Yahoo Tuning Groups Ultimate Backup tuning Golden ratio scale: has this been done before?

I will tell you this much. I tried the scale and the following steps from the 11 notes listed in the scala file

root
2nd)
3rd)
4th)
6th)
8th)
10th)
...make an ABSOLUTELY fantastic sounding 7-note scale.
    And...that scale can be transposed within the 11-note "golden ratio" tuning and still sounds great.
****************************
   I wonder why this has not been suggested...as an alternative to 7-note scales under 12TET?!.
   To my ear the above scale sound's much better than the usual "do-re-mi-fa-so-la-ti" and all the grammar issues of transposition seem to be resolved by using the full 11 notes as a tuning.   It may not get more notes in the scale for melodic possibilities (IE it
seems limited as 7 notes rather than the 9 I was hoping for...but it sure is ace for 7-note
scale. And who knows, maybe Sethares-ian overtone alignment could enable it to make a consonant 9-note scale.
-------------------------
    Any ideas anyone, why this has not overtaken 12TET? I say...shoot, we should try to get more people to adopt use of this "golden ratio tuning"! :-)

--- On Tue, 2/3/09, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] Golden ratio scale: has this been done before?
To: tuning@yahoogroups.com
Date: Tuesday, February 3, 2009, 2:34 PM

Thanks Petr, I'll give it a go.

Wasn't there also a gentleman's site with a bunch of scale built on spirals and Phi?

I remember I didn't know how to translate it into a scale, at least, not at the time.

Chris

On Tue, Feb 3, 2009 at 5:15 PM, Petr Parízek <p.parizek@chello. cz> wrote:

Michael
wrote:
>
Or...perhaps most importantly, has this sort of thing been done
> before
(in which case I might as well just research it and stop trying
> to
re-invent the wheel)?
Do
you mean
this?
http://groups. yahoo.com/ group/tuning/ message/65869
Petr

🔗Petr Parízek <p.parizek@...>

🔗Graham Breed <gbreed@...>

djtrancendance wrote:

> This is formed by a "circle of the golden ratio" (which is why it
> is not stated in fractional form).
> > Far as I have tried, I can take any of the numbers in scale,
> multiply or divide them by 1.61803 then (if needed) multiply them or
> divide them by 2 and get them to fit back into the octave...and the
> result will always be another note in the scale.
> > Does this have any special mathematical significance and/or can it
> be used to help create beautiful scales?

You can use it with a golden ratio timbre, in which has the property that difference tones are always in tune with the timbre. I've tried it, it's not that great, but it works well with distortion.

> Or...perhaps most importantly, has this sort of thing been done
> before (in which case I might as well just research it and stop trying
> to re-invent the wheel)?

O'Connell, Walter. "The Tonality of the Golden Section: The Symmetrization of Tone Space", Xenharmonikï¿½n vol. 15, 1993, pp. 3-18.

Graham

🔗Kraig Grady <kraiggrady@...>

🔗Chris Vaisvil <chrisvaisvil@...>

Thank you - horograms. This is what I was looking for earlier.

Perhaps I know enough now to program this in scala.

Again, thank you very much!

Chris

On Tue, Feb 3, 2009 at 11:43 PM, Kraig Grady <kraiggrady@...>wrote:

> Loren Temes then Walter O'connell where the first to investigate it.
> see temes article here in here http://anaphoria.com/library.html
> All the horograms eventually make scales where the large to small is the
> golden ratio. There is the horogram Wilson numbers 1 that is based on phi.
> see toward the bottom
> http://anaphoria.com/wilson.html
> then you can also look at the scales of Mt. Meru which also starts with
> phi and looks at other recurrent sequences.
>
> --
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Petr Parízek <p.parizek@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "djtrancendance" <djtrancendance@...>
wrote:
>
> 1) 1
> 2) 1.0589866
> 3) 1.1214765537
> 4) 1.18765394581
> 5) 1.3089827
> 6) 1.3862246729
> 7) 1.468024629720
> 8) >>1.61803<<<
> 9) 1.713472151
> 10) 1.814582708257
> 11) 1.921659713944
>
> This is formed by a "circle of the golden ratio" (which is why it
> is not stated in fractional form).
>
> Far as I have tried, I can take any of the numbers in scale,
> multiply or divide them by 1.61803 then (if needed) multiply them or
> divide them by 2 and get them to fit back into the octave...and the
> result will always be another note in the scale.
>
> Does this have any special mathematical significance and/or can it
> be used to help create beautiful scales?
>
>
> Or...perhaps most importantly, has this sort of thing been done
> before (in which case I might as well just research it and stop
trying
> to re-invent the wheel)?
>
Attempts to apply the Golden mean or ratio to music is well
documented. Bela Bartok devised a whole compositional system by
applying the Fibbonachi series (which approaches the Golden ratio as
its upper limit) in both harmony and tempo. True, the ratio has
certain recursive properties which are geometrically unique (It is
the only number where its inverse equals itself minus 1). Yet at the
end of the day, this really has little to do with music, which comes
with its own mathematical system dating right back to the very
beginnings of maths/science with Pythagoras. In fact music has its
own version of recursion in tempered intervals like the square root
of 2. Since octaves are equivalent in music and sqr2/1 = 2/sqr2, then
it is impossible to determine which is the tonic, 1: sqr2 = sqr2: 2,
such that the denominator and numerator plays precisely the same role
(i.e. Given C and F#, F# is the b5 to C and C is the b5 to F# so that
the ear can't tell which is the tonic).
I don't think the eg you gave is more in tune than "do re mi" by any
means and I suspect that you've probably listened to it too much and
are reading into it.

- Rick

🔗Kraig Grady <kraiggrady@...>

rick said

"Yet at the
end of the day, this really has little to do with music, which comes
with its own mathematical system dating right back to the very
beginnings of maths/science with Pythagoras."

it is these type of recurrent sequences i have been working with for the past 10 years. although they even appeared earlier in scattered applications.
not only have i found many of them musically rewarding and useful. i have also stumbled upon quite a few scales in the world that fit well into these sequences.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

🔗Mark Rankin <markrankin95511@...>

How about trying "with it's own mathematical system dating right back to the very beginning of math(s)/science with The Babylonians (a couple thousand years before Pythagoras)...".

-- Mark Rankin

--- On Wed, 2/4/09, Kraig Grady <kraiggrady@...> wrote:

From: Kraig Grady <kraiggrady@...>
Subject: [tuning] Re: Golden ratio scale: has this been done before?
To: tuning@yahoogroups.com
Date: Wednesday, February 4, 2009, 5:09 PM

rick said

"Yet at the
end of the day, this really has little to do with music, which comes
with its own mathematical system dating right back to the very
beginnings of maths/science with Pythagoras."

it is these type of recurrent sequences i have been working with for
the past 10 years. although they even appeared earlier in scattered
applications.
not only have i found many of them musically rewarding and useful. i
have also stumbled upon quite a few scales in the world that fit well
into these sequences.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria. com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasou th.blogspot. com/>

',',',',',', ',',',',' ,',',',', ',',',',' ,',',',', ',',',',' ,

🔗Danny Wier <dawiertx@...>

Sorry for the late reply, but since I am writing something using a chain
of "Golden sixths", I'll chime in.

This is an overture to something that starts off with a long chord in
the strings that starts on C1 in the double basses and adds the
harmonics of the Fibonacci series one by one: C2, G2, E-3, C4, Ab++4,
F--5, C#6, A++6, F#--7 and finally D8 (can you play a harmonic that high
on violin?). Plusses and minuses alter a 12-equal pitch by a 72-edo
comma, 16 2/3 cents.

Notice that every pitch beginning with middle C has another one 25 steps
in 36-et higher; these approximate the ratios 13/8, 21/13, 34/21, 55/34,
89/55 and 144/89. So a Golden tuning approximates the primes 7, 13, 17
and 23 best, while 55/54 is a type of sixth tone. Since 36 is a multiple
of 12, you get the good fifths and fourths as well.

If you want to use fewer tones in an octave, 13 is the next lower number
that's maximally even. That would produce a tuning with steps of
L=99.271 and s=69.097, with the symmetrical scale being LLsLLLsLLLsLL.

You should write us something that uses this scale in that case. ;)

~D.

On Tue, 2009-02-03 at 14:44 -0800, djtrancendance@... wrote:
> I will tell you this much. I tried the scale and the following steps
> from the 11 notes listed in the scala file
>
> root
> 2nd)
> 3rd)
> 4th)
> 6th)
> 8th)
> 10th)
> ...make an ABSOLUTELY fantastic sounding 7-note scale.
> And...that scale can be transposed within the 11-note "golden
> ratio" tuning and still sounds great.
> ****************************
> I wonder why this has not been suggested...as an alternative to
> 7-note scales under 12TET?!.
> To my ear the above scale sound's much better than the usual
> "do-re-mi-fa-so-la-ti" and all the grammar issues of transposition
> seem to be resolved by using the full 11 notes as a tuning. It may
> not get more notes in the scale for melodic possibilities (IE it seems
> limited as 7 notes rather than the 9 I was hoping for...but it sure is
> ace for 7-note scale. And who knows, maybe Sethares-ian overtone
> alignment could enable it to make a consonant 9-note scale.

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...>
wrote:
>
> How about trying "with it's own mathematical system dating right
back to the very beginning of math(s)/science with The Babylonians (a
couple thousand years before Pythagoras)...".
>
> -- Mark Rankin
>
> --- On Wed, 2/4/09, Kraig Grady <kraiggrady@...> wrote:
>
> From: Kraig Grady <kraiggrady@...>
> Subject: [tuning] Re: Golden ratio scale: has this been done before?
> To: tuning@yahoogroups.com
> Date: Wednesday, February 4, 2009, 5:09 PM
>
> But Mark, it is important not to rewrite history to support modern
(political??) agendas. It wasn't the Babylonians who found the
connection between music and maths, it was the Greeks. Our word logic
comes from the Greek logos which to the Pythagoreans meant ratio,
with the implication that the universe (Gk Kosmos) is ordered and
harmonic. The Latin translation is ratio and is the root of words
like rational etc... Try to imagine how wonderful it must have seemed
to them to discover that the previously two separate worlds of the
(Babylonian) series of natural numbers and (Greek) natural harmonies
seemed to be saying the same thing in different ways. This led to the
idea of the music of the spheres, which although incorrect, did
provide the belief-system which eventually led to modern astronomy
etc. It is here and nowhere else where the very idea of maths and
physics come together i.e. the birth of science, which did after all
begin as a western institution. And in fact a study of modern wave
theory does show that Pythagoras was only incorrect in assuming that
we are restricted to the small whole-numbers; the rational numbers DO
lead to periodic waves, and the GCD of these waves produce the basis
of modern tonality. The tradition has managed to survive underground
in musical harmony and wave theory (It HAS been discovered that
matter, sound and light are all forms of waves and that energy-
frequency are quantised in whole numbers...is this not just the Greek
Kosmos in another form?).
>
> Rick
>
>
> rick said
>
> "Yet at the
> end of the day, this really has little to do with music, which comes
> with its own mathematical system dating right back to the very
> beginnings of maths/science with Pythagoras."
>
> it is these type of recurrent sequences i have been working with
for
> the past 10 years. although they even appeared earlier in scattered
> applications.
> not only have i found many of them musically rewarding and useful.
i
> have also stumbled upon quite a few scales in the world that fit
well
> into these sequences.
> --
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria. com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasou th.blogspot.
com/>
>
> ',',',',',', ',',',',' ,',',',', ',',',',' ,',',',', ',',',',' ,
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Mark Rankin <markrankin95511@...>
wrote:
>
> How about trying "with it's own mathematical system dating right
back to the very beginning of math(s)/science with The Babylonians (a
couple thousand years before Pythagoras)...".
>
> -- Mark Rankin
>
> --- On Wed, 2/4/09, Kraig Grady <kraiggrady@...> wrote:
>
> From: Kraig Grady <kraiggrady@...>
> Subject: [tuning] Re: Golden ratio scale: has this been done before?
> To: tuning@yahoogroups.com
> Date: Wednesday, February 4, 2009, 5:09 PM
>
>
>
>
>
>
> rick said
>
> "Yet at the
> end of the day, this really has little to do with music, which comes
> with its own mathematical system dating right back to the very
> beginnings of maths/science with Pythagoras."
>
> it is these type of recurrent sequences i have been working with
for
> the past 10 years. although they even appeared earlier in scattered
> applications.
> not only have i found many of them musically rewarding and useful.
i
> have also stumbled upon quite a few scales in the world that fit
well
> into these sequences.
> --
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria. com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasou th.blogspot.
com/>
>
> ',',',',',', ',',',',' ,',',',', ',',',',' ,',',',', ',',',',' ,
>Hi Kraig, I too find recursive sequence very interesting and would
like very much to hear some of your examples. I'm not saying that the
rational numbers and other systems are necessarilly mutually
exclusive (The example I heard on this post sounded badly out-of-tune
to me). What I do suspect is that there are other things going on in
the background to do with rationality/periodicity which at the very
least needs to be researched with some concerted effort. eg; the
tempered minor 3rd, 2 to the power of 1/3, is usually considered
irrational therefore aperiodic. But can we honestly hear an infinite
number of digits? And if it is irrational, then how is it that we can
use this as a tonality (i.e. as a minor key)? On the other hand, the
diminished chord does not contain a tonic.

One possibility I am exploring is that these intervals are not
irrational at all but are rational numbers high up in the harmonic
series (and that Pythagoras' restriction to small whole numbers is a
bias). 609/512 you will find seems to be a very good 'tempered' minor
third. And the fact that 512 is octave equiv. to 1 produces the
correct tonic (512 = 2 power of 9, ninth octave). Also, the
difference between the irrational and rational form seems to be below
the schisma. Squaring the interval gives 1.4147986, very close to
sqr2, cubing gives 1.6828...close to min6, and power of 4 gives
2.0016...close to octave. If we can't hear beyond (say) two decimal
places then 2.0016 = 2, and whole 'chunks' of rational numbers can be
assigned to the same interval (and even change due to harmonic
context).

So perhaps there is something similar going on with the golden ratio,
which strictly speaking is an irrational number? Anyway, I'd love to
hear your recursive sequences (and sorry I could'nt make your
concert),

Rick

🔗Michael Sheiman <djtrancendance@...>

---If you want to use fewer tones in an octave, 13 is the next lower number
---that's maximally even. That would produce a tuning with steps of
---L=99.271 and s=69.097, with the symmetrical scale being LLsLLLsLLLsLL.

Coincidence, these ("69" and "99") are EXACTLY the intervals I came up with by ear out of all the possibilities in the golden tuning (and the number of notes in the scale was approximately 13)!

My short example/melodic test of a scale under this "13 note per 2/1 octave" (and a transposition of it) is below:
http://www.geocities.com/djtrancendance/micro/goldenscaletrans.mp3

Let me know what you think... :-)

--- On Thu, 2/5/09, Danny Wier <dawiertx@...> wrote:

From: Danny Wier <dawiertx@...>
Subject: Re: [tuning] Golden ratio scale: has this been done before?
To: tuning@yahoogroups.com
Date: Thursday, February 5, 2009, 4:44 PM

Sorry for the late reply, but since I am writing something using a chain

of "Golden sixths", I'll chime in.

This is an overture to something that starts off with a long chord in

the strings that starts on C1 in the double basses and adds the

harmonics of the Fibonacci series one by one: C2, G2, E-3, C4, Ab++4,

F--5, C#6, A++6, F#--7 and finally D8 (can you play a harmonic that high

on violin?). Plusses and minuses alter a 12-equal pitch by a 72-edo

comma, 16 2/3 cents.

Notice that every pitch beginning with middle C has another one 25 steps

in 36-et higher; these approximate the ratios 13/8, 21/13, 34/21, 55/34,

89/55 and 144/89. So a Golden tuning approximates the primes 7, 13, 17

and 23 best, while 55/54 is a type of sixth tone. Since 36 is a multiple

of 12, you get the good fifths and fourths as well.

If you want to use fewer tones in an octave, 13 is the next lower number

that's maximally even. That would produce a tuning with steps of

L=99.271 and s=69.097, with the symmetrical scale being LLsLLLsLLLsLL.

You should write us something that uses this scale in that case. ;)

~D.

On Tue, 2009-02-03 at 14:44 -0800, djtrancendance@ yahoo.com wrote:

> I will tell you this much. I tried the scale and the following steps

> from the 11 notes listed in the scala file

> root

> 2nd)

> 3rd)

> 4th)

> 6th)

> 8th)

> 10th)

> ...make an ABSOLUTELY fantastic sounding 7-note scale.

> And...that scale can be transposed within the 11-note "golden

> ratio" tuning and still sounds great.

> ************ ********* *******

> I wonder why this has not been suggested... as an alternative to

> 7-note scales under 12TET?!.

> To my ear the above scale sound's much better than the usual

> "do-re-mi-fa- so-la-ti" and all the grammar issues of transposition

> seem to be resolved by using the full 11 notes as a tuning. It may

> not get more notes in the scale for melodic possibilities (IE it seems

> limited as 7 notes rather than the 9 I was hoping for...but it sure is

> ace for 7-note scale. And who knows, maybe Sethares-ian overtone

> alignment could enable it to make a consonant 9-note scale.

Golden ratio scale: has this been done before?

🔗djtrancendance <djtrancendance@...>

🔗Petr Parízek <p.parizek@...>

🔗Michael Sheiman <djtrancendance@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗djtrancendance@...

🔗Petr Parízek <p.parizek@...>

🔗Graham Breed <gbreed@...>

🔗Kraig Grady <kraiggrady@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Petr Parízek <p.parizek@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗rick_ballan <rick_ballan@...>

🔗Kraig Grady <kraiggrady@...>

🔗Mark Rankin <markrankin95511@...>

🔗Danny Wier <dawiertx@...>

🔗rick_ballan <rick_ballan@...>

🔗rick_ballan <rick_ballan@...>

🔗Michael Sheiman <djtrancendance@...>