Yahoo Tuning Groups Ultimate Backup tuning Bobs one true tuning for the rest of his life...

I mused and Monz responded...

> >
> > ... It may well be that by the time I get to a
> > 14-o-o-131ED(22/3), that I'll be playing with a
> > random collection of just intervals, but thats
> > a few years out.
>
>
> This looked like a good tuning puzzle which I
> couldn't resist... but my question is: which 14
> pitches would you choose as a subset, Bob?
>

Monz,

You honored my comments much more than they merited. Now
I am compelled to look more deeply into this, however...

This scale is a few years awaw on my personal/musical
path and there are only a few things that I know about
it.

1) it cannot be expressed numerically
2) it cannot be analyzed using any mechanical means
3) it is natural
3b) (the natural laws governing it are also relevent
across subatomic physics and cosmology and may
apply to nail polish as well)

But to show you that I did honor your efforts, this
"shimmery diminished stack" is based on the
little know musical identity that

15/7 + 8/5 + 15/7 = 22/3.

9 10 12 10 9 10 11 10 9 10 12 10 9
0 9 19 31 41 50 61 72 82 91 101 113 123 131

Note a very good 6/5 at 12/131 ED(22/3)
4/3 at 19/131 ED(22/3)
okay 7/6 at 10/131...

hmmm...wheres that fret saw...

Bob Valentine

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> ----- Original Message -----
> From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 11, 2001 3:41 AM
> Subject: [tuning] Bobs one true tuning for the rest of his life...
>

>
> But to show you that I did honor your efforts, this
> "shimmery diminished stack" is based on the
> little know[n] musical identity that
>
> 15/7 + 8/5 + 15/7 = 22/3.

Hi Bob,

It looks like you're using approximations here
without saying so.

The only way this expression makes any sense is
if one assumes:

- that these fractions are musical interval ratios
(since they supposedly add up to the ratio you're
dividing to create your scale),

- that (as Graham hinted at) your plus signs really
mean multiplication, and

- that your equal sign is accomodating an error
of ~3.208961158 cents in the final result.

15/7 * 8/5 * 15/7 = 360/49 = ~7.346938776 = ~3452.571903 cents

22/3 = ~3449.362941 cents

(22/3)^(50/131) is ~2.892067231 cents smaller than 15:7.
(22/3)^(31/131) is ~2.575173304 cents larger than 8:5.

So (22/3)^(50/131) * (22/3)^(31/131) * (22/3)^(50/131)
*does* equal 22:3. That's easy to see by doing vector
addition on the exponents: (50 + 31 + 50) / 131 = 1,
and so (22/3)^1 = 22/3.

>
>
> 9 10 12 10 9 10 11 10 9 10 12 10 9
> 0 9 19 31 41 50 61 72 82 91 101 113 123 131

Er... Bob, your arithmetic is off here.

The actual interval size between the scale steps you give,
in 131-ED(22/3) degrees, is:

9 10 12 10 9 11 11 10 9 10 12 10 8
0 9 19 31 41 50 61 72 82 91 101 113 123 131

But I don't think this is where you made the error, as it
destroys the beautiful symmetry of the between-step sizes
you did give. So I think the scale you meant was:

9 10 12 10 9 10 11 10 9 10 12 10 9
0 9 19 31 41 50 60 71 81 90 100 112 122 131

>
> Note a very good 6/5 at 12/131 ED(22/3)
> 4/3 at 19/131 ED(22/3)
> okay 7/6 at 10/131...

Interesting.

(22/3)^(100/131) and (22/3)^(122/131) are nearly identical to
(22/3)^(9/131) and (22/3)^(31/131), respectively.

So since you're not substituting the near-"octave"-equivalents,
by "shimmery diminished stack" it appears that you're
thinking of this in terms of these actual intervals, so
that it's a non-"octave" (and non-"octave"-equivalent) scale
made up of a series of various-sized "diminished 3rds".

The nearest low-integer rational approximations for these
"diminished 3rds" are:

131-ED(22/3) ratio ~cents ratio ~cents ratio ~cents
degree error error error

(22/3)^(12/131) ~= 6:5 -0.33
(22/3)^(11/131) ~= 13:11 -0.43
(22/3)^(10/131) ~= 7:6 +3.56 78:67 -0.13
(22/3)^( 9/131) ~= 8:7 -5.81 39:34 +0.55 47:41 -0.53

And here's the interval matrix for the non-"octave" version
of the scale (broken in half to fit on screen), in Semitones:

131 122 112 100 90 81 71
131 0.00
122 2.37 0.00
112 5.00 2.63 0.00
100 8.16 5.79 3.16 0.00
90 10.80 8.43 5.79 2.63 0.00
81 13.17 10.80 8.16 5.00 2.37 0.00
71 15.80 13.43 10.80 7.64 5.00 2.63 0.00
60 18.70 16.33 13.69 10.53 7.90 5.53 2.90
50 21.33 18.96 16.33 13.17 10.53 8.16 5.53
41 23.70 21.33 18.70 15.54 12.90 10.53 7.90
31 26.33 23.96 21.33 18.17 15.54 13.17 10.53
19 29.49 27.12 24.49 21.33 18.70 16.33 13.69
9 32.12 29.75 27.12 23.96 21.33 18.96 16.33
0 34.49 32.12 29.49 26.33 23.70 21.33 18.70

60 50 41 31 19 9 0
60 0.00
50 2.63 0.00
41 5.00 2.37 0.00
31 7.64 5.00 2.63 0.00
19 10.80 8.16 5.79 3.16 0.00
9 13.43 10.80 8.43 5.79 2.63 0.00
0 15.80 13.17 10.80 8.16 5.00 2.37 0.00

So in addition to the ubiquitous approximations of
6:5s, 13:11s, 7:6s, and 8:7s between all the degrees
of your scale (as described above), you've also got
terrific approximations of many other familiar
"within the octave" ratios:

~ratio ED(22/3) interval ~cents error

14:9 (22/3)^(100/131) / (22/3)^(71/131) -1.316474941
(22/3)^(60/131) / (22/3)^(31/131)

11:8 (22/3)^(81/131) / (22/3)^(60/131) +1.633368868
(22/3)^(71/131) / (22/3)^(50/131)

13:8 (22/3)^(41/131) / (22/3)^(9/131) +2.06481249
(22/3)^(122/131) / (22/3)^(90/131)

8:5 (22/3)^(131/131) / (22/3)^(100/131) +2.575173304
(22/3)^(112/131) / (22/3)^(81/131)
(22/3)^(81/131) / (22/3)^(50/131)
(22/3)^(50/131) / (22/3)^(19/131)
(22/3)^(31/131) / (22/3)^(0/131)

(Of course the error would be less by a skhisma if
approximating 6561/4096 [== 3^8] instead of 8:5, so
it would be only ~0.621452516 cents.)

128:81 (22/3)^(90/131) / (22/3)^(60/131) -2.24955192
[3^-4] (22/3)^(7/131) / (22/3)^(41/131)

4:3 (22/3)^(131/131) / (22/3)^(112/131) +2.244282457
(22/3)^(100/131) / (22/3)^(81/131)
(22/3)^(90/131) / (22/3)^(71/131)
(22/3)^(60/131) / (22/3)^(41/131)
(22/3)^(50/131) / (22/3)^(31/131)
(22/3)^(19/131) / (22/3)^(0/131)

7:5 (22/3)^(122/131) / (22/3)^(100/131) -3.229866551
(22/3)^(112/131) / (22/3)^(90/131)
(22/3)^(41/131) / (22/3)^(19/131)
(22/3)^(31/131) / (22/3)^(9/131)

(22/3)^(131/131) / (22/3)^(100/131)

11:6 (22/3)^(100/131) / (22/3)^(60/131) +3.877651325
(22/3)^(90/131) / (22/3)^(50/131)
(22/3)^(81/131) / (22/3)^(41/131)
(22/3)^(71/131) / (22/3)^(31/131)

Of course, "octave"-reducing the basic 22:3 interval
you're dividing gives you none other than 11:6.

And several instances of the same decent approximation
to the 5-limit "major 7th":

15:8 (22/3)^(131/131) / (22/3)^(90/131) -8.697107085
(22/3)^(122/131) / (22/3)^(81/131)
(22/3)^(112/131) / (22/3)^(71/131)
(22/3)^(60/131) / (22/3)^(19/131)
(22/3)^(50/131) / (22/3)^(9/131)
(22/3)^(41/131) / (22/3)^(0/131)

Now Bob, since I've accomplished all this numerical
analysis, I can only imagine that when you write:

> 1) it cannot be expressed numerically
> 2) it cannot be analyzed using any mechanical means
> 3) it is natural
> 3b) (the natural laws governing it are also relevent
> across subatomic physics and cosmology and may
> apply to nail polish as well)

you're really saying that 131-ED(22/3) is *also* merely
an approximation to the tuning you actually have in mind.

No sweat... take a few years to get to it. I think I'll
play around with *this* one for a bit. :)

-monz
http://www.monz.org
"All roads lead to n^0"

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🔗monz <joemonz@yahoo.com>

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

In discussion between > > Bob and > Monz
>
> >
> > 15/7 + 8/5 + 15/7 = 22/3.
>
>
> Hi Bob,
>
> It looks like you're using approximations here
> without saying so.
>

Yes, as you and Graham surmised I was thinking
something more like

~15/7 * ~8/5 * ~15/7 = 22/3

You correctly fixed my "off by one" error in
the "symmetric sort of diminished stack".

9 10 12 10 9 10 11 10 9 10 12 10 9
0 9 19 31 41 50 60 71 81 90 100 112 122 131

> So since you're not substituting the near-"octave"-equivalents,
> by "shimmery diminished stack" it appears that you're
> thinking of this in terms of these actual intervals, so
> that it's a non-"octave" (and non-"octave"-equivalent) scale
> made up of a series of various-sized "diminished 3rds".
>
> The nearest low-integer rational approximations for these
> "diminished 3rds" are:
>
> 131-ED(22/3) ratio ~cents ratio ~cents ratio ~cents
> degree error error error
>
> (22/3)^(12/131) ~= 6:5 -0.33
> (22/3)^(11/131) ~= 13:11 -0.43
> (22/3)^(10/131) ~= 7:6 +3.56 78:67 -0.13
> (22/3)^( 9/131) ~= 8:7 -5.81 39:34 +0.55 47:41 -0.53
>

Yes this was exactly my process. I guess I accepted the "offness"
of the 7:6 and 8:7 approximations because the 4:3 resultant was
pretty good (12ED2 to a half a cent).

In fact, the first thing I noticed was that the sequence

0 19 31 50 60

gave a

30:40:48:64:75

(sort of a Dbmaj7#9 with a bit of voicing magic going on)

C : 1 4 b6 b9 10 = Db : 7 3 5 1 #9

and then I tried to fill it pretilly to 14 notes. This
almost tells you everything about the way I
think at this point in my extremely limited microtonal
career.

I made the rather non-startling realization during this
exchange that a non-octave based tuning won't necessarily
have the inversion of an interval present... kool. So
I was surprised to see lots of good minor thirds, an ~8/5
but nothing that I wanted to use for a ~5/4 (395c isn't
all that awful, but it was beyond the error bounds that
I seemed to be using for no good reason).

Perhaps the great spirit took over my hands when I wrote
what I thought was a somewhat senseless scale description,
and really HAS handed me the one true path to follow!

Follow-ups to my new newsgroup. It has no name, but you'll
know when you're posting there.

Bob Valentine