LucyTuning with pi-based spectra?

Hey Charles Lucy:

I've been reading Tuning, Timbre, Spectrum, Scale (which seems like a
tour de force; I should have read it long ago), and I got an idea.

You always say LucyTuning is not just another meantone, but a unique
system closely related to the special properties of pi. Right? Have
you ever tried using a special instrument (most likely synthesized),
whose overtones have factors of pi, to bring out this aspect?

Keenan Pepper

Sounds like an interesting idea.
So you are proposing that by inputting frequencies having LucyTuned
intervals, it will generate a particular timbre which represents the
tuning.
I have mainly been experimenting with samples and real fretted instruments.
I Mac OSX running; what application do you suggest would be suitable?
What do you mean by factors of pi?

On Thu, September 20, 2007 11:15 pm, Keenan Pepper wrote:
> Hey Charles Lucy:
>
>
> I've been reading Tuning, Timbre, Spectrum, Scale (which seems like a
> tour de force; I should have read it long ago), and I got an idea.
>
> You always say LucyTuning is not just another meantone, but a unique
> system closely related to the special properties of pi. Right? Have you
> ever tried using a special instrument (most likely synthesized), whose
> overtones have factors of pi, to bring out this aspect?
>
> Keenan Pepper
>
>

Charles Lucy - lucy@harmonics.com
-- Promoting global harmony through LucyTuning --
for information on LucyTuning go to: http://www.lucytune.com
for LucyTuned Lullabies go to http://www.lullabies.co.uk

Charles Lucy wrote:
> Sounds like an interesting idea.
> So you are proposing that by inputting frequencies having LucyTuned
> intervals, it will generate a particular timbre which represents the
> tuning.
> I have mainly been experimenting with samples and real fretted instruments.
> I Mac OSX running; what application do you suggest would be suitable?

You can do it with Csound but I don't know if Csound would be to your taste.

> What do you mean by factors of pi?

That's what Keenan said, of course. So whatever he meant to say, what you *could* do is take a harmonic timbre and retune each partial to a LucyTuned equivalent. So the 3 would be a fifth above the 2, the 5 a major third above the 4, and so on.

Graham

It's a fun idea. For what it's worth, tuning - timbre
relationships become very evident on a Tonal Plexus
keyboard, where depending on the timbre of a given
patch, the most consonant sounding intervals are not
always the same fingering in the default JND tuning. This
goes for any interval, including the octave! I found right
away that a patch which uses stretched harmonics can
sound smoother when the interval patterns used on the
keyboard stretch by an extra key = one JND larger.
Sometimes two, or sometimes compressing in the
opposite direction, for example with meantone fifths.
Anyway, playing around with timbre and tuning on the
TP is a lot of fun and for me it was a revelation, being
able to experiment with it right there at the keyboard
in real time.

Yours,
Aaron Hunt
H-Pi Instruments

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Charles Lucy wrote:
> > Sounds like an interesting idea.
> > So you are proposing that by inputting frequencies having LucyTuned
> > intervals, it will generate a particular timbre which represents the
> > tuning.
> > I have mainly been experimenting with samples and real fretted instruments.
> > I Mac OSX running; what application do you suggest would be suitable?
>
> You can do it with Csound but I don't know if Csound would
> be to your taste.
>
> > What do you mean by factors of pi?
>
> That's what Keenan said, of course. So whatever he meant to
> say, what you *could* do is take a harmonic timbre and
> retune each partial to a LucyTuned equivalent. So the 3
> would be a fifth above the 2, the 5 a major third above the
> 4, and so on.
>
>
> Graham
>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Charles Lucy wrote:
> > Sounds like an interesting idea.
> > So you are proposing that by inputting frequencies having
LucyTuned
> > intervals, it will generate a particular timbre which represents
the
> > tuning.
> > I have mainly been experimenting with samples and real fretted
instruments.
> > I Mac OSX running; what application do you suggest would be
suitable?
>
> You can do it with Csound but I don't know if Csound would
> be to your taste.

Csound is certainly to my taste, and I work with a number of
pi-based tunings and timbres. But I think Csound is the bee's
knees anyway. And it runs like a rock live, by the way.
>
> > What do you mean by factors of pi?
>
> That's what Keenan said, of course. So whatever he meant to
> say, what you *could* do is take a harmonic timbre and
> retune each partial to a LucyTuned equivalent. So the 3
> would be a fifth above the 2, the 5 a major third above the
> 4, and so on.

Speaking of pi and phi and all that, the "meantony" fifth which
lies at the golden cut of 1:1 and 5:9 is a beauty. It is very
close to the Lucy fifth.

-Cameron Bobro

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

> Speaking of pi and phi and all that, the "meantony" fifth which
> lies at the golden cut of 1:1 and 5:9 is a beauty. It is very
> close to the Lucy fifth.

Presumably you don't mean either (9/5)/phi or
cents(9/5)/phi. What do you mean? Why is it a beauty?

On 9/21/07, Gene Ward Smith <genewardsmith@sbcglobal.net> wrote:
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> > Speaking of pi and phi and all that, the "meantony" fifth which
> > lies at the golden cut of 1:1 and 5:9 is a beauty. It is very
> > close to the Lucy fifth.
>
> Presumably you don't mean either (9/5)/phi or
> cents(9/5)/phi. What do you mean? Why is it a beauty?

I think the "golden cut" between A and B is A + (B-A)/phi (or the
other cut, B + (A-B)/phi = A + (B-A)/phi^2).

So the fifth Cameron's talking about is 1 + (9/5-1)/phi = (4*phi+1)/5
= 1.4944 = 695.5 cents. I don't know what makes it a "beauty", though;
Cameron will have to explain that.

Keenan

This page shows the figures for the phi (Kornerup) tuning.

http://www.lucytune.com/tuning/mean_tone.html

The idea is that the ratio between the Large and small intervals is phi.

i.e. the only number which is one greater than its reciprocal.

Maybe these "meantone-type" tunings should be called 5L+2s,

which will also solve the scalecoding and mapping using the system which John Gibbon, Jonathan Glasier and I put together about 15 years ago, to categorise meantone-type scales, chords, etc.

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 21 Sep 2007, at 23:30, Keenan Pepper wrote:

> On 9/21/07, Gene Ward Smith <genewardsmith@sbcglobal.net> wrote:
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> > wrote:
> >
> > > Speaking of pi and phi and all that, the "meantony" fifth which
> > > lies at the golden cut of 1:1 and 5:9 is a beauty. It is very
> > > close to the Lucy fifth.
> >
> > Presumably you don't mean either (9/5)/phi or
> > cents(9/5)/phi. What do you mean? Why is it a beauty?
>
> I think the "golden cut" between A and B is A + (B-A)/phi (or the
> other cut, B + (A-B)/phi = A + (B-A)/phi^2).
>
> So the fifth Cameron's talking about is 1 + (9/5-1)/phi = (4*phi+1)/5
> = 1.4944 = 695.5 cents. I don't know what makes it a "beauty", though;
> Cameron will have to explain that.
>
> Keenan
>
>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> On 9/21/07, Gene Ward Smith <genewardsmith@...> wrote:
> > --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@>
wrote:
> >
> > > Speaking of pi and phi and all that, the "meantony" fifth which
> > > lies at the golden cut of 1:1 and 5:9 is a beauty. It is very
> > > close to the Lucy fifth.
> >
> > Presumably you don't mean either (9/5)/phi or
> > cents(9/5)/phi. What do you mean? Why is it a beauty?
>
> I think the "golden cut" between A and B is A + (B-A)/phi (or the
> other cut, B + (A-B)/phi = A + (B-A)/phi^2).
>
> So the fifth Cameron's talking about is 1 + (9/5-1)/phi =
(4*phi+1)/5
> = 1.4944 = 695.5 cents.

Yes, that's the one. I use 2cos(Pi/5) to do Phi because I can
plug it into other things, and other things into it, and so
integrate it easily into Csound: 2cos(Pi/4) is the square root
of two and so on.

>I don't know what makes it a "beauty", though;
> Cameron will have to explain that.

Same thing that makes any other interval a beauty- the sound.

Unlike the harmonic mean, which I consider a "simple proportion"
when it's found between more or less simple ratios, or Pi itself,
which I also consider a "simple proportion" to a certain extent,
I'm sceptical about the golden cut in sound, but whatever,
this interval sounds very good, even if it were found in the
tea leaves.

-Cameron Bobro

On 9/21/07, Charles Lucy <lucy@harmonics.com> wrote:
> This page shows the figures for the phi (Kornerup) tuning.
>
>
>
>
> http://www.lucytune.com/tuning/mean_tone.html
>
>
>
>
> The idea is that the ratio between the Large and small intervals is phi.

Right, this is what people usually mean when they say "golden
meantone". The actual value for the fifth is 2^((8-phi)/11), and it
has the special property that the MOSes make a kind of Fibonacci
sequence: 5, 7, 12, 19, 31... where each number is the sum of the
previous two. Most meantone fifths will match this sequence for the
first few numbers, but the golden fifth is the only one that makes the
pattern continue forever.

This is quite different from the fifth Cameron was talking about,
which is (1+4*phi)/5 (note the lack of any exponent). This has no
special MOS-related properties AFAICT.

The golden fifth comes from making the golden cut in logarithmic pitch
space (cents), and Cameron's fifth comes from making it in linear
frequency space (Hz).

Keenan

> Right, this is what people usually mean when they say "golden
> meantone". The actual value for the fifth is 2^((8-phi)/11), and it
> has the special property that the MOSes make a kind of Fibonacci
> sequence: 5, 7, 12, 19, 31... where each number is the sum of the
> previous two.

That's not a special property of meantone. Most MOS series
I've seen obey some sort of recurrence relation.

> Most meantone fifths will match this sequence for the
> first few numbers, but the golden fifth is the only one
> that makes the pattern continue forever.

That's interesting. In some MOS series, there seems to be
a constant adder for 2 or 3 terms, then a new constant adder
for 2 or 3 terms...

> The golden fifth comes from making the golden cut in
> logarithmic pitch space (cents), and Cameron's fifth comes
> from making it in linear frequency space (Hz).

I'm still waiting to see how adaptive timbres for LucyTuning
would be related to pi.

-Carl

On 9/22/07, Carl Lumma <carl@lumma.org> wrote:
> That's not a special property of meantone. Most MOS series
> I've seen obey some sort of recurrence relation.

It's not a special property of meantone, but it is a special property
of the golden fifth 2^((8-phi)/11). For example, if you take the
1/4-comma meantone fifth, 5^(1/4), the sequence of MOSes goes 5, 7,
12, 19, 31, 50, 81, *112*... The 112 is where the pattern breaks down.

Of course, this is of little more than academic interest, because
using more than 100 notes of meantone would be pretty silly (you'd
have notes closer to 3/2 than your "fifths"!).

> That's interesting. In some MOS series, there seems to be
> a constant adder for 2 or 3 terms, then a new constant adder
> for 2 or 3 terms...

I'm not sure what you mean by this. Are you basically just talking
about the continued fraction expansion of the size of the generator in
octaves? It only repeats if it's the root of some quadratic equation
(like phi).

> I'm still waiting to see how adaptive timbres for LucyTuning
> would be related to pi.

Yeah, that is the original question I asked... How did we end up
talking about all this golden mean stuff anyway?

Keenan

Keenan wrote...
> > That's not a special property of meantone. Most MOS series
> > I've seen obey some sort of recurrence relation.
>
> It's not a special property of meantone, but it is a special
> property of the golden fifth 2^((8-phi)/11). For example, if
> you take the 1/4-comma meantone fifth, 5^(1/4), the sequence
> of MOSes goes 5, 7, 12, 19, 31, 50, 81, *112*... The 112 is
> where the pattern breaks down.
//
> > That's interesting. In some MOS series, there seems to be
> > a constant adder for 2 or 3 terms, then a new constant adder
> > for 2 or 3 terms...
>
> I'm not sure what you mean by this. Are you basically just
> talking about the continued fraction expansion of the size of
> the generator in octaves? It only repeats if it's the root
> of some quadratic equation (like phi).

112 is 81 + 31.

> > I'm still waiting to see how adaptive timbres for LucyTuning
> > would be related to pi.
>
> Yeah, that is the original question I asked... How did we end up
> talking about all this golden mean stuff anyway?

The timbres would just be LucyTuned. They wouldn't have
anything more to do with pi than LucyTuning itself.

-Carl

On 9/22/07, Carl Lumma <carl@lumma.org> wrote:
> > I'm not sure what you mean by this. Are you basically just
> > talking about the continued fraction expansion of the size of
> > the generator in octaves? It only repeats if it's the root
> > of some quadratic equation (like phi).
>
> 112 is 81 + 31.

Right... which is why the pattern is broken, because those are not the
two preceding numbers. The two preceding numbers are 81 and 50, so the
next MOS should be 131 to continue the pattern.

> > Yeah, that is the original question I asked... How did we end up
> > talking about all this golden mean stuff anyway?
>
> The timbres would just be LucyTuned. They wouldn't have
> anything more to do with pi than LucyTuning itself.

I'm just saying that, according to Bill Sethares's book, if you use
sounds with harmonic spectra, you cannot dismiss the effect of just
intonation. If Charles Lucy says his tuning is not "just" (no pun
intended) another approximation to JI, but instead has something
special to do with pi, he should use anharmonic sounds to match that.

Keenan

> > > I'm not sure what you mean by this. Are you basically just
> > > talking about the continued fraction expansion of the size
> > > of the generator in octaves? It only repeats if it's the
> > > root of some quadratic equation (like phi).
> >
> > 112 is 81 + 31.
>
> Right... which is why the pattern is broken, because those are
> not the two preceding numbers. The two preceding numbers are
> 81 and 50, so the next MOS should be 131 to continue the pattern.

It's still a recurrence relation, because 31 was earlier in
the series... and all MOS series I've seen have these patterns.

> > > Yeah, that is the original question I asked... How did we
> > > end up talking about all this golden mean stuff anyway?
> >
> > The timbres would just be LucyTuned. They wouldn't have
> > anything more to do with pi than LucyTuning itself.
>
> I'm just saying that, according to Bill Sethares's book, if you
> use sounds with harmonic spectra, you cannot dismiss the effect
> of just intonation. If Charles Lucy says his tuning is not "just"
> (no pun intended) another approximation to JI, but instead has
> something special to do with pi, he should use anharmonic sounds
> to match that.

One of Charles' only decipherable claims is that the beat rates
in LucyTuning (as apart from other meantones, presumably) are the
right ones for entraining the brain into alpha rhythms.
Zero-beating the intervals is exactly what he doesn't want.
But even if he did, the spectra would not be any more pi-based
than the tuning -- which is to say not very much -- it would be
nearly harmonic with offsets given by the errors of LucyTuning.
[Granted, it could be any wide-enough intervals in LucyTuning,
but you'll get bell-like results if you don't use roughly
harmonic spacing.]

-Carl

Thanks for clarifying that Keenan;

I'll play with the numbers;-)

As a matter of mathematical interest, in the mid 1970's, the mathematical connection between pi and phi was found.

Details here:

http://www.lucytune.com/academic/pi_phi.html

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 22 Sep 2007, at 22:06, Keenan Pepper wrote:

> On 9/21/07, Charles Lucy <lucy@harmonics.com> wrote:
> > This page shows the figures for the phi (Kornerup) tuning.
> >
> >
> >
> >
> > http://www.lucytune.com/tuning/mean_tone.html
> >
> >
> >
> >
> > The idea is that the ratio between the Large and small intervals > is phi.
>
> Right, this is what people usually mean when they say "golden
> meantone". The actual value for the fifth is 2^((8-phi)/11), and it
> has the special property that the MOSes make a kind of Fibonacci
> sequence: 5, 7, 12, 19, 31... where each number is the sum of the
> previous two. Most meantone fifths will match this sequence for the
> first few numbers, but the golden fifth is the only one that makes the
> pattern continue forever.
>
> This is quite different from the fifth Cameron was talking about,
> which is (1+4*phi)/5 (note the lack of any exponent). This has no
> special MOS-related properties AFAICT.
>
> The golden fifth comes from making the golden cut in logarithmic pitch
> space (cents), and Cameron's fifth comes from making it in linear
> frequency space (Hz).
>
> Keenan
>
>

Yes, Carl that's what I would have thought too.

I probably do have the technology to produce the sounds (timbres), yet need to learn and experiment how to generate the frequencies using CSound or MetaSynth.

(which I have yet to get running on Intel)

Tony Salinas suggested that I might be able to produce them using Spear. (Praat might also be able do it).

At the moment I am busy attempting to meet a film deadline, and converting my systems to run Logic Pro 8,

as the "improvements" in L8 should make the mechanics of completing the dialogue and audio salvage much easier,

now that I have "solved" the sampling rate incompatibility problems between 48 and 44.1 of DAT and camera recordings.

Is it just my ageing brain, or has anyone else noticed that the rate of technology "improvements" seem to be accelerating?

I seem to spend more and more time re/learning to keep up with the latest developments ;-)

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

http://www.myspace.com/lucytuning

On 22 Sep 2007, at 23:19, Carl Lumma wrote:

> Keenan wrote...
>
>
> > > I'm still waiting to see how adaptive timbres for LucyTuning
> > > would be related to pi.
> >
> > Yeah, that is the original question I asked... How did we end up
> > talking about all this golden mean stuff anyway?
>
> The timbres would just be LucyTuned. They wouldn't have
> anything more to do with pi than LucyTuning itself.
>
> -Carl
>
>
>

meta meantone gives you a fifth at 695.63
see http://anaphoria.com/meantone-mavila.PDF
This is the recurrent sequence i used in my "beyond the windows" CD

>
> I think the "golden cut" between A and B is A + (B-A)/phi (or the
> other cut, B + (A-B)/phi = A + (B-A)/phi^2).
>
> So the fifth Cameron's talking about is 1 + (9/5-1)/phi =
(4*phi+1)/5
> = 1.4944 = 695.5 cents.

Yes, that's the one. I use 2cos(Pi/5) to do Phi because I can
plug it into other things, and other things into it, and so
integrate it easily into Csound: 2cos(Pi/4) is the square root
of two and so on.

>I don't know what makes it a "beauty", though;
> Cameron will have to explain that.

Same thing that makes any other interval a beauty- the sound.

Unlike the harmonic mean, which I consider a "simple proportion"
when it's found between more or less simple ratios, or Pi itself,
which I also consider a "simple proportion" to a certain extent,
I'm sceptical about the golden cut in sound, but whatever,
this interval sounds very good, even if it were found in the
tea leaves.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

This is quite fascinating. thanks for calling attention to it. Basically what holds phi derived scales together is difference tones where the other tones of the scale are generated. So it becomes a self contained structure. Hence we have an acoustical basis to view them in the parallel light as simple ratios.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Thanks for clarifying that Keenan;
>
> I'll play with the numbers;-)
>
> As a matter of mathematical interest, in the mid 1970's, the
> mathematical connection between pi and phi was found.
>
> Details here:
>
> http://www.lucytune.com/academic/pi_phi.html

I don't see the most obvious pi/circle connection,
2cos(Pi/5), which is especially odd since, if I'm not
mistaken, 600 cents is used in Lucy Tuning and 600 cents
can be found exactly the same way (from equal division of
Pi): 2cos(Pi/4).

-Cameron Bobro