Tuning-Math Archive Section 2: 1875

Message: 1875 - Contents - Hide Contents 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. :)

Message: 1876 - Contents - Hide Contents 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?

Message: 1877 - Contents - Hide Contents 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.

Message: 1878 - Contents - Hide Contents 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?

Message: 1880 - Contents - Hide Contents 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!

Message: 1881 - Contents - Hide Contents 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.

Message: 1882 - Contents - Hide Contents 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.

Message: 1883 - Contents - Hide Contents 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 . . . ?

Message: 1884 - Contents - Hide Contents 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!).

Message: 1885 - Contents - Hide Contents 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.

Message: 1886 - Contents - Hide Contents 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?

Message: 1887 - Contents - Hide Contents 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?

Message: 1889 - Contents - Hide Contents 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.

Message: 1891 - Contents - Hide Contents 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.

Message: 1892 - Contents - Hide Contents 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.

Message: 1893 - Contents - Hide Contents 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.

Message: 1895 - Contents - Hide Contents 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.

Message: 1897 - Contents - Hide Contents 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.

Message: 1899 - Contents - Hide Contents 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!