Tuning-Math Digests messages 1875 - 1899

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 2

Previous Next

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950

1850 - 1875 -



top of page bottom of page down


Message: 1875

Date: Mon, 29 Oct 2001 21:29:57

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Hey Gene, any thoughts on this year-old tuning list post? I think 
Dan 
> Stearns thinks he understands the phenomena, but I didn't follow 
his 
> answers too closely . . .

Despite its name, the linear recurrence I gave is for musical 
purposes more like the Fibonacci recurrence than the Tribonacci 
recurrence is. From the point of view of generators, we would be 
taking a generalized mediant, not an ordinary one, using 
M(p1/q1,p2/q2,p3/q3)=(p1+p2+p3)/(q1+q2+q3) to get the successive 
terms of a Tribonacci generator. We can make the Tribonnaci sequence 
Dan gave into a Tribonacci-mediant sequence by

1/2,1/2,2/3,4/7,7/12,13/22,24/41,44/75,81/138...

Note that 81/138=27/46; we don't reduce fractions for the mediants, 
or in other words, the numerators and denominators are linear 
recurrences. The result is a slightly sharp (about 2 cents) 
Tribonnaci fifth. I don't see anything hyper-MOS, but I never was 
clear what that meant. One *could* reduce fractions and see where 
that leads, but I don't see why it leads anywhere beyond a paper for 
the Fibonnaci Quarterly.

I wouldn't mess with ternary continued fractions if I were you. :)


top of page bottom of page up down


Message: 1876

Date: Mon, 29 Oct 2001 23:35:34

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Hey Gene, any thoughts on this year-old tuning list post? I think 
> Dan 
> > Stearns thinks he understands the phenomena, but I didn't follow 
> his 
> > answers too closely . . .
> 
> Despite its name, the linear recurrence I gave is for musical 
> purposes more like the Fibonacci recurrence than the Tribonacci 
> recurrence is.

Yes, I see the difference between a two-term recurrence and a three-
term recurrence.

> From the point of view of generators, we would be 
> taking a generalized mediant, not an ordinary one, using 
> M(p1/q1,p2/q2,p3/q3)=(p1+p2+p3)/(q1+q2+q3) to get the successive 
> terms of a Tribonacci generator. We can make the Tribonnaci 
sequence 
> Dan gave

Hmm . . . where did Dan give this? I thought it was original to me in 
the post I just forwarded.

> into a Tribonacci-mediant sequence by
> 
> 1/2,1/2,2/3,4/7,7/12,13/22,24/41,44/75,81/138...
> 
> Note that 81/138=27/46; we don't reduce fractions for the mediants,

Of course.

> or in other words, the numerators and denominators are linear 
> recurrences. The result is a slightly sharp (about 2 cents) 
> Tribonnaci fifth. I don't see anything hyper-MOS, but I never was 
> clear what that meant.

Well, it's not the same thing as "hyper-MOS" has meant on this list, 
or at least not in an obvious way. But I guess the thinking behind 
the question should be clear . . . ?

Would Dan like to chime in here?


top of page bottom of page up down


Message: 1877

Date: Tue, 30 Oct 2001 19:56:21

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> 
> > 7-out-of-12-out-of-22:
> > 
> > 0 175 383 496 671 879  992 1200
> > 0 208 321 496 704 817 1025 1200
> > 0 113 288 496 609 817  992 1200
> > 0 175 383 496 704 879 1087 1200
> > 0 208 321 529 704 912 1025 1200
> > 0 113 321 496 704 817  992 1200
> > 0 208 383 591 704 879 1087 1200
> 
> These don't seem to be 22-et intervals, though some are close.
> 
> > So here's the Tribonacci
> > 
> > 2, 3, 7, 12, 22, 41, ...
> > 
> > 12-out-of-22-out-of-41:
> 
> When you say "12 out of 22" do you mean 12;17+5 (circle of 12 
fifths 
> of a 22 et) or something else (such as some version of 12;12+10?)
> 
> this is the
> > junction in this story when I always expect a Gene, or a Robert
> > Walker, or anybody else with the math know-how to ride on in and 
> save
> > the day!
> 
> I have to figure out what you are doing first.

Look at the original post again. It looks like there should be a 
simple generating rule that converts the steps of one scale in the 
series to the steps of the next scale in the series, kind of like 
the "Fibonacci" MOS generating rule illustrated in the original 
message. One would think that there would be three step sizes in 
these scales.


top of page bottom of page up down


Message: 1878

Date: Tue, 30 Oct 2001 20:02:12

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
> 
> <<You still need to show me that the Tribonacci constant is the 
right
> constant. What are some other scales besides the 12-tone one? If 
this
> is truly like the Fibonacci model wrt the Golden Ratio, you should 
see
> all the notes in the 12-tone scale preserved without alteration in 
the
> 22-tone scale, etc.>>
> 
> Right, the other one was actually the
> 
> 2, 2, 3, 7, 12, 22, ...
> 
> 7-out-of-12-out-of-22:
> 
> 0 175 383 496 671 879  992 1200
> 0 208 321 496 704 817 1025 1200
> 0 113 288 496 609 817  992 1200
> 0 175 383 496 704 879 1087 1200
> 0 208 321 529 704 912 1025 1200
> 0 113 321 496 704 817  992 1200
> 0 208 383 591 704 879 1087 1200
> 
> So here's the Tribonacci
> 
> 2, 3, 7, 12, 22, 41, ...
> 
> 12-out-of-22-out-of-41:
> 
> 0  95 208 321 416 529 591 704 817 912 1025 1087 1200
> 0 113 226 321 434 496 609 722 817 930  992 1105 1200
> 0 113 208 321 383 496 609 704 817 879  992 1087 1200
> 0  95 208 270 383 496 591 704 766 879  974 1087 1200
> 0 113 175 288 401 496 609 671 784 879  992 1105 1200
> 0  62 175 288 383 496 557 671 766 879  992 1087 1200
> 0 113 226 321 434 496 609 704 817 930 1025 1138 1200
> 0 113 208 321 383 496 591 704 817 912 1025 1087 1200
> 0  95 208 270 383 478 591 704 799 912  974 1087 1200
> 0 113 175 288 383 496 609 704 817 879  992 1105 1200
> 0  62 175 270 383 496 591 704 766 879  992 1087 1200
> 0 113 208 321 434 529 643 704 817 930 1025 1138 1200

Aha!! So the Tribonacci constant really works, Dan! I don't think I 
ever fully realized this before -- great work, Dan! Gene, take note! 
And don't let Dan's "L-out-of-M-out-of-N" notation confuse you into 
thinking that N-tET has anything to do with this -- think instead of 
the analogy of MOS meantone scales with exactly the Kornerup golden 
generator, and how each scale in the series can be generated from the 
previous scale by the rule

s(old) -> L(new)
L(old) -> L(new) + s(new)

So the question is, what's the analogous rule for the Tribonacci 
case, and is there anything that serves as analogous to the generator?


top of page bottom of page up down


Message: 1880

Date: Tue, 30 Oct 2001 20:43:21

Subject: Re: Tribonacci

From: paul@xxxxxxxxxxxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> The method I use appears to do this. I looked a ton of different
> Tribonacci A,B,C,...  scales,

How are these defined, and where did you get the idea to use the 
Tribonacci constant??

> and while I remember a few cases where I
> had to slightly alter the method, these were rare and I think they
> were tied to some slight bug in the way I did this. After a while I
> just got sick of working on it--I do this "by hand", so mistakes
> brought on by tedium are inevitable too--and I decided that someone
> else with a math background would eventually have to come along and
> tidy things up.

Gene, this is your chance!


top of page bottom of page up down


Message: 1881

Date: Tue, 30 Oct 2001 01:08:06

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> What I was interested in was this: If you take the first two terms 
in
> a Fibonacci series to be the number of two stepsizes in given n-tet
> where the fourth term is the n-tet and the third term is the number 
of
> notes in the scale or subset of the n-tet, then the third term is
> always the generator that renders the scale (and this of course is
> consistent wherever you might be in the series).

In the notation I have suggested I think you are saying that, for 
instance, 5,7,12,19,31,50... leads to the temperaments/scales 
7;7+5, 12;12+7, 19;19+12, 31;31+19 etc. This is Yasser's sequence for 
the music of the future, I understand.

 Some rotation of this
> will also always agree with an L-out-of-M where L is the third term
> and M is the fourth term.

Rotation?

> So with all that in mind, my question basically was, "how then does 
a
> three term Tribonacci analogue shake down?" I found and posted some 
of
> my own answers, but I'd definitely be interested to hear others' as
> well

In the same notation, I presume the sequence 2,2,3,7,12,22 ... leads 
to 7;7+5, 12;12+10, 22;22+19, 41;41+34 ... Here the white keys are 
one term of the denominator sequence, and the black keys are the sum 
of the two previous terms. This hops all over the place as far as 
generators go--4/22, 13/41, 11/75 etc.


top of page bottom of page up down


Message: 1882

Date: Tue, 30 Oct 2001 22:03:21

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> So the question is, what's the analogous rule for the Tribonacci 
> case, and is there anything that serves as analogous to the 
generator?

I don't see that this has anything to do with MOS. If I take Dan at 
his word about the 7-12-22 business, it seems to me you just get 
1/t = t^2-t-1 (where t is the Tribonacci constant) as a generator. On 
the other hand, if you do as I suggested, and look at 
1/2,1/2,1/3,3/7,5/12,9/22,17/41... you get that the denominators tend 
to ((8*t^2-7*t+9)/22)*t^n and the numerators to ((3*t^2-4*t+2)/11)*t^n
with the result that the generator approaches a slightly flat (two 
cents worth) Tribonnaci fourth of (t^2-4*t+11)*(1200/17) cents. If 
you look at muddles, if both parts of the muddle grow with n then the 
whole thing approaches a MOS. It looks like the thing to do is to 
toss all this out the window and look at what Dan is doing on its own 
terms.


top of page bottom of page up down


Message: 1883

Date: Tue, 30 Oct 2001 01:21:26

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> In the same notation, I presume the sequence 2,2,3,7,12,22 ... 
leads 
> to 7;7+5, 12;12+10, 22;22+19, 41;41+34 ... Here the white keys are 
> one term of the denominator sequence, and the black keys are the 
sum 
> of the two previous terms. This hops all over the place as far as 
> generators go--4/22, 13/41, 11/75 etc.

Well I think the idea was that, while the Fibonacci case corresponds 
to a series of MOS scales where the generator is the limit of the 
sequence, the Tribonacci case might instead correspond in some way to 
some sort of "hyper-MOS" scale, generated not 1-dimensionally by a 
single interval but rather (in some sense) perhaps 2-dimensionally by 
a triad or something . . . ?


top of page bottom of page up down


Message: 1884

Date: Tue, 30 Oct 2001 22:28:13

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > So the question is, what's the analogous rule for the Tribonacci 
> > case, and is there anything that serves as analogous to the 
> generator?
> 
> I don't see that this has anything to do with MOS. If I take Dan at 
> his word about the 7-12-22 business, it seems to me you just get 
> 1/t = t^2-t-1 (where t is the Tribonacci constant) as a generator.

Huh? How do you see a single generator operating here? These are 
three-step-size scales, while a single generator would produce two-
step-size scales.

> On 
> the other hand, if you do as I suggested, and look at 
> 1/2,1/2,1/3,3/7,5/12,9/22,17/41... you get that the denominators 
tend 
> to ((8*t^2-7*t+9)/22)*t^n and the numerators to ((3*t^2-4*t+2)/11)
*t^n
> with the result that the generator approaches a slightly flat (two 
> cents worth) Tribonnaci fourth of (t^2-4*t+11)*(1200/17) cents.

Such a fourth cannot generate the scales Dan posted, nor can any 
single interval.

> If 
> you look at muddles, if both parts of the muddle grow with n then 
the 
> whole thing approaches a MOS.

Here we are most definitely not approaching an MOS.

> It looks like the thing to do is to 
> toss all this out the window and look at what Dan is doing on its 
own 
> terms.

I believe I am asking the same questions Dan is, which are the same 
as the ones in my original year-old tuning list post with which I 
started this thread (though Dan has made some progress over that by 
actually constructing the relevant scales!).


top of page bottom of page up down


Message: 1885

Date: Tue, 30 Oct 2001 01:33:45

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:

> > In the same notation, I presume the sequence 2,2,3,7,12,22 ... 
> leads 
> > to 7;7+5, 12;12+10, 22;22+19, 41;41+34 ... Here the white keys 
are 
> > one term of the denominator sequence, and the black keys are the 
> sum 
> > of the two previous terms. This hops all over the place as far as 
> > generators go--4/22, 13/41, 11/75 etc.
 
> Well I think the idea was that, while the Fibonacci case 
corresponds 
> to a series of MOS scales where the generator is the limit of the 
> sequence, the Tribonacci case might instead correspond in some way 
to 
> some sort of "hyper-MOS" scale, generated not 1-dimensionally by a 
> single interval but rather (in some sense) perhaps 2-dimensionally 
by 
> a triad or something . . . ?

The characteristic polynomial, x^3-x^2-x-1, defines a Pisot number, 
meaning a real algebraic integer greater than one all of whose 
conjugates are less than one in absolute value (another example would 
be the golden ratio.) In consequence, the ratios of successive terms 
converge, and the generator should settle down to something definite.


top of page bottom of page up down


Message: 1886

Date: Tue, 30 Oct 2001 23:40:00

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Gene,
> 
> <<When you do that for 12/22,>>
> 
> But I wrote N/M, so for a 7-out-of-12-out-of-22 that would be 22/12 
or
> 0 1 3 5 7 9 11 12 14 16 18 20 22. Then the L-tet (7-tet here) inside
> of that would be 0 3 7 9 12 16 18 22.
> 
> The idea is that this should work for any arbitrary A, B, C, ...
> Tribonacci series.

This is a scale out of the 7;7+5;12+10 muddle--why do you need to 
bring in the Tribonacci constant at all?


top of page bottom of page up down


Message: 1887

Date: Tue, 30 Oct 2001 02:20:04

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > > In the same notation, I presume the sequence 2,2,3,7,12,22 ... 
> > leads 
> > > to 7;7+5, 12;12+10, 22;22+19, 41;41+34 ... Here the white keys 
> are 
> > > one term of the denominator sequence, and the black keys are 
the 
> > sum 
> > > of the two previous terms. This hops all over the place as far 
as 
> > > generators go--4/22, 13/41, 11/75 etc.
>  
> > Well I think the idea was that, while the Fibonacci case 
> corresponds 
> > to a series of MOS scales where the generator is the limit of the 
> > sequence, the Tribonacci case might instead correspond in some 
way 
> to 
> > some sort of "hyper-MOS" scale, generated not 1-dimensionally by 
a 
> > single interval but rather (in some sense) perhaps 2-
dimensionally 
> by 
> > a triad or something . . . ?
> 
> The characteristic polynomial, x^3-x^2-x-1, defines a Pisot number, 
> meaning a real algebraic integer greater than one all of whose 
> conjugates are less than one in absolute value (another example 
would 
> be the golden ratio.) In consequence, the ratios of successive 
terms 
> converge, and the generator should settle down to something 
definite.

The ratio of successive terms converges to the "Tribonacci constant", 
yes, but I don't think you're catching my drift. Don't the scales 
have three, rather than two, step sizes, therefore not being MOS 
scales, or scales with a single generator at all? Dan, can you help 
here?


top of page bottom of page up down


Message: 1889

Date: Tue, 30 Oct 2001 02:49:06

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> What does seem to
> work is taking the M-out-of-N idea as an L-out-of-M-out-of-N idea.

It sounds like what you are describing is what I was calling 
a "muddle" in some recent postings.

> So take the Tribonacci syntonic diatonic sequence again:
> 
> 2, 2, 3, 7, 12, 22, 41, ...
> 
> 12-out-of-22-out-of-41 gives 0 3 7 9 12 16 18 22. 

You could get for instance the sequence of muddles 7;12+10;22+19,
12;22+19;41+34; 22;41+34;75+63..., though what you give looks like
it is out of 22, not 41.

If you let the
> Tribonacci constant be X and the three terms (or stepsizes) be A, B
> and C, then converting this so that C/A=X and (C+A)/B=X gives:
> 
> 0 175 383 496 671 879  992 1200
> 0 208 321 496 704 817 1025 1200
> 0 113 288 496 609 817  992 1200
> 0 175 383 496 704 879 1087 1200
> 0 208 321 529 704 912 1025 1200
> 0 113 321 496 704 817  992 1200
> 0 208 383 591 704 879 1087 1200

The way I think of it a muddle is not just one scale, but the whole 
set.


top of page bottom of page up down


Message: 1891

Date: Tue, 30 Oct 2001 02:56:33

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> The ratio of successive terms converges to the "Tribonacci 
constant", 
> yes, but I don't think you're catching my drift. Don't the scales 
> have three, rather than two, step sizes, therefore not being MOS 
> scales, or scales with a single generator at all? Dan, can you help 
> here?

What I gave were MOS by definition, but muddles are quite another 
matter.


top of page bottom of page up down


Message: 1892

Date: Tue, 30 Oct 2001 03:15:20

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> I'm not quite getting exactly what a "muddle" is yet, but I'll take 
a
> look in the archives.

It was over on the tuning list, I'm afraid. By p;q+r with q and r 
relatively prime I mean a scale of p notes with a generator 
(a+b)/(q+r), where the numerators are defined by taking the 
pentultimate semiconvergent for q/r. If q and r are not relatively 
prime, then for example 10;12+10 would refer to the various paultone 
scales, with 5+5;12+10 and 6+4;12+10 being his two main kinds.

By a "muddle" I mean the set of related scales defined by a 
temperament of a temperament--for instance 7;19+12;41+31 is the set 
of scales one gets by treating Canasta as if it were 31-equal and 
using it to define diatonic scales. One gets things very close to 
major, minor and Indian diatonic scales out of it, all related by key 
relationships.


top of page bottom of page up down


Message: 1893

Date: Tue, 30 Oct 2001 03:29:38

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> So take the Tribonacci syntonic diatonic sequence again:
> 
> 2, 2, 3, 7, 12, 22, 41, ...
> 
> 12-out-of-22-out-of-41 gives 0 3 7 9 12 16 18 22. If you let the
> Tribonacci constant be X and the three terms (or stepsizes) be A, B
> and C, then converting this so that C/A=X and (C+A)/B=X gives:
> 
> 0 175 383 496 671 879  992 1200
> 0 208 321 496 704 817 1025 1200
> 0 113 288 496 609 817  992 1200
> 0 175 383 496 704 879 1087 1200
> 0 208 321 529 704 912 1025 1200
> 0 113 321 496 704 817  992 1200
> 0 208 383 591 704 879 1087 1200
> 
> Unlike the 2D (triadic) idea, I believe this works for any arbitrary
> A,B,C Tribonacci series. However, unlike the two term series scales,
> these three term scales do not guarantee trivalence (if you allow 
that
> Myhill is equals "bivalence").

Dan, this is great, thanks.

You still need to show me that the Tribonacci constant is the right 
constant. What are some other scales besides the 12-tone one? If this 
is truly like the Fibonacci model wrt the Golden Ratio, you should 
see all the notes in the 12-tone scale preserved without alteration 
in the 22-tone scale, etc.


top of page bottom of page up down


Message: 1895

Date: Tue, 30 Oct 2001 05:09:09

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> 7-out-of-12-out-of-22:
> 
> 0 175 383 496 671 879  992 1200
> 0 208 321 496 704 817 1025 1200
> 0 113 288 496 609 817  992 1200
> 0 175 383 496 704 879 1087 1200
> 0 208 321 529 704 912 1025 1200
> 0 113 321 496 704 817  992 1200
> 0 208 383 591 704 879 1087 1200

These don't seem to be 22-et intervals, though some are close.

> So here's the Tribonacci
> 
> 2, 3, 7, 12, 22, 41, ...
> 
> 12-out-of-22-out-of-41:

When you say "12 out of 22" do you mean 12;17+5 (circle of 12 fifths 
of a 22 et) or something else (such as some version of 12;12+10?)

this is the
> junction in this story when I always expect a Gene, or a Robert
> Walker, or anybody else with the math know-how to ride on in and 
save
> the day!

I have to figure out what you are doing first.


top of page bottom of page up down


Message: 1897

Date: Tue, 30 Oct 2001 07:02:06

Subject: Re: Tribonacci

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> <<When you say "12 out of 22" do you mean 12;17+5 (circle of 12 
fifths
> of a 22 et) or something else (such as some version of 12;12+10?)>>

> Hmm, I mean N/M multiplied by zero through M and rounded down to the
> nearest integer, and from that I take the nearest L-tet.

When you do that for 12/22, you get 
0,0,1,1,2,2,3,3,4,4,5,6,6,7,7,8,8,9,9,10,10,11,12, which I doubt is 
what you mean.


top of page bottom of page up down


Message: 1899

Date: Wed, 31 Oct 2001 21:21:03

Subject: Re: Tribonacci

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> where C/A=X and (A+C)/B=X
> (and X is the Tribonacci constant)

That's what I was looking for! Now, how did you come upon the idea to 
do it this way? It certainly ends up working just wonderfully!


top of page bottom of page up

Previous Next

1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950

1850 - 1875 -

top of page