Cubic lattice of chords

Hi Gene,

About your chords:

> No, but that wouldn't be a bad idea. If we multiply all the notes of a
> (major or minor) JI tetrad together and ignore 2, we get a rational number

> q = 3^a 5^b 7^c

Okay, so for instance 1/1 5/4 3/2 -> 3*5, and 1/1 6/5 3/2 -> 3^2*5^-1

How is it unique though, couldn't that also be
1/1 9/8 8/5?

I know that isn't a major or minor chord, the question is, what is the
way you determine if it is to count as a major or minor chord, and
how you decide what way to distribute the numbers amongst the component
notes of the triad.

Also I can't see yet how this is a lattice rather than just
a geometrical shape - does say 3^21*5^11*7^13 or whatever
correspond to any kind of a musical chord? Or are they just the
chords near the centre of the latticd around the 1/1 that are
of interest.

Can't help but feel I'm missing something here.

> which uniqely determines that chord. Transforming coordinates by using
> instead [(b+c-2)/4,(a+c-2)/4,(a+b-2)/4] exhibits the tetrads as a cubic
> lattice; every triple [a,b,c] corresponds to a tetrad, and two tetrads
> which share a common interval are one unit apart. If a+b+c
> is even, the tetrad is major, if odd, minor. The existence of this
> cubic lattice has to do with special facts about the lattice of notes
> in the 7-limit which do not obtain in other prime limits.

I'll wait until I understand the first part before getting onto understanding
how you get a cubic lattice out of it.

Thanks,

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> How is it unique though, couldn't that also be
> 1/1 9/8 8/5?

Not if we are only talking about major and minor tetrads, which are the lattice points of the lattice in question.

> I know that isn't a major or minor chord, the question is, what is the
> way you determine if it is to count as a major or minor chord, and
> how you decide what way to distribute the numbers amongst the component
> notes of the triad.

1:3:5:7 major
1:1/3:1/5:1/7 minor

Hi Gene

Sorry, I just don't understand yet what you are saying.

Not the specific point, but the basic idea of how the lattice as a whole works

Can you give a few concrete examples perhaps, and explain how
you derive them?

E.g. 3^21*5^11*7^13 is going to be in your lattice somewhere.
So what is that as a minor triad?

Then perhaps I'll understand what you mean when you say that
the chords tile.

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> Can you give a few concrete examples perhaps, and explain how
> you derive them?

For a major tetrad, from q = 3^a 5^b 7^c we would have
3^(a-1)/4 5^(b-1)/4 7^(c-1)/4 as the root of the chord. For a minor chord, simply find the root of 1/q as a major chord, and that will be the fifth of the chord.

> E.g. 3^21*5^11*7^13 is going to be in your lattice somewhere.

It isn't. 3^1 5^1 7^1 correponds to a major tetrad, and any major tetrad will come from this tetrad (1:3:5:7) multiplied by some
7-limit number r; multiplying each of the four notes of the chord by r multiplies our representing number q by r^4. Hence, the exponents modulo four of the representing number q are all congruent to 1 for a major tetrad, and to -1 for a minor tetrad.

Hi Gene

> > E.g. 3^21*5^11*7^13 is going to be in your lattice somewhere.

> It isn't. 3^1 5^1 7^1 correponds to a major tetrad, and any major tetrad will
> come from this tetrad (1:3:5:7) multiplied by some
> 7-limit number r; multiplying each of the four notes of the chord by r
> multiplies our representing number q by r^4. Hence, the exponents modulo four
> of the representing number q are all congruent to 1 for a major tetrad, and to
> -1 for a minor tetrad.

Ah right, that explains it! Thanks!

Robert

hi Gene and Robert,

> From: "Robert Walker" <robertwalker@ntlworld.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, November 07, 2002 4:03 PM
> Subject: [tuning] Re: Cubic lattice of chords
>
>
> Hi Gene
>
> Sorry, I just don't understand yet what you are saying.
>
> Not the specific point, but the basic idea of how the
> lattice as a whole works

i'm with Robert on this one. i'd be really happy to create
a webpage with lattices, showing what you're talking about,
but i don't really know what you're saying.

-monz

--- In tuning@y..., "monz" <monz@a...> wrote:

> i'm with Robert on this one. i'd be really happy to create
> a webpage with lattices, showing what you're talking about,
> but i don't really know what you're saying.

Did you see my latest exchange with Robert? He's tracking me now.

Here's another way to look at it:

Let [a,b,c] be a 3-tuple of integers. If a+b+c is even, then define
m = 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2). The major tetrad
represented by [a,b,c] is then m:3m:5m:7m. Now suppose a+b+c is odd, we define m in this case as
m = 3^((-a+b+c+1)/2) 5^((a-b+c+1)/2) 7^((a+b-c+1)/2)
Now the minor tetrad the 3-tuple represents is m:m/3:m/5:m/7.

i've been making frequent updates to my
1/4-comma meantone page recently.

http://sonic-arts.org/dict/1-4cmt.htm

the latest additions are a couple of tables
with graphs, showing how closely intervals
of 1/4-comma meantone approximate JI chord
identities.

my position is that the proximity of many
1/4-meantone intervals to high-odd JI identities
may have given composers who used these meantone
chord-structures a "window" into higher-limit JI.

-monz
"all roads lead to n^0"

Hi Paul,

Thanks. I'm now completely clear about it.

It's related to that way of drawing a stack of cubes using hexagons isn't
it - the alternating sixty degree rhomb tiling shaded in three colours
you often see shwoing a stack of cubes in puzzle books.

That shows that if you slice through a cubic tiling at the correct
angle you get the honeycomb tiling.

Then in that puzzle book type rhombi tiling, if you now place a point
in the centre of every tile, those points are going to form a tiling
by regular triangles - in fact the same one as the 3 5 lattice.

The phrase springs to mind now, face centred cubic, which I think
is what this must be.

So, now, you could instead take the 3 5 7 lattice, and if you
turn it aroudn a bit and slice through it at the correct angle
you'll get a square tiling. (Corresponding to one of the
planes in the cubic tiling)

So would be interesting to do a gif of that 2D square tiling,
and all its chords, and see how they fit together.

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> The phrase springs to mind now, face centred cubic, which I think
> is what this must be.

It's much simpler--it is simply the cubic lattice, all points [a,b,c]
with a, b, and c integers.

> So, now, you could instead take the 3 5 7 lattice, and if you
> turn it aroudn a bit and slice through it at the correct angle
> you'll get a square tiling. (Corresponding to one of the
> planes in the cubic tiling)

Don't mix up the lattice of notes with the lattice of chords!

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> The phrase springs to mind now, face centred cubic, which I
think
> is what this must be.

the face centered cubic lattice is the same thing as the
octahedral-tetrahedral lattice, which is what we use for the 3-5-7
lattice of notes.

Hi everyone,

I've added a picture of one layer of the cubic lattice of chords to
my hexany etc page:

http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm#cubic_latt

While doing this, discovered that my search and replace routine to
transform e.g.

1 3/7 1/7 5/7.mid
to 1_3o7_1o7_5o7.mid

to make the file name suitable for use in a url, isn't replacing spaces with
underlines - fixed for next upload of FTS later today, if anyone is making
other pictures like this. Of course you can just enter underlines instead
of spaces yourself for the file names, which is what I did before, and
why I didn't discover it before.

Unfortunately if you use Quicktime then when you click to hear a clip
it will probably take you to a new page, which isn't what we want here.

You can change your settings to play them on Windows Media player
in Windows, bt with a caveat:

Start | Programs | Accessories | Entertainement | Windows Media Player
| Tools | Formats | and if you then select Midi file there you
will hear them in Media Player instead.

However, that will override the quicktime settings.
One thing about quicktime - it does do the pitch bends while
some soundcards (such as on-board sound) don't.
Also another thing, if you then want to go back to quicktime
then the only way I've found to do that is to set it so
quicktime handles _all_ your internet file associations
such as .gif, .jpg etc. as well.

I'm looking into possibilities for htis - one idea is to use a java
applet for the image maps instead of doing them in html. Then
the clips would need to be in the sun au 8Khz format suitable for java
1.1 or earlier applets.

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi everyone,
>
> I've added a picture of one layer of the cubic lattice of chords to
> my hexany etc page:
>
> http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm#cubic_latt

Totally cool. Now how do we get a 3D pushbutton array of chords--stacked slices? I see a program possibility lurking here--"Seven Limit Harmony for Dummies", with notes for the right hand and chords for the left, like those cheap chord organs for dummies. Or if we are willing to restrict ourselves to a temperament, say miracle, just two keybords, one for chords, one for notes.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

/tuning/topicId_40737.html#40832

> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> > Hi everyone,
> >
> > I've added a picture of one layer of the cubic lattice of chords
to
> > my hexany etc page:
> >
> >
http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm#cub
ic_latt
>
> Totally cool. Now how do we get a 3D pushbutton array of chords--
stacked slices? I see a program possibility lurking here--"Seven
Limit Harmony for Dummies", with notes for the right hand and chords
for the left, like those cheap chord organs for dummies. Or if we are
willing to restrict ourselves to a temperament, say miracle, just two
keybords, one for chords, one for notes.

****Congrats to Robert Walker for such a nice page! I'm still
enjoying the hexany chords on this page. Nice to have a tuning page
that's so *interactive...*

J. Pehrson

hi Robert,

an easy way to eliminate the change-of-window problem
with Quicktime is to add the following code:

TARGET="_blank"

to your A HREF tags. i know a lot of people bitch
about the way i've set up my website to do this,
but for this particular situation it's ideal. this
way Quicktime will open in a separate window and
your original "musical geometry" page will stay
open in its own window.

BTW, a job well done!

-monz

----- Original Message -----
From: "Robert Walker" <robertwalker@ntlworld.com>
To: <tuning@yahoogroups.com>
Sent: Monday, November 11, 2002 7:08 AM
Subject: [tuning] Cubic lattice of chords

> Hi everyone,
>
> I've added a picture of one layer of the cubic lattice of chords to
> my hexany etc page:
>
>
http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm#cubic_lat
t
>
> While doing this, discovered that my search and replace routine to
> transform e.g.
>
> 1 3/7 1/7 5/7.mid
> to 1_3o7_1o7_5o7.mid
>
> to make the file name suitable for use in a url, isn't replacing spaces
with
> underlines - fixed for next upload of FTS later today, if anyone is making
> other pictures like this. Of course you can just enter underlines instead
> of spaces yourself for the file names, which is what I did before, and
> why I didn't discover it before.
>
> Unfortunately if you use Quicktime then when you click to hear a clip
> it will probably take you to a new page, which isn't what we want here.
>
> You can change your settings to play them on Windows Media player
> in Windows, bt with a caveat:
>
> Start | Programs | Accessories | Entertainement | Windows Media Player
> | Tools | Formats | and if you then select Midi file there you
> will hear them in Media Player instead.
>
> However, that will override the quicktime settings.
> One thing about quicktime - it does do the pitch bends while
> some soundcards (such as on-board sound) don't.
> Also another thing, if you then want to go back to quicktime
> then the only way I've found to do that is to set it so
> quicktime handles _all_ your internet file associations
> such as .gif, .jpg etc. as well.
>
> I'm looking into possibilities for htis - one idea is to use a java
> applet for the image maps instead of doing them in html. Then
> the clips would need to be in the sun au 8Khz format suitable for java
> 1.1 or earlier applets.
>
> Robert
>
>
>
>
> You do not need web access to participate. You may subscribe through
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--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

/tuning/topicId_40737.html#40825

>
> You can change your settings to play them on Windows Media player
> in Windows, bt with a caveat:
>
> Start | Programs | Accessories | Entertainement | Windows Media
Player
> | Tools | Formats | and if you then select Midi file there you
> will hear them in Media Player instead.
>
> However, that will override the quicktime settings.
> One thing about quicktime - it does do the pitch bends while
> some soundcards (such as on-board sound) don't.
> Also another thing, if you then want to go back to quicktime
> then the only way I've found to do that is to set it so
> quicktime handles _all_ your internet file associations
> such as .gif, .jpg etc. as well.
>
> I'm looking into possibilities for htis - one idea is to use a java
> applet for the image maps instead of doing them in html. Then
> the clips would need to be in the sun au 8Khz format suitable for
java
> 1.1 or earlier applets.
>
> Robert

***Hi Robert,

It really doesn't seem so bad to go to a "different page" with
Quicktime... Supposedly one can remember what one was clicking
on... :) At least the pitch bends are accurate, or so they seem to
be.

J. Pehrson

Hi Joseph,

Yes, quicktime is reasonable enough pitch bend wise.
The thing about it is that it translates all pitch bends
you give it into fractional semitones (7 bits / semitone)
i.e. 128 units per semitone, so the pitch bend resolution is a little
under a cent. So not so good as some but reasonable
enough for many purposes.

http://developer.apple.com/techpubs/quicktime/qtdevdocs/REF/tp_qtma_qtmaref.a.htm

Of course, that means if you are happy with that level of resolution
and also specify it in your midi file, you could have a single
pitch bend swoop through the entire midi range of pitches :-).
That's why they do it I think.

I think opening in a new window is okay for quicktime, but it would be
nice if one could specify the size of it and make it just the right
size for the midi clip play controls and no more - then one could also see
the original page.

Would be even nicer if one could open them all in the same window.

I'll do that as soon as I can figure out how to do it, assuming it
can be done. I ran into something there - that the javascript code
that works in an ordinary link doesn't work in the links used for
the image maps, so that needs investigating first. Instead it
opens a blank new window with the url field at the top shown as
"Javascript:onClick=OpenInNewWin('1.mid')" or whatever the javascript
code is.

If anyone who is reading this happens to know what I need to put into the
ulr field there to get it to work, do let me know :-).

Thanks,

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Joseph,
>
> Yes, quicktime is reasonable enough pitch bend wise.
> The thing about it is that it translates all pitch bends
> you give it into fractional semitones (7 bits / semitone)
> i.e. 128 units per semitone, so the pitch bend resolution is a
little
> under a cent. So not so good as some but reasonable
> enough for many purposes.

Monz's eqtemp dictionary page lists quicktime under 3072-equal (256
units / semitone). is monz incorrect? if so, that would be a shame,
because 3072-equal's 5-limit deviations from JI are mere thousanths
of a cent!

Hi Paul,

Seems pretty clear from the page:

http://developer.apple.com/techpubs/quicktime/qtdevdocs/REF/tp_qtma_qtmaref.a.htm

ControllerPitchBend
This controller bends the pitch. Pitch bend is specified in positive and
negative semitones, with 7 bits per fraction.

7 bits = 128
8 bits = 256

I haven't tried to measure it to see how well it does.

Robert

Hi Paul,

E-mail truncated somehow in Yahoo - on-line anyway. If this one also gets truncated,
try view source to see it complete.

Anyway what I said was: seems pretty clear from the page:

http://developer.apple.com/techpubs/quicktime/qtdevdocs/REF/tp_qtma_qtmaref.a.htm

ControllerPitchBend
This controller bends the pitch. Pitch bend is specified in positive and
negative semitones, with 7 bits per fraction.

7 bits = 128
8 bits = 256

I haven't tried to measure it to see how well it does.

Robert

Hi There,

I've now got the musical geometry page working as it should with quicktime :-).

http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry_new_win.htm

One of these things that turned out to be simple once you knew how :-).

What I'd missed before is that you have to put the OnClick in a separate field
for image maps - and in fact you don't need it at all, just use HREF="javascript:popup(...)"

I found the solution here:
http://zorak.best.vwh.net/imagemap/

Then here is an even easier way of doing it which will work even if you disable
javascript (does anyone do that?).

With this one, first time you show the window you have to resize it to make it small.

It just uses the exact method you suggest, Monz, but instead of putting
target="_blank" you put target="image_map_popup" - use any name one
likes there - then all those links open into the same window - the
one you have just named by doing that.

For the example done that way:

http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry_qt.htm

I haven't yet updated FTS to make the links this way - it may be
possible to improve it as at present you need two pages - one for use
with quicktime and one without - if you open the quicktime versions of the
page with say Media Player as your player, then a blank browser window pops up as
well as the midi player - and you can't get rid of the blank page, even though you
don't need it, because it will pop up again whenever you click on one of the clips.

So for those with Windows Media Player, it still is

http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm

- I'll think this over and see if there is any way they can be
combined.

Thanks,

Robert