24-note lemba keyboard mapping

Lemba is one of the "middle path" temperaments mentioned in Paul Erlich's paper, with a half-octave period and a generator of approximately 5 steps of 26-ET. I've been using a keyboard mapping with 16 notes per octave, centered on D, but I haven't been very happy with that. It's too hard to find octaves, and too easy to run across mistuned fifths, among other things. So the other day I decided to try a 24-note keyboard mapping similar to the one suggested on Graham Breed's page for miracle temperament (http://x31eq.com/miracle/keyboard.html). The notes of the basic 10-note lemba scale are mapped to the black keys, and the "sharps" and "flats" to the white keys.

I'll use the letters D L T U N O P I J R to represent the 10-note basic scale, with two chains of generators a half-octave apart (P-J-D-T-N, L-U-O-I-R). This already has some good tetrads, and the 8-note subset which includes these tetrads is also interesting to use melodically.

Step sizes:
D..L.T..U.N..O..P.I..J.R..D
L s L s L L s L s L

Diatonic scale for comparison:
D...E..F...G...A...B..C...D
L s L L L s L

Basic harmonic structure: (--- 3/2, / 5/4, \ 6/5)
O D
/ /
P---T L---I
/L\N/ /P\R/
U---R J---N
/ /
D O

Notes beyond this 10-note scale are represented with the addition of Sagittal accidentals )||( and )!!(, which for convenience I'll represent with generic "up" and "down" symbols: ^ v.

I settled on a keyboard mapping with the pitch D mapped to the central F# on a 5-octave keyboard. This turns out to work pretty well: the small steps L-T, U-N, P-I, and J-R have one white key between the black keys, while the large steps T-U, N-O, I-J, and R-D have two white keys. I even found a contiguous range of 12 generator steps that works with this mapping.

D Lv L Tv T T^ Uv U Nv N N^ Ov O Pv P Iv I I^ Jv J Rv R R^ Dv
+0 +3 -2 +6 +1 -4 +4 -1 +7 +2 -3 +5 +0 +3 -2 +6 +1 -4 +4 -1 +7 +2 -3 +5
F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C C# D Eb E F

Interestingly, in the TOP tuning that I'm using (period = 601.7 cents, generator = 230.875 cents), the "wolf" fifths (tempered approximations of 40:27) sound better than the actual approximation of 3:2 (2.2 cents flat as opposed to 9.33 cents flat).

I'm already finding that I like this keyboard arrangement better than the 16 note per octave version. The limited range is not as much a problem as the irregularity of the 16-note mapping, and if I want to restrict myself to using the 16-note subset, it's easy to remember to play only the E, G, and B keys on the keyboard in addition to the black notes, avoiding the other white keys.

Cool deal! I'm still happy to snail-mail a copy of the paper to
anyone who wants it. I'll have to try this mapping on my Ensoniq
sometime . . .

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Lemba is one of the "middle path" temperaments mentioned in Paul
> Erlich's paper, with a half-octave period and a generator of
> approximately 5 steps of 26-ET. I've been using a keyboard mapping
with
> 16 notes per octave, centered on D, but I haven't been very happy
with
> that. It's too hard to find octaves, and too easy to run across
mistuned
> fifths, among other things. So the other day I decided to try a 24-
note
> keyboard mapping similar to the one suggested on Graham Breed's
page for
> miracle temperament
(http://x31eq.com/miracle/keyboard.html).
> The notes of the basic 10-note lemba scale are mapped to the black
keys,
> and the "sharps" and "flats" to the white keys.
>
> I'll use the letters D L T U N O P I J R to represent the 10-note
basic
> scale, with two chains of generators a half-octave apart (P-J-D-T-
N,
> L-U-O-I-R). This already has some good tetrads, and the 8-note
subset
> which includes these tetrads is also interesting to use melodically.
>
> Step sizes:
> D..L.T..U.N..O..P.I..J.R..D
> L s L s L L s L s L
>
> Diatonic scale for comparison:
> D...E..F...G...A...B..C...D
> L s L L L s L
>
> Basic harmonic structure: (--- 3/2, / 5/4, \ 6/5)
> O D
> / /
> P---T L---I
> /L\N/ /P\R/
> U---R J---N
> / /
> D O
>
> Notes beyond this 10-note scale are represented with the addition
of
> Sagittal accidentals )||( and )!!(, which for convenience I'll
represent
> with generic "up" and "down" symbols: ^ v.
>
> I settled on a keyboard mapping with the pitch D mapped to the
central
> F# on a 5-octave keyboard. This turns out to work pretty well: the
small
> steps L-T, U-N, P-I, and J-R have one white key between the black
keys,
> while the large steps T-U, N-O, I-J, and R-D have two white keys. I
even
> found a contiguous range of 12 generator steps that works with this
mapping.
>
> D Lv L Tv T T^ Uv U Nv N N^ Ov O Pv P Iv I I^ Jv J Rv R
R^ Dv
> +0 +3 -2 +6 +1 -4 +4 -1 +7 +2 -3 +5 +0 +3 -2 +6 +1 -4 +4 -1 +7 +2 -
3 +5
> F# G G# A Bb B C C# D Eb E F F# G G# A Bb B C C# D Eb
E F
>
> Interestingly, in the TOP tuning that I'm using (period = 601.7
cents,
> generator = 230.875 cents), the "wolf" fifths (tempered
approximations
> of 40:27) sound better than the actual approximation of 3:2 (2.2
cents
> flat as opposed to 9.33 cents flat).
>
> I'm already finding that I like this keyboard arrangement better
than
> the 16 note per octave version. The limited range is not as much a
> problem as the irregularity of the 16-note mapping, and if I want
to
> restrict myself to using the 16-note subset, it's easy to remember
to
> play only the E, G, and B keys on the keyboard in addition to the
black
> notes, avoiding the other white keys.