PHI-th of an octave---dat-sa-nice-ah!

Hey all,

There's been lots and lots of PHI talk around here lately.

Sorry if this has been mentioned, I haven't dug too deeply into some of
these threads...but:

I am interested in an alternative interpretation of PHI--instead of the
acoustic factor of ~1.618 as a generator, we could look at intervals which
are derived from 0.618/oct (i.e. 2^phi....). This way, the octave becomes
our natural period, and when pitches are viewed as on a circle, we are
seeing a literal analog to the way leaves on many plants grow to maximize
their exposure to sunlight...via the Fibonacci sequence. IOW, many plant
leaves grow at angles which relate to each other by 2*3.1415/phi radians,
maximizing their potential NOT to be blocked by higher layers as the plant
grows. It's called the 'golden angle' relationship.

http://library.thinkquest.org/27890/applications5.html

One can derive scales from 'Fibonacci EDOs' : 1,1,2,3,5,8,13,21,34,55...
I like 34. Interestingly, MOS scales occur when the subset is also a
fibonacci number.

My current favorite is '8of34 MOS':
5 5 3 5 5 3 5 3 or L L s L L s L s

You can of course play around with the various rotations (modes) and get
different moods and effects.

Having 8 notes in the scale is nice: more possibilities than 5, but less
tension and information than 13 and up....it seems to be a sweet spot, just
slightly more information than a typical diatonic scale. Try it!

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

Aaron,

  Your scale seems to translate to
2^(x/34) where x =
5   = 1.107 (d)
10  = 1.226 (near 1.23606 in my PHI scale)
13  = 1.3034 (very near 2^(14/36) in Rick's 2^(x/36) PHI scale)
18  = 1.4433
23  = 1.59822   (g#)
26  = 1.6990235 (a)
31  = 1.88134 (b)
34  = 2/1 (c)
  Note: my latest PHI scale is based on the formula 1+(0.618034^x)...so it also utilizes the "PHI inverse" of 1/1.618034 = 0.618034.

    Oddly enough, the only number your scale has nearly common with my scale is 1.226 near my scale's 1.23606 and your scale seems to miss the 1.618034 "phi-tave" (the period in my scale) entirely. 

    What is interesting, however, is how it completely avoids the standard 5th, 3rd, and other 12TET notes: it definitely qualifies as a scale that does not imitate
12TET and/or diatonic intervals and yet still works pretty well, IMVHO...even if it is "simply an MOS scale" (note I encourage you to try methods beyond MOS for scale generation using fractal numbers such as PHI). 
 
     I will have to try composing with this to see how well it fairs practically...and will, of course, post sound examples. 

Aaron> "Having 8 notes in the scale is nice: more possibilities than 5"

   I will say this much...I have done too many different scale experiments to count and think the ideal # of notes for a scale to balance consonance and compositional flexibility is somewhere between 8 and 9 per octave (including both 8 and 9, of course).  Even my 'monster' 10-tone per 2/1 interval "silver ratio" scale really has only 8 to 9 notes usable at once in harmony.

   So, in short, I think your coming up with 8-tones is a good sign and in a very "good ballpark" far as tones-per-octave and very interesting work on your part. :-)

-Michael

Can I talk anyone into posting a scala file for the 8 note inverse Phi tuning?

Thks

Chris
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: djtrancendance@yahoo.com

Date: Tue, 28 Apr 2009 09:19:26
To: <tuning@yahoogroups.com>
Subject: Re: [tuning] PHI-th of an octave---dat-sa-nice-ah!

Aaron,

  Your scale seems to translate to
2^(x/34) where x =
5   = 1.107 (d)
10  = 1.226 (near 1.23606 in my PHI scale)
13  = 1.3034 (very near 2^(14/36) in Rick's 2^(x/36) PHI scale)
18  = 1.4433
23  = 1.59822   (g#)
26  = 1.6990235 (a)
31  = 1.88134 (b)
34  = 2/1 (c)
  Note: my latest PHI scale is based on the formula 1+(0.618034^x)...so it also utilizes the "PHI inverse" of 1/1.618034 = 0.618034.

    Oddly enough, the only number your scale has nearly common with my scale is 1.226 near my scale's 1.23606 and your scale seems to miss the 1.618034 "phi-tave" (the period in my scale) entirely. 

    What is interesting, however, is how it completely avoids the standard 5th, 3rd, and other 12TET notes: it definitely qualifies as a scale that does not imitate
12TET and/or diatonic intervals and yet still works pretty well, IMVHO...even if it is "simply an MOS scale" (note I encourage you to try methods beyond MOS for scale generation using fractal numbers such as PHI). 
 
     I will have to try composing with this to see how well it fairs practically...and will, of course, post sound examples. 

Aaron> "Having 8 notes in the scale is nice: more possibilities than 5"

   I will say this much...I have done too many different scale experiments to count and think the ideal # of notes for a scale to balance consonance and compositional flexibility is somewhere between 8 and 9 per octave (including both 8 and 9, of course).  Even my 'monster' 10-tone per 2/1 interval "silver ratio" scale really has only 8 to 9 notes usable at once in harmony.

   So, in short, I think your coming up with 8-tones is a good sign and in a very "good ballpark" far as tones-per-octave and very interesting work on your part. :-)

-Michael

Mike> "Note: my latest PHI scale is based on the formula"
>" 1+(0.618034^ x)...so
it also utilizes the "PHI inverse" of"
>"1/1.618034= 0.618034."
  Well, apparently both Aaron's scale and mine both utilize the "PHI inverse".
  So, which version would you like?

  Furthermore, Aaron, if I did indeed document/interpret your scale correctly I would like to try and release the following "consonant PHI scales" in SCALA format:
A) My own new 1+(0.618034^ x) "inverse PHI" scale
B) My old PHI^x/2^y PHI scale
C) Aaron's 2^(x/34) inverse PHI MOS scale
D) Rick's 2^(x/36) scale (I'll have to make sure I get this one correctly documented as well)
E) PHI-mean-tone (http://www.rev.net/~aloe/music/golden.html_ as
    developed by Thorvald Kornerup and advocated by Jacques Dudon

...and release these in a "tuning pack" on my site to compare various methods of devising scales using PHI.
   Because, let's face it...I think it's pretty clear a whole bunch of us are having trouble making head or tail of the PHI scales floating around...not to mention realizing how different many of them are despite having the same or similar generators. 

  What I am very happy about though...is how many people here are finally making the effort to approach PHI-based scales as a way of maximizing consonance, rather than dissonance (which seems to,sadly, be the prevailing tradition view about PHI). Keep it up!  :-)

-Michael

P.S. - Is dat-sa-nice supposed to comically replace do-re-mi?!  Haha...I love it.

Is "all" an option?

The non-12tet imitation consonance sounds very intriguing.
Sent via BlackBerry from T-Mobile

--- In tuning@yahoogroups.com, djtrancendance@... wrote:

>   Your scale seems to translate to
> 2^(x/34) where x =
> 5   = 1.107 (d)
> 10  = 1.226 (near 1.23606 in my PHI scale)
> 13  = 1.3034 (very near 2^(14/36) in Rick's 2^(x/36) PHI scale)
> 18  = 1.4433
> 23  = 1.59822   (g#)
> 26  = 1.6990235 (a)
> 31  = 1.88134 (b)
> 34  = 2/1 (c)

Yup...looks right--in the incorrect, non-canonical mode I gave you....see below. BTW, IIRC, PHI and 1/PHI are both still called 'PHI' by mathematicians.

>     Oddly enough, the only number your scale has nearly common with > my scale is 1.226 near my scale's 1.23606 and your scale seems to
> miss the 1.618034 "phi-tave" (the period in my scale) entirely. 

Actually, it's there in a mode of the scale. If you look at this mode, for instance:

3 5 3 5 5 3 5 5

which translates to

3 8 11 16 21 24 29 34

you'll see that 21/34 is an approximation to a PHI-th of an octave (0.618)..

I should have presented the original scale as "3 5 3 5 5 3 5 5", so that it was immediatly apparent, my bad! I was fooling around with my python script using a different variable, and mistakenly notated the original the 'non canonical' way!

here's a correct .scl file, where the generator is 21/34 oct, and period is an octave:

! 8of34fibo-MOS.scl
!
8 out of 34 Fibonacci Moment of Symmetry scale
8
!
105.882
282.353
388.235
564.706
741.176
847.059
1023.529
1200

>     What is interesting, however, is how it completely avoids the
> standard 5th, 3rd, and other 12TET notes: it definitely qualifies
> as a scale that does not imitate

It still has some things like a pretty good major third. (~388.235 cents)

> 12TET and/or diatonic intervals and yet still works pretty well, >IMVHO...even if it is "simply an MOS scale" (note I encourage you to >try methods beyond MOS for scale generation using fractal numbers >such as PHI). 

The MOS properties of this scale are a simple side-effect, and I generated the scale using PHI. I just happens that I test for MOS properties after the fact, and discovered that any Fibonacci EDO has a MOS subset whose size is also a Fibonacci number.

MOS scales are a good bet musically. Although MOS theory is new, thanks to Wilson, it can be shown that most historically used scales are MOS scales.

>      I will have to try composing with this to see how well it
> fairs practically...and will, of course, post sound examples. 

Good...looking forward to hearing what you'd do.

> > "Having 8 notes in the scale is nice: more possibilities than 5"
>
>    I will say this much...I have done too many different scale experiments to count and think the ideal # of notes for a scale to balance consonance and compositional flexibility is somewhere between 8 and 9 per octave (including both 8 and 9, of course).  Even my 'monster' 10-tone per 2/1 interval "silver ratio" scale really has only 8 to 9 notes usable at once in harmony.
>
>    So, in short, I think your coming up with 8-tones is a good sign and in a very "good ballpark" far as tones-per-octave and very interesting work on your part. :-)

Well, interesting things can be done in any scale/tuning...other than that, congratulations on congratulating yourself by congratuating me for liking octatonic scales!

What's up with the way you've wrapped the lines with returns in your response? Sure is hard to read!

-AKJ

--- In tuning@yahoogroups.com, djtrancendance@... wrote:

>   Furthermore, Aaron, if I did indeed document/interpret your scale correctly I would like to try and release the following "consonant PHI scales" in SCALA format:
> A) My own new 1+(0.618034^ x) "inverse PHI" scale
> B) My old PHI^x/2^y PHI scale
> C) Aaron's 2^(x/34) inverse PHI MOS scale
> D) Rick's 2^(x/36) scale (I'll have to make sure I get this one correctly documented as well)
> E) PHI-mean-tone (http://www.rev.net/~aloe/music/golden.html_ as
>     developed by Thorvald Kornerup and advocated by Jacques Dudon
>
> ...and release these in a "tuning pack" on my site to compare various methods of devising scales using PHI.
>    Because, let's face it...I think it's pretty clear a whole bunch of us are having trouble making head or tail of the PHI scales floating around...not to mention realizing how different many of them are despite having the same or similar generators. 

Michael, those of us who have been here for years remember also things like Keenan Pepper's PHI-based tuning which later was called "Peppermint" by Margo Schulter. Google the archives....

As for PHI based scales, there are sure plenty already in the scala archives.

Here's the list I got from grepping my directory:

cet119.scl:7th root of phi
cet167.scl:5th root of phi
clampitt-phi.scl:! clampitt-phi.scl
clampitt-phi.scl:David Clampitt, phi+1 mod 3phi+2, from "Pairwise Well-Formed Scales", 1997
mcgoogy_phi.scl:! mcgoogy_phi.scl
mcgoogy_phi.scl:Brink McGoogy's Phinocchio tuning, mix of 5th (black keys) and 7th (white keys) root of phi
mcgoogy_phi2.scl:! mcgoogy_phi2.scl
meangold.scl:Meantone scale with Blackwood's R = phi, and diat./chrom. ST = phi, ~4/15-comma
phi1_13.scl:! phi1_13.scl
phi_10.scl:! phi_10.scl
phi_11.scl:! phi_11.scl
phi_12.scl:! phi_12.scl
phi_13.scl:! phi_13.scl
phi_13a.scl:! phi_13a.scl
phi_13b.scl:! phi_13b.scl
phi_17.scl:! phi_17.scl
phi_7b.scl:! phi_7b.scl
phi_7be.scl:! phi_7be.scl
phi_7be.scl:36-tET approximation of phi_7b
phi_8.scl:! phi_8.scl
phi_8a.scl:! phi_8a.scl
temes.scl:Lorne Temes' 5-tone phi scale (1970)
wilson_gh1.scl:Golden Horagram nr.1: 1phi+0 / 7phi+1
wilson_gh11.scl:Golden Horagram nr.11: 1phi+0 / 3phi+1
wilson_gh2.scl:Golden Horagram nr.2: 1phi+0 / 6phi+1
wilson_gh50.scl:Golden Horagram nr.50: 7phi+2 / 17phi+5
iter20.scl:Binary PHI Scale #2
iter21.scl:Binary PHI Scale 5+2 #2
iter32.scl:Iterated PHI scale, IE= 1.61803339, PD=8, SD=0

Knowing that there might be others which don't get noticed by a simple "grep 'phi' *" command in my directory, but have PHI-based reasoning behind them, I'm sure there's lots for you to discover and learn from your predecessors! (true for us all)

>   What I am very happy about though...is how many people here are finally making the effort to approach PHI-based scales as a way of maximizing consonance, rather than dissonance (which seems to,sadly, be the prevailing tradition view about PHI). Keep it up!  :-)

I, for one, am *not* making any such effort to maximize consonance in my example, to be honest--I'm just exploring what numerical properties map to tunings in potentially musical useful ways....

Best,
AKJ

Aaron> "PHI and 1/PHI are both still called 'PHI' by mathematicians."
    Right...especially since the symmetrical relationship holds for both...since they are in fact the two results of solving the equation x = 2x + 1 because of the + or - sqrt(b^2 - 4ac) in the quadratic formula.

>"Actually, it's there in a mode of the scale (cut...which contains a"
>"0.618034 PHI-tave...cut)" If you look at this mode, for instance:"

>"3 5 3 5 5 3 5 5 which translates to "

>"3 8 11 *16* 21 24 29 34"
>"you'll see that 21/34 is an approximation to a PHI-th of an octave" >"(0.618).."

    Yes, however, that's still the exponential part of the result and not the whole, evaluated result.  2^(21/34) = 1.53437.  My point is that our ears don't 'hear' the exponential of how your formula was derived...but they do hear the result (a slightly sharp 5th).

>"I should have presented the original scale as "3 5 3 5 5 3 5 5","
>" so that it was immediately apparent, my bad!"
    Well, apparently now you did and we all have access to the correct version of your scale, so there's no reason to whine about formalities AKA don't worry about it. :-)

>"here's a correct .scl file, where the generator is 21/34 oct, and"
>" period is an octave:"
    Thank you! :-)  Simple, from the author scale postings like these help stop the ambiguity between PHI scales (and the nasty idea some people still have that all PHI scales are either the same or very similar).

Aaron> "It still has some things like a pretty good major third."
>" (~388.235 cents)"
    It sure does, to an extent...  But, just as a warning from experience, I would strongly recommend not rating PHI-based scales by how many intervals they have in common with standard diatonic intervals.    
    Ironically I have found...with PHI or other fractal scales trying to force things to fit diatonic makes consonance worse unless you make virtually everything fit diatonic...and, in that case, you would basically just have a badly approximated/"poor man's" version of JI...which kind of kills the point IMVHO.

>"The MOS properties of this scale are a simple side-effect,"
>"and I generated the scale using PHI."
   I understand that...but it looks to me also to be a side-effect of your supposedly deliberate move of using 34TET, which fits the 2/1 octave perfectly and the PHI-tave (as in the actual ratio 1.618034, not the exponential) less so.

>"MOS scales are a good bet musically. Although MOS theory is new,"
>"thanks
to Wilson, it can be shown that most historically used"
>" scales are MOS
scales."
    Right, but Moment of Symmetry involves symmetry to the octave, not the PHI-tave...which pretty much sums up the issue I have with it.  And, of course, MOS scales work well...but PHI is a different animal and, IMVHO, it at least begs the challenge "is achieving symmetry at the octave really the best way to push consonance out of scales based on PHI?".  And, in my tests (believe me, I've done hundreds in the 4-5 months I've been somewhat obsessed with fractal-number-generate scales)...the octave does not work so well as the PHI-tave for PHI.  However, as a side note and nod to JI and the use of "odd limits", I did notice that rounding 1.618 to 1.61 so the ear thinks it is 1.6 IE 8/5 does an some additional consonance while mostly preserving the symmetry: so that makes a very good "compromise" version of the period.

>"What's up with the way you've wrapped the lines with returns in"
>" your response? Sure is hard to read!"
    Oh, what the hell... CARL ARE YOU READING THIS?! 

    Basically Carl and I had a long debate about how supposedly bad my quoting habits are and he pretty much demanded IE both use quote marks for every line I quote and use carriage returns for every line..or face having many of my messages removed from discussion.

   I'm sorry Carl but, from here on in (IE starting after this message), I'm quoting without your carriage return advice.  I can't satisfy everyone and, for those who think I'm "not listening" simply because I can't take 2+ people's conflicting advice on how I quote, well, tough...I can't part the Red Sea and take two conflicting pieces of advice.  And in this case, Aaron, I agree with your advice.

-Michael

Aaron> "phi_7be.scl: 36-tET approximation of phi_7b"
...Sounds like it might be Rick's scale, which is also based on 36-TET, or something a good deal like it...I'll defintely have to check on that one.

>"I'm sure there's lots for you to discover and learn from your predecessors! (true for us all)"

   Of course...but here's a (perhaps a bit stubborn) question: how many of these scales
A) actually include 1.618034 or a very close estimate as one of their ratios
B) also include 0.618034 (IE the "inverse PHI-tave")
  Of course...I'll try all the scales but, I am betting in advance from the hundreds of little sound tests I have done...anything symmetric about the octave, but not PHI, is either doing to be less consonant than a scale based on the "PHI-tave" or so much like diatonic intervals, it might as well be a poor-man's JI scale.  I'm fairly convinced the "monopoly of the octave", while it's great for JI-type symmetry (which is what a huge majority of historical scales are based upon), is not ideal for fractal numbered scales.

>"I, for one, am *not* making any such effort to maximize consonance in my example, to be honest--I'm just exploring what numerical properties map to tunings in potentially musical useful ways...."

   Well, in that case...I'll just have to try the scale and see how it fits in musically (be that it helping consonance, tension, or both).   
    BTW, when I said "maximize consonance"...maybe I took it too far...what I meant is approach PHI without the intent of maximizing dissonance (which is what the noble mediant  does: maximize dissonance).

   Even I am not "maximizing consonance" (as we all know diatonic JI does that best, on the average)...but rather making as many notes as possible fit together in such a way that they teeter on the edge of consonance and dissonance so they are "just good enough".
  Hence why my scales virtually all end up having "passable consonance" and being 8 or 9-tones and >>not<< 7 or below "consonant but with little additional flexibility" or 10 or above  "tons of flexibility but too little consonance".  When I say "passable consonance", I mean something like the sound of a minor 13th (but not diminished) chord.

-Michael