still looking for "white key" scales

Dan said:

> Here's the 1/6 comma meantone version with the tonic scale in the
> X-VII-IV-I-IX position,
>

Yowsers, I ran into exactly this scale over the
weekend as part of finally sitting down to determine
(in part)
1) what do I like to hear/play with
2) does my program "like" the same things

So this scale sort of appeared in the process.

Very very cool. For 19ED2-ers, a pretty functional version
can be had by 31131131131. (This is an [11,4] right? I'm
sorry if I haven't got this notation correct, but its what
I've been using and will use in the remainder of this post).

I can't remember we discussed the similar [9,4] of

BaBaBaBaa B~=156, a~=116.

It has much of the same feeling with many thirds
(major and minor) sort of crawling all over each
other. However, the thirds in this scale are a little
more exotic.

Ba ~= 7/6
aBa ~= 5/4
BaB ~= 9/7
aa ~= 8/7

(Of course in 31ED2 it is very well represented as
434343433).

Unlike the [11.4], the "fifth and fourth" are
badly bruised (although quite close to 16/11 and
11/8 respectively. I found that the 16/11 fifth
was useable and if I really wanted, a "Picardy
third" analog seems is both available and
justifiable to sharpen the fifth for greater
repose).

This [9.4] mapped very well to the piano keyboard,
where I was able to fit four complete scales onto
the 12 notes even though my tuning tables are
constrained to +- 64c per step. Woop-dee-doo,
I can modulate!

Another one I found that looks interesting is an
11.3

BaaaBaaaBaa with B ~= 57, a ~= 129.

aaBaaa ~= 3/2
Baa ~= 6/5
aaa ~= 5/4

I haven't given it too much study so far but, again,
it has lots of tightly approximated low integer
JI crawling around like a pile of snakes.

And its another good fit to 19, 12221222122.

Bob Valentine

> 0 181 249 317 498 566 634 815 883 951 1132 1200
> 0 68 136 317 385 453 634 702 770 951 1019 1200
> 0 68 249 317 385 566 634 702 883 951 1132 1200
> 0 181 249 317 498 566 634 815 883 1064 1132 1200
> 0 68 136 317 385 453 634 702 883 951 1019 1200
> 0 68 249 317 385 566 634 815 883 951 1132 1200
> 0 181 249 317 498 566 747 815 883 1064 1132 1200
> 0 68 136 317 385 566 634 702 883 951 1019 1200
> 0 68 249 317 498 566 634 815 883 951 1132 1200
> 0 181 249 430 498 566 747 815 883 1064 1132 1200
> 0 68 249 317 385 566 634 702 883 951 1019 1200
>
> --Dan Stearns
>

Ooops, I've figured out the [q,r] nomenclature now.

It means diatonic with q of one size and r of
another (I hope it doesn't care about ordering of
sizes).

So, in my post...

> Very very cool. For 19ED2-ers, a pretty functional version
> can be had by 31131131131. (This is an [11,4] right?

...wrong Bob, its a [7,4] scale.

>
> I can't remember <if> we discussed the similar [9,4] of
>
> BaBaBaBaa B~=156, a~=116.
>

...which is a [5,4] scale.

Meanwhile, I went back to look at the [7,2] we were
discussing last week. The [n,2] scales (n != 3, 5...)
seem to give my program problems, there are too many local
minima for it to provide a good guide.

Bob Valentine

Hi Robert!

You should post these scales to the tuning-math list -- I wonder if
any or all of them came out of Graham's search . . . I envision a big
book with all of these scales (as well as diatonic, MIRACLE, etc.),
deriving each of them from a colorful JI lattice with certain unison
vectors being tempered out, as we've been discussing . . .

-Paul

Hi Robert,

<<Ooops, I've figured out the [q,r] nomenclature now. It means
diatonic with q of one size and r of another (I hope it doesn't care
about ordering of sizes).>>

Right, "diatonic with q of one size and r of another". But the
ordering of sizes definitely does matter -- it's always smallest to
largest.

So a two-term index always coincides with a Fibonacci series built on
that index where the two terms are converted into adjacent fractions
(think Yasser and Kornerup combined).

So here's the familiar [2,5] diatonic index:

1 3 4 7 11 18 29
-, -, -, --, --, --, --, ..., oo
2 5 7 12 19 31 50

And here's the [2,5] index:

3 1 4 5 9 14 23
-, -, -, -, --, --, --, ..., oo
5 2 7 9 16 25 41

--Dan Stearns

> From: "D.Stearns" <STEARNS@CAPECOD.NET>
> Subject: Re: Re: still looking for "white key" scales
>
> Hi Robert,
>
> <<Ooops, I've figured out the [q,r] nomenclature now. It means
> diatonic with q of one size and r of another (I hope it doesn't care
> about ordering of sizes).>>
>
> Right, "diatonic with q of one size and r of another". But the
> ordering of sizes definitely does matter -- it's always smallest to
> largest.
>

Okay, I'll try to communicate using that convention.

As you know, my approach has been to completely ignore which was
bigger and which was smaller. The nomenclature I've been using in
my notebook where I've been grinding away at this stuff is
[q,r] where q is the total number of notes and r is the division
of the octave that the MOS is 'sort of' repeating at.

So [10,3] = BaaBaaBaaa
[12,3] = BaaBaBaaBaBa

and I don't care if B or a is larger (I suppose I should just use
x and y but then I'd HAVE to take it to the tuning-math list).

Henceforth I'll use q.r to refer to what I refer to so...

10.3 = { [3,7], [7,3] }

etc...

> So a two-term index always coincides with a Fibonacci series built on
> that index where the two terms are converted into adjacent fractions
> (think Yasser and Kornerup combined).
>
> So here's the familiar [2,5] diatonic index:
>
> 1 3 4 7 11 18 29
> -, -, -, --, --, --, --, ..., oo
> 2 5 7 12 19 31 50
>
> And here's the [2,5] index:
>
> 3 1 4 5 9 14 23
> -, -, -, -, --, --, --, ..., oo
> 5 2 7 9 16 25 41
>

And just when I thought I understood... I feel like there must be a
typo here. This seems to be producing the [2,7] flavor of 9.2.

I have no idea why all you had to do was reverse the
order of the first two terms...

> --Dan Stearns