Re: Intro to Hexany page

HI Gene,

Yes, I can do a cubic lattice one too. The last one would be the octony instead
of the hexany. Probably start a new page for that one and just follow the
same general plan, and show the octony as a cube at the end. I could then
go on and mention the way you can find the hexany also as a subset of the
octony, and so lead into the CPS sets and the pascal triangle.

BTW I've just done an update of the page mentioning the 1 3 7 9 hexany and the
septimal minor chord. I was interested to find out that the septimal minor
chord is actually an otonal chord, 3 7 9 in the harmonic series :-).

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

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> HI Gene,
>
> Yes, I can do a cubic lattice one too. The last one would be the octony instead
> of the hexany.

A cube of chords in the cubic lattice leads to a stellated octahedron of 14 notes.

> BTW I've just done an update of the page mentioning the 1 3 7 9 hexany and the
> septimal minor chord. I was interested to find out that the septimal minor
> chord is actually an otonal chord, 3 7 9 in the harmonic series :-).

I still think 3:5:7:9 is the "obvious" meaning of subminor tetrad, given that the above is called a subminor triad. Carl and Paul think I'm all wet, but produce chords less closely related and higher up on the overtone/undertone series, which hardly makes sense.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> HI Gene,
>
> Yes, I can do a cubic lattice one too. The last one would be the
octony instead
> of the hexany.

that's not what gene meant -- he meant the cubic lattice of chords,
not of notes. notes, gene and i agree, are best viewed in the
octahedral-tetrahedral lattice, not the cubic lattice.

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

> that's not what gene meant -- he meant the cubic lattice of chords,
> not of notes. notes, gene and i agree, are best viewed in the
> octahedral-tetrahedral lattice, not the cubic lattice.

Exactly. The 7-limit has the unique (unless you count things like the
{2,5,7,11} group) and extremely nifty property of chords which form a cubic lattice.

Hi Paul and Gene,

Thanks for the correction. I haven't heard of the cubic lattice of chords,
- sounds interesting, are there any web pages describing it?

Anyway someone else can maybe do it? I think perhaps some of these
lattices that get posted to the TL would be really nice with musical
clips so that you can actually hear what they sound like!

It's really easy to make this type of a page now with FTS
- you just do all the file names in the page as the notes you want in
the chord, e.g.

href="1/1 7/6 3/2.mid" for the septimal minor (subminor) chord.

Then FTS will go through the page and make them all with a single click
of a button. It changes them to url safe file names of course, this one to

href="1o1_7o6_3o2.mid"

which FTS reads as the same chord.

That's all there is to it.

See Help | FAQ | Music Making | How do I use the feature to make midi clips
for all the file names in a web page?
- and you are best with the most recent update of FTS for this option, from:
http://tunesmithy.co.uk

So, most of the work is already done once you've done the graphics :-).
Well, you have to draw the clickable regions for the pictures, and you have
to type in all the midi clip names, which can take a little while if you
have lots of those to do.

As for the pictures, to make the type of picture where you click on a particular
position to hear a note - you need a program to make / edit these types of clickable
images (technically called client side image maps). I use Map This! which is
freeware and an excellent tool:

http://www.abdn.ac.uk/tools/ibmpc/mapthis
when I got it, but I think it is now shareware.

With some types of set ups then you get taken to a new page when you click on
the picture. This page of course works best if you have the set up where
the clip gets played in the background, and I've added a note about that to
it.

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul and Gene,
>
> Thanks for the correction. I haven't heard of the cubic lattice of chords,
> - sounds interesting, are there any web pages describing it?

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

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.

>Exactly. The 7-limit has the unique (unless you count things
>like the {2,5,7,11} group) and extremely nifty property of
>chords which form a cubic lattice.

Did I wake up in a parallel universe? Where did this come
from?

-Carl

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
> >Exactly. The 7-limit has the unique (unless you count things
> >like the {2,5,7,11} group) and extremely nifty property of
> >chords which form a cubic lattice.
>
> Did I wake up in a parallel universe? Where did this come
> from?

Eh? It's part of the mathematics of lattices.

>>>Exactly. The 7-limit has the unique (unless you count things
>>>like the {2,5,7,11} group) and extremely nifty property of
>>>chords which form a cubic lattice.
>>
>>Did I wake up in a parallel universe? Where did this come
>>from?
>
>Eh? It's part of the mathematics of lattices.

I mean, when was this concept introduced on these lists?

Lattice of chords? Are there any graphics?

-Carl

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:

> I mean, when was this concept introduced on these lists?

I posted on it on tuning-math a while back.

> Lattice of chords? Are there any graphics?

Not yet--I don't recall even seeing the hexagonal tiling of 5-limit chords, in fact.