naughty and nice [12 note scales]

OUCH!

What was I going to say?? No, I just sat on my keyboard...

I wanted to thank both Paul Erlich and Kraig Grady for their interesting
scales which, comforting to me, included 12 notes per octave (whew!)

I'm assuming, Paul that the original 22 note scale that you are
extracting your 12-note scale from is the following:

0: 1/1 0.000
1: 50.250 cents 50.250
2: 105.750 cents 105.750
3: 161.250 cents 161.250
4: 211.500 cents 211.500
5: 272.250 cents 272.250
6: 322.500 cents 322.500
7: 383.250 cents 383.250
8: 428.250 cents 428.250
9: 494.250 cents 494.250
10: 539.250 cents 539.250
11: 594.750 cents 594.750
12: 650.250 cents 650.250
13: 705.750 cents 705.750
14: 761.250 cents 761.250
15: 816.750 cents 816.750
16: 872.250 cents 872.250
17: 917.250 cents 917.250
18: 983.250 cents 983.250
19: 1028.250 cents 1028.250
20: 1089.000 cents 1089.000
21: 1139.250 cents 1139.250
22: 2/1 1200.000 octave

Please correct me if I'm wrong. (Very possible)

Then we apply the correctly spelled erl22.kbm to get our just 7-limit
scale with repeating octave at the 12th pitch...

BTW the other two files were two22.kbm and two22b.kbm to get 22 notes
from the decatonics... there is no two22a.kbm. My bad, as JM says...

Now this compares very nicely with Kraig Grady's, "Centaur" scale, which
I tuned up.

Hey, you know these two scales are really fairly close to our
traditional 12-tET... at least the way my ears are on at the moment...
(Maybe wrong) And if I'm doing everything right. Well, I mean, of
course they are in 7-limit just... but closer than some of the other
scales I have been "messing" with...

As I mentioned, there is something really comforting about 12 notes per
octave and repeating octaves at 12... particularly among the
"keyboard-centric" set, in which I, for better or worse, number... It
also makes it easier to transcribe such pieces for traditional
instruments, if one cares to do that. (Not, obviously, for everybody).

HOWEVER, at the moment I am fixated on the following... a scale which
has 12 notes per octave but which is constructed of hexanys! I guess
these would be "stellated hexanies," yes??

The scale is in Scala, and it is called "hexanys.scl." It is as
follows:

Hexanys 1 3 5 7 9
0: 1/1 0.000 unison, perfect prime
1: 35/32 155.140 septimal neutral second
2: 9/8 203.910 major whole tone
3: 5/4 386.314 major third
4: 21/16 470.781 narrow fourth
5: 45/32 590.224 tritone
6: 3/2 701.955 perfect fifth
7: 105/64 857.095 septimal neutral sixth
8: 27/16 905.865 Pythagorean major sixth
9: 7/4 968.826 harmonic seventh
10: 15/8 1088.269 classic major seventh
11: 63/32 1172.736 octave - septimal comma
12: 2/1 1200.000 octave

It seems there are MANY interesting things about this scale. For one
thing, there are FIVE hexanys contained within it!

John Chalmers showed me this... I was going to make you believe I
figured it out for myself, but "chickened out" :-)

Hexany 1.3.5.7: 32/35 5/4 21/16 3/2 7/4 15/8

Hexany 1.3.5.9: 9/8 5/4 45/32 3/2 27/16 15/8

Hexany 1.3.7.9: 9/8 21/16 3/2 27/16 7/4 63/32

Hexany 1.5.7.9: 35/32 9/8 5/4 45/32 7/4 63/32

Hexany 3.5.7.9: 35/32 21/16 45/32 27/16 15/8 63/32

Whoopie!

This means, of course, that the pre-compositional composer can have a
"field day..." One can begin with a certain hexany and then "mutate" it
to another near form... one can "transpose" by LINKED common-notes
between hexanys, etc.

Or, one could just listen to the sounds... not entirely prohibited, but
for some, perhaps unadvisable.

A funny property of this scale for the "keyboard fixated" is the fact
that our keyboard "major chord" C-E-G-C comes out as the subdominant
(using a "narrow fourth" and a "septimal neutral sixth") and the "major
triad" of the root can be easily made by C-Eb-Gb-C (a just major third
and perfect fifth!)... so there can be a kind of "plagal cadence" action
going on... although I doubt seriously I would use the scale like
that...

It also doesn't seem that hard to "transcribe" to traditional 12-tET
staff notation, using quartertones and cents deviation... but I know
there are people on this list who feel this is very "naughty, naughty!"

____________ ______ ___ __ __
Joseph Pehrson

--- In tuning@egroups.com, Joseph Pehrson <josephpehrson@c...> wrote:
> OUCH!

> I'm assuming, Paul that the original 22 note scale that you are
> extracting your 12-note scale from is the following:
>
> 0: 1/1 0.000
> 1: 50.250 cents 50.250
> 2: 105.750 cents 105.750
> 3: 161.250 cents 161.250
> 4: 211.500 cents 211.500
> 5: 272.250 cents 272.250
> 6: 322.500 cents 322.500
> 7: 383.250 cents 383.250
> 8: 428.250 cents 428.250
> 9: 494.250 cents 494.250
> 10: 539.250 cents 539.250
> 11: 594.750 cents 594.750
> 12: 650.250 cents 650.250
> 13: 705.750 cents 705.750
> 14: 761.250 cents 761.250
> 15: 816.750 cents 816.750
> 16: 872.250 cents 872.250
> 17: 917.250 cents 917.250
> 18: 983.250 cents 983.250
> 19: 1028.250 cents 1028.250
> 20: 1089.000 cents 1089.000
> 21: 1139.250 cents 1139.250
> 22: 2/1 1200.000 octave
>
> Please correct me if I'm wrong. (Very possible)
>
I'm afraid you're wrong. It's supposed to be 22-tET, or 22-tone equal
temperament.

> Then we apply the correctly spelled erl22.kbm to get our just 7-
limit
> scale with repeating octave at the 12th pitch...
>
> BTW the other two files were two22.kbm and two22b.kbm to get 22
notes
> from the decatonics... there is no two22a.kbm. My bad, as JM
says...
>
> Now this compares very nicely with Kraig Grady's, "Centaur" scale,
which
> I tuned up.
>
> Hey, you know these two scales are really fairly close to our
> traditional 12-tET... at least the way my ears are on at the
moment...
> (Maybe wrong) And if I'm doing everything right. Well, I mean, of
> course they are in 7-limit just...

Well mine is not just of course, with its 17-cent errors and all.
Once you have the erl.kbm scale tuned right, try dominant sevenths
with the root, fifth, and seventh on one color and the third on the
other color (I'm talking about white keys and black keys as the two
colors). Then try the mirror versions of those chords (i.e., the way
they would appear in a mirror placed next to your keyboard). What is
your reaction? Do you like them better than in 12-tET? Try the other
tetrads -- do some sound more dissonant to you? Do a few of them
sound almost consonant?

--- In tuning@egroups.com, Joseph Pehrson <josephpehrson@c...> wrote:

> HOWEVER, at the moment I am fixated on the following... a scale
which
> has 12 notes per octave but which is constructed of hexanys! I
guess
> these would be "stellated hexanies," yes??

Not exactly -- hexanies are stellated with tetrads, constructed from
the same set of factors.
>
> The scale is in Scala, and it is called "hexanys.scl." It is as
> follows:
> [snip]

Let me lattice this scale so that we can see it harmonically in the 7-
limit (9-limit is ignored for now):

35/32----105/64
,'/ \`. ,'/ \`.
5/4-/---\15/8-/---\45/32
/|\/ \/|\/ \/|\
/ |/\ /\|/\ /\| \
/ 7/4------21/16-----63/32\
/,' `.\ /,' `.\ /,' `.\
1/1-------3/2-------9/8------27/16

So only one note is different from either of Paul Hahn's scales that
I posted earlier, up to a transposition (which is arbitrary anyway).
Again, there are not one but _two_ 1.3.5.7 hexanies.

Back at the end of the Mills era we were discussing 12-tone subsets
of 7-limit JI and how many 7-limit-consonant intervals they could
have. Paul Hahn was the winner -- he found 24 scales with 32 7-limit-
consonant intervals each. They are the 24 possible
rotations/reflections of this:

5:3---------5:4
/|\ /|\
/ | \ / | \
/ | \ / | \
/ 7:6---------7:4 \
/.-'/ \'-.\ /.-'/ \'-.\
4:3--/---\--1:1--/---\--3:2
|\ / \ /|\ / \ /|
| / \ | / \ |
|/ \ / \|/ \ / \|
28:15--------7:5--------21:20
'-.\ /.-' '-.\ /.-'
8:5---------6:5

As with most of the JI scales in this thread, these each have two
hexanies and four tetrads. But they are all fairly uneven -- probably
none of them have the Constant Structures property.