Families mod 12 of circulating temperaments

I decided to look 205 and 289, and to my surprise the patterns turned
out to be exactly the same. If I take fifths of size 118, 119, 120,
and 121 in 205, or fifths of size 167, 168, 169 and 170 in 289, we get
precisely the same 24 possibilities. Using the same pattern in 270
would always result in seven octaves plus a step, and for 311, seven
octaves plus two steps.

The reason 205 and 289 have the same pattern is that they are the same
modulo 12, both being congruent to 1. If n is the number of steps to
the octave and a is the number of steps to a fifth, we are asking for
weights w1, w2, w3 such that

w1*(a-2) + w2*(a-1) + w3*a + (12-w1-w2-w3)*(a+1) = 7*n,

which reduces to

3*w1 + 2*w2 + w3 = -7*n + 12*a + 12,

which mod 12 depends only on n. If we calculate 12*a+12-7*n for n
which are congruent to 1 mod 12 in the range 1 to 553, we get 17. In
fact, a standard val hn has the property that 12*hn(3/2)+12-7*n = 17
if and only if n is from 1 to 553 and congruent to 1 mod 12. Hence the
weights will satisfy a single linear dependency,

3*w1 + 2*w2 + w3 = 17

and the conditions w1, w2, w3 >= 0 and w1+w2+w3<=12.

Obviously there is a large amount of theory to all this, and I wonder
if any of it has been noticed before. Does this analysis ring any
bells for anyone? We end up with parametrized familties of tunings,
with the parameter determining the tuning.

1 mod 12 family temperament weights

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

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

> If we calculate 12*a+12-7*n for n
> which are congruent to 1 mod 12 in the range 1 to 553, we get 17. In
> fact, a standard val hn has the property that 12*hn(3/2)+12-7*n = 17
> if and only if n is from 1 to 553 and congruent to 1 mod 12.

Here are similar ranges for the other mod 12 congruences. We would get
different families for each of these ranges and congruences.

12*Fifth + 12 - 7*Octave for mod 12 families

0 mod 12

12 to 300, 12
312 to 912, 24

1 mod 12

1 to 553, 17

2 mod 12

2 to 194, 10
206 to 818, 22

3 mod 12

3 to 459, 15

4 mod 12

4 to 100, 8
112 to 712, 20

5 mod 12

5 to 353, 13

6 mod 12

6 to 606, 18

7 mod 12

7 to 247, 11
259 to 859, 23

8 mod 12

8 to 500, 16

9 mod 12

9 to 153, 9
165 to 765, 21

10 mod 12

10 to 406, 10

11 mod 12

11 to 47, 7
59 to 659, 19

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I decided to look 205 and 289, and to my surprise the patterns
turned
> out to be exactly the same. If I take fifths of size 118, 119, 120,
> and 121 in 205, or fifths of size 167, 168, 169 and 170 in 289, we
get
> precisely the same 24 possibilities. Using the same pattern in 270
> would always result in seven octaves plus a step, and for 311, seven
> octaves plus two steps.
>
> The reason 205 and 289 have the same pattern is that they are the
same
> modulo 12, both being congruent to 1. If n is the number of steps to
> the octave and a is the number of steps to a fifth, we are asking
for
> weights w1, w2, w3 such that
>
> w1*(a-2) + w2*(a-1) + w3*a + (12-w1-w2-w3)*(a+1) = 7*n,
>
> which reduces to
>
> 3*w1 + 2*w2 + w3 = -7*n + 12*a + 12,
>
> which mod 12 depends only on n. If we calculate 12*a+12-7*n for n
> which are congruent to 1 mod 12 in the range 1 to 553, we get 17. In
> fact, a standard val hn has the property that 12*hn(3/2)+12-7*n = 17
> if and only if n is from 1 to 553 and congruent to 1 mod 12. Hence
the
> weights will satisfy a single linear dependency,
>
> 3*w1 + 2*w2 + w3 = 17
>
> and the conditions w1, w2, w3 >= 0 and w1+w2+w3<=12.
>
> Obviously there is a large amount of theory to all this, and I
wonder
> if any of it has been noticed before. Does this analysis ring any
> bells for anyone? We end up with parametrized familties of tunings,
> with the parameter determining the tuning.
>
> 1 mod 12 family temperament weights
>
> [[0, 6, 5, 1], [0, 7, 3, 2], [0, 8, 1, 3], [1, 3, 8, 0],
> [1, 4, 6, 1], [1, 5, 4, 2], [1, 6, 2, 3], [1, 7, 0, 4],
> [2, 2, 7, 1], [2, 3, 5, 2], [2, 4, 3, 3], [2, 5, 1, 4],
> [3, 0, 8, 1], [3, 1, 6, 2], [3, 3, 2, 4], [3, 4, 0, 5],
> [4, 0, 5, 3], [4, 1, 3, 4], [4, 2, 1, 5], [5, 0, 2, 5],
> [5, 1, 0, 6], [3, 2, 4, 3], [2, 1, 9, 0], [0, 5, 7, 0]]

The last 3 of these 24 are out of order. Any reason? Nice to be back.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

> > 1 mod 12 family temperament weights
> >
> > [[0, 6, 5, 1], [0, 7, 3, 2], [0, 8, 1, 3], [1, 3, 8, 0],
> > [1, 4, 6, 1], [1, 5, 4, 2], [1, 6, 2, 3], [1, 7, 0, 4],
> > [2, 2, 7, 1], [2, 3, 5, 2], [2, 4, 3, 3], [2, 5, 1, 4],
> > [3, 0, 8, 1], [3, 1, 6, 2], [3, 3, 2, 4], [3, 4, 0, 5],
> > [4, 0, 5, 3], [4, 1, 3, 4], [4, 2, 1, 5], [5, 0, 2, 5],
> > [5, 1, 0, 6], [3, 2, 4, 3], [2, 1, 9, 0], [0, 5, 7, 0]]
>
> The last 3 of these 24 are out of order. Any reason? Nice to be back.

Hi. No reason; I didn't try to order them at all.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
>
> > If we calculate 12*a+12-7*n for n
> > which are congruent to 1 mod 12 in the range 1 to 553, we get 17.
In
> > fact, a standard val hn has the property that 12*hn(3/2)+12-7*n =
17
> > if and only if n is from 1 to 553 and congruent to 1 mod 12.
>
> Here are similar ranges for the other mod 12 congruences. We would
get
> different families for each of these ranges and congruences.
>
> 12*Fifth + 12 - 7*Octave for mod 12 families
>
> 0 mod 12
>
> 12 to 300, 12
> 312 to 912, 24
>
> 1 mod 12
>
> 1 to 553, 17
>
> 2 mod 12
>
> 2 to 194, 10
> 206 to 818, 22
>
> 3 mod 12
>
> 3 to 459, 15
>
> 4 mod 12
>
> 4 to 100, 8
> 112 to 712, 20
>
> 5 mod 12
>
> 5 to 353, 13
>
> 6 mod 12
>
> 6 to 606, 18
>
> 7 mod 12
>
> 7 to 247, 11
> 259 to 859, 23
>
> 8 mod 12
>
> 8 to 500, 16
>
> 9 mod 12
>
> 9 to 153, 9
> 165 to 765, 21
>
> 10 mod 12
>
> 10 to 406, 10
>
> 11 mod 12
>
> 11 to 47, 7
> 59 to 659, 19

I get 10 mod 12, 10 to 406, 14 if the pattern holds...

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> >
> > > If we calculate 12*a+12-7*n for n
> > > which are congruent to 1 mod 12 in the range 1 to 553, we get
17.
> In
> > > fact, a standard val hn has the property that 12*hn(3/2)+12-7*n
=
> 17
> > > if and only if n is from 1 to 553 and congruent to 1 mod 12.
> >
> > Here are similar ranges for the other mod 12 congruences. We
would
> get
> > different families for each of these ranges and congruences.
> >
> > 12*Fifth + 12 - 7*Octave for mod 12 families
> >
> > 0 mod 12
> >
> > 12 to 300, 12
> > 312 to 912, 24
> >
> > 1 mod 12
> >
> > 1 to 553, 17
> >
> > 2 mod 12
> >
> > 2 to 194, 10
> > 206 to 818, 22
> >
> > 3 mod 12
> >
> > 3 to 459, 15
> >
> > 4 mod 12
> >
> > 4 to 100, 8
> > 112 to 712, 20
> >
> > 5 mod 12
> >
> > 5 to 353, 13
> >
> > 6 mod 12
> >
> > 6 to 606, 18
> >
> > 7 mod 12
> >
> > 7 to 247, 11
> > 259 to 859, 23
> >
> > 8 mod 12
> >
> > 8 to 500, 16
> >
> > 9 mod 12
> >
> > 9 to 153, 9
> > 165 to 765, 21
> >
> > 10 mod 12
> >
> > 10 to 406, 10
> >
> > 11 mod 12
> >
> > 11 to 47, 7
> > 59 to 659, 19
>
> I get 10 mod 12, 10 to 406, 14 if the pattern holds...

Distances between top of each range, by value are given below. Is
there any significance to these numbers?

47 53 53 41 53 53 53 53 53
41 53 53 53 53 53 53 41 53

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:

> Distances between top of each range, by value are given below. Is
> there any significance to these numbers?
>
> 47 53 53 41 53 53 53 53 53
> 41 53 53 53 53 53 53 41 53

41 and 53 are the next denominators after 12 for convergents to
log2(3/2). 47??