how my pseudo-code works?" -- so what was it in that
> paragraph that you think your pseudocode does?

My code transforms the prime-axes to a right-angled unit cube,
transforms the primary lattice metrics along the 3 and 5 axes
to the unit metrics along those new axes, then iterates thru
the unit cube to fill it with coordinates x,y, always bouncing
to the other side (i.e., modulo) when it goes beyond the
floor or ceiling values (i.e., 1/2 > x,y > -1/2), then
transforms back to the original lattice coordinates.

This is exactly how I understood your paragraph. Please correct.

> That paragraph was meant to demonstrate the truth of my
> hypothesis about the relationship between PBs and MOS,
> which appears to be original to me and not something
> Fokker ever mentioned.

OK. I really wasn't even concerning myself directly with MOS,
just trying to figure out how to have Excel calculate not only
the boundaries of the periodicity-block, but all of the
coordinates within it as well.

(I was quite impressed with myself for getting the job done
on my own, even tho it's still quite crude.)

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, December 26, 2001 3:26 PM
> > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> >
> >
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > From: paulerlich <paul@s...>
> > > > To: <tuning-math@y...>
> > > > Sent: Wednesday, December 26, 2001 2:58 PM
> > > > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> > > > Right -- but since the 81:80 _is_ tempered out, your lattices
> > should
> > > > be proceeding infinitely in the direction of the 81:80.
> > >
> > >
> > > Well... this is the part of your post that I was least sure
about.
> > > My lattices obviously proceed infinitely in the direction of
> > > the interval that's not tempered out.
> >
> > If you're latticing a periodicity block, they shouldn't -- you
should
> > hit a "wolf" at some point in that direction.
>
>
> Right, of course... they continue infinitely in the direction
> of the meantone chain if you don't close the chain somewhere.
> I *am* interested in closing it so that I get a periodicity-block.

No, sir, I'm afraid you're completely misunderstanding me. If 81:80
is tempered out, then you can keep moving by as many 81:80s as you
want in the lattice, and you're still within the strip! In terms of
the cylinder, all you're doing is making a full circle around the
cylinder in the same direction over and over again.

> So you mean that on your ideal lattice you'd have long
> (or I probably should say wide) strips of cylinders, right?

Wide strips, _or_ a single cylinder.

> > > > > > Anyhow, each occurence of the vector will cross either
> > > > > > floor(M/N) or ceiling(M/N) boundaries between
> > > > > > strips/layers/hyperlayers. Now, each time one crosses
> > > > > > one of these boundaries in a given direction, one shifts
> > > > > > by a chromatic unison vector. Hence each specific
occurence
> > > > > > of the generic interval in question will be shifted by
> > > > > > either floor(M/N) or ceiling(M/N) chromatic unison
vectors.
> > > > > > Thus there will be only two specific sizes of the interval
> > > > > > in question, and their difference will be exactly 1 of the
> > > > > > chromatic unison vector. And since the vectors in the
chain
> > > > > > are equally spaced and the boundaries are equally spaced,
> > > > > > the pattern of these two sizes will be an MOS pattern.
> > > > >
> > > > > Isn't this exactly how my pseudo-code works? posted here:
> /tuning-math/messages/2069?expand=1
> > > >
> > > > Monz, I don't see anything in your pseudocode that would give
> > > > you any of this -- have you actually managed to produce MOSs
> > > > with it?
> > >
> > >
> > > Um... well... I never actually checked that anything I my code
> > > produced was MOS or had any other scalar property. I simply
> > > compared my periodicity-block coordinates (and those of all
> > > their contents) with the ones you, Gene, and Fokker produced
> > > using the same unison-vectors and kept working on the code
> > > until it produced the same results.
> >
> > Right, but you quoted a paragraph of mine and then asked "Isn't
this
> > exactly how my pseudo-code works?" -- so what was it in that
> > paragraph that you think your pseudocode does?
>
>
> My code transforms the prime-axes to a right-angled unit cube,
> transforms the primary lattice metrics along the 3 and 5 axes
> to the unit metrics along those new axes, then iterates thru
> the unit cube to fill it with coordinates x,y, always bouncing
> to the other side (i.e., modulo) when it goes beyond the
> floor or ceiling values (i.e., 1/2 > x,y > -1/2), then
> transforms back to the original lattice coordinates.
>
> This is exactly how I understood your paragraph. Please correct.

You appear to have the correct picture of how to create periodicity
blocks. I was saying much more than that, but if that's all you were
looking for, then you're fine.

> > That paragraph was meant to demonstrate the truth of my
> > hypothesis about the relationship between PBs and MOS,
> > which appears to be original to me and not something
> > Fokker ever mentioned.
>
>
> OK. I really wasn't even concerning myself directly with MOS,
> just trying to figure out how to have Excel calculate not only
> the boundaries of the periodicity-block, but all of the
> coordinates within it as well.
>
> (I was quite impressed with myself for getting the job done
> on my own, even tho it's still quite crude.)

If there's still any confusion, part 3 of the Gentle Introduction
should clear it up.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 26, 2001 9:29 PM
> Subject: [tuning-math] Re: Paul's lattice math and my diagrams
>

> > [me, monz]
> > Right, of course... they continue infinitely in the direction
> > of the meantone chain if you don't close the chain somewhere.
> > I *am* interested in closing it so that I get a periodicity-block.
>
> No, sir, I'm afraid you're completely misunderstanding me. If 81:80
> is tempered out, then you can keep moving by as many 81:80s as you
> want in the lattice, and you're still within the strip! In terms of
> the cylinder, all you're doing is making a full circle around the
> cylinder in the same direction over and over again.

Oh, OK Paul, I've got you now. My description really is based
on the planar representation, while you were talking about the
cylindrical representation.

>
> > So you mean that on your ideal lattice you'd have long
> > (or I probably should say wide) strips of cylinders, right?
>
> Wide strips, _or_ a single cylinder.

Right... got it.

> > My code transforms the prime-axes to a right-angled unit cube,
> > transforms the primary lattice metrics along the 3 and 5 axes
> > to the unit metrics along those new axes, then iterates thru
> > the unit cube to fill it with coordinates x,y, always bouncing
> > to the other side (i.e., modulo) when it goes beyond the
> > floor or ceiling values (i.e., 1/2 > x,y > -1/2), then
> > transforms back to the original lattice coordinates.
> >
> > This is exactly how I understood your paragraph. Please correct.
>
> You appear to have the correct picture of how to create periodicity
> blocks. I was saying much more than that, but if that's all you were
> looking for, then you're fine.

Cool. But even tho it works, there still is something wrong
with the mathematics in my spreadsheet. I'd appreciate some
error correction.

> If there's still any confusion, part 3 of the Gentle Introduction
> should clear it up.

Yes, I've since taken another look at that. When I have time
I'll go over my spreadsheet with a fine-tooth comb and compare
it to your webpage description.

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, December 26, 2001 9:29 PM
> > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> >
>
> > > [me, monz]
> > > Right, of course... they continue infinitely in the direction
> > > of the meantone chain if you don't close the chain somewhere.
> > > I *am* interested in closing it so that I get a periodicity-
block.
> >
> > No, sir, I'm afraid you're completely misunderstanding me. If
81:80
> > is tempered out, then you can keep moving by as many 81:80s as
you
> > want in the lattice, and you're still within the strip! In terms
of
> > the cylinder, all you're doing is making a full circle around the
> > cylinder in the same direction over and over again.
>
>
> Oh, OK Paul, I've got you now. My description really is based
> on the planar representation,

The wrong planar representation, in my opinion.

> while you were talking about the
> cylindrical representation.

_Or_ a planar representation, like the ones in _The Forms Of
Tonality_.

> Cool. But even tho it works, there still is something wrong
> with the mathematics in my spreadsheet. I'd appreciate some
> error correction.

Since I think part 3 of the Gentle Introduction should answer the
mathematics part of your question, I'm not currently inclined to
decipher the meaning of the mathematical method you've come up with.
I'd be happy to work with you on understanding and implementing the
method in part 3 of the GI so that you may do what you're trying to
do.

P.S. How can you include W. A. Mozart under 55-EDO on your Equal
Temperament definition page? I could understand if you wanted to put
Mozart on a meantone page, but 55? Totally unjustified. Come on,
let's not just make things up.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, December 27, 2001 1:51 PM
> Subject: [tuning-math] Re: Paul's lattice math and my diagrams
>
>
> > Oh, OK Paul, I've got you now.

Hope you didn't take that the wrong way... I meant that
I understand (I think...)

> > My description really is based on the planar representation,
>
> The wrong planar representation, in my opinion.

Even after emphasizing the equivalence of tiled
periodicity-blocks? I don't get it!

> > while you were talking about the cylindrical representation.
>
> _Or_ a planar representation, like the ones in _The Forms Of
> Tonality_.

Yes, well... I no longer have that available for consultation,
and am eagerly awaiting a new copy...

> > Cool. But even tho it works, there still is something wrong
> > with the mathematics in my spreadsheet. I'd appreciate some
> > error correction.
>
> Since I think part 3 of the Gentle Introduction should answer the
> mathematics part of your question, I'm not currently inclined to
> decipher the meaning of the mathematical method you've come up with.
> I'd be happy to work with you on understanding and implementing the
> method in part 3 of the GI so that you may do what you're trying to
> do.

OK... when I have time, I'll have to sit down with my spreadsheet
and create a section that works strictly according to your method.
If I have trouble then, I'll ask for help.

> P.S. How can you include W. A. Mozart under 55-EDO on your Equal
> Temperament definition page? I could understand if you wanted to put
> Mozart on a meantone page, but 55? Totally unjustified. Come on,
> let's not just make things up.

Well... his conception was clearly based on the
"9 commas per whole-tone, 5 commas per diatonic semitone" idea.
So his teaching of intonation definitely implied a *subset* of
55-EDO. I suppose amending my webpage to that effect would be best.

-monz

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🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Friday, December 28, 2001 12:02 PM
> > Subject: [tuning-math] Re: Paul's lattice math and my diagrams
> >
> >
> > > > [Paul]
> > > > P.S. How can you include W. A. Mozart under 55-EDO on your
Equal
> > > > Temperament definition page? I could understand if you wanted
to
> > > > put Mozart on a meantone page, but 55? Totally unjustified.
> > > > Come on, let's not just make things up.
> > >
> > >
> > > Well... his conception was clearly based on the
> > > "9 commas per whole-tone, 5 commas per diatonic semitone" idea.
> >
> > There is no evidence for that. All we know is that he taught
sharps
> > lower than the "equivalent" flats.
>
>
> Hmmm... I'll have to find some time to dig back into this... too
> preoccupied with periodicity-block stuff math right now. So
> essentially what you're saying is that Chesnut, in his article
> on Mozart, *extrapolates* from Tosi's description of a 55-EDO
> conception, to Leopold Mozart's praise of Tosi, to W. A. Mozart,
> and that I have mistakenly accepted that as evidence?

Nowhere does Chesnut claim that Mozart used 55-EDO or "9 commas per
whole-tone, 5 commas per diatonic semitone". He simply provides a
historical context in which Mozart's preferences can be understood.
Daniel Wold, for example, advocated 1/4-comma meantone for
Mozart . . . who can say?

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Friday, December 28, 2001 2:21 PM
> Subject: [tuning-math] Re: Paul's lattice math and my diagrams
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > So essentially what you're saying is that Chesnut, in his article
> > on Mozart, *extrapolates* from Tosi's description of a 55-EDO
> > conception, to Leopold Mozart's praise of Tosi, to W. A. Mozart,
> > and that I have mistakenly accepted that as evidence?
>
> Nowhere does Chesnut claim that Mozart used 55-EDO or "9 commas per
> whole-tone, 5 commas per diatonic semitone". He simply provides a
> historical context in which Mozart's preferences can be understood.
> Daniel Wold, for example, advocated 1/4-comma meantone for
> Mozart . . . who can say?

Hmmm... what about my ears telling me that my 55-EDO rendition of
the beginning of Mozart's 40th Symphony, on my webpage
http://www.ixpres.com/interval/monzo/55edo/55edo.htm
sounds so much like the great old recording of it from 78s
that I loved as a kid? That was one of the most startling
things that came out of this webpage, for me.

OK, so that still doesn't weigh all that much... I haven't
listened to versions in other meantones yet, and in any case
it's only a few measures.

-monz

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