Phi MOS

Dear friends,

I wanted to take an evening and drop in to give another humble
offering of Phi scales. Just reading some of the exchanges about the
Golden Horograms, and appreciating the noble efforts of David
Finnamore in working to bring deeper understanding of Wilson's Golden
Structures, has shaken me out of my busy mode of musical production,
prompting me to offer what may be for me, a final transposition of
the logic behind the Self Similar Phi scales I have posted in the
past.

For those who have followed and contributed to the thread about the
Self Similar scales, you will recall that the structures posted, had
scale degrees which were obtained by a scheme of iterative division
of the Interval of Equivalence (or "repeat ratio" in cents) by the
constant itself, which yielded scales with small steps near the outer
edges and larger intervals in the center of the scale; all in
relation to Phi itself - this way, one could actually play an
expanding set of pitches based on the constant proportions.

Here, let's look at a few more examples of the wonderful properties
of Phi, where we will see these large scale steps divided further by
the Phi constant, which invariably will generate either a Moment of
Symmetry, or else scales with exactly 2 different step sizes, and
which are always Constant Structures.

As a refresher of this concept, let's look at a basic Self Similar
Phi Scale, so that we may see how the large scale steps may be
divided. If we take the ratio Phi, and divide the cents values
iteratively by Phi; stopping where we see the appearance of the first
semitone, we get the following structure:

833.090
514.878
318.212
196.666
121.546

And when we multiply Phi by itself, then subtract each interval above
from this, we get the following structure ascending above the ratio
Phi:

1029.756
1151.302
1226.422
1347.968

Now we can see that the large steps are as follows:

Phi Steps Consecutive Large intervals.
514.878 196.666
833.090 318.212
1029.756 196.666

So here if we again iteratively divide these large intervals by Phi,
we get the following Phi Non-Octave MOS:

Phi Non-Octave MOS
Cents Consecutive
0.000
121.546 121.546
196.666 75.120
318.212 121.546
393.332 75.120
514.878 121.546
636.424 121.546
711.544 75.120
833.090 121.546
908.210 75.120
1029.756 121.546
1151.302 121.546
1226.422 75.120
1347.968 121.546

This is probably one of my favorite Phi scales I've ever created!
This Phi MOS is so full of Phi relationships, that it's bursting at
the seams with Golden properties.

And now, as if by some Golden Magic, we use the 514.878 and 711.544
cents generators of this "primal" Phi scale, to create two Golden Non-
Octave MOS scales, we get the following:

Phi 4th MOS
Cents Consecutive
0.000
121.546 121.546
243.092 121.546
318.212 75.120
439.758 121.546
514.878 75.120
636.424 121.546
757.970 121.546
833.090 75.120
954.636 121.546
1029.756 75.120
1151.302 121.546
1226.422 75.120

Phi Fifth MOS
Cents Consecutive
0.000
75.120 75.120
196.666 121.546
271.786 75.120
393.332 121.546
468.452 75.120
589.998 121.546
711.544 121.546
786.664 75.120
908.210 121.546
983.330 75.120
1104.876 121.546
1179.996 75.120

One can easily see how the same Large and Small Phi steps, make up
both scales, yet yield quite different structures. These are quite
lovely sounding too and I've enjoyed playing them on a piano timbre
to blissful Golden effect.

And now for something completely different. Quite a while ago, I
recognized that one could create Phi MOS scales which are a "hybrid"
of rationals and Phi. Using the same logic as above, if we insert a
rational interval (in the place of Phi), then iteratively divide the
intervals by Phi, we get some truly remarkable scales. Here we'll
look at a series of scales based on Superparticular Ratios and Phi,
as follows:

2/1 Phi Scale
Cents Consecutive
0.000
108.204 108.204
175.078 66.874
283.282 108.204
350.155 66.874
458.359 108.204
566.563 108.204
674.767 108.204
741.641 66.874
849.845 108.204
916.718 66.874
1024.922 108.204
1091.796 66.874
1200.000 108.204
1308.204 108.204
1416.408 108.204
1483.282 66.874
1550.155 66.874
1658.359 108.204
1766.563 108.204
1833.437 66.874
1941.641 108.204

Please note that this scale is not technically a MOS, but has 2 step
sizes, and is a Constant Structure. All that will follow are MOS
though.

3/2 Phi MOS
Cents Consecutive
0.000
102.414 102.414
165.709 63.295
268.123 102.414
331.418 63.295
433.832 102.414
536.246 102.414
599.541 63.295
701.955 102.414
765.250 63.295
867.664 102.414
970.078 102.414
1033.373 63.295
1135.787 102.414

This remarkable scale has 12 3/2 fifths!

4/3 Phi MOS
Cents Consecutive
0.000
117.572 117.572
190.236 72.664
307.809 117.572
380.473 72.664
498.045 117.572
615.617 117.572
688.281 72.664
805.854 117.572

5/4 Phi MOS
Cents Consecutive
0.000
91.196 91.196
147.559 56.362
238.755 91.196
295.117 56.362
386.314 91.196
477.510 91.196
533.872 56.362
625.069 91.196

6/5 Phi MOS
Cents Consecutive
0.000
120.564 120.564
195.077 74.513
315.641 120.564
390.154 74.513
510.718 120.564

7/6 Phi MOS
Cents Consecutive
0.000
101.936 101.936
164.935 63.000
266.871 101.936
329.871 63.000
431.806 101.936

8/7 Phi MOS
Cents Consecutive
0.000
88.301 88.301
142.873 54.573
231.174 88.301
285.747 54.573
374.048 88.301

9/8 Phi MOS
Cents Consecutive
0.000
126.023 126.023
203.910 77.887
329.933 126.023

I'll leave it to the creative and analytical ones out there to
recognize the many occurrences of "near rationals", with this
procedure. Notice for instance the scale based on 3/2 and Phi, and
observe how neatly a near 7/6, 17/14, and 9/7 came forth. Of course,
which ratio one chooses to use in the scale, will be woven throughout
the pattern of the MOS.

Obviously we may use Phi in this manner with any other ratios we
choose, and I've created many more than I'm showing here, but this is
just to inspire discussion and creativity. The infinite potential
possibilities for generation of Phi/Rational Hybrid scales, is quite
an interesting twist on the theme of Phi structures, and in the case
of the Phi/Rational Hybrid, we gain the benefits of MOS, while being
able to create scales with desired rational flavors.

Thanks,

Jacky Ligon

Jacky!

ligonj@northstate.net wrote:

> Here, let's look at a few more examples of the wonderful properties
> of Phi, where we will see these large scale steps divided further by
> the Phi constant, which invariably will generate either a Moment of
> Symmetry, or else scales with exactly 2 different step sizes, and
> which are always Constant Structures.

Moment of Symmetry scales will produce scales with two different step size where as Constant
structures will have 3 or more.

All the Horagrams are Moment of Symmetry scales, in fact you could view them as the most
archetypal ones!

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Kraig wrote,

>Moment of Symmetry scales will produce scales with two different step size
where as Constant structures will >have 3 or more.

Hi Kraig -- perhaps Jacky was going by the instance when you wrote:

constantconstant structure
A tuning system where each interval occurs always subtended by the same
number of steps. (THAT IS ALL, NO OTHER RESTRICTIONS)

[from Erv Wilson, via Kraig Grady, " <http://www./> Onlist Tuning Digest #
340, message 15 ]

Perhaps we need a new term for the specific type of Constant Structure
you're thinking of here (having 3 or more step sizes).

I doubt it

"Paul H. Erlich" wrote:

> Kraig wrote,
>
> >Moment of Symmetry scales will produce scales with two different step size
> where as Constant structures will >have 3 or more.
>
> Hi Kraig -- perhaps Jacky was going by the instance when you wrote:
>
>
>
>
> constantconstant structure
> A tuning system where each interval occurs always subtended by the same
> number of steps. (THAT IS ALL, NO OTHER RESTRICTIONS)
>
> [from Erv Wilson, via Kraig Grady, " <http://www./> Onlist Tuning Digest #
> 340, message 15 ]
>
>
>
>
>
> Perhaps we need a new term for the specific type of Constant Structure
> you're thinking of here (having 3 or more step sizes).
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold for the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest mode.
> tuning-normal@yahoogroups.com - change your subscription to individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm