Lemba[12] and Lemba[22]

I give Scala scl files for the 12 and 22 note DE scales of lemba
below. The tuning used is 270-equal, which is poptimal; good old 270
being another contender for the throne of universal temperament.

! lemba12.scl
Lemba[12] in 270-et (poptimal)
12
!
93.333333
137.777778
231.111111
368.888889
462.222222
600.000000
693.333333
737.777778
831.111111
968.888889
1062.222222
1200.000000

! lemba22.scl
Lemba[22] in 270-et (poptimal)
22
!
44.444444
93.333333
137.777778
231.111111
275.555556
324.444444
368.888889
462.222222
506.666667
555.555556
600.000000
644.444444
693.333333
737.777778
831.111111
875.555556
924.444444
968.888889
1062.222222
1106.666667
1155.555556
1200.000000

Gene Ward Smith wrote:

> I give Scala scl files for the 12 and 22 note DE scales of lemba
> below. The tuning used is 270-equal, which is poptimal; good old 270
> being another contender for the throne of universal temperament.

This tuning of lemba[12] is a proper scale, but not DE; lemba[16] is DE but not proper with this tuning, according to Scala. TOP lemba[16] is DE and strictly proper. The next DE scale is lemba[26]; beyond that, it depends on which tuning you use (but 26 is about the practical limit for lemba, since you start getting better approximations for some of the intervals, which is one reason I settled on using 26-ET notation for lemba).

I started out with lemba[10] (which I was calling "zirdec" for lack of a better name, from "zireen decatonic", and thought of as a 5-limit temperament, with TOP tuning of [1203.571465, 1896.294363, 2778.021029]); more recently I've been using lemba[16], which is much more interesting (although the 10-note scale has some nice qualities). There's just something about scales in that size range: kleismic[15], porcupine[15], mavila[16], schismic[17].... They're not too big to be unwieldy, but just big enough to have a useful range of harmonic and melodic resources.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Gene Ward Smith wrote:
>
> > I give Scala scl files for the 12 and 22 note DE scales of lemba
> > below. The tuning used is 270-equal, which is poptimal; good old 270
> > being another contender for the throne of universal temperament.
>
> This tuning of lemba[12] is a proper scale, but not DE; lemba[16] is DE
> but not proper with this tuning, according to Scala.

I don't know what definition Scala is using, but I thought we had
agreed something with a period of 1/n of an octave counted as DE if it
is the union of n scales, each of which is a MOS with the given
generator, and which are the period translates of the same scale. So,
lemba has period 1/2 octave, or 135 in 270-equal, and generator 52 in
270-equal, the size of 8/7 in 270-et. Then a 6-note scale with
generator of 52 is 52-52-52-52-52-10; if I take this and its translate
by 135, and reduce to the octave, I get a 12-note scale I was calling
DE, though it has three sizes of steps. The 16-note scale you have in
mind I would count as only half of a DE scale; taking its translate by
a half-octave together with it gives a 32 note scale.

I'm not here.

Neither of these is a DE scale.

A scale is DE if and only if, for any whole N, traversing N steps in
the scale can result in at most two specific sizes of interval.

The term comes from the late John Clough, who published a paper with
Nora Engebretsen on the topic.

Right off the bat, both scales fail for N=1, since they both have
three step sizes.

My hypothesis states that any Fokker periodicity block, when all but
one of the unison vectors are uniformly tempered out, results in a DE
scale.

I'm not here.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I give Scala scl files for the 12 and 22 note DE scales of lemba
> below. The tuning used is 270-equal, which is poptimal; good old 270
> being another contender for the throne of universal temperament.
>
> ! lemba12.scl
> Lemba[12] in 270-et (poptimal)
> 12
> !
> 93.333333
> 137.777778
> 231.111111
> 368.888889
> 462.222222
> 600.000000
> 693.333333
> 737.777778
> 831.111111
> 968.888889
> 1062.222222
> 1200.000000
>
> ! lemba22.scl
> Lemba[22] in 270-et (poptimal)
> 22
> !
> 44.444444
> 93.333333
> 137.777778
> 231.111111
> 275.555556
> 324.444444
> 368.888889
> 462.222222
> 506.666667
> 555.555556
> 600.000000
> 644.444444
> 693.333333
> 737.777778
> 831.111111
> 875.555556
> 924.444444
> 968.888889
> 1062.222222
> 1106.666667
> 1155.555556
> 1200.000000

wallyesterpaulrus wrote:

> I'm not here.
>
> Neither of these is a DE scale.

once again , if ever stated what does DE stand for and/or what are it
equivalents.

>
>
> A scale is DE if and only if, for any whole N, traversing N steps in
> the scale can result in at most two specific sizes of interval.

In this sense the disjunction is not considered?

>
>
> The term comes from the late John Clough, who published a paper with
> Nora Engebretsen on the topic.
>
> Right off the bat, both scales fail for N=1, since they both have
> three step sizes.
>
> My hypothesis states that any Fokker periodicity block, when all but
> one of the unison vectors are uniformly tempered out, results in a DE
> scale.

>
>
> I'm not here.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> I'm not here.
>
> Neither of these is a DE scale.
>
> A scale is DE if and only if, for any whole N, traversing N steps in
> the scale can result in at most two specific sizes of interval.

This won't do. I gave up MOS because MOS was supposed to mean this. We
need a word which means what I said DE meant, and formerly you, I
thought, told us that DE was it. If not DE, then what?

> My hypothesis states that any Fokker periodicity block, when all but
> one of the unison vectors are uniformly tempered out, results in a DE
> scale.

A 16-note Fokker block based on 50/49, 525/512 and 3125/3072 is

128/125, 35/32, 8/7, 6/5, 5/4, 32/25, 175/128, 10/7, 256/175,
25/16, 8/5, 42/25, 7/4, 64/35, 125/64, 2

Mapping that to (period, generator) pairs in lemba leads to

{[0, 2], [2, -3], [2, -2], [2, -1], [2, 0], [3, -3], [1, 2],
[1, 1], [1, 0], [-1, 3], [1, -2], [-1, 4], [0, 4], [1, -1],
[0, 3], [0, 1]}

The periods range from -1 to 3 and the generators from -3 to 4, so
this is constructed in a completely different way than what I was
calling a DE. In 270-equal it has two sizes of scale steps, 10 and 21,
something impossible to achieve with a single generator of 270.

Similarly, a 26-note Fokker block based on 50/49, 525/512 and 875/864 is

25/24, 21/20, 35/32, 192/175, 8/7, 7/6, 6/5, 5/4, 32/25, 125/96,
4/3, 48/35, 10/7, 35/24, 3/2, 192/125, 25/16, 8/5, 5/3, 12/7, 7/4,
175/96, 64/35, 40/21, 48/25, 2

Mapping that to lemba gives

{[0, 2], [2, -3], [-1, 5], [2, -2], [2, -1], [2, 0], [3, -3],
[3, -4], [1, 2], [2, -5], [-2, 6], [2, -4], [3, -6], [3, -5],
[1, 1], [1, 0], [4, -6],[-1, 3], [1, -2], [-1, 6], [-1, 4],
[0, 4], [1, -1], [0, 3], [0, 1], [0, 5]}

This has step sizes 10 and 11 in 270-equal tuning. I give Scala files
for both of these below.

We've got two different concepts at work here, both important, so we'd
better agree on names; apparently the difference between MOS and DE,
about which I now have no clue, isn't it.

> I'm not here.

Too bad, you are missing an interesting discussion with noted tuning
theorist Paul Erlich.

! lemba16.scl
16 note lemba scale in 270 tuning (poptimal)
16
!
93.333333
137.777778
231.111111
324.444444
368.888889
462.222222
506.666667
600.000000
693.333333
737.777778
831.111111
924.444444
968.888889
1062.222222
1106.666667
1200.000000

! lemba26.scl
26 note lemba scale in 270 tuning (poptimal)
26
!
44.444444
93.333333
137.777778
186.666667
231.111111
275.555556
324.444444
368.888889
413.333333
462.222222
506.666667
555.555556
600.000000
644.444444
693.333333
737.777778
786.666667
831.111111
875.555556
924.444444
968.888889
1013.333333
1062.222222
1106.666667
1155.555556
1200.000000

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

The 26 note scale we've discussed for lemba is actually an example of
an even scale according to my definition; instead of 52/270 as the
generator, if works for 187/270 as a generator (52+135=187.) This
still leaves 16 unaccounted for.

Anyone want to name my definition for evenness before I tag it with
another neologism?

kraig grady wrote:

> > wallyesterpaulrus wrote:
> > >>I'm not here.
>>
>>Neither of these is a DE scale.
> > > once again , if ever stated what does DE stand for and/or what are it
> equivalents.

"distributionally even". I think it's more or less equivalent to Myhill's property.

http://tonalsoft.com/enc/disteven.htm
http://tonalsoft.com/enc/myhill.htm

>> once again , if ever stated what does DE stand for and/or what
>> are it equivalents.
>
>"distributionally even". I think it's more or less equivalent to
>Myhill's property.

Hi Herman. Not quite: the former requires at least two sizes
of 2nd, the latter requires exactly two. The latter is thus
eqivalent to MOS, while the former allows half-octave scales
and the like.

(I don't recognize the difference, though, since 2:1 octaves
are wrongly fixed by the traditional way of counting 2nds. If
you use the actual period instead, there are still exactly two
sizes of 2nd.)

-Carl

>>> once again , if ever stated what does DE stand for and/or what
>>> are it equivalents.
>>
>>"distributionally even". I think it's more or less equivalent to
>>Myhill's property.
>
>Hi Herman. Not quite: the former requires at least two sizes
>of 2nd, the latter requires exactly two. The latter is thus
>eqivalent to MOS, while the former allows half-octave scales
>and the like.

Drat, I meant the former requires *no more* than two sizes *in
each generic class*, while the latter exactly two.

>(I don't recognize the difference, though, since 2:1 octaves
>are wrongly fixed by the traditional way of counting 2nds. If
>you use the actual period instead, there are still exactly two
>sizes of 2nd.)

In other words, in classes which have only one interval size
in them, that interval is some multiple of the period.

-Carl

>We need a word which means what I said DE meant, and formerly you,
>I thought, told us that DE was it. If not DE, then what?

What you said was:

>I don't know what definition Scala is using, but I thought we had
>agreed something with a period of 1/n of an octave counted as DE
>if it is the union of n scales, each of which is a MOS with the
>given generator, and which are the period translates of the same
>scale.

Oh my goodness, this is the most convoluted way I can imagine to
say what you're saying.

These things are DE, but not MOS. So I think your original thought
was right. See the replies I just posted to Herman.

>So, lemba has period 1/2 octave, or 135 in 270-equal, and
>generator 52 in 270-equal, the size of 8/7 in 270-et. Then a
>6-note scale with generator of 52 is 52-52-52-52-52-10; if
>I take this and its translate by 135, and reduce to the
>octave, I get a 12-note scale I was calling DE, though it
>has three sizes of steps. The 16-note scale you have in mind
>I would count as only half of a DE scale; taking its translate
>by a half-octave together with it gives a 32 note scale.

Hrm, how'd you get three sizes here. . . is the mapping to 270
somehow inconsistent?

Oh wait, you're using an octave period AND a half-octave
period at the same time. That won't work. Your 6-note scale
has to be wrapped at the same "period" that you later
translate by. Namely 135.

Presumably you know all this. Why do you think this is
important, though? DE are nice because they correspond to
linear temperaments via the Hypothesis (and because they
have few step sizes).

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> These things are DE, but not MOS. So I think your original thought
> was right. See the replies I just posted to Herman.

OK.

> Hrm, how'd you get three sizes here. . . is the mapping to 270
> somehow inconsistent?
>
> Oh wait, you're using an octave period AND a half-octave
> period at the same time. That won't work. Your 6-note scale
> has to be wrapped at the same "period" that you later
> translate by. Namely 135.

That's not the problem; the problem is that it's a 2MOS, not a MOS.
The MOS are of size 8 and 13, leading to the 16 and 26 scales, and I
mistakenly got a 2MOS instead, of size 6 and 11.

> Presumably you know all this. Why do you think this is
> important, though?

You want something nicely rectangular in terms of periods and
generators in order to maximize the number of chords. The DE part
isn't all that important, but a MOS (or 2MOS, for that matter) makes
it a little nicer scale-wise.

Carl Lumma wrote:

> The latter is thus
> eqivalent to MOS, while the former allows half-octave scales
> and the like.

MOS allows half octave scales, in fact the recurring interval can be any
interval

>
>
>
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

>> Oh wait, you're using an octave period AND a half-octave
>> period at the same time. That won't work. Your 6-note scale
>> has to be wrapped at the same "period" that you later
>> translate by. Namely 135.
>
>That's not the problem; the problem is that it's a 2MOS, not a
>MOS. The MOS are of size 8 and 13, leading to the 16 and 26
>scales, and I mistakenly got a 2MOS instead, of size 6 and 11.

What's a 2MOS?

-Carl

Hi Kraig.

In my limited understanding of your language, you've gone back and
forth twice on this issue over the last few years. I have caused my
tuning-math colleagues a lot of consternation as a result.

But maybe you're being consistent all along.

One of your more recent missives on the topic was

/specmus/topicId_unknown.html#1210

You said that LsssLsss is not an MOS.

In that case, we need a term for scales like LsssLsss.

There already is such a term, as I mentioned: Distributionally Even
scales.

Is this OK?

-Paul

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
>
>
> Carl Lumma wrote:
>
> > The latter is thus
> > eqivalent to MOS, while the former allows half-octave scales
> > and the like.
>
> MOS allows half octave scales, in fact the recurring interval can
be any
> interval
>
> >
> >
> >
> >
> >
>
> -- -Kraig Grady
> North American Embassy of Anaphoria Island
> http://www.anaphoria.com
> The Wandering Medicine Show
> KXLU 88.9 FM WED 8-9PM PST