"Plastic" tetrachord and temperament

I've been playing around with numbers again. I came up with this idea last night; I don't know if it's been proposed before, and I found nothing in a Web search.

We're probably all aware of Golden meantone, which defines the major second as being ~1.618034 times the size of a minor second in scale terms; if the period is 2/1, the generator is 696.214470 cents. But there's another consonant found as a limit in a number series similar to Fibonacci's but different, called the Padovan sequence:

http://en.wikipedia.org/wiki/Padovan_sequence
http://en.wikipedia.org/wiki/Plastic_number
http://members.fortunecity.com/templarser/padovan.html

The first numbers in the series are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200. The ratio betwen a member and the previous approaches the plastic number (p), approximately 1.324718.

I came up with a tetrachord which consists of a major second, then a neutral second 1/p the scale size of the major second, then a minor second 1/p the scale size of the neutral second. The entire series of intervals, each 1/p the size of the previous:

Perfect fourth: 498.044999 cents
"Major" third: 375.963047
Minor third: 283.806107
Major second: 214.238892
Neutral second: 161.724155
Minor second: 122.081953
Augmented prime (sharp/flat): 92.156939
Third tone: 69.567215
Quarter tone: 52.514737
... etc.

The "major" third is better thought of as a semi-major third lying in-between 16/13 and 5/4, approximated very well by 41/33. I nicknamed the 0-214-376-498 tetrachord "plastic Rast".

For the octave-period linear temperament, the fifths-generator is 706.205294 cents (or 493.794706 if you prefer fourths). The list of equal temperaments with their fifths:

7/4, 10/6, 12/7, 17/10, 22/13, 29/17, 39/23, 51/30, 68/40, 90/53, 119/70, 158/93, 209/123, 277/163, 367/216, 644/379, 853/502, 1130/665...

I strongly recommend 209-EDO which has a P4 of 86 steps, a sM3 of 65, a m3 of 49, a M2 of 37, a n2 of 28, a m2 of 21, and so on (see the list of Padovan numbers again).

I'm going to play around with these for a while; you can do the same.

~Danny~

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@...> wrote:
>
> I've been playing around with numbers again. I came up with this
idea last
> night; I don't know if it's been proposed before, and I found
nothing in a
> Web search.

We've discussed "Osmium" (I hate the word "plastic" in this
connection) recurrences among others.

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@...> wrote:
>>
>> I've been playing around with numbers again. I came up with this
> idea last
>> night; I don't know if it's been proposed before, and I found
> nothing in a
>> Web search.
>
> We've discussed "Osmium" (I hate the word "plastic" in this
> connection) recurrences among others.

I checked the Tuning list archives and found a couple of posts on Osmium. It's similar, but not identical, unless I'm misreading. (I haven't looked at the tuning-math archives yet; I need to actually sign up to that list.)

My "plastic" tuning--I personally have no problem with the name, as p ~ 1.324718 is indeed called the plastic number--is based on three step sizes (L, m, s), the large and medium, being p times the size of the next smaller, but I came up with a fifth that's _higher_ than 3/2, not lower, so 19-tone is not a plastic temperament.

The "major" scale has the steps L-m-s-L-L-m-s, so the size of the small step in octaves is 2x + 2xp + 3xp� = 1. Since I didn't give the step sizes in the octave-based scale:

s = 121.040111 cents
m = 160.344008 cents
L = 212.410587 cents

That produces a fifth of 706.205293 cents, 4.250293 sharp of just. (Compare to the fifth of 17-EDO: 705.882353.) But this tuning can't be expressed as a linear temperament, because even 1531-tone fails to produce the exact small and medium steps, only the large.

--- In tuning@yahoogroups.com, "Daniel A. Wier" <dawiertx@...> wrote:

> The "major" scale has the steps L-m-s-L-L-m-s, so the size of the
small step
> in octaves is 2x + 2xp + 3xp� = 1. Since I didn't give the step
sizes in the
> octave-based scale:

From this, we find that if s, m and L are given in parts of an octave,
then:

23s^3 - 23s^2 + 12s - 1 = 0
23m^3 + 23m^2 + 4m - 1 = 0
23L^3 - 23L^2 + 9L - 1 = 0

> That produces a fifth of 706.205293 cents, 4.250293 sharp of just.
(Compare
> to the fifth of 17-EDO: 705.882353.) But this tuning can't be
expressed as a
> linear temperament, because even 1531-tone fails to produce the
exact small
> and medium steps, only the large.

You are jumping ahead here. To start out with, we have a regular
tuning. The generators can be given as 2^s, 2^m and 2^L, so we have a
group of rank 3. To get a temperament, we want approximations. To
start out with, we have the octave and the fifth approximated. If you
also give approximations for 5/4 and 7/4, you have a rank four group
being approximated by a rank three tuning, so now you can analyze it
as a temperament.