re: Good 5-limit scale generators

>Single chain: No. gene- Notes in m/n
> No. rators in smallest Gen. is
> triads Min Min interval proper approx
>Generator in 8 7-limit 7-limit 2 4 5 MOS with m steps
>(+-1c) notes RMS err MA err. 3 5 6 >4 notes of n-tET
>---------------------------------------------------------------------------
>317c m3 * 4 1.0c 1.4c 6 5 1 15 5/19
>380c M3 6 4.6c 6.0c 5 1 4 16 or 19 6/19, 13/41

Groovy, Dave!

>Most of them suck melodically, as the second-last column shows, and 2 or
>4 triads in 8 notes isn't really harmonically good.

Above, I see 4 and 6 triads in 8 notes, which can be very good! The 317c
scale, while not MOS and not very stable, is proper. As for the 380c
one...

At http://lumma.org/diatonic.txt the latest version of my diatonicity
rules may be found. I think very highly of them. In fact, there are
only two properties I would call very useful which don't appear on the
chart: tetrachordality and efficiency. (These are two properties that our
traditional diatonic scale has, but which I don't think are needed for or
related to what I'd call diatonic effects.)

Anyway, you'll notice that I list Rothenberg stability _or_ chord coverage
as sufficient for interval coding. I haven't checked yet, but with 6
triads in 8 notes, the 380c scale has probably got the coverage thing down.

>>>It only has 2 major and 2 minor triads.
>>
>>Yeah -- a really good amount for a 7-tone scale!
>
>I assume you're being facetious.

I don't think he was. Anyway, the scale looks cool to me.

-Carl

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Thank you
Kevin

>>Single chain: No. gene- Notes in m/n
>> No. rators in smallest Gen. is
>> triads Min Min interval proper approx
>>Generator in 8 7-limit 7-limit 2 4 5 MOS with m steps
>>(+-1c) notes RMS err MA err. 3 5 6 >4 notes of n-tET
>>---------------------------------------------------------------------------
>>317c m3 * 4 1.0c 1.4c 6 5 1 15 5/19
>>380c M3 6 4.6c 6.0c 5 1 4 16 or 19 6/19, 13/41
>
>Groovy, Dave!
>
>>Most of them suck melodically, as the second-last column shows, and 2 or
>>4 triads in 8 notes isn't really harmonically good.
>
>Above, I see 4 and 6 triads in 8 notes, which can be very good! The 317c
>scale, while not MOS and not very stable, is proper. As for the 380c
>one...

Dave Keenan! I just realized that the 8-tone 317-cent scale above is
my 8-tone subset of your 11-tone chain-of-minor-thirds scale from last
year! Four triads, on 1-3-5! I don't think I ever realized that!!

-Carl

David C Keenan wrote,

> I was mistaken in calling it a MOS, but it _is_ proper. Note that
the generator here can be either a fifth or a major third. They end up
as the same 8 note scale. In 12-tET it is 12121212. Outside of 12-tET
the minor thirds must stay at 300 cents but you are free to narrow the
fifths to improve the major thirds.

I don't see why you can't look at these as MOS scales... I do!
Periodicity is periodicity with MOS structures no? That's why in this
case "outside of 12-tET the minor thirds must stay at 300 cents", no?
Here's the 0/1 1/1 "golden diminished scale" version:

0 185 300 485 600 785 900 1085 1200
0 115 300 415 600 715 900 1015 1200

> Here's the 9 note proper scale with 12 triads, based on 3 chains of
fifths (or minor thirds) 1/3 octave apart. In 12-tET it's 121121121.
The major thirds are stuck at 400 cents but you can widen the fifths
to improve the
minor thirds.

Here's the P = 400, 1/1 2/3 "golden Tcherepnin scale":

0 179 289 400 579 689 800 979 1089 1200
0 111 221 400 511 621 800 911 1021 1200
0 111 289 400 511 689 800 911 1089 1200

> Do these symmetrical 8 and 9 note scales have standard names?

The first is common enough (I gave it as a rotation of the plain ol'
diminished scale), and I brought up the second one a while ago in the
"9-tone Liszt" thread. Here's a bit of an informative reply from Kyle
Gann who attributed the scale to Alexander Tcherepnin... "I wouldn't
be surprised if Liszt had a late piano piece or two in that scale, and
I'll check around. But Tcherepnin's music is loaded with it. Leon
Botstein conducted Tcherepnin's Rhapsodie Georgienne here last year,
an early work based on Georgian folk tunes, and it was loaded with
Tcherepnin scale, every page. Easy to find, because there's always one
augmented triad missing from it. If Tcherepnin's interest was piqued
by Georgian folk music, there may be some folk use of it that Liszt
got hold of, as omniverously as he studied every folk music he could
find out about."

--Dan Stearns

Dan Stearns wrote,

>I don't see why you can't look at these as MOS scales...

Erv Wilson invented the term MOS to refer to scales with a _single_
generator, and it's a very strange term . . . if you're after a different
meaning, then why not just coin a different term?

Paul H. Erlich wrote,

> Erv Wilson invented the term MOS to refer to scales with a _single_
generator, and it's a very strange term . . . if you're after a
different meaning, then why not just coin a different term?

Well I try to use accepted terms to help cut down on troublesome
communication problems... anyway, I am using "a _single_ generator";
in both those scales the generator and the moment of symmetry exists
within a fraction of the octave and not the octave (same goes for the
2L8s decatonics)... that was my whole point about periodicity. Can you
think of a better term for what I'm doing here?

--Dan Stearns

Dan wrote,

>anyway, I am using "a _single_ generator";
>in both those scales

I don't think Erv would see it that way . . . you're using an additional
generator which is a fraction of an octave.

>Can you
>think of a better term for what I'm doing here?

Uhh . . . something may have come up when we were talking about Myhill's
property . . . John Chalmers?

Paul H. Erlich wrote,

> I don't think Erv would see it that way . . . you're using an
additional generator which is a fraction of an octave.

Hmm, I see it as a single generator and two periodic pitch spaces; the
second "periodicity" being a fraction of an octave. Maybe there's a
more suitable term than MOS for this, but only one generator is
creating the MOS pattern... By using not only the octave as "P", you
can easily generalize all manner of usual and unusual scales with an
Ls, two-stepsize cardinality.

--Dan Stearns

Dan Stearns wrote...

>I don't see why you can't look at these as MOS scales... I do!
>Periodicity is periodicity with MOS structures no?

Paul Erlich replied...

>Erv Wilson invented the term MOS to refer to scales with a _single_
>generator, and it's a very strange term . . . if you're after a different
>meaning, then why not just coin a different term?

Good point (be creative, Dan!). Also, other authors since Wilson have
recognized Myhill's property as a sufficient and necessary condition
for MOS (no multi-chain scales have Myhill's property).

>Well I try to use accepted terms to help cut down on troublesome
>communication problems... anyway, I am using "a _single_ generator";
>in both those scales the generator and the moment of symmetry exists
>within a fraction of the octave and not the octave (same goes for the
>2L8s decatonics)... that was my whole point about periodicity. Can you
>think of a better term for what I'm doing here?

Wait a minute, Dan... are you doing non-octave MOSs? They are still
MOS, but it isn't fair to use an octave to talk about them (i.e.
describe them as the superposition of two octave-periodic chains).

-Carl

Carl Lumma wrote,

> Wait a minute, Dan... are you doing non-octave MOSs? They are still
MOS,

Right. The generalized "Golden scale" algorithm I use only defines the
periodic space in which the MOS pattern exists:

X = P/(N*D)

"P" = periodicity, "N" = (b+phi*d) and "D" = (a+phi*c) where a/b, c/d
are the adjacent fractions (by "adjacent" I mean fractions that always
differ by 1 when cross-multiplied), and "X" = the corresponding
phi-weighted interval, or "Golden generator"

So it's completely generalized. "P" only implies the octave because
that's usually the deal. But all usual and unusual scales with an Ls,
two-stepsize cardinality can be given a unique "Golden measure" with
this method.

> but it isn't fair to use an octave to talk about them (i.e. describe
them as the superposition of two octave-periodic chains).

I mentioned the octave in the context of yesterdays post because the
particular scales that were being discussed were ones where "P" also
repeats a given amount of times within the octave.

--Dan Stearns

>>Wait a minute, Dan... are you doing non-octave MOSs? They are still
>>MOS,
>
>Right. The generalized "Golden scale" algorithm I use only defines the
>periodic space in which the MOS pattern exists:

Eeep! You're way over my head here, Dan. I'm not really quick with math,
and I have very little time to spend on tuning issues (I can barely track
the threads as it is now). Is there a chance you could explain what you're
up to and why in language for simple folk?

I'll do us the favor of showing how a non-octave MOS can be viewed as
a superposition of multiple, octave-based MOSs. Take the wholetone scale
in 12-tET (0 2 4 6 8 10 12). It doesn't have Myhill's property, so it
isn't an MOS. But it can be described as the superposition of
(0 2 4 6 8 12) and (0 4 6 8 10 12), both of which have Myhill's property.
Now, call 1300 cents the interval of equivalence. The set
(0 2 4 6 8 10 12) _is_ a single MOS here. Did I make a mistake?

In short, a set is MOS if and only if it has Myhill's property at the
interval of equivalence specified; scales can be MOS at a given IE, but
not at another.

-Carl

I wrote...

>The set (0 2 4 6 8 10 12) _is_ a single MOS here. Did I make a mistake?

Yup. That should read: (0 2 4 6 8 10 13).

-Carl

Carl Lumma wrote,

> In short, a set is MOS if and only if it has Myhill's property at
the interval of equivalence specified; scales can be MOS at a given
IE, but not at another.

Yes, this is all I was saying; though I wasn't aware of the definition
of Myhill's property. So "P" in the generalized Golden scale algorithm
I use is just a given "interval of equivalence"... and in the 4L4s
diminished scale example, the IE or "P" was 1/4th of an octave, and in
the 3L6s Tcherepnin scale example, the IE (or P) was 1/3rd of an
octave.

--Dan Stearns