23-and-29 only, and sagittal ET notation

First of all, has anyone noticed that 29/16 is one cent away from six steps of 7 equal?
Ain't that something? Two implications I see:

1. 7 equal could be notated with ABCDEFG defined as a "circle" of sevenths, or
seconds (32/29's), and its lower multiples with accidentals in accordance with some
non-3 lattice.

2. In particular, a 21-note periodicity block in 23,29 only, defined by "unison vectors"
23^(-3)2*29^3 and 23^3 29^(-4):

0: 1/1 0.000 unison, perfect prime
1: 24389/23552 60.457
2: 560947/524288 117.006
3: 594823321/536870912 177.463
4: 841/736 230.880
5: 19343/16384 287.429
6: 20511149/16777216 347.886
7: 29/23 401.303
8: 707281/541696 461.760
9: 707281/524288 518.309
10: 374151649/268435456 574.857
11: 24389/16928 632.183
12: 24389/16384 688.732
13: 12901781/8388608 745.280
14: 841/529 802.606
15: 841/512 859.154
16: 20511149/12058624 919.612
17: 471756427/268435456 976.160
18: 29/16 1029.577 29th harmonic
19: 707281/376832 1090.034
20: 16267463/8388608 1146.583
21: 2/1 1200.000 octave

Interesting. No more than 6 cents away from 21-equal. Also points out a nice way of
doing multiples of 3-equal...29/23.

Fun,
Jacob

hi Jacob,

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:

> First of all, has anyone noticed that 29/16 is one cent
> away from six steps of 7 equal?
> Ain't that something? Two implications I see:
>
> 1. 7 equal could be notated with ABCDEFG defined as a
> "circle" of sevenths, or seconds (32/29's), and its lower
> multiples with accidentals in accordance with some
> non-3 lattice.
>
> 2. In particular, a 21-note periodicity block in 23,29 only,
> defined by "unison vectors" 23^(-3)2*29^3 and 23^3 29^(-4):
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 24389/23552 60.457
> 2: 560947/524288 117.006
> 3: 594823321/536870912 177.463
> 4: 841/736 230.880
> 5: 19343/16384 287.429
> 6: 20511149/16777216 347.886
> 7: 29/23 401.303
> 8: 707281/541696 461.760
> 9: 707281/524288 518.309
> 10: 374151649/268435456 574.857
> 11: 24389/16928 632.183
> 12: 24389/16384 688.732
> 13: 12901781/8388608 745.280
> 14: 841/529 802.606
> 15: 841/512 859.154
> 16: 20511149/12058624 919.612
> 17: 471756427/268435456 976.160
> 18: 29/16 1029.577 29th harmonic
> 19: 707281/376832 1090.034
> 20: 16267463/8388608 1146.583
> 21: 2/1 1200.000 octave
>
> Interesting. No more than 6 cents away from 21-equal.
> Also points out a nice way of doing multiples of 3-equal...29/23.
>
> Fun,
> Jacob

i made graphs a long time ago of chains of prime intervals.
i.e., a chain of 3/2s, of 5/4s, of 7/4s, of 11/8s, etc.

looks like i never put these on my website ... maybe if
i find them i can. they're in my book.

anyway, here's the list of monzos for your scale:
(in descending order as usual with me; on the Yahoo
interface, click "Reply" to view properly)

2,23,29-monzo

21: [ 1, 0, 0 >
20: [-23, 1, 4 >
19: [-14, -1, 4 >
18: [ -4, 0, 1 >
17: [-28, 1, 5 >
16: [-19, -1, 5 >
15: [ -9, 0, 2 >
14: [ 0, -2, 2 >
13: [-23, 2, 3 >
12: [-14, 0, 3 >
11: [ -5, -2, 3 >
10: [-28, 2, 4 >
9: [-19, 0, 4 >
8: [-10, -2, 4 >
7: [ 0, -1, 1 >
6: [-24, 0, 5 >
5: [-14, 1, 2 >
4: [ -5, -1, 2 >
3: [-29, 0, 6 >
2: [-19, 1, 3 >
1: [-10, -1, 3 >
0: [ 0, 0, 0 >

8ve-equivalent 23,29-primespace lattice:
(numbers are scale-degrees)

<<< 29 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
0 1 2 3 4 5 6

^ 2 ------------------13---10-------------
^ | | | | | | |
^ 1 --------------5----2---20---17-------
^ | | | | | | |
23 0 --0=21--18---15---12----9----6----3---
v | | | | | | |
v -1 ---------7----4----1---19---16--------
v | | | | | | |
v -2 -------------14---11----8-------------

-monz

hi Jacob,

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
>
> > 2. In particular, a 21-note periodicity block in 23,29 only,
> > defined by "unison vectors" 23^(-3)2*29^3 and 23^3 29^(-4):

are you sure about those unison-vectors?

the first one is OK, but the second one looks a
little strange. for one thing, it's missing 2^6,
which it needs in order to be close to a unison.
for another thing, it's bigger than 166 cents,
which isn't much of a unison anyway.

here's my tabulation of them:

2,23,29-monzo ratio ~cents

[-1 -3 3 > 24389 / 24334 3.908540654
[ 6 3 -4 > 778688 / 707281 166.5142652

-monz

hi Jacob,

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> --- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> >
> > > 2. In particular, a 21-note periodicity block in 23,29 only,
> > > defined by "unison vectors" 23^(-3)2*29^3 and 23^3 29^(-4):
>
>
> are you sure about those unison-vectors?
>
> the first one is OK, but the second one looks a
> little strange. for one thing, it's missing 2^6,
> which it needs in order to be close to a unison.
> for another thing, it's bigger than 166 cents,
> which isn't much of a unison anyway.
>
> here's my tabulation of them:
>
> 2,23,29-monzo ratio ~cents
>
> [-1 -3 3 > 24389 / 24334 3.908540654
> [ 6 3 -4 > 778688 / 707281 166.5142652

you meant for the second one to be this:

[-33 3 4 > 7470507 / 7457005 3.131818418

or in your notation: 2^(-33) 23^3 29^4

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> hi Jacob,
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > --- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> > >
> > > > 2. In particular, a 21-note periodicity block in 23,29 only,
> > > > defined by "unison vectors" 23^(-3)2*29^3 and 23^3 29^(-4):
> >
> >
> > are you sure about those unison-vectors?
> >
> > the first one is OK, but the second one looks a
> > little strange. for one thing, it's missing 2^6,
> > which it needs in order to be close to a unison.
> > for another thing, it's bigger than 166 cents,
> > which isn't much of a unison anyway.
> >
> > here's my tabulation of them:
> >
> > 2,23,29-monzo ratio ~cents
> >
> > [-1 -3 3 > 24389 / 24334 3.908540654
> > [ 6 3 -4 > 778688 / 707281 166.5142652
>
>
>
> you meant for the second one to be this:
>
> [-33 3 4 > 7470507 / 7457005 3.131818418
>
>
> or in your notation: 2^(-33) 23^3 29^4

so the lattice would show the unison-vectors like this:

8ve-equivalent 23,29-primespace lattice:
(numbers are scale-degrees)

<<< 29 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
0 1 2 3 4 5 6

^ 3 ---------------------(0=21)-----------
^ | | | | | | |
^ 2 ------------------13---10-------------
^ | | | | | | |
^ 1 --------------5----2---20---17-------
^ | | | | | | |
23 0 --0=21--18---15---12----9----6----3---
v | | | | | | |
v -1 ---------7----4----1---19---16--------
v | | | | | | |
v -2 -------------14---11----8-------------
v | | | | | | |
v -3 ----------------(0=21)----------------

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Jacob,
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > --- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> > >
> > > > 2. In particular, a 21-note periodicity block in 23,29 only,
> > > > defined by "unison vectors" 23^(-3)2*29^3 and 23^3 29^(-4):
> >
> >
> > are you sure about those unison-vectors?
> >
> > the first one is OK, but the second one looks a
> > little strange. for one thing, it's missing 2^6,
> > which it needs in order to be close to a unison.
> > for another thing, it's bigger than 166 cents,
> > which isn't much of a unison anyway.
> >
> > here's my tabulation of them:
> >
> > 2,23,29-monzo ratio ~cents
> >
> > [-1 -3 3 > 24389 / 24334 3.908540654
> > [ 6 3 -4 > 778688 / 707281 166.5142652
>
>
>
> you meant for the second one to be this:
>
> [-33 3 4 >

I sure did. But I get 8605487927/8589934592, not

> 7470507 / 7457005
>

Weird. But, yeah. Typo.

Jacob

From: "Jacob" <jbarton@r>
> First of all, has anyone noticed that 29/16 is one cent away from six
steps of 7 equal?
> Ain't that something?

Great. It's just as you said. 32/29 is about a cent flatter than one step of
7-equal, and therefore, of course, (32/29)^14 is 14 cents flatter than 2
octaves. And what about the fact that 39/32 is close to two steps of
7-equal? If you're interested, I can tell you that (39/32)^7 is only ~2.6
cents flatter than 2 octaves (or 13^7 * 3^7 / 2^37). So if I wanted to make
something like a 7-tone well-temperament, I'd definitely go this way.
Incidentally, your idea of approximating 7-equal made me write a new scale
approximating 12-equal (maybe Monz could be interested). Unlike my previous
scales approximating 12-equal, this one does not use such primes as 17 or
19. On the other hand, I managed to use the primes of 7 and 11 here, which
are not found in any of the previous rational well-temps. How did I get it?
Just like this:

! 11lwt.scl
!
11-limit Rational Well-temperament
12
!
1323/1250
55/49
297/250
63/50
4/3
99/70
220/147
100/63
42/25
5500/3087
66/35
2/1

Petr