Coherent49

For fans of slightly sharp fifths and differential coherence, here's a 49 note MOS with generator the positive root of x^4-x^2-1, aka a 14/11 superthird. In terms of the regular mapping paradigm, we can consider this to be the 72&121 temperament. Jacques will have to tell us if he thinks it has the correct -c mojo, but at least it has triads and stuff. Generators of the form x^a +- x^b +- 1 seem to me to be good candidates for this -c stuff, presuming I really understand it.

! coherent49.scl
Generator is the positive root of x^4 - x^2 - 1
! 72&121 temperament <<30 25 38 39 ... ||
49
!
30.17385
49.63544
79.80930
99.27089
129.44474
148.90633
179.08019
198.54178
228.71563
248.17722
278.35108
297.81267
317.27426
347.44811
366.90970
397.08356
416.54515
446.71900
466.18059
496.35445
515.81604
545.98989
565.45148
595.62533
615.08693
645.26078
664.72237
694.89622
714.35782
733.81941
763.99326
783.45485
813.62870
833.09030
863.26415
882.72574
912.89959
932.36119
962.53504
981.99663
1012.17048
1031.63207
1061.80593
1081.26752
1111.44137
1130.90296
1150.36456
1180.53841
1200.00000

It's too many notes to digest quickly, but it's got the -c mojo. In fact it sounds a lot like a phi-based tuning. I work a lot with -c tunings, but I came at them completely backwards. A few years ago, here at Yee Olde Liste, I described how- by taking a few tones that caught my fancy, calculating their difference tones, and using the octaves of the difference tones in the scale.

Simple Just structures have strong -c qualities, and somehow a strong -c structure in a tuning that it definitely not simple Just can make it smoother and oddly "Just" sounding.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> For fans of slightly sharp fifths and differential coherence, here's a 49 note MOS with generator the positive root of x^4-x^2-1, aka a 14/11 superthird. In terms of the regular mapping paradigm, we can consider this to be the 72&121 temperament. Jacques will have to tell us if he thinks it has the correct -c mojo, but at least it has triads and stuff. Generators of the form x^a +- x^b +- 1 seem to me to be good candidates for this -c stuff, presuming I really understand it.
>
> ! coherent49.scl
> Generator is the positive root of x^4 - x^2 - 1
> ! 72&121 temperament <<30 25 38 39 ... ||
> 49
> !
> 30.17385
> 49.63544
> 79.80930
> 99.27089
> 129.44474
> 148.90633
> 179.08019
> 198.54178
> 228.71563
> 248.17722
> 278.35108
> 297.81267
> 317.27426
> 347.44811
> 366.90970
> 397.08356
> 416.54515
> 446.71900
> 466.18059
> 496.35445
> 515.81604
> 545.98989
> 565.45148
> 595.62533
> 615.08693
> 645.26078
> 664.72237
> 694.89622
> 714.35782
> 733.81941
> 763.99326
> 783.45485
> 813.62870
> 833.09030
> 863.26415
> 882.72574
> 912.89959
> 932.36119
> 962.53504
> 981.99663
> 1012.17048
> 1031.63207
> 1061.80593
> 1081.26752
> 1111.44137
> 1130.90296
> 1150.36456
> 1180.53841
> 1200.00000
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> It's too many notes to digest quickly, but it's got the -c mojo. In fact it sounds a lot like a phi-based tuning. I work a lot with -c tunings, but I came at them completely backwards.

You might say this was arrived at backwards, as I started from a three-term monic polynomial with only +-1 or 0 coefficients and then looked to see if its root could be understood as the generator of a linear temperament in good standing in the regular mapping paradigm.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> It's too many notes to digest quickly, but it's got the -c mojo. In fact it sounds a lot like a phi-based tuning.

I should point out that the generator is sqrt(phi).

Hey hey, that's Raph ! in other terms, square root of Phi, or half a Phi interval.
I would like to understand what means "72 & 121 temperament" and <<30 25 38 39 ... || and what interest you find to this one.
As far as I know, if it has as -c qualities, it borrows it from Phi :
1.618 - 1 = 0.618 = 1/1.618 everywhere (the 833.09030 c. has its differential 1 octave lower 366.90970 c.)
but perhaps more interesting, as I suppose it is octave-periodic, it shares the quality you have with a Iph series (that I have been discussing in parallel with Margo), based on a generator of 2/Phi, or sqrt of 5 - 1, this sort of Turkish third you have in here with 366.90970 c.
Iph's 1st order difference tone is 3 octaves lower than three Iphs :
3*366.9097 = 1100.7291 c.
For some reason I don't see it here so it may not be present everywhere but I see its reverse = 99.27089 c.
and one Iph from 99.27089 c, to 466.18059 c., generates 1/1.
But I just find that the generator is almost -c : it seems to generate a 146.13933 c., where you have 148.90633 c., interesting.
It seems you get 148.9 c. after 9 repetitions, so it might be an idea for another algorithm, if you like that chord.
This is very much like 14 - 11 = 3, which I have done before : unusual harmony but it works.
I think this must be x^9 = 32x - 32 and I find 1.27154919 = 415.9047286476 c.

Here is one of the simplest series I found for the Raph recurrent sequence, based on series Phi17 & Phi 93 :
45 57 73 93 118 150 191 243 309 393 500 636 809 1029 1309 1665 2118 2694...
The sequence is convergent and it can go up as much as you want - x^49 is even accepted by my pocket calculator !
Here is a taste with 12 notes for a start :

! raph.scl
!
Raph recurrent sequence, series Phi17 & Phi93
12
!
31/30
191/180
131/120
19/15
59/45
27/20
25/18
73/45
5/3
103/60
53/30
2/1
! x^4 - x^2 = 1, generator x = sqrt of Phi = 1.2720196495
! series here 45 57 73 93 118 150 191 243 309 393 500 636
! (Iph -c : 393 - 318 = 75 ; 309 - 250 = 59, etc.)

- - - - - - -
Jacques

Gene wrote :

> For fans of slightly sharp fifths and differential coherence, > here's a 49 note MOS with generator the positive root of x^4-x^2-1, > aka a 14/11 superthird. In terms of the regular mapping paradigm, > we can consider this to be the 72&121 temperament. Jacques will > have to tell us if he thinks it has the correct -c mojo, but at > least it has triads and stuff. Generators of the form x^a +- x^b +- > 1 seem to me to be good candidates for this -c stuff, presuming I > really understand it.
>
> ! coherent49.scl
> Generator is the positive root of x^4 - x^2 - 1
> ! 72&121 temperament <<30 25 38 39 ... ||
> 49
> !
> 30.17385
> 49.63544
> 79.80930
> 99.27089
> 129.44474
> 148.90633
> 179.08019
> 198.54178
> 228.71563
> 248.17722
> 278.35108
> 297.81267
> 317.27426
> 347.44811
> 366.90970
> 397.08356
> 416.54515
> 446.71900
> 466.18059
> 496.35445
> 515.81604
> 545.98989
> 565.45148
> 595.62533
> 615.08693
> 645.26078
> 664.72237
> 694.89622
> 714.35782
> 733.81941
> 763.99326
> 783.45485
> 813.62870
> 833.09030
> 863.26415
> 882.72574
> 912.89959
> 932.36119
> 962.53504
> 981.99663
> 1012.17048
> 1031.63207
> 1061.80593
> 1081.26752
> 1111.44137
> 1130.90296
> 1150.36456
> 1180.53841
> 1200.00000

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> I would like to understand what means "72 & 121 temperament" and <<30
> 25 38 39 ... || and what interest you find to this one.

If we call the root "raph", then 42/121 < raph < 25/72. Both 121et and 72et are interesting divisions of the octave from the point of view of approximating just intervals, though the good qualities of 121 aren't so apparent if we stop at the 11-limit instead of going on to the 19-limit. If we go here

http://x31eq.com/temper/net.html

and put both 72 and 121 into Graham's magic box, and 11 in the box below, we see that the temperament supported by both 72 and 121 makes for a relatively reasonable linear temperament. <<30 25 38 39 ... || is the initial segment of a "wedgie" and tells us that to get to the class for 3, 30 generator steps are needed, 5 requires 25 and so forth. Since 11 takes 39 and 7 38, 11/7 takes 1 generator step, meaning it is a generator, so 14/11 is a generator also, and that's approximated by "raph". At 703.6 cents, the fifth isn't quite sharp enough to make Margo happy, but it's a good fifth, and the major third is almost pure. Moreover, the interval class for major thirds contains two perfectly functional major thirds, one nearly pure and the other of 397 cents. There's a slightly sharp 7/4 and a flat 7/4 distributed between two classes, and similarly for 11/8. So there are some nice tetrads showing a slight sharp tendency, plus all sort of other stuff. To the extent phi approximates 13/8, we have boatloads of those, of course, plus functional 17's and very accurate 19's. The point is, if you are willing to go so far as to have 49 notes, you can have both your -c and your just intonation approximations.

> But I just find that the generator is almost -c : it seems to
> generate a 146.13933 c., where you have 148.90633 c., interesting.

I'm not sure what you are saying, as sqrt(phi), which I thought was "raph", is 416.545 cents.

> Here is one of the simplest series I found for the Raph recurrent
> sequence, based on series Phi17 & Phi 93 :

I have no idea what those are.

That would explain it, haha! I was just going by the sound of the tuning as I noodled about in it a bit, it has "that sound". Strange tall vertical sonorities sound far more still and smooth than they "should", that's a sign of a -c tuning.

It's a feeling different from, but kind of a cousin of, proportional beating temperaments. Both very tangible when performing with acoustic instruments that require continual adjustment of pitch, it's simply easier to tune unusual intervals accurately.

--- In tuning@...m, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > It's too many notes to digest quickly, but it's got the -c mojo. In fact it sounds a lot like a phi-based tuning.
>
> I should point out that the generator is sqrt(phi).
>

On the 21st of September 2010 Gene wrote :

> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> > I would like to understand what means "72 & 121 temperament" and > <<30
> > 25 38 39 ... || and what interest you find to this one.
>
> If we call the root "raph", then 42/121 < raph < 25/72. Both 121et > and 72et are interesting divisions of the octave from the point of > view of approximating just intervals, though the good qualities of > 121 aren't so apparent if we stop at the 11-limit instead of going > on to the 19-limit. If we go here
>
> http://x31eq.com/temper/net.html
>
> and put both 72 and 121 into Graham's magic box, and 11 in the box > below, we see that the temperament supported by both 72 and 121 > makes for a relatively reasonable linear temperament. <<30 25 38 > 39 ... || is the initial segment of a "wedgie" and tells us that to > get to the class for 3, 30 generator steps are needed, 5 requires > 25 and so forth. Since 11 takes 39 and 7 38, 11/7 takes 1 generator > step, meaning it is a generator, so 14/11 is a generator also, and > that's approximated by "raph". At 703.6 cents, the fifth isn't > quite sharp enough to make Margo happy, but it's a good fifth, and > the major third is almost pure. Moreover, the interval class for > major thirds contains two perfectly functional major thirds, one > nearly pure and the other of 397 cents. There's a slightly sharp > 7/4 and a flat 7/4 distributed between two classes, and similarly > for 11/8. So there are some nice tetrads showing a slight sharp > tendency, plus all sort of other stuff. To the extent phi > approximates 13/8, we have boatloads of those, of course, plus > functional 17's and very accurate 19's. The point is, if you are > willing to go so far as to have 49 notes, you can have both your -c > and your just intonation approximations.

Thanks. Just one more question, the numbers of generators inside the wedgie <<30 25 38 39 ... || seem to generate rather 4/3, 8/5, 8/7, 16/11 - could they not be negative values in order to generate 3, 5, 7, 11 ?

> > But I just find that the generator is almost -c : it seems to
> > generate a 146.13933 c., where you have 148.90633 c., interesting.
>
> I'm not sure what you are saying, as sqrt(phi), which I thought was > "raph", is 416.545 cents.

Yes and as a ratio it is 1.2720196495 whose difference tone with 1/1 = 0.2720196495
and 4*0.2720196495 = 1.088078598 > 146.13933 c.

> > Here is one of the simplest series I found for the Raph recurrent
> > sequence, based on series Phi17 & Phi93 :
>
> I have no idea what those are.

Two Phi recurrent series, "proprietary" or "prioritary", as you prefer, from 17 and 93.
This is a convenient code for Phi series I proposed on this list before, that indicates that from "17" the best Phi series will be :
17 28 45 73 118 ... because 17*Phi will be rounded to 28, while precedent numbers in the full series :
1 5 6 11 17 28 45 73 118 ...
are already part of simpler series, ex. : 11*Phi will be rounded not to 17, but to 18
and 11 : 18 belongs to the series 1 3 4 7 11 18 29... which is Phi7 (but more known as the "P series").

BTW I also extend this concept of "P series" to all ratio generators whose powers converge towards whole numbers, ex : (sqrt13 + 3) / 2

Still a mystery for me how this is possible, though I can predict by their convergence why certain fractals have a P series and others don't.
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> and 11 : 18 belongs to the series 1 3 4 7 11 18 29... which is
> Phi7 (but more known as the "P series").
>
> BTW I also extend this concept of "P series" to all ratio generators
> whose powers converge towards whole numbers, ex : (sqrt13 + 3) / 2
>
> Still a mystery for me how this is possible, though I can predict by
> their convergence why certain fractals have a P series and others don't.

I might be able to say something about that if I knew what the concept was and how "P series" is defined.

Gene wrote :

> > (Jacques) : and 11 : 18 belongs to the series 1 3 4 7 11 18 29... > which is
> > Phi7 (but more known as the "P series").
> >
> > BTW I also extend this concept of "P series" to all ratio generators
> > whose powers converge towards whole numbers, ex : (sqrt13 + 3) / 2
> >
> > Still a mystery for me how this is possible, though I can predict by
> > their convergence why certain fractals have a P series and others > don't.
>
> I might be able to say something about that if I knew what the > concept was and how "P series" is defined.

Leaving all concepts aside, let us say it simply this way : generally, powers of non-whole numbers do not tend toward whole numbers. But there are some (non-whole numbers) whose successive powers tend toward whole numbers. How can we explain it ?
By "tend toward" I don't mean getting always closer and closer necessarily, but getting closer and closer to whole numbers as an average tendancy.
When it happens, they tend toward a series of whole numbers that I call the "P series" of those numbers.
Ex : Phi^n and [2 1 3 4 7 11 18 29 47 76...

I am sure this type question as been resolved long ago by specialists of recurrent series etc. But my question is more : can it be explained to neophytes (like me), in simple words, with a minimum of mathematical charabia ? ;-)

The question is not without relation with linear temperaments, because when a generator ratio has a P series, you can see immediately what possible harmonics it may temper (though they rarely care about our usual musical preoccupations)
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> > I might be able to say something about that if I knew what the
> > concept was and how "P series" is defined.
>
>
>
> Leaving all concepts aside, let us say it simply this way :
> generally, powers of non-whole numbers do not tend toward whole
> numbers. But there are some (non-whole numbers) whose successive
> powers tend toward whole numbers. How can we explain it ?

This should help:

http://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number

Gene wrote :

> (Jacques) : let us say it simply this way :
> generally, powers of non-whole numbers do not tend toward whole
> numbers. But there are some (non-whole numbers) whose successive
> powers tend toward whole numbers. How can we explain it ?

This should help:

http://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number

Thanks, Gene ! Interesting page. I don't have exactly an explicit
answer to my question, but I know where to learn more about it now.
I'm glad that I was correct about the smallest of those :
1.324717957244746025960908854
and to know a new name for it : the plastic number ! Never heard of
that one before.
It is known also as "Psi", and the "number of the architects"
and of course "Natté" (in french with an accent) and "Meta-slendro".
A very musical generator, that would also work for 15/37 and 43/106
temperaments :

! natte.scl
!
Natte series from 28 to 151, pentatonic sequence out of 7 consecutive
tones
12
!
86/57
65/57
98/57
74/57
151/114
2/1
86/57
65/57
98/57
74/57
112/57
2/1
! White keys play the seven tones (with two commas 148:151 and 56:57),
! Black keys play only plain pentatonic 43 : 49 : 57 : 65 : 74
! Natte triple eq-b sequence (Dudon), known also as "Meta-
Slendro" (Wilson)
! x^3 - x = x^5 - x^4 = 1 (x = 1.3247179572447 or 486.822277651
c.)

- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Thanks, Gene ! Interesting page. I don't have exactly an explicit
> answer to my question, but I know where to learn more about it now.

What questions are left unanswered for you? Pisot's theorem goes a long way towards saying all "P series" are from PV numbers.

> I'm glad that I was correct about the smallest of those :
> 1.324717957244746025960908854
> and to know a new name for it : the plastic number ! Never heard of
> that one before.

I've tried to promote the idea of calling it the Osmium number, because of the density of Osmium.

Gene wrote :

> What questions are left unanswered for you? Pisot's theorem goes a > long way towards saying all "P series" are from PV numbers.

May be I need to find what's the Pisot's theorem then, or to find someone to translate it for me. Now I can only marvel at this phenomena, without being able to understanding it.
Why the powers of these numbers, whatever they are, stick around whole numbers, more and more and without end ?
And subsidiary question, why is there no simple words to explain it ?

> > I'm glad that I was correct about the smallest of those :
> > 1.324717957244746025960908854
> > and to know a new name for it : the plastic number ! Never heard of
> > that one before.
>
> I've tried to promote the idea of calling it the Osmium number, > because of the density of Osmium.

Funny, because Osmium's number of neutrons is 114, part of P-V 1 main series :
0 1 1 1 2 2 3 4 5 7 9 12 16 21 28 37 49 65 86 114...

But not its atomic number, 76 - that's part of the second smaller P-V number,
Amaz = 1.380277569098
main series 0 1 1 2 2 3 4 6 8 11 15 21 29 40 55 76 105 145 200...
(that would do BTW for a 20/43 temperament)
- - - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Gene wrote :
>
> > What questions are left unanswered for you? Pisot's theorem goes a
> > long way towards saying all "P series" are from PV numbers.
>
> May be I need to find what's the Pisot's theorem then, or to find
> someone to translate it for me. Now I can only marvel at this
> phenomena, without being able to understanding it.

Pisot's theorem says that if a "P series" approaches integers fast enough it consists of powers of a PV number. Specifically, if the square of the distance to the nearest integer sums to some number, it is a PV number. Hence anything approaching integers at a rate bounded by K/n, where K is a constant and we are looking at the nth term must be a PV number, but bounded by K/log(n) but not J/n would not be, if such a thing exists.

> Why the powers of these numbers, whatever they are, stick around
> whole numbers, more and more and without end ?

For any algebraic integer, the sum of the nth power of all of its algebraic conjugates is a symmetric polynomial and hence is an ordinary (rational) integer. For a PV number, all the other conjugates are less than 1 in absolute value, and hence their sum approaches 0 exponentially, so the nth power of the PV number approaches integers exponentially.

Gene wrote :

> Pisot's theorem says that if a "P series" approaches integers fast > enough it
> consists of powers of a PV number. Specifically, if the square of > the distance
> to the nearest integer sums to some number, it is a PV number. > Hence anything
> approaching integers at a rate bounded by K/n, where K is a > constant and we are
> looking at the nth term must be a PV number, but bounded by K/log> (n) but not J/n
> would not be, if such a thing exists.
>
> > Why the powers of these numbers, whatever they are, stick around
> > whole numbers, more and more and without end ?
>
> For any algebraic integer, the sum of the nth power of all of its > algebraic
> conjugates is a symmetric polynomial and hence is an ordinary > (rational)
> integer. For a PV number, all the other conjugates are less than 1 > in absolute
> value, and hence their sum approaches 0 exponentially, so the nth > power of the
> PV number approaches integers exponentially.

Thanks for the effort Gene, but apart from your first two sentences, I do not catch much of it and it would be more reasonable for me first to take mathematical lessons.
Let's see if I understood your second sentence :
"Specifically, if the square of the distance to the nearest integer sums to some number, it is a PV number."
I take an example we all can find with our pocket calculators : Phi = (sqrtof 5 + 1)/2
The "distances" of its powers to the nearest integers, in my sense, are :
2 - Phi = 0.382
3 - Phi^2 = 0.382
Phi^3 - 4 = 0.236
7 - Phi^4 = 0.146
Phi^5 - 11 = .090 ... etc.
First I have a doubt with the first line, because 2 is the nearest integer of Phi but is not relevant to Phi's "Pseries"
1 3 4 7 11 18...
So Phi - 1 = .618 could be a more logical exception.
What I notice is that these distances (frequencies beats BTW) here are also equal to :
Phi - 1 = 1 /Phi
3 - Phi^2 = 1 /Phi^2
Phi^3 - 4 = 1 /Phi^3
7 - Phi^4 = 1 /Phi^4
Phi^5 - 11 = 1 /Phi^5 ... etc.
(which comforts the Phi - 1 exception for the 1st line)
Where do we go from here ?
It seems that the sum of all those tends toward Phi,
and that the sum of their squares tends toward Phi - 1.
Am I getting closer ?
But things are far from being as simple with 1.324717957244746 and other P-V numbers under Phi...