Ls scales, LMs, transposability, (Dan, Paul, Neil?)

Just some comments on diatonic scales.

Transposability is accomplished by having
the changing of an L into an s (and the
concurrent changing of an s into an L)
cause a rotation of the scale.

I've been playing with this in a sort of
vector notation similar to the following.

L L s L L L s
+ 0 0 L-s s-L 0 0 0
---------------------
L L L s L L s

So, as far as I can see, the necessary
conditions for an exactly transposable
system are that it be representable as

[ M * [ N * L ] ] s [ (N-1) * L ] s

So LLsLs is a transposable system with
N=2 and M=1. The diatonic system above
is transposable with N=3 and M=1.

As long as M=1, N>1, I think you'll
always be rotating to someplace
'at or near' the fourth or fifth.

So I was wonderring, why go to decatonic?

I was thinking of a system with seventh
chords (or, tetrads) as the basic consonance
and spelling out a transposable scale that
seemed to fit nicely I got this interesting
set.

LLL sLL LsL Ls
1 3 5 7 1

This system transposes by 'sharpenning the
root and root goes up a third'. (Of course,
there may be more interesting rotations which
behave somewhat differently, but still
transpose).

L LL sLL LsL L s
+ s-L 00 000 000 0 L-s
-----------------------------
sL Ls LLL sLL L
7 1 3 5

I sought out some EDOs where one could pun the
'major triad' in this system with that in the
normal diatonic. 19 works great!

3 3 2 3 3 3 2
0 3 6 8 11 14 17

2 2 2 1 2 2 2 1 2 2 1
0 2 4 6 7 9 11 13 14 16 18

27, 46, 65 and 84 also work nice, though they have
diatonics of LMsLMLs format (which are not
exactly transposable).*

I guess the question that arrises is why decatonic
instead of eleptatonic (??? 11 ???)

* regarding the transposability of LMsLMLs type
scales, for instance, 53, and apply the usual
# = L-s calculation, then apply the vectors around
the circle of fifths, the following happens...

9 8 5 9 8 9 5 diatonic scale
4 -4 F# moves C->G
9 8 9 5 8 9 5
-4 4 C# moves G->D
5 8 9 5 8 9 9
4 -4 G# moces D->A
5 8 9 9 4 9 9
4 -4 D# moves A->E
9 4 9 9 4 9 9

Look at that! A normal diatonic system, and its
hypermeantone! In 53 all we've done is walk backwards
from the schismic thirds to the pythagorean!

4 -4 A#
9 4 9 9 8 5 9
4 -4 E#
9 8 5 9 8 5 9
4 -4 B#
9 8 5 9 8 9 5 key of C# looks like key of C.

This gives a somewhat 'trad' approach to these sorts
of scales which includes 34 which was mentioned lately.

Bob Valentine

Robert Valentine wrote,

>I was thinking of a system with seventh
>chords (or, tetrads) as the basic consonance
>and spelling out a transposable scale that
>seemed to fit nicely I got this interesting
>set.

> LLL sLL LsL Ls
>1 3 5 7 1

>19 works great!

That's the scale with the major third generator. It was discussed quite a
bit here some time ago. Anyone have any references? I believe Dave Keenan
has a web page on this.

Let's see . . . assuming you're allowing a comma-sized error, the 7-limit
major tetrad in 19 would be 0 6 11 15. But your chord is 0 6 11 16, more
like a dissonant 5-prime-limit dominant seventh chord. Assuming that's what
you want, let's see what other chords this pattern gets you in your scale:

2. LLs LLL sLL sL -> 0 5 11 16 -- 5-limit minor seventh chord
3. LsL LLs LLs LL -> 0 5 10 15 -- 5-limit diminished seventh chord
4. sLL LsL LsL LL -> 0 5 10 15 -- 5-limit diminished seventh chord
5. LLL sLL sLL Ls -> 0 6 11 16 -- 5-limit dominant seventh chord
6. LLs LLs LLL sL -> 0 5 10 16 -- 5-limit half-diminished seventh
chord
7. LsL LsL LLs LL -> 0 5 10 15 -- 5-limit diminished seventh chord
8. sLL sLL LsL LL -> 0 5 10 15 -- 5-limit diminished seventh chord
9. LLs LLL sLL Ls -> 0 5 11 16 -- 5-limit minor seventh chord
10. LsL LLs LLL sL -> 0 5 10 16 -- 5-limit half-diminished seventh
chord
11. sLL LsL LLs LL -> 0 5 10 15 -- 5-limit diminished seventh chord

>I guess the question that arrises is why decatonic
>instead of eleptatonic (??? 11 ???)

The decatonic scales I devised in 22-tET have a chord pattern that
approximates _7-limit tetrads_ in either six or eight positions in the scale
(depending on which variety of the scale you uses).

Robert Valentine wrote:

>>I was thinking of a system with seventh
>>chords (or, tetrads) as the basic consonance
>>and spelling out a transposable scale that
>>seemed to fit nicely I got this interesting
>>set.
>
>> LLL sLL LsL Ls
>>1 3 5 7 1
>
>>19 works great!

Paul Erlich wrote:
>That's the scale with the major third generator. It was discussed quite a
>bit here some time ago. Anyone have any references? I believe Dave Keenan
>has a web page on this.

No. It's a supermajor-third/diminished-fourth generator, an approximate 7:9.

In 19-tET the generator is 1200 * 7/19 = 442.1 cents. As the noble mediant
of 4/11 octave and 7/19 octave (which is the same as the noble mediant of
1/3 and 3/8 octave) it would be 1200*(1+3phi)/(3+8phi)= 440.6 cents. A 7:9
interval is 435 cents.

I don't have a web page on either this or major third MOS's. Maybe you're
thinking about the minor third MOS, which also has 11 tones.

Incidentally, the above 11-tone chain-of-7:9's scale has an 8 tone strictly
proper MOS as a subset. It doesn't have many consonant intervals.

Regards,
-- Dave Keenan
http://dkeenan.com

0 2 4 6 7 9 11 13 14 16 18 19

>>That's the scale with the major third generator.
>
>No. It's a supermajor-third/diminished-fourth
>generator, an approximate 7:9.

I'm not a big fan of the 5:4 as an MOS generator,
because the 128:125 comma is rather small and
hard to avoid. About a year ago I wrote...

>The smallest proper MOS available built from anything
>like a 5/4 is decatonic. The closest to 5/4 you can
>get is 369.2 cents, an error of 17.1 cents. Fortunately,
>no other interval in the scale comes as close to 5/4.
>Unfortunately, 17 cents is really bad, especially on
>the flat side of the 5/4.
>
>I just realized this is 10 out of 13tET!

0 2 3 4 6 7 8 10 11 12 13

Though proper, the scale is wildly unstable. It does
have good 7:6's, 9:5's, and 11:8's, but all of these
occur over instabilities in the scale. Paul E. suggested
the scale could be harmonized in 26-tET...

-Carl

Carl Lumma wrote,

> I just realized this is 10 out of 13tET!
>
> 0 2 3 4 6 7 8 10 11 12 13

I know it must seem like I'm madly bent on turning everyone's scales
into L/s = Phi scales at the moment <!>, but just in case you or
anyone else is interested, here's the 3L7s generalized "golden scale".

0 164 265 366 530 631 733 896 998 1099 1200
0 101 202 366 467 569 733 834 935 1036 1200
0 101 265 366 467 631 733 834 935 1099 1200
0 164 265 366 530 631 733 834 998 1099 1200
0 101 202 366 467 569 670 834 935 1036 1200
0 101 265 366 467 569 733 834 935 1099 1200
0 164 265 366 467 631 733 834 998 1099 1200
0 101 202 304 467 569 670 834 935 1036 1200
0 101 202 366 467 569 733 834 935 1099 1200
0 101 265 366 467 631 733 834 998 1099 1200

My feeling is that these golden scales one overriding characteristic
is that they put a given Ls index into an arrangement where scale
class distinction is uniquely "optimized". So if clear, uniquely
defined scale classes are your thing, these generalized golden scales
would seem to be of interest.

--Dan Stearns