Re: [tuning] Applying Jake's octave PHI sections idea to Silver sections (was 12th root of Phi)

Jake,

   Correction: your scale divides the octave into PHI sections...mine divides the "PHI-tave" into PHI sections an mine divides exponentially, not additively.  But I ditched all my old PHI compositions after intense scrutiny from this list...though I believe Chris Vaisvil made a few compositions with that scale which did surprisingly well...

  It makes me wonder how your theory of calculation would apply to the Silver Ratio IE where 2a + b / a = a / b when applied to the octave? 

    So if I have it right (for the first note)...

a + b = 1200 then b = 1200 - a so

2a + (1200 - a) / a = a / (1200 - a)

((1200 - a) + 2a) / a = (a / (1200-a))

((1200 - a) + 2a) = a(a / (1200-a))

1200 + a = a^2 / (1200 - a)

(1200 + a)(1200 - a) = a^2

1440000 - a^2 = a^2

1440000 = 2a^2

a = 848.52 cents

b = 1200 - 848.52 = 351.38 cents

.......thought surely there must be an easier way to calculate this...
Interestingly enough, this hits 11/9 and 18/11 almost dead-on.

> Correction: your scale divides the octave into PHI sections...mine
> divides the "PHI-tave" into PHI sections an mine divides exponentially,
> not additively.

Right, those are pretty different.

> It makes me wonder how your theory of calculation would apply to the
> Silver Ratio IE where 2a + b / a = a / b when applied to the octave?

Your math (omitted) is right, but there are two things to consider.

1. In the phi case, Gene pointed out that what I was really doing was
using 1/phi as a generator: log((1/phi),2)*1200 = 741.64 cents.
Instead of doing the math the way I did it, I could have just stacked
that interval on top of itself repeatedly and reduced to an octave. In
that case, though, I think it used the peculiar properties of phi to
work out so easily.

In this case, the numbers aren't nearly as straightforward, and the
more you divide, the more interval values you get. The result is that
I'm not "generating" anything in the tuning-list sense, and the scale
I create below doesn't have clean structural properties: It's not
proper, isn't distributionally even, and has a lot of different scale
step sizes. It might still be musically useful -- I haven't tried it
-- but it's not obvious to me that it will be.

2. You don't have to solve (2a+b)/a = a/b. Since the silver ratio =
1+sqrt(2), you can just set one of the two ratios equal to 1+sqrt(2).
For any interval i, then:

i. a+b = i ==> b=i-a.
ii. (2a + (i-a)) / a = 1+sqrt(2)
iii. (a+i)/a = 1+sqrt(2)
iv. 1 + (i/a) = 1+sqrt(2)
v. i/a = sqrt(2)
vi. i/sqrt(2) = a

Suddenly things are really easy: To get the silver ratio for any
number, just divide it by the square root of two.

When i = 1200, a = 848.5281 and b = 351.4719. Dividing each of these
by their own silver ratios, and doing it again for each smaller
segment, would lead to the following set of intervals:

interval a+b
--------
a
b

1200
--------
848.5281
351.4719

848.5281
--------
600
248.5281

351.4718
--------
248.5281
102.9437

600
--------
424.2640
175.7359

248.5281
--------
175.7359
72.79220

102.9437
--------
72.79220
30.15151

424.2640
--------
300
124.2640

175.7359
--------
124.2640
51.47186

300
--------
212.1320
87.86796

124.2640
--------
87.86796
36.39610

212.1320
--------
150
62.13203

150
--------
106.0660
43.93398

By subdividing intervals only when both a and b would be more than 50
cents, I get the following scale:

! C:\Program Files (x86)\Scala22\Silver Ratio.scl
!
Silver Ratio scale
13
!
150.00000
212.13200
300.00000
424.26400
548.52800
600.00000
724.26400
775.73600
848.52800
972.79200
1024.26400
1097.05600
1200.00000

Regards,
Jake

By subdividing intervals only when both a and b would be more than 50

cents, I get the following scale:

! C:\Program Files (x86)\Scala22\Silver Ratio.scl

!

Silver Ratio scale

13

!

150.00000

212.13200

300.00000

424.26400

548.52800

600.00000

724.26400

775.73600

848.52800

972.79200

1024.26400

1097.05600

1200.00000

>"It's not proper, isn't distributionally even, and has a lot of different scale step sizes."

Right so I'm tempted to take a subset of it...to get such properties...

150 1.0909 (150)                  10/9
212 1.13   (62)                      9/8
300 1.189  (88)                     6/5
424 1.27777 (124)                 14/11
548 1.372 (124))                  11/8
600.00000 1.414  (50)          10/7 7/5       
724.26400 1.519 (124)        
775.73600 1.56 (73)            14/9
848          1.632 (124)          18/11
972           1.75 (124)           7/4
1097          1.88448 (76)       15/8

So a fairly stable subset seems to be

150   10/9  (150 step size)
300   6/5    (150 step size)
600   10/7 or 7/5 (300 step size)
848   18/11  (about 248 step size)
1024  9/5  (176 step size)
1200  2/1 (200 step size)

    This way no step size is more than twice any other step size...although that scale is only 6 tones in size.

> I'm tempted to take a subset of it...to get such properties...
>
> 150 1.0909 (150)                  10/9
[snip]

Okay, but why would you use the silver ratio to create JI
approximations? The silver ratio is about proportions between two
numbers in one way, and the ratios you're using are about proportions
between two numbers in a different way. It looks like a mismatch, or a
shoehorning of one structure into another.

By the way, I posted this to the MMM list, but I don't know if you saw
it: Based on an attempt to do some phi-based chords and melodies, I
think the phi-based scale has potential. It has a very good major
third, but I ignored it except in places where it was the result of a
golden division. There's a little chunk of sound (about 45 seconds)
here:
http://www.freivald.org/~jake/documents/phi-sample.mp3

Regards,
Jake

Jake>"The silver ratio is about proportions between two numbers in one way, and the ratios you're using are about proportions between two numbers in a different way. It looks like a mismatch, or a shoehorning of one structure into another."

I'm looking at taking the Silver Ratio tuning parts that are "not too far from JI", so the resulting tuning can be used in both repeating-sequence ways and JI ways...best of both options.  BTW, the fractions were just for mathematical reference...the part of the scale I meant to use was in cents IE copies from the scale you posted and not "perfectly in JI".

--- On Thu, 4/14/11, Jake Freivald <jdfreivald@...> wrote:

From: Jake Freivald <jdfreivald@...>
Subject: Re: [tuning] Applying Jake's octave PHI sections idea to Silver sections (was 12th root of Phi)
To: tuning@yahoogroups.com
Date: Thursday, April 14, 2011, 11:15 AM

> I'm tempted to take a subset of it...to get such properties...

>

> 150 1.0909 (150)                  10/9

[snip]

Okay, but why would you use the silver ratio to create JI

approximations? The silver ratio is about proportions between two

numbers in one way, and the ratios you're using are about proportions

between two numbers in a different way. It looks like a mismatch, or a

shoehorning of one structure into another.

By the way, I posted this to the MMM list, but I don't know if you saw

it: Based on an attempt to do some phi-based chords and melodies, I

think the phi-based scale has potential. It has a very good major

third, but I ignored it except in places where it was the result of a

golden division. There's a little chunk of sound (about 45 seconds)

here:

http://www.freivald.org/~jake/documents/phi-sample.mp3

Regards,

Jake