Yahoo Tuning Groups Ultimate Backup tuning Fwd: Serial music

I now see I just sent this to Cameron.

I never did get around to working on a score. However, I was asked..
yesterday I think, how I calculated the 12 th root of Phi tuning -
and after two years I can't remember what I did and the cent values
are not falling into place in a relationship with Phi easily - this is
kind of embarrassing.

There was someone on the list who said they saw the 12th root of Phi
in the tuning - now would be a good time to speak up because a
mathematician I'm not....

Thanks,

Chris

! C:\Cakewalk\scales\12throotofphi12.scl
!
12th root pf phi in 12 steps
11
!
69.42116
138.84232
208.26348
277.68464
347.10580
416.52696
485.94813
555.36929
624.79045
694.21161
763.63277

On Fri, Apr 8, 2011 at 2:29 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Thank you Cameron.
>
> I plan to work on scoring this afternoon in Sibelius with GPO et al and it will be microtonal and serial. Right now I'm off to spend some time with my daughter. (spring break).
>
> Chris

🔗Jake Freivald <jdfreivald@...>

Chris,

It looks like the scale you have is 12-EDPhi, or close to it.

According to Wikipedia, Phi is (1+sqrt(5))/2. That's 1.618034, or 833.09030 cents.

The 12th root of Phi is thus 833.09030 cents / 12, or 69.42419 cents. That's your semitone. For some reason, your calculation seems a little off, possibly because you were using a weaker approximation of Phi; however, we're talking about a few thousandths of a cent per semitone, so what you have is close enough over the span of a few octaves.

Here's the precise set of tones, from what I can see:

! C:\Program Files (x86)\Scala22\12th root of phi.scl
!
12th root of phi tuning (Chris Vaisvil's idea)
12
!
69.42419
138.84838
208.27257
277.69677
347.12096
416.54515
485.96934
555.39353
624.81772
694.24191
763.66610
833.09030

The major difference between yours and mine is that I have the pseudo-octave represented -- the 833.09030 that represents Phi itself -- but unless you're trying to work with notation, that won't matter; as an equal division of a pseudo-octave, adding one semitone would still give the same scale step.

Regards,
Jake

> ! C:\Cakewalk\scales\12throotofphi12.scl
> !
> 12th root pf phi in 12 steps
> 11
> !
> 69.42116
> 138.84232
> 208.26348
> 277.68464
> 347.10580
> 416.52696
> 485.94813
> 555.36929
> 624.79045
> 694.21161
> 763.63277
>

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Jake - since the numbers work out obviously this is what I did.

Thank you so very much!!
Last night I sat here for 2 hours trying to remember what on earth I did as
none of the numbers worked out and then gave up and played the piano.

I do intend to make a "stick" guitar with this tuning since it is pretty
straight forward with this tool that MatC shared with me:

http://www.ekips.org/tools/guitar/fretfind2d/

Thanks again!!

Chris

On Sat, Apr 9, 2011 at 7:36 PM, Jake Freivald <jdfreivald@...> wrote:

>
>
> Chris,
>
> It looks like the scale you have is 12-EDPhi, or close to it.
>
> According to Wikipedia, Phi is (1+sqrt(5))/2. That's 1.618034, or
> 833.09030 cents.
>
> The 12th root of Phi is thus 833.09030 cents / 12, or 69.42419 cents.
> That's your semitone. For some reason, your calculation seems a little
> off, possibly because you were using a weaker approximation of Phi;
> however, we're talking about a few thousandths of a cent per semitone,
> so what you have is close enough over the span of a few octaves.
>
> Here's the precise set of tones, from what I can see:
>
> ! C:\Program Files (x86)\Scala22\12th root of phi.scl
> !
> 12th root of phi tuning (Chris Vaisvil's idea)
> 12
> !
> 69.42419
> 138.84838
> 208.27257
> 277.69677
> 347.12096
> 416.54515
> 485.96934
> 555.39353
> 624.81772
> 694.24191
> 763.66610
> 833.09030
>
> The major difference between yours and mine is that I have the
> pseudo-octave represented -- the 833.09030 that represents Phi itself --
> but unless you're trying to work with notation, that won't matter; as an
> equal division of a pseudo-octave, adding one semitone would still give
> the same scale step.
>
> Regards,
> Jake
>
> > ! C:\Cakewalk\scales\12throotofphi12.scl
> > !
> > 12th root pf phi in 12 steps
> > 11
> > !
> > 69.42116
> > 138.84232
> > 208.26348
> > 277.68464
> > 347.10580
> > 416.52696
> > 485.94813
> > 555.36929
> > 624.79045
> > 694.21161
> > 763.63277
> >
>
>
>

🔗Jake Freivald <jdfreivald@...>

You're welcome, Chris. I've barely even touched the tuning itself, just did the math, so I'll be interested in knowing what you do with it and seeing / hearing the stick.

This got me thinking, though -- never a good idea -- that we hear logarithmically, and therefore it might be worth experimenting with the golden ratio directly in cents. If we take 1200 cents as a+b, and we want (a+b)/a = a/b, then a = 741.640786499872 cents and b = 458.359213500125 cents. (I'm not sure how precise I'd need to be here, so I'm giving you the precision at Excel's limit.)

I barely know what I'm even saying here, but perhaps some tuning wizards can explain how to use these two numbers as generators for some sort of interesting scale?

The 742-cent interval comes very close to circulating at 377 iterations, at 1198.6 cents octave-reduced, and the 458-cent interval comes even closer at 610 iterations, at 1199.1 cents octave-reduced. Is that interesting? Seems too high to be worth anything, but what do I know? :)

Regards,
Jake

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> The 742-cent interval comes very close to circulating at 377 iterations,
> at 1198.6 cents octave-reduced, and the 458-cent interval comes even
> closer at 610 iterations, at 1199.1 cents octave-reduced. Is that
> interesting? Seems too high to be worth anything, but what do I know? :)

377 and 610 are Fibonacci numbers, and it comes increasingly closer to closing at one of those. You are proposing a 1/phi = phi-1 generator measured in terms of octaves; there are others such as the Golden Meantone generator of (8-phi)/11 octaves which have attracted more interest even though they are more complicated. They have, however, the same or similar recurrence properties.

🔗Jake Freivald <jdfreivald@...>

>> The 742-cent interval comes very close to circulating at 377 iterations,
>> at 1198.6 cents octave-reduced, and the 458-cent interval comes even
>> closer at 610 iterations, at 1199.1 cents octave-reduced. Is that
>> interesting? Seems too high to be worth anything, but what do I know? :)
> 377 and 610 are Fibonacci numbers, and it comes increasingly closer to closing at one of those. You are proposing a 1/phi = phi-1 generator measured in terms of octaves; there are others such as the Golden Meantone generator of (8-phi)/11 octaves which have attracted more interest even though they are more complicated. They have, however, the same or similar recurrence properties.

How about 233 and 144?

If I temper a=741.640786499872 to A=741.6309013 and b=458.359213500125 to B=458.3690987, then:

1. A+B still = 1200 cents precisely
2. A and B both circulate at 233 iterations.

That strikes me as really weird -- isn't that too coincidental? Am I doing something wrong?

Or I can temper a to A=741.6666666666667 (i.e., 741 and 2/3) and B to 458.333333333 (i.e., 458 and 1/3) to get circulation at 144 iterations, again with A+B = 1200.

So 233 EDO and 144 EDO seem to be good temperaments for the golden ratio in cents, if I understand what I'm doing at all. (Big "if".)

Thanks,
Jake

🔗Callahan White <cortaigne@...>

I always knew this day would come. ;-)

http://cortaigne.mitb.net/phifths/

http://cortaigne.mitb.net/phifth1.mp3

http://cortaigne.mitb.net/phifth2.mp3

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> You're welcome, Chris. I've barely even touched the tuning itself, just
> did the math, so I'll be interested in knowing what you do with it and
> seeing / hearing the stick.
>
> This got me thinking, though -- never a good idea -- that we hear
> logarithmically, and therefore it might be worth experimenting with the
> golden ratio directly in cents. If we take 1200 cents as a+b, and we
> want (a+b)/a = a/b, then a = 741.640786499872 cents and b =
> 458.359213500125 cents. (I'm not sure how precise I'd need to be here,
> so I'm giving you the precision at Excel's limit.)
>
> I barely know what I'm even saying here, but perhaps some tuning wizards
> can explain how to use these two numbers as generators for some sort of
> interesting scale?
>
> The 742-cent interval comes very close to circulating at 377 iterations,
> at 1198.6 cents octave-reduced, and the 458-cent interval comes even
> closer at 610 iterations, at 1199.1 cents octave-reduced. Is that
> interesting? Seems too high to be worth anything, but what do I know? :)
>
> Regards,
> Jake
>

🔗Jake Freivald <jdfreivald@...>

Callahan, how did you derive your tuning? It looks like you did some divisions by the Golden Ratio and then reflected them within the octave.

On a lark I did something similar, but without the reflection. Here's the process:

1200 cents divided by the Golden Ratio gives you 742 & 458 cents.
742 divided gives you 458 & 283.
458 divided gives you 283 & 175.
283 divided gives you 175 & 108.
175 divided gives you 108 & 67.

(I did the math with higher precision, of course.)

That's as far as I divided things. When you stack them up "properly", with each division in its place and the smaller number on top for each division, I get this scale:

108.20393
175.07764
283.28157
391.48551
458.35921
566.56315
633.43685
741.64079
849.84472
916.71843
1024.92236
1133.12629
1200.00000

It's a 13-note strictly proper scale made of two step sizes, arranged as LSLLSLSLLSLLS. "Show Data" in Scala shows things I haven't seen before in other scales I've contstructed (or slapped together, as things may be): "distributional even" (does that mean MOS?), constant structure, rothenberg stability = 1, Myhill's property, etc. But I don't know what these things mean, so I'm not sure if they matter.

If you're trying to approximate JI, it looks like you're out of luck (though the high-prime-limit people around here may tell me otherwise), but the scale sounds surprisingly stable and kind of interesting anyway.

Regards,
Jake

> I always knew this day would come. ;-)
>
> http://cortaigne.mitb.net/phifths/
>
> http://cortaigne.mitb.net/phifth1.mp3
>
> http://cortaigne.mitb.net/phifth2.mp3
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

Is it possible to have longer versions of the music ? I'm intrigued but it
is so hard to "grok" such a short segment.

Thanks,

Chris

On Sun, Apr 10, 2011 at 1:21 AM, Callahan White <cortaigne@...> wrote:

>
>
> I always knew this day would come. ;-)
>
> http://cortaigne.mitb.net/phifths/
>
> http://cortaigne.mitb.net/phifth1.mp3
>
> http://cortaigne.mitb.net/phifth2.mp3
>
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> >
> > You're welcome, Chris. I've barely even touched the tuning itself, just
> > did the math, so I'll be interested in knowing what you do with it and
> > seeing / hearing the stick.
> >
> > This got me thinking, though -- never a good idea -- that we hear
> > logarithmically, and therefore it might be worth experimenting with the
> > golden ratio directly in cents. If we take 1200 cents as a+b, and we
> > want (a+b)/a = a/b, then a = 741.640786499872 cents and b =
> > 458.359213500125 cents. (I'm not sure how precise I'd need to be here,
> > so I'm giving you the precision at Excel's limit.)
> >
> > I barely know what I'm even saying here, but perhaps some tuning wizards
> > can explain how to use these two numbers as generators for some sort of
> > interesting scale?
> >
> > The 742-cent interval comes very close to circulating at 377 iterations,
> > at 1198.6 cents octave-reduced, and the 458-cent interval comes even
> > closer at 610 iterations, at 1199.1 cents octave-reduced. Is that
> > interesting? Seems too high to be worth anything, but what do I know? :)
> >
> > Regards,
> > Jake
> >
>
>
>

🔗Callahan White <cortaigne@...>

I approached it as simply as possible, without any regard for how close the result might be to just intonation or any EDO, or for maximal evenness or any other theory; I just wanted to let phi do what it does. The basic version is the simplest (I think of it like a nautilus shell), but of course it's completely empty above the "fifth". (I nearly dismissed the whole idea when I heard how the tonic triad ended up sounding; the more I played around, though, the more I got used to it.) To fill out the top I looked at reflecting the divisions (I think of this one like a peacock tail), then at repeating the pattern from the tonic shifted up to the "fifth" (inspired by tetrachords), and I noticed they both had 916.7 and 1024.9, so I merged them into a "full" tuning (with the 350.2 interval a reflection of the 849.8 interval in the repeated version). Initially, I hadn't stopped dividing at 108.2, but when I realized doing so would give me this nifty 12-tone scale with which I could retune my keyboard, I stopped crunching numbers and started making music.

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
>
> Callahan, how did you derive your tuning? It looks like you did some
> divisions by the Golden Ratio and then reflected them within the octave.
>
> On a lark I did something similar, but without the reflection. Here's
> the process:
>
> 1200 cents divided by the Golden Ratio gives you 742 & 458 cents.
> 742 divided gives you 458 & 283.
> 458 divided gives you 283 & 175.
> 283 divided gives you 175 & 108.
> 175 divided gives you 108 & 67.
>
> (I did the math with higher precision, of course.)
>
> That's as far as I divided things. When you stack them up "properly",
> with each division in its place and the smaller number on top for each
> division, I get this scale:
>
> 108.20393
> 175.07764
> 283.28157
> 391.48551
> 458.35921
> 566.56315
> 633.43685
> 741.64079
> 849.84472
> 916.71843
> 1024.92236
> 1133.12629
> 1200.00000
>
> It's a 13-note strictly proper scale made of two step sizes, arranged as
> LSLLSLSLLSLLS. "Show Data" in Scala shows things I haven't seen before
> in other scales I've contstructed (or slapped together, as things may
> be): "distributional even" (does that mean MOS?), constant structure,
> rothenberg stability = 1, Myhill's property, etc. But I don't know what
> these things mean, so I'm not sure if they matter.
>
> If you're trying to approximate JI, it looks like you're out of luck
> (though the high-prime-limit people around here may tell me otherwise),
> but the scale sounds surprisingly stable and kind of interesting anyway.
>
> Regards,
> Jake
>
>
>
> > I always knew this day would come. ;-)
> >
> > http://cortaigne.mitb.net/phifths/
> >
> > http://cortaigne.mitb.net/phifth1.mp3
> >
> > http://cortaigne.mitb.net/phifth2.mp3
> >
> >
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:

> It's a 13-note strictly proper scale made of two step sizes, arranged as
> LSLLSLSLLSLLS. "Show Data" in Scala shows things I haven't seen before
> in other scales I've contstructed (or slapped together, as things may
> be): "distributional even" (does that mean MOS?), constant structure,
> rothenberg stability = 1, Myhill's property, etc. But I don't know what
> these things mean, so I'm not sure if they matter.

The reason for all this is that you've constructed a MOS with a Fibonacci number of notes using a generator of 1/phi octaves. Hence the ratio of step sizes is phi, approximated in edos which are Fibonacci numbers by two successive Fibonacci numbers; thus 34edo would approximate it with L=3, s=2, 55 with L=5, s=3 etc. If you look at the 5-limit 34&55 temperament (two successive Fibonacci numbers) you find it tempers out |39 -7 -12> and has a POTE generator of 741.365 cents, so while not much is going on in the 13-note MOS in the 5-limit, it could be used in the 21-note MOS. Other approximated ratios I'll leave to someone else to consider.

🔗Jake Freivald <jdfreivald@...>

Me:

Gene:
> 377 and 610 are Fibonacci numbers, and it comes increasingly closer to
> closing at one of those. You are proposing a 1/phi = phi-1 generator
> measured in terms of octaves; there are others such as the Golden
> Meantone generator of (8-phi)/11 octaves which have attracted more
> interest even though they are more complicated. They have, however,
> the same or similar recurrence properties.

Also:
> The reason for all this is that you've constructed a MOS with a Fibonacci
> number of notes using a generator of 1/phi octaves. Hence the ratio of
> step sizes is phi, approximated in edos which are Fibonacci numbers by
> two successive Fibonacci numbers; thus 34edo would approximate it with
> L=3, s=2, 55 with L=5, s=3 etc.

That certainly wasn't my intent, but I think I see how it works now:
because phi is the ratio of (a+b)/a, there's a natural affinity
between it and the Fibonacci numbers. Weird phi properties: I'm
pushing the "I believe" button on that one.

If I understand correctly, the reason to temper this to an EDO instead
of using the exact scale are the same as the reason for tempering to
an EDO for JI scales: You get guaranteed circulation, easy
transposition, and "close enough" sonic properties.

The approximations for the scale in 13 are up to 25 cents off. The
approximations in 21 are up to almost 10 cents off. The approximations
in 34 are all under 5 cents off. Since the scale's not based on JI,
the proportions may be close enough even in 13 or 21 to give the
desired effect, but 34 makes them, to my ear, indistinguishable from
perfect.

I ran the pure intervals through my testing piece, and it's an
interesting-sounding scale. One of these weeks / months / years I'll
try to play with it more fully.

Thanks, Gene, for explaining it, and Chris for sparking the idea.

Regards,
Jake

🔗chrisvaisvil@...

So the goal of the 34th root of phi tuning is to line up with the fibonacci series?
-----Original Message-----
From: Jake Freivald <jdfreivald@gmail.com>
Sender: tuning@yahoogroups.com
Date: Mon, 11 Apr 2011 22:36:47
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: 12th root of Phi

Me:

I ran the pure intervals through my testing piece, and it's an
interesting-sounding scale. One of these weeks / months / years I'll
try to play with it more fully.

Thanks, Gene, for explaining it, and Chris for sparking the idea.

Regards,
Jake

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> This got me thinking, though -- never a good idea -- that we hear
> logarithmically, and therefore it might be worth experimenting with the
> golden ratio directly in cents. If we take 1200 cents as a+b, and we
> want (a+b)/a = a/b, then a = 741.640786499872 cents and b =
> 458.359213500125 cents. (I'm not sure how precise I'd need to be here,
> so I'm giving you the precision at Excel's limit.)

This looks like some variety of Father temperament. What are the audible sonic properties that one could expect from such a scale? Will such a relationship to phi lead to anything musically important?

-Igs

🔗Jake Freivald <jdfreivald@...>

Chris,

> So the goal of the 34th root of phi tuning is to line up with the fibonacci series?

No, on two counts. First, this isn't the 34th root of phi, which would mean that we're dividing phi into 34 equal parts. (By "equal" I mean equal in cents.) Second, the Fibonacci connection seems to be a happy accident, perhaps resulting from the special relationship phi has with the Fibonacci series; I didn't plan on using Fibonacci numbers at all when I started this.

Also, note that "phi" means something different compared to the 12th root of phi scale.

In that scale, I used phi directly, as a ratio: Just as 3/2 = 1.5 = 702 cents, phi = 1.61803399..., or 833 cents.

In this scale, I found the Golden Section of the 1200-cent octave. It might be easier to think of in spatial terms, so think of finding Golden sections in the width of a 1200mm-wide canvas. That means (a+b)/a = a/b, so (1200 mm)/a mm = a mm / b mm. Solving for a and b gives you about 742 mm and about 458 mm. Then I divided each section of the painting into its golden section: 742 mm has a Golden Section at 458 mm and 283 mm, 458 has a Golden section at 283 mm and 175 mm, and so on, until the entire canvas is divided up into strips that are either 67mm or 108 mm wide.

Igs,

> This looks like some variety of Father temperament.

As you can see from the procedure above, the first scale I gave wasn't a temperament of anything. It was "just intonation", except the factor used was an irrational number instead of a product of primes.

When I started talking about approximating it by using some EDO or another that I started tempering it.

> What are the audible sonic properties that one could expect from such a scale?
> Will such a relationship to phi lead to anything musically important?

Good questions.

Since the procedure leads to a very regular scale, it may be usable because it has the properties of propriety, only two step sizes, and so on. This may be true regardless of the fact that it's related to phi.

If the ear likes phi the way the eye does, it's possible that these intervals will sound pleasing enough even though they don't approximate many JI intervals. Triads or melodies based on the relationship (a+b)/a = a/b might sound good: 0-741-1200, 0-283-458, 0-108-175, etc. Tetrads might include breaking these elements down into other phi-based ratios, like 0-548-741-1200. These might be a sort of alternate harmony that "works" despite not being related to prime intervals.

Then again, there's no physical reason for me to think the ear will like these relationships. If the ear doesn't like phi, these may sound like junk. :)

Playing around with it in a preliminary way suggests that some of this sounds okay, but whether that's related to phi or not isn't clear to me.

Regards,
Jake

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Michael <djtrancendance@...>

🔗Jake Freivald <jdfreivald@...>

> If possible could you post a scala file of the 13 note / Phi and 34 note /
Phi tuning you mentioned?

Here's the "pure" scale...

! C:\Program Files (x86)\Scala22\Golden Ratio.scl
!
Successive divisions of the octave by the Golden Section
13
!
108.20393
175.07764
283.28157
391.48551
458.35921
566.56315
633.43685
741.64079
849.84472
916.71843
1024.92236
1133.12629
1200.00000

...and here's the scale tempered to 34 EDO.

! C:\Program Files (x86)\Scala22\Golden Ratio (in 34-EDO).scl
!
Successive divisions of the octave by the Golden Section (tempered to 34
EDO)
13
!
105.88235
176.47059
282.35294
388.23529
458.82353
564.70588
635.29412
741.17647
847.05882
917.64706
1023.52941
1129.41176
1200.00000

As Gene points out, I could have tempered it in any Fibonacci-numbered EDO;
I guess I'm using 34 because its values aren't very far off, but it seems
small enough to manage -- which may just be superstition on my part. :) I
could have used 89 or 144 EDO for higher accuracy.

I test some scales by plugging them into a little "composition" in Csound,
and that's all I've done with the scale so far. Not all tones are
represented, but most are. You can here it here:
http://www.freivald.org/~jake/documents/phi_bells.mp3

That's the untempered scale. It's not meant to be particularly musical, just
to get some of the main notes out there.

For comparison, you can listen to Cantonpenta or [a semi-random jumble of]
JI intervals:
http://www.freivald.org/~jake/documents/canton_bells.mp3
http://www.freivald.org/~jake/documents/ji_bells.mp3

Regards,
Jake

🔗Jake Freivald <jdfreivald@...>

🔗Mike Battaglia <battaglia01@...>

On Mon, Apr 11, 2011 at 11:44 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Jake Freivald <jdfreivald@...> wrote:
> > This got me thinking, though -- never a good idea -- that we hear
> > logarithmically, and therefore it might be worth experimenting with the
> > golden ratio directly in cents. If we take 1200 cents as a+b, and we
> > want (a+b)/a = a/b, then a = 741.640786499872 cents and b =
> > 458.359213500125 cents. (I'm not sure how precise I'd need to be here,
> > so I'm giving you the precision at Excel's limit.)
>
> This looks like some variety of Father temperament. What are the audible sonic properties that one could expect from such a scale? Will such a relationship to phi lead to anything musically important?

I think it'll lead to an MOS structure that looks roughly like this:

http://farm2.static.flickr.com/1278/694780262_8874b4f225.jpg

-Mike