The unexpected 13-limit resources of lemba temperament

I'm resending this because Yahoo thinks my email account was bouncing messages. I hope I didn't miss much.

Now I have an idea why certain 9-note scales of lemba temperament are so good.

The lemba temperament <<6, -2, -2, -17, -20, 1]] has a period of 601.7004928 cents and a generator of 230.8749260 cents (in the 7-limit TOP tuning). The map is [[2, 2, 5, 6][0, 3, -1, -1]], so an approximation of the minor sixth 8/5 is (1 * 601.7...) + (1 * 230.87...), or 832.5754188 cents. This is pretty sharp for an 8/5, and actually closer to a 13/8 (about 8 cents flat).

The 11/8 approximation is even better: (-1 * 601.7...) + (5 * 230.87...), which works out to 552.6741372 cents (1.356 cents sharp)! This is the same note which approximates 45/32.

Both of these are slightly better than the 26-ET approximations. One of the 9-note lemba scales actually includes all 8 harmonics from 8 to 15, plus a minor third. In lemba notation it goes F G Ab A B C Db Dx E (generators: +1 +7 +0 +6 +4 +2 +0 +3). Since lemba is a half-octave scale, there's another copy of the overtone scale a tritone away: B# Cx D# Dx Ex Fx G# A Ax.

This suggests an 11-limit extension of lemba:

map: [[2, 2, 5, 6, 5][0, 3, -1, -1, 5]]
wedgie: <<6, -2, -2, 10, -17, -20, -5, 1, 30, 35]]

and in the 13-limit:

map: [[2, 2, 5, 6, 5, 7][0, 3, -1, -1, 5, 1]]
wedgie: <<6, -2, -2, 10, 2, -17, -20, -5, -19, 1, 30, 12, 35, 13, -30]]

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> The lemba temperament <<6, -2, -2, -17, -20, 1]] has a period of
> 601.7004928 cents and a generator of 230.8749260 cents (in the 7-limit
> TOP tuning). The map is [[2, 2, 5, 6][0, 3, -1, -1]], so an
> approximation of the minor sixth 8/5 is (1 * 601.7...) + (1 *
> 230.87...), or 832.5754188 cents. This is pretty sharp for an 8/5, and
> actually closer to a 13/8 (about 8 cents flat).

Fee phi foo fum...(13/8)/(8/5) = 65/64. If we keep the squared
Fibonnaci comma thing going, the next is (13/8)/(21/13) = 169/168.

> The 11/8 approximation is even better: (-1 * 601.7...) + (5 *
> 230.87...), which works out to 552.6741372 cents (1.356 cents sharp)!
> This is the same note which approximates 45/32.

(45/32)/(11/8) = 45/44.

> This suggests an 11-limit extension of lemba:
>
> map: [[2, 2, 5, 6, 5][0, 3, -1, -1, 5]]
> wedgie: <<6, -2, -2, 10, -17, -20, -5, 1, 30, 35]]

If we add 45/44 to the commas of 7-limit lemba, this is indeed what we
get. TM basis {45/44, 50/49, 385/384}

> and in the 13-limit:
>
> map: [[2, 2, 5, 6, 5, 7][0, 3, -1, -1, 5, 1]]
> wedgie: <<6, -2, -2, 10, 2, -17, -20, -5, -19, 1, 30, 12, 35, 13, -30]]

If now we add 65/64, this is again what we get. TM basis {45/44,
50/49, 65/64, 78/77}, which looks pretty darned reasonable, and the
Fibonnaci comma 169/168 is indeed one of its commas. 2^(37/192) comes
amazingly close to the rms generator, flatter by a mere 0.000146
cents. Poptimal generators are 2^(21/109), 2^(26/135), and, rather
nifty, 8/7.

Lemba[16] (Lemba[8] in the tritone) is super-awesome, I've decided.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> Both of these are slightly better than the 26-ET approximations. One of
> the 9-note lemba scales actually includes all 8 harmonics from 8 to 15,
> plus a minor third. In lemba notation it goes F G Ab A B C Db Dx E
> (generators: +1 +7 +0 +6 +4 +2 +0 +3). Since lemba is a half-octave
> scale, there's another copy of the overtone scale a tritone away: B# Cx
> D# Dx Ex Fx G# A Ax.
>

I was intuitively hoping for a subharmonic series in there too, but I can't find one. Is
there one? If not, why not?

Jacob

--- In tuning@yahoogroups.com, "Jacob" <jbarton@r...> wrote:
> Lemba[16] (Lemba[8] in the tritone) is super-awesome, I've decided.
>
> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >
> > Both of these are slightly better than the 26-ET approximations.
One of
> > the 9-note lemba scales actually includes all 8 harmonics from 8
to 15,
> > plus a minor third. In lemba notation it goes F G Ab A B C Db Dx
E
> > (generators: +1 +7 +0 +6 +4 +2 +0 +3). Since lemba is a half-
octave
> > scale, there's another copy of the overtone scale a tritone away:
B# Cx
> > D# Dx Ex Fx G# A Ax.
> >
>
> I was intuitively hoping for a subharmonic series in there too, but
I can't find one. Is
> there one?
> If not, why not?

The "straight" Lemba scales have 10 and 16 notes, symmetric at the
half-octave -- and mirror-symmetric too. If you're choosing 9, you're
ruining the mirror symmetry, so maybe harmonic fits while subharmonic
doesn't.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> The "straight" Lemba scales have 10 and 16 notes, symmetric at the
> half-octave -- and mirror-symmetric too. If you're choosing 9, you're
> ruining the mirror symmetry, so maybe harmonic fits while subharmonic
> doesn't.

I am using all 16 notes, so it should be possible. And there it is. If the overtones
were +1 +7 +0 +6 +4 +2 +0 +3 then the undertones can be generated by just the
opposite procedure: -1 -7 -0 -6 -4 -2 -0 -3! And there it is. I still don't get this
note naming procedure, but the underfundamental is somewhere on the sharp side of
things.

Jacob

Jacob wrote:

> Lemba[16] (Lemba[8] in the tritone) is super-awesome, I've decided.
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>Both of these are slightly better than the 26-ET approximations. One of >>the 9-note lemba scales actually includes all 8 harmonics from 8 to 15, >>plus a minor third. In lemba notation it goes F G Ab A B C Db Dx E >>(generators: +1 +7 +0 +6 +4 +2 +0 +3). Since lemba is a half-octave >>scale, there's another copy of the overtone scale a tritone away: B# Cx >>D# Dx Ex Fx G# A Ax.
>>
> > > I was intuitively hoping for a subharmonic series in there too, but I can't find one. Is > there one? If not, why not?

You can take the negative of the generators (-1 -3 -0 -2 -4 -6 -0 -7 -1), and if you add 7 to keep them in the range 0-7, you end up with +6 +4 +7 +5 +3 +1 +7 +0 +6. In lemba notation that would be B C Cx D# E F G A B or Ex Fx G Ab Ax B# Cx Dx Ex. Here's the whole lemba[16] scale in 26-ET notation:

0 3 6 1 4 7 2 5 0 3 6 1 4 7 2 5 0
A Ax B B# C CxDb D#Dx E ExF Fx G G# AbA
(Bbb)(Cbb) (Ebb) (Fb) (Gbb)