12 of 22 EDO MOS

Paul,

I tried the 12 of 22 scale. I think if I were looking for a tuning
that would have to remain set, like a refretted guitar, I would
strongly consider 22 EDO. It suits my tastes better than 19. Or if
I needed to roam all over the lattice in a single piece of music, or
roam around a bit of a path, using mostly tetrads, I could see the
value of an EDO compromise. But it doesn't provide the strong
colors of unequal scales. It's tamer, less evocative. Since my
ideals are more in line with medieval, Gothic, and early Renaissance
music, colorful intervals have become more important to me than
approximating JI chords well, or traveling around the lattice as a
means of harmonic motion. With the restrictions I place on myself
for most of the music I make, I can get all JI intervals when I want
them without any temperamental compromises. IOW, I seldom need more
than 5 to 12 (octave equivalent) tones in a single piece of music,
usually only about 7.

The noble generators seem to provide a fairly consistent way of
sorting out some striking but sensible tunings from the nearly
endless range of possibilities. By "sensible" I mean that they make
sense to the ear. I imagine there are some other methods that would
return equally appealing sets. I might give the "silver" and phi
sets a go if ever I exhaust the noble ones. Dan Stearns suggested
those same divisions of IOE other than 2:1. But that opens up a can
of worms that I'm not ready for yet.

Thanks for the insight.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> I might give the "silver" and phi
> sets a go if ever I exhaust the noble ones.

Aren't the "phi sets" the same as the "noble ones" in this context?

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> The noble generators seem to provide a fairly consistent way of
> sorting out some striking but sensible tunings from the nearly
> endless range of possibilities.

Be sure to try a generator near 164 cents -- it's come up a lot lately
and I'll be playing its 7-tone MOS at the Microthon.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:
>
> > I might give the "silver" and phi
> > sets a go if ever I exhaust the noble ones.
>
> Aren't the "phi sets" the same as the "noble ones" in this context?

I could be confused but I'm refering to the scales in which the ratio
of the large to small steps is phi.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:
>
> > The noble generators seem to provide a fairly consistent way of
> > sorting out some striking but sensible tunings from the nearly
> > endless range of possibilities.
>
> Be sure to try a generator near 164 cents -- it's come up a lot
lately
> and I'll be playing its 7-tone MOS at the Microthon.

I'll do that. Wish I could be there to hear you! (That's both a
singular and plural "you.")

David Finnamore

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:

> > Aren't the "phi sets" the same as the "noble ones" in this
context?
>
> I could be confused but I'm refering to the scales in which the
ratio
> of the large to small steps is phi.

Which is exactly the one "magical" property of the noble generators
that I was pointing out to you, isn't it?

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:
>
> > > Aren't the "phi sets" the same as the "noble ones" in this
> context?
> >
> > I could be confused but I'm refering to the scales in which the
> ratio
> > of the large to small steps is phi.
>
> Which is exactly the one "magical" property of the noble generators
> that I was pointing out to you, isn't it?

Oh, I see. I was thrown by the fact that some of these have phi and
some phi+1. Either that or it's phi+1 and phi+2. I can't remember
whether phi is 0.618... or 1.618.... Since some were clearly
different from others, I assumed that they couldn't both be phi.
Which, strictly speaking, is true. Unless the definition of phi only
takes the decimal part into account?

Per your suggestion of a scale with a generator near 164c, I looked
at
the 3/22 member of the scale tree. It's generator is 163.6363c.
Seems you have to have at least 15 tones per octave to take advantage
of the MOS properties (to make it twice around the horagram). "Too
many notes" for me.

David Finnamore

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:
> --- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> > --- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:
> >
> > > > Aren't the "phi sets" the same as the "noble ones" in this
> > context?
> > >
> > > I could be confused but I'm refering to the scales in which the
> > ratio
> > > of the large to small steps is phi.
> >
> > Which is exactly the one "magical" property of the noble
generators
> > that I was pointing out to you, isn't it?
>
> Oh, I see. I was thrown by the fact that some of these have phi
and
> some phi+1. Either that or it's phi+1 and phi+2. I can't remember
> whether phi is 0.618... or 1.618....

It depends who you ask, but . . .

> Since some were clearly
> different from others, I assumed that they couldn't both be phi.
> Which, strictly speaking, is true. Unless the definition of phi
only
> takes the decimal part into account?

I think what you're missing, David, is that 1:0.618... is exactly
equal to 1.618...:1. Remember that equation for phi: x-1=1/x (or for
the "other" phi, x+1=1/x). So either 0.618... or 1.618... would be
the correct ratio.

Wait a minute -- are you saying some of the scales you're talking
about have L:s = 2.618 . . . ?

> Per your suggestion of a scale with a generator near 164c, I looked
> at
> the 3/22 member of the scale tree. It's generator is 163.6363c.

Umm . . . that's not noble, that's 3-out-of-22-tET, exactly as I use
it.

> Seems you have to have at least 15 tones per octave to take
advantage
> of the MOS properties (to make it twice around the horagram). "Too
> many notes" for me.

You must be looking at the wrong horagram (which would make sense
because there are no horagrams for 22-tET generators). This generator
produces a 7-tone MOS.

I wrote,

> Wait a minute -- are you saying some of the scales you're talking
> about have L:s = 2.618 . . . ?
>
Of course you are -- the 12-of-22 - like one, for example. Well,
these are the "extra" MOSs you get from a given nible generator
before the Fibonacci-like sequence kicks in (in this case, beginning
with 17, 22, 39, 61, 100 . . .). So the MOS scales with a phi:1 ratio
of step sizes is a subset (and an almost exhaustive one at that) of
the MOS scales with a noble generator.

As for the 3/22 MOS, I see you were probably not referring to a
Wilson horagram at all, but looked at it yourself. The 7-tone MOS I
referred to is not that special in this regard -- every number of
notes from 1 to 8 gives an MOS with this generator -- so maybe that's
why you jumped right to 15. You can see that same pattern for a noble
generator at http://www.anaphoria.com/hrgm07.html.

Paul Erlich wrote,

> You must be looking at the wrong horagram (which would make sense
because there are no horagrams for 22-tET generators). This generator
produces a 7-tone MOS.

Hey Paul, nice scale. Do you see this as some variation and tempering
of these?

5/3-----5/4
\ / \
11/6-/-11/8
\ / / \
1/1--/--3/2
\ /
11/10

5/3-----5/4
\ / \
20/11-/-15/11
\ / / \
1/1--/--3/2
\ /
12/11

Generalizing it as 6s1L and running through the apical algorithm as
1/6 0/1 Ls index gives an L/s = phi Golden scale of:

0 158 315 473 630 788 945 1200
0 158 315 473 630 788 1042 1200
0 158 315 473 630 885 1042 1200
0 158 315 473 727 885 1042 1200
0 158 315 570 727 885 1042 1200
0 158 412 570 727 885 1042 1200
0 255 412 570 727 885 1042 1200

An L/s = 2 Equal scale of:

0 150 300 450 600 750 900 1200
0 150 300 450 600 750 1050 1200
0 150 300 450 600 900 1050 1200
0 150 300 450 750 900 1050 1200
0 150 300 600 750 900 1050 1200
0 150 450 600 750 900 1050 1200
0 300 450 600 750 900 1050 1200

And an L/s = sqrt(2)+1 Silver scale of:

0 143 285 428 570 713 856 1200
0 143 285 428 570 713 1057 1200
0 143 285 428 570 915 1057 1200
0 143 285 428 772 915 1057 1200
0 143 285 630 772 915 1057 1200
0 143 487 630 772 915 1057 1200
0 344 487 630 772 915 1057 1200

These are probably not what you had in mind! In this case the 22-tET
version is probably best seen in terms of the Stern-Brocot or Wilson
Tree, where the apical 8-tET Equal scale is also the simplest mapping:

1/6 0/1
1/7
1/8
2/15 1/9
3/22 3/23 2/17 1/10

The 4/29ths seems like another nice generator if in fact a tempering
of something like the 1/1 12/11 5/4 15/11 3/2 5/3 20/11 2/1 is what
you had in mind.

--Dan Stearns

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> I wrote,
>
> > Wait a minute -- are you saying some of the scales you're talking
> > about have L:s = 2.618 . . . ?
> >
> Of course you are -- the 12-of-22 - like one, for example. Well,
> these are the "extra" MOSs you get from a given nible generator
> before the Fibonacci-like sequence kicks

I see. The scales from these same horagrams, but from higher rings,
have L:s=1.618.... My scale tone number limit kept me from getting
that far. Do those 2.618 scales still display MOS properties, but
not golden properties?

> As for the 3/22 MOS, I see you were probably not referring to a
> Wilson horagram at all, but looked at it yourself. The 7-tone MOS I
> referred to is not that special in this regard -- every number of
> notes from 1 to 8 gives an MOS with this generator -- so maybe
that's
> why you jumped right to 15. You can see that same pattern for a
noble
> generator at http://www.anaphoria.com/hrgm07.html.

'Zakly. There was no golden scale with a generator near 164c so I
plucked forbidden fruit from the scale-tree of the knowlege of good
and evil. :-) Anyone know where I can find a fig leaf with a 36
waist? Ahem, anyway...

If the notes of successive rings don't interlace somewhere in the
scale, you don't get the benefit of MOS, right? It's just a string
of same sized steps. I suppose you could make a 5 to 9 step scale
out of large and small steps in the first 15, but wouldn't the
choices be arbitrary?

Woah! So the scale-tree is showing logarithmic ratios, not frequency
ratios? He's not using the lambdoma as a set of JI intervals, then,
after all. This is interesting.

David Finnamore

David Finnamore,

> Per your suggestion of a scale with a generator near 164c, I looked
at the 3/22 member of the scale tree. It's generator is 163.6363c.

Paul Erlich,

> Umm . . . that's not noble, that's 3-out-of-22-tET, exactly as I use
it.

I think David meant the 1/8 1/7, 162.5583 phi generator.

David Finnamore,

> Seems you have to have at least 15 tones per octave to take
advantage of the MOS properties (to make it twice around the
horagram).

Paul Erlich,

> You must be looking at the wrong horagram (which would make sense
because there are no horagrams for 22-tET generators).

Hmm..., 1:2^(3/22) is the apical Equal generator for the 15-tone 1/8
1/7 Ls index. Here's the Golden, Equal and Silver scales:

0
1037 163
875 325
712 488
550 650
387 813
225 975
62 1138
1100 100
937 263
774 426
612 588
449 751
287 913
124 1076

0
1036 164
873 327
709 491
545 655
382 818
218 982
55 1145
1091 109
927 273
764 436
600 600
436 764
273 927
109 1091

0
1035 165
871 329
706 494
542 658
377 823
213 987
48 1152
1084 116
919 281
755 445
590 610
425 775
261 939
96 1104

Using a 6s1L subset of the 1/8 1/7 Ls index would give an L/s phi
relation of 3-((sqrt(5)+1)/2). Here's the resulting scale as a
theoretical:

5/3-----5/4
\ / \
11/6-/-11/8
\ / / \
1/1--/--3/2
\ /
11/10

0 163 387 550 712 875 1037 1200
0 225 387 550 712 875 1037 1200
0 163 325 488 650 813 975 1200
0 163 325 488 650 813 1037 1200
0 163 325 488 650 875 1037 1200
0 163 325 488 712 875 1037 1200
0 163 325 550 712 875 1037 1200

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

> Hey Paul, nice scale. Do you see this as some variation and
tempering
> of these?
>
> 5/3-----5/4
> \ / \
> 11/6-/-11/8
> \ / / \
> 1/1--/--3/2
> \ /
> 11/10
>
> 5/3-----5/4
> \ / \
> 20/11-/-15/11
> \ / / \
> 1/1--/--3/2
> \ /
> 12/11

I think you're on the right track -- if you call the scale

G A B C D E F G
4 3 3 3 3 3 3

then I'm using GABCD as an 8:9:10:11:12, FABC as an 8:10:11:12, and
DFA and EGB as 10:12:15. So that would be

D------,A
\ ,-'/ \
E---/--,B
\ /,-'/ \
F------,C
\ /,-' \
G-------D------,A
\ ,-'/ \
E------,B
\ /,-'/ \
F---/--,C
\ /,-' \
G-------D-etc.

In conjunction with the 7-tone hyper-Pythagorean scale centered on
Ab, tuned halfway between G and A, I also get a 4:5:6:7:9:11 as Bb-D-
F-Ab-C-E and a 4:5:6:7:11 as F-A-C-Eb-B.

> The 4/29ths seems like another nice generator if in fact a tempering
> of something like the 1/1 12/11 5/4 15/11 3/2 5/3 20/11 2/1 is what
> you had in mind.

Perhaps!

--- In tuning@egroups.com, "David Finnamore" <daeron@b...> wrote:
> --- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> > I wrote,
> >
> > > Wait a minute -- are you saying some of the scales you're
talking
> > > about have L:s = 2.618 . . . ?
> > >
> > Of course you are -- the 12-of-22 - like one, for example. Well,
> > these are the "extra" MOSs you get from a given nible generator
> > before the Fibonacci-like sequence kicks
>
> I see. The scales from these same horagrams, but from higher
rings,
> have L:s=1.618.... My scale tone number limit kept me from getting
> that far. Do those 2.618 scales still display MOS properties, but
> not golden properties?

Every ring in the horagram is an MOS.

> If the notes of successive rings don't interlace somewhere in the
> scale, you don't get the benefit of MOS, right?

I don't see what you mean.

> It's just a string
> of same sized steps.

Almost -- one step is different. That's OK, though, you still get
the "benefit" of MOS. There's nothing wrong with a string of same
sized steps -- the diatonic scale has a string of three whole
steps . . .
>
> Woah! So the scale-tree is showing logarithmic ratios, not
frequency
> ratios? He's not using the lambdoma as a set of JI intervals,
then,
> after all.

Not here, but in other applications, he does (according to Kraig
Grady).