Latest effort: a PHI-based scale "directly rounded to JI"

I think this is my best scale yet...at least a good bit better than my original PHI scale for instruments with intense higher harmonics, such as harpsichords. :-)
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Reading through these messages I found comments on how using the harpsichord on a scale can serve as an "ultimate test of dissonance". So I tried it and, guess what, a few of the notes in my own PHI scale came out obviously sour (even though those same notes sounded fine with a piano).
In other words, there was an obvious sense to my ear of "this note needs to move slightly down/upwards"...as if someone "bent" the notes by using the harpsichord.
-------------------------
So I messed around using a decimal to fraction calculator to round the notes in my PHI scale to rational numbers. And I got the following scale (using 1.61803399 as the octave):

1
9/8
15/8
4/3
22/15
14/9
1.618033 (octave...which just happens to be nearest to 13/8)
----------------------------------

The odd thing I noticed (beside the obvious improvement of the sound of the scale when used with the harpsichord) is the fact that in ALL of these fractions either
A) The numerator is dividable by 3 and the denominator by 2
OR
B) The denominator is dividable by 3 and the numerator by 2
-------------------------------------------------------------

Maybe some of you can see some deeper meaning in this pattern or other patterns. But it seems pretty obvious to me now...that irrational generator rounded to nearest ration numbers actually seems to provide a good option for generating scales.

djtrancendance wrote:

> So I messed around using a decimal to fraction calculator to round the notes in my PHI scale to rational numbers. And I got the following scale (using 1.61803399 as the octave):
> > 1
> 9/8
> 15/8
> 4/3
> 22/15
> 14/9
> 1.618033 (octave...which just happens to be nearest to 13/8)

15/8 is out of place there.

> The odd thing I noticed (beside the obvious improvement of the sound of the scale when used with the harpsichord) is the fact that in ALL of these fractions either
> A) The numerator is dividable by 3 and the denominator by 2
> OR
> B) The denominator is dividable by 3 and the numerator by 2
> > Maybe some of you can see some deeper meaning in this pattern or other patterns. But it seems pretty obvious to me now...that irrational generator rounded to nearest ration numbers actually seems to provide a good option for generating scales.

It tends to give simple intervals between notes. For example, (4/3):(14/9) simplifies to 2:(7/3) or 6:7 because of the common factors.

A good rule of thumb to get simple ratios between notes is to keep a given prime in *either* the numerator *or* the denominator. The biggest problems here are likely to be {9/8, 14/9} and {15/8, 22/15} but maybe an implicit approximation cleans them up. (Have you tried 10/9 instead of 9/8?)

Graham

-------------15/8 is out of place there.
 Sorry...I mis-copied, please put 19/16 in place of 15/8.
 I also found a few other tones that needed slight correction...so the final scale is
A) 1               1    
C) 1.111111   10/9      (clean up)              
D) 1.1875       19/16    (clean up)
F) 1.33333     4/3
H) 1.45454     16/11    (clean up..but still not 100% sure on this one, any suggestions?)                        
I) 1.5625       
25/16                                                
**********
A) 1.61803 (octave)    13/8 = 1.625

--keep a given prime in *either* the numerator *or* the
--denominator.
   Ah, got it.
   By coincidence, the ratios my ears found as wrong turned out to be the same ones your method did. :-)

-Michael

Try the Fibonacci series, which converges on phi. 89/55 is the first "indistinguishable" rational phi I guess, but I find 21/13 (at 830.25 cents) very nice, as I link harmonic partials/JI with phi and 21/13 is a great link for both the 7th and 13th partials.

BTW Michael, sorry for not getting back to you earlier on other stuff, I'll PM you asap!

--- In tuning@yahoogroups.com, djtrancendance@... wrote:
>
>
> -------------15/8 is out of place there.
>  Sorry...I mis-copied, please put 19/16 in place of 15/8.
>  I also found a few other tones that needed slight correction...so the final scale is
> A) 1               1    
> C) 1.111111   10/9      (clean up)              
> D) 1.1875       19/16    (clean up)
> F) 1.33333     4/3
> H) 1.45454     16/11    (clean up..but still not 100% sure on this one, any suggestions?)                        
> I) 1.5625       
> 25/16                                                
> **********
> A) 1.61803 (octave)    13/8 = 1.625
>
> --keep a given prime in *either* the numerator *or* the
> --denominator.
>    Ah, got it.
>    By coincidence, the ratios my ears found as wrong turned out to be the same ones your method did. :-)
>
> -Michael
>

Here's a quick very rough example, I'd have to pick at it for a couple of hours before using it, and it would probably wind up some changes here and there and with either fewer or more tones, 10 or 17 I'd guess from experience:

1/1
14/13
147/130
16/13
17/13
7/5
189/130
98/65
21/13
7/4
13/7
126/65
2/1

The complex intervals are simple intervals in relation to other previous intervals in the tuning. If you noodle around and play some tall sonorities, perhaps you'll agree that it is overall much less disonant sounding than it "should be", and if you've worked with either synthesis or higher-prime JI you might agree that it is pretty dang 7+13 sounding. The possibilities are, of course, endless.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> Try the Fibonacci series, which converges on phi. 89/55 is the first "indistinguishable" rational phi I guess, but I find 21/13 (at 830.25 cents) very nice, as I link harmonic partials/JI with phi and 21/13 is a great link for both the 7th and 13th partials.
>
> BTW Michael, sorry for not getting back to you earlier on other stuff, I'll PM you asap!
>
> --- In tuning@yahoogroups.com, djtrancendance@ wrote:
> >
> >
> > -------------15/8 is out of place there.
> >  Sorry...I mis-copied, please put 19/16 in place of 15/8.
> >  I also found a few other tones that needed slight correction...so the final scale is
> > A) 1               1    
> > C) 1.111111   10/9      (clean up)              
> > D) 1.1875       19/16    (clean up)
> > F) 1.33333     4/3
> > H) 1.45454     16/11    (clean up..but still not 100% sure on this one, any suggestions?)                        
> > I) 1.5625       
> > 25/16                                                
> > **********
> > A) 1.61803 (octave)    13/8 = 1.625
> >
> > --keep a given prime in *either* the numerator *or* the
> > --denominator.
> >    Ah, got it.
> >    By coincidence, the ratios my ears found as wrong turned out to be the same ones your method did. :-)
> >
> > -Michael
> >
>

djtrancendance@... wrote:
> -------------15/8 is out of place there.
> Sorry...I mis-copied, please put 19/16 in place of 15/8.
> I also found a few other tones that needed slight correction...so the final > scale is
> A) 1 1 > C) 1.111111 10/9 (clean up) > D) 1.1875 19/16 (clean up)
> F) 1.33333 4/3
> H) 1.45454 16/11 (clean up..but still not 100% sure on this one, any > suggestions?) How about 13/9? (1.44444)

> I) 1.5625 25/16 > **********
> A) 1.61803 (octave) 13/8 = 1.625
> > --keep a given prime in *either* the numerator *or* the
> --denominator.
> Ah, got it.
> By coincidence, the ratios my ears found as wrong turned out to be the same > ones your method did. :-)

That's good! Keep it up and you'll be accused of putting numbers ahead of listening like the rest of us. ;-)

Graham

---How about 13/9? (1.44444)
    I could try it...it seems good but you have to remember I'm trying to approximate the 1.46808 from the "circle of PHI" with 1.45454.  It seems numerically sound, now I will try it with my ears and see how it pans out. :-)

----That's good! Keep it up and you'll be accused of putting
----numbers ahead of listening like the rest of us. ;-)
   Haha, exactly!  It seems like no matter how much you lean toward either numbers or listening there's always someone politely reminding you to lean more the other way. :-)  
   Let's just say using PHI tuning to guide rationals is the latest effort on my part to try and do both.

-Michael

--- On Thu, 3/19/09, Graham Breed <gbreed@...> wrote:

From: Graham Breed <gbreed@...>
Subject: Re: [tuning] Latest effort: a PHI-based scale "directly rounded to JI"
To: tuning@yahoogroups.com
Date: Thursday, March 19, 2009, 2:45 AM

djtrancendance@ yahoo.com wrote:

> ------------ -15/8 is out of place there.

> Sorry...I mis-copied, please put 19/16 in place of 15/8.

> I also found a few other tones that needed slight correction.. .so the final

> scale is

> A) 1 1

> C) 1.111111 10/9 (clean up)

> D) 1.1875 19/16 (clean up)

> F) 1.33333 4/3

> H) 1.45454 16/11 (clean up..but still not 100% sure on this one, any

> suggestions? )

How about 13/9? (1.44444)

> I) 1.5625 25/16

> **********

> A) 1.61803 (octave) 13/8 = 1.625

>

> --keep a given prime in *either* the numerator *or* the

> --denominator.

> Ah, got it.

> By coincidence, the ratios my ears found as wrong turned out to be the same

> ones your method did. :-)

That's good! Keep it up and you'll be accused of putting

numbers ahead of listening like the rest of us. ;-)

Graham

   It also turns out...the revision of the scale is VERY well rounded by using harmonic series fractions x/9 and x/16 to get

1                 1    
1.111111     10/9                     
1.1875        19/16
1.33333      12/9
1.4444444  13/9          
1.5625     25/16        
 1.625 = 13/8   (26/16)
-----------------------------------------------------------------------------------------
   Note also that when you compare the sound of it to my original PHI-based scale below, the moods and intervals sound relatively alike, but a good deal clearer

1
1.12152
1.18771
1.2578
1.33333
1.46807
1.55473
1.618033
*************************************************************
-Michael

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

Hi Cameron,

> 1/1 C
> 14/13 C#
> 147/130 D
> 16/13 Eb
> 17/13 E
> 7/5 F
> 189/130 F#
> 98/65 G
> 21/13 G#
> 7/4 A
> 13/7 Bb
> 126/65 B
> 2/1 C'

in order to get from that the corresponding absolute-pitches,
simply multiply all the 12 ratios by the common factor 130:

c 130 low_C3
# 140
d 147
# 160
e 170
f 182
# 189
g 196
# 210
a 227.5
# 241 + 3/7 ? or better 241.5 instead of that?
b 252
c'260 middle_C4

or when considered as bumpy cycle of craggy 5hts:

C 65 'c130'
G 49 98 'g196' (> 195 = C*3)
D (19 38 72 < 73 146 <) 'd147' = G*3
A (57 104 <) 105 'a210' (> 209 = 73*3)
E (21 42 84 <) 85 'e170' (< 171 = 57*3 )
B 63 126 'b252'
F# (47 94 188 <) 189 = B*3
C# 35 70 'c#140' (< 141 = 47*3)
G# (53 106 >) 105 'g#210' = C#*3
D# 5 10 20 40 80 'd#160' (< 159 = 53*3)
A# (11^2=121 242 484 < ) 483 'a#?241.5' (> 240 = 80*3
F (45 90 < ) 91 'f182' 364 ( > 363 = 121*3)
C 65 'c130' ( < 135 = 45*3)

Sorry, but:
Some modulations in that sound in my ears all to much harsh
for my personal taste.

bye
A.S.