As I reported in a revious article, the 7-limit tetrads form a cubic

lattice. I haven't gotten much feedback on this stuff, and wonder if this

is well known, not known, or somewhere in between.) By identifying the

major tetrad with root q, q a 7-limit octave equivalence class, with q

itself, we may represent the note-lattice of classes as a sublattice of

the cubic lattice of chords. The basis is [0 1 1] representing 3/2, [0 1

0] 5/4, and [1 1 0] 7/4, so that 3^a 5^b 7^c is represented by

[b+c,a+c,a+b]. In this form, the *usual* Euclidean metric applies to the

note lattice.

Using this representation, we may define a block in a way entirely

analogous to note-class blocks. If for instance we take

<9/8, 15/14, 25/24>, the TM-reduced basis for the kernel of h4, we obtain

upon transforming to the cubic lattice coordinates <[0 2 2], [0 0 2], [2

-1 -1]>. Taking the adjoint matrix M of the matrix (of determinant +-8 =

2*4) defined by these as rows, we may construct a corresponding block by

requiring that if [p q r] = [a b c]M, then -4<p<5, -4<q<5, -4<r<5. This

gives us the following set of eight (= 2*4) chords:

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

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

The notes of these give the following scale:

[1, 25/24, 15/14, 35/32, 9/8, 75/64, 5/4, 21/16, 75/56, 45/32, 35/24,

3/2, 25/16, 45/28, 5/3, 7/4, 25/14, 15/8]

From a brief look this question arose: are these chord

analogues a subset of real periodicity blocks in some

special way?

Your scale reminded me of the 4:5:6:7 double tied circular

mirroring, but it's bigger and not a superset.

Manuel

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:

> From a brief look this question arose: are these chord

> analogues a subset of real periodicity blocks in some

> special way?

There's a close relationship with periodicity blocks. Pick either major or minor tetrads, and with those, one particular chord element--for instance the roots of the minor tetrads, or the major third element of the major tetrads. These will all individually be corresponding Fokker blocks.

> Your scale reminded me of the 4:5:6:7 double tied circular

> mirroring, but it's bigger and not a superset.

Where is the 4:5:6:7 double tied circular mirroring discussed?

Gene wrote:

>There's a close relationship with periodicity blocks. Pick either major or

minor tetrads, and with those, one >particular chord element--for instance

the roots of the minor tetrads, or the major third element of the major

>tetrads. These will all individually be corresponding Fokker blocks.

I see, sort of a Carthesian product of a Fokker block with a chord then?

>Where is the 4:5:6:7 double tied circular mirroring discussed?

I wrote a posting to the Tuning List several years ago, I can't find

the specific date at the moment. In the scala archive they are the

scales *kring*.scl.

In a double tied circular mirroring a chord is inverted repeatedly

with two tones in common each time with the next inversion, until

coming back to the original.

They are closely related to Partch diamonds.

Manuel