Tempered blocks and semiblocks

As I mentioned at the end of my last posting here, we can define a
tempered block by omitting some of the vals (other than for 2) for
the notation used to define the block. Since JI scales proliferate,
and since we probably want to temper anyway, it seems like a good
plan to look at the tempered blocks directly, rather than always
first constructing a JI block and then tempering.

For instance, if we omit h3 from [h7, h2, h3] we are tempering out
the interval associated to h3, namely 81/80. The diatonic scale then
becomes, in terms of the notation [h7, 7 h2 - 2 h7],
[0 0], [1 2], [2 4], [3 -1], [4 1], [5 3], [6 5]. This is a tempered
block--in both coordinates, the maximum difference is 6, less than 7.
The generator is easily found by sorting according to the second
coordinate, giving us [3 -1], [0 0], [4 1], [1 2], [5 3], [2 4],
[6 5]; the starting value is [3 -1], corresponding to 4/3, and the
generator is [4 1]--the note after [0 0] in this sorted order,
corresponding to 3/2. The diatonic scale therefore turns out to be
(hold on to your hats!) the scale one gets by starting from 4/3,
iterating meantone fifths, and reducing modulo octaves.

We also can usefully add a definition of semiblock to our definition
of block; we do this by relaxing the condition that the diameter be
less than 1 to a condition that it be less than or equal to 1, but
less than 1 on the first coordinate; and adding a condition that if
n, m are distinct notes in S then the first val gives them distinct
values. Let's look again at the ten-note scales where semiblocks came
up.

Recall that these scales arose from the notation
(21/20, 28/27, 64/63, 225/224)^(-1) = [h10, h2, g7, h5] where g7
differs from h7 by the fact that g7(7) = 19. We found from this the
JI block 1-15/14-7/6-5/4-4/3-45/32-3/2-5/3-7/4-15/8-(2), in terms of
the notation [h10, 10h2 - 2h10, 10g7 - 7h10, 10h5 - 5 h10] this
becomes

[0 0 0 0]
[1 -2 3 5]
[2 6 -4 0]
[3 4 -1 5]
[4 2 2 0]
[5 0 -5 5]
[6 -2 -2 0]
[7 6 1 5]
[8 4 -6 0]
[9 2 -3 5]

Note that since the second coordinate comes from the val 10h2-2h10 it
is divisible by 2, and since the last coordinate comes from 10h5-510
it is divisible by 5. Hence we will not be able to completely sort
the notes of the scale using these coordinates. However we easily
check that this is a block, and by dropping the last two vals we get
a tempered block which tempers out 64/63 and 225/224. We can sort by
the last coordinate into two groups, giving us [7 6], [8 4], [9 2],
[0 0], and [1 -2] in the first group, and [2 6], [3 4], [4 2], [5 0]
and [6 -2] in the second group. The generator is [1 -2],corresponding
to a semitone of 15/14, 16/15 or 21/20, and we start the first group
at [7 6], corresponding to 5/3, and the second at [2 6],
corresponding to 7/6.

The other JI scale we considered was 1-21/20-9/8-5/4-4/3-7/5-3/2-5/3-
7/4-15/8-(2), corresponding to

[0 0 0 0]
[1 -2 -7 -5]
[2 -4 -4 0]
[3 4 -1 5]
[4 2 2 0]
[5 0 -5 5]
[6 -2 -2 0]
[7 6 1 5]
[8 4 -6 0]
[9 2 -3 5]

We see that the diameter is 10, since we go from -4 to 6 according to
the second val, and from -5 to 5 according to the last. However, it
is a semiblock since we can't add a note without the first val giving
us a difference of at least 10, which is not allowed. If we temper
out 64/63 and 225/224 and sort according to the second coordinate, we
get [7 6], [8 4], [9 2], [0 0], [1 -2], and [2 -4] in one group, and
[3 4], [4 2], [5 0], [6 -2] in the second group; we have the same
generator as before and start the first group from [7 6],
corresponding to 5/3 as before, but extend the chain by one more
note, to [2 -4] corresponding to 9/8. We make up for this by dropping
7/6 at the start of the second group, and beginning instead from 4/3.