Chord lattice generators for 7-limit planars

In just the same way as for note-classes, we can project the 3D
lattice of 7-limit chords to a 2D lattice, and find a Minkowski
reduced basis for it, with a shortest and second-shortest size for
the generators. I don't see a convincing way to decide between the +
and - version of a generator and so normalize as I did before, but I
suppose some definition could be given.

For examples, consider 225/224 and 2401/2400; the first line is the
reduced generator pair, and the second a unimodular lattice which
transforms a chord in the usual basis to the two generators, plus a
major tetrad on 225/224 or 2401/2400 respectively.

This kind of basis can be used to define scales, by taking square
arrays of chords in terms of the generators. Below I give the 3x3
square for 2401/2400, which boils 25 JI notes down to 18. The 18
notes to the octave give five major and six minor tetrads, and three
supermajor and three subminor tetrads. Inverting, of course, gives
six major and five minor tetrads. All of this, of course, is with
2401/2400-planar accuracy, which means effectively JI.

225/224

[[0,0,1], [0,1,2]]
[[-2,-2,1], [-1,1,0], [1,0,0]]

2401/2400

[[1,1,-1],[1,1,-2]]

[[3,-1,1], [0,-1,-1], [-1,1,0]]

! sqoo.scl
3x3 chord square, 2401/2400 projection of tetrad lattice (612-et
tuning)
18
!
35.294118
84.313725
119.607843
266.666667
350.980392
386.274510
470.588235
582.352941
617.647059
701.960784
737.254902
849.019608
884.313726
933.333333
968.627451
1052.941176
1088.235294
1200.000000

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> In just the same way as for note-classes, we can project the 3D
> lattice of 7-limit chords to a 2D lattice, and find a Minkowski
> reduced basis for it, with a shortest and second-shortest size for
> the generators. I don't see a convincing way to decide between the
+
> and - version of a generator and so normalize as I did before, but
I
> suppose some definition could be given.
>
> For examples, consider 225/224 and 2401/2400; the first line is the
> reduced generator pair, and the second a unimodular lattice which
> transforms a chord in the usual basis to the two generators, plus a
> major tetrad on 225/224 or 2401/2400 respectively.

Could someone give me an example of "a chord in the usual basis"
Is that chord then multiplied by the unimodular lattice to obtain
the gens/tetrad?

>
> This kind of basis can be used to define scales, by taking square
> arrays of chords in terms of the generators. Below I give the 3x3
> square for 2401/2400, which boils 25 JI notes down to 18. The 18
> notes to the octave give five major and six minor tetrads, and
three
> supermajor and three subminor tetrads. Inverting, of course, gives
> six major and five minor tetrads. All of this, of course, is with
> 2401/2400-planar accuracy, which means effectively JI.

If its not too much to ask, could I get the 3x3 square? Thanks.
>
> 225/224
>
> [[0,0,1], [0,1,2]]
> [[-2,-2,1], [-1,1,0], [1,0,0]]
>
> 2401/2400
>
> [[1,1,-1],[1,1,-2]]
>
> [[3,-1,1], [0,-1,-1], [-1,1,0]]
>
> ! sqoo.scl
> 3x3 chord square, 2401/2400 projection of tetrad lattice (612-et
> tuning)
> 18
> !
> 35.294118
> 84.313725
> 119.607843
> 266.666667
> 350.980392
> 386.274510
> 470.588235
> 582.352941
> 617.647059
> 701.960784
> 737.254902
> 849.019608
> 884.313726
> 933.333333
> 968.627451
> 1052.941176
> 1088.235294
> 1200.000000