Ten notes to the octave

Starting from a basis <T(4),T(5),S(4),T(6)> =
<10/9,15/14,16/15,21/20> we defined another basis
<21/20,28/27,64/63,225/224>, the corresponding notation of which is
N(5), defined by the matrix

[10 2 7 5]
[16 3 11 8]
[23 5 16 12]
[28 6 19 14]

which we may write as [h10,h2,g7,h5], where we write g7 and not h7
since we have defined h7 so that h7(7) = 20, the nearest integer to
7 log_2(7). By choosing various base points, we may produce a variety
of PBs, for instance

(21/20)^n * (28/27)^ceil(-.3+ 2.N) * (64/63)^ceil(-.65 +.7 n) *
(225/224)^ceil(-.25+.5n)

leading to

1 – 15/14 – 7/6 – 5/4 – 4/3 – 45/32 – 3/2 – 5/3 – 7/4 – 15/8 – (2).

Since we have already shown the JI diatonic scale to be a PB, we
might wonder if we can obtain a PB containing it using this basis,
and perhaps also one with epimoric intervals. For instance, we might
consider

1 – 21/20 – 9/8 – 5/4 – 4/3 – 7/5 – 3/2 – 5/3 – 7/4 – 15/8 – (2).

The question of whether this is a PB turns out to be a matter of
definitions. If instead of partitioning into half-open and non-
intersecting regions to define blocks, we could use closed regions
which intersect on their boundaries. We then can have more than one
candidate for a given scale degree; if we allow ourselves to select
any such candidate we choose then the above can be seen as a block,
but not otherwise. We might call such a thing a periodic semiblock,
or PS. Its scale steps in any case are rather marginal, given the
fact that (21/20)^2 < 10/9.

In the first of these ten-note scales we had the interval 45/32,
which is approximately 7/5; in fact 45/32 = (7/5)(225/224),
illustrative of the fact that we might want to temper 225/224 out for
this scale. Ets doing this can be expressed as combinations
p h10 + q h2 + r g7, and ets n with cons(7, n) < 1.1 and 225/224~1
are 12, 19, 22, 31, 41, 53, and 72. Also of interest with these
scales are ets with 64/63~1, if we again look for examples with cons
(7,n)<1.1 we obtain 12, 15, 22, and 27.

The 22 et, which appears on both lists, would seem especially well
suited to these scales. If we look at our two scales in the 22et, the
first has steps 2232223222, and the second 2322223222; we see that
these are in fact distinct rather than modal variants. If Paul will
send me his paper or put it into a folder on tuning-math, I'll try to
collect on a bet I made with myself that these scales are both
discussed in it.

--- In tuning-math@y..., genewardsmith@j... wrote:
> Starting from a basis <T(4),T(5),S(4),T(6)> =
> <10/9,15/14,16/15,21/20> we defined another basis
> <21/20,28/27,64/63,225/224>, the corresponding notation of which is
> N(5), defined by the matrix
>
> [10 2 7 5]
> [16 3 11 8]
> [23 5 16 12]
> [28 6 19 14]
>
> which we may write as [h10,h2,g7,h5], where we write g7 and not h7
> since we have defined h7 so that h7(7) = 20, the nearest integer to
> 7 log_2(7). By choosing various base points, we may produce a
variety
> of PBs, for instance
>
> (21/20)^n * (28/27)^ceil(-.3+ 2.N) * (64/63)^ceil(-.65 +.7 n) *
> (225/224)^ceil(-.25+.5n)
>
> leading to
>
> 1 – 15/14 – 7/6 – 5/4 – 4/3 – 45/32 – 3/2 – 5/3 – 7/4 – 15/8 – (2).
>
> Since we have already shown the JI diatonic scale to be a PB, we
> might wonder if we can obtain a PB containing it using this basis,
> and perhaps also one with epimoric intervals. For instance, we
might
> consider
>
> 1 – 21/20 – 9/8 – 5/4 – 4/3 – 7/5 – 3/2 – 5/3 – 7/4 – 15/8 – (2).
>
> The question of whether this is a PB turns out to be a matter of
> definitions. If instead of partitioning into half-open and non-
> intersecting regions to define blocks, we could use closed regions
> which intersect on their boundaries.

I'm unclear about the distinction here. Can you be more explicit? I
don't see any grounds for considering this _not_ a PB under any
reasonable definitions.

Anyway, please read this:

/tuning/topicId_20642.html#20642
/tuning/topicId_20642.html#20694

> We then can have more than one
> candidate for a given scale degree; if we allow ourselves to select
> any such candidate we choose then the above can be seen as a block,
> but not otherwise. We might call such a thing a periodic semiblock,
> or PS. Its scale steps in any case are rather marginal, given the
> fact that (21/20)^2 < 10/9.

I don't get this . . . scale steps rather marginal?
>
> In the first of these ten-note scales we had the interval 45/32,
> which is approximately 7/5; in fact 45/32 = (7/5)(225/224),
> illustrative of the fact that we might want to temper 225/224 out
for
> this scale. Ets doing this can be expressed as combinations
> p h10 + q h2 + r g7, and ets n with cons(7, n) < 1.1 and 225/224~1
> are 12, 19, 22, 31, 41, 53, and 72. Also of interest with these
> scales are ets with 64/63~1, if we again look for examples with cons
> (7,n)<1.1 we obtain 12, 15, 22, and 27.
>
> The 22 et, which appears on both lists, would seem especially well
> suited to these scales. If we look at our two scales in the 22et,
the
> first has steps 2232223222, and the second 2322223222; we see that
> these are in fact distinct rather than modal variants. If Paul will
> send me his paper or put it into a folder on tuning-math, I'll try
to
> collect on a bet I made with myself that these scales are both
> discussed in it.

You win! They are the main topic of the paper. But I already
mentioned both of these scales to you here on this list, as "my"
decatonic scales. I came up with them in 1991, age 19 -- and that's
why I'm interested in microtonality today!

--- In tuning-math@y..., genewardsmith@j... wrote:

> The question of whether this is a PB turns out to be a matter of
> definitions. If instead of partitioning into half-open and non-
> intersecting regions to define blocks, we could use closed regions
> which intersect on their boundaries. We then can have more than one
> candidate for a given scale degree; if we allow ourselves to select
> any such candidate we choose then the above can be seen as a block,
> but not otherwise. We might call such a thing a periodic semiblock,
> or PS.

I think I understand this now . . . sorry I didn't at first, Gene.
You're talking about the _Fokker_ partitioning into parallelepipeds,
right? As you can see in the _Gentle Introduction_, periodicity
blocks can have boundaries of different shapes, not just the Fokker
specification . . . but I don't mind your suggestion for something
like the "PS" terminology . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

As you can see in the _Gentle Introduction_, periodicity
> blocks can have boundaries of different shapes, not just the Fokker
> specification . . . but I don't mind your suggestion for something
> like the "PS" terminology . . .

Interesting--I thought we were committed to paralleopipeds, though
any convex region would seem good enough. I still get PBs as "scales
along a line", but need to adapt my definition of closeness from its
implicit L-infinity norm to other possibilities.