Squaring the octave

Hi Gene, and Kelly,

I truncated the scale at 16/9, and made a scale with all the steps square / (square - 1)
Then reflected it to fill in the 4/3 step at the start with square
steps too:

http://www.tunesmithy.netfirms.com/tunes/tunes.htm#squaring_the_octave

:-)

9/8 8/7 7/6 6/5 5/4 4/3 3/2 8/5 5/3 12/7 7/4 16/9 2/1

As steps: 9/8 64/63 49/48 36/35 25/24 16/15 9/8 16/15 25/24 36/35 49/48 64/63 9/8

The smallest symmetrical square steps scale is

1/1 4/3 3/2 2/1.

Robert

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

> 9/8 8/7 7/6 6/5 5/4 4/3 3/2 8/5 5/3 12/7 7/4 16/9 2/1

This scale contains two otontal quintads:

1-9/8-5/4-3/2-7/4 and 1-7/6-4/3-3/2-5/3

and two utonal quintads:

1-6/5-4/3-3/2-12/7 and 1-8/7-4/3-8/5-16/9

One way to describe it is as simply the union of these qunitads. In
consequence, it is well-supplied with 9-limit chords of various kinds.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> One way to describe it is as simply the union of these qunitads. In
> consequence, it is well-supplied with 9-limit chords of various kinds.

It is also a square scale in another sense--the qunitads, considered
as in the cubic lattice of quintads, form a square. They are the
quintads associated to

[-1 0 0] [0 0 0]
[-1 -1 -1][0 -1 -1]

The horizontally related (major/minor pairings) and vertically related
(a fifth apart) qunitads have two notes in common. [-1 0 0] and
[0 -1 -1] have three notes in common, and [0 0 0] and [-1 -1 -1]
(inverse to each other) one common note. All of the quintads, in fact,
contain 1 as a note; this is a consequence of the fact that it is a
subscale of the 9-limit tonality diamond, which can be described as
the union of all quintads containing 1.

>> One way to describe it is as simply the union of these qunitads. In
>> consequence, it is well-supplied with 9-limit chords of various kinds.
>
>It is also a square scale in another sense--the qunitads, considered
>as in the cubic lattice of quintads,

Hey; I thought only the 7-limit formed a cubic chord lattice!

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> One way to describe it is as simply the union of these qunitads. In
> >> consequence, it is well-supplied with 9-limit chords of various
kinds.
> >
> >It is also a square scale in another sense--the qunitads, considered
> >as in the cubic lattice of quintads,
>
> Hey; I thought only the 7-limit formed a cubic chord lattice!

Stick on a 9/8 or 16/9 to the tetrad, and you have a quintad; then
just use the same lattice.

>> >It is also a square scale in another sense--the qunitads, considered
>> >as in the cubic lattice of quintads,
>>
>> Hey; I thought only the 7-limit formed a cubic chord lattice!
>
>Stick on a 9/8 or 16/9 to the tetrad, and you have a quintad; then
>just use the same lattice.

I remember suggesting that, and you shot it down. For the very
good reason, IIRC, that you don't get all the relationships
between chords.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I remember suggesting that, and you shot it down. For the very
> good reason, IIRC, that you don't get all the relationships
> between chords.

You don't get a lattice with all the shared-interval chord
relationships, it's true, but I thought what I didn't like was trying
to stick in the shared-interval chord relationships and then claim the
resulting graph as a lattice. It would be interesting to sort that
graph out, but each vertex connects with 11 others, and the way major
vertexes and minor vertexes look is no longer the same: both get
connected by the eight edges [-1 0 0], [0 -1 0], [0 0 -1],
[1 0 0], [0 1 0], [0 0 1], [0 -1 -1], [0 1 1]. However, the otonal
verticies are connected to utonal also by [0 1 2], [0 2 1], [-1 2 2],
whereas the utonal are connected to otonal also by [0 -1 -2],
[0 -2 -1] and [1 -2 -2].

Anyway, it does work for 9-limit chords and I've been using it for a
long time now. Another example--you can take the 2x2x2 tetrad cube,
aka the stellated hexany, and consider it as a quintad cube. The
result is the 9-limit diamond, and you get two extra qunitads for
free--[-1 1 1] representing 1-9/8-9/7-3/2-9/5, and [0 -2 -2],
representing its inverse, 1-10/9-4/3-14/9-16/9.

>Anyway, it does work for 9-limit chords and I've been using it
>for a long time now. Another example--you can take the 2x2x2
>tetrad cube, aka the stellated hexany, and consider it as a
>quintad cube. The result is the 9-limit diamond,

This doesn't sound right for some reason... can you print out
a projection? Maybe I'll see if I can pick it out on one of
Erv's diagrams. . .

-Carl