color lattice of 19-tone scale

Paul Erlich just did a terrific color lattice of the 19-tone scale I
am going to be using, a la Dave Keenan, Pierre Lamothe and Robert
Walker.

I have posted it to the files section so that everyone can see it. I
believe the relationships are really made clear in this process!

Here it is: "pehrcolor.bmp"

/tuning/files/Pehrson/

____________ ______ _____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> Paul Erlich just did a terrific color lattice of the 19-tone scale
I
> am going to be using, a la Dave Keenan, Pierre Lamothe and Robert
> Walker.
>
> I have posted it to the files section so that everyone can see it.
I
> believe the relationships are really made clear in this process!
>
> Here it is: "pehrcolor.bmp"
>
> /tuning/files/Pehrson/
>
> ____________ ______ _____ _
> Joseph Pehrson

Someone enjoys inversional symmetry besides me!

JL

Jacky Ligon wrote,

<<Someone enjoys inversional symmetry besides me!>>

The 7-tone, 1 b2 3 4 5 b6 7 8 is also a subset of 19-tET. In this case
a [4,1,2] trivalent three-stepsize subset:

6----17
/ \ /
/ \ /
8-----0----11
/ \ /
/ \ /
2----13

4, 1, 2, 7, 10, 19, ...

Interestingly enough, if this scale is weighted with the Tribonacci
constant, the fourth and the fifth both round to a familiar 500 and
700 cents.

0 130 369 500 700 831 1070 1200
0 239 369 570 700 940 1070 1200
0 130 331 461 700 831 961 1200
0 201 331 570 700 831 1070 1200
0 130 369 500 630 869 999 1200
0 239 369 500 739 869 1070 1200
0 130 260 500 630 831 961 1200

This scale has quite a different character than the

*-----*
/ \ /
/ \ /
*-----*-----*
/ \ /
/ \ /
*-----*

JI version, as it kind of shifts all the implied 7-limit ratios by a
49/48.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Jacky Ligon wrote,
>
> <<Someone enjoys inversional symmetry besides me!>>
>
> The 7-tone, 1 b2 3 4 5 b6 7 8 is also a subset of 19-tET. In this
case
> a [4,1,2] trivalent three-stepsize subset:
>
> 6----17
> / \ /
> / \ /
> 8-----0----11
> / \ /
> / \ /
> 2----13
>
> 4, 1, 2, 7, 10, 19, ...
>
> Interestingly enough, if this scale is weighted with the Tribonacci
> constant, the fourth and the fifth both round to a familiar 500 and
> 700 cents.

Dan,

Could you kindly explain in simple terms "the Tribonacci constant"?

Thanks,

JL

>
> 0 130 369 500 700 831 1070 1200
> 0 239 369 570 700 940 1070 1200
> 0 130 331 461 700 831 961 1200
> 0 201 331 570 700 831 1070 1200
> 0 130 369 500 630 869 999 1200
> 0 239 369 500 739 869 1070 1200
> 0 130 260 500 630 831 961 1200
>
> This scale has quite a different character than the
>
> *-----*
> / \ /
> / \ /
> *-----*-----*
> / \ /
> / \ /
> *-----*
>
> JI version, as it kind of shifts all the implied 7-limit ratios by a
> 49/48.
>
> --Dan Stearns

Hi Jacky,

I don't know any simple little closed expression that gives the
Tribonacci constant such as the (1+sqrt(5))/2 for the Fibonacci
constant. I just take a sufficiently high enough Tribonacci number and
divide it by its preceding term.

The Tribonacci scales I give are set so that C/A=t. In other words, so
that the largest step divided by the smallest equals the Tribonacci
constant. But this is really just an intuitive best guess on my part
as to how this should be done so it will parallel the Fibonacci
series. To the best of my knowledge no one has either refuted or
'proven' this yet.

--Dan Stearns

Maybe this can't be called simple or little, but it is:

(19/27 + sqrt(11/27)) ^ 1/3 + 4/9 * (19/27 + sqrt(11/27)) ^ -1/3 + 1/3

I did a page on this before I knew it was called 'Tribonacci'

http://www.kees.cc/gldsec.html

Kees

----- Original Message -----
From: "D.Stearns" <STEARNS@CAPECOD.NET>
To: <tuning@yahoogroups.com>
Sent: Friday, March 23, 2001 9:33 PM
Subject: Re: [tuning] Re: color lattice of 19-tone scale

> Hi Jacky,
>
> I don't know any simple little closed expression that gives the
> Tribonacci constant such as the (1+sqrt(5))/2 for the Fibonacci
> constant. I just take a sufficiently high enough Tribonacci number and
> divide it by its preceding term.
>
> The Tribonacci scales I give are set so that C/A=t. In other words, so
> that the largest step divided by the smallest equals the Tribonacci
> constant. But this is really just an intuitive best guess on my part
> as to how this should be done so it will parallel the Fibonacci
> series. To the best of my knowledge no one has either refuted or
> 'proven' this yet.
>
> --Dan Stearns
>
>
>
> You do not need web access to participate. You may subscribe through
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>

Hi Dan,

I wrote in 20022 (Tribonacci and 3D lattices) :

<<
We know that the ratio, at the limit, of successive numbers in the
Tribonacci series is the Psi number. Using rt2 for square root
and rt3 for cubic root, we may calculate its value with this formula :

Psi = (1/3)(1 + rt3(19 + rt2(33)) + rt3(19 - rt2(33)))

= 1.839 286 755 ...
>>

Pierre

Thanks Kees (and sorry Pierre, I forgot that you recently gave this as
well),

I had seen this before as well at:

<http://www.lacim.uqam.ca/piDATA/tribo.txt>

--Dan Stearns

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20376.html#20376

> Paul Erlich just did a terrific color lattice of the 19-tone scale
I
> am going to be using, a la Dave Keenan, Pierre Lamothe and Robert
> Walker.
>
> I have posted it to the files section so that everyone can see it.
I
> believe the relationships are really made clear in this process!
>
> Here it is: "pehrcolor.bmp"
>
> /tuning/files/Pehrson/
>
> ____________ ______ _____ _
> Joseph Pehrson

Paul Erlich's lattice of the 19-tone scale has now been change to a
.gif file to save space in the arichives.

It is now called, naturally, "pehrcolor.gif"

Thanks!

_________ ______ _____ _
Joseph Pehrson

P.S. I realize there are several outstanding issues on the TL that I
have not had time yet to address... but such discussion will be
forthcoming!