Yahoo Tuning Groups Ultimate Backup tuning the apical BP "Lambda scales"

The Bohlen-Pierce scale structures site gives the following diatonic
scale -- the "Lambda scale":

1/1 25/21 9/7 7/5 5/3 9/5 15/7 7/3 25/9 3/1

<http://members.aol.com/bpsite/scales.html#anchor245013>

Looking at this scale as a generalized Apical mapping you'd have a
5s4L scale where "P" = 1:3. Running this through the following
generalized algorithm:

X = P/((b+A*d))*(a+A*c)
X = P/((d+A*b))*(c+A*a)

Where:

"P" = any given periodicity

"A" = one of three Apical constants ('a bud which terminates a stem')
taken from the following Golden, Equal, and Silver, series:

1/1, 1/2, 2/3, 3/5, 5/8, ...

1/1.5, 1.5/3.25, 3.25/6.375, 6.375/12.8125, 12.8125/25.59375, ...

1/2, 2/5, 5/12, 12/29, 29/70, ...

"a"/"b", "c"/"d" and "c"/"d", "a"/"b" = the two adjacent fractions of
a given Ls index

"X" = the resulting generator

would give the following three scales...

the Golden Lambda scale

0 268 434 600 868 1034 1302 1468 1736 1902
0 166 332 600 766 1034 1200 1468 1634 1902
0 166 434 600 868 1034 1302 1468 1736 1902
0 268 434 702 868 1136 1302 1570 1736 1902
0 166 434 600 868 1034 1302 1468 1634 1902
0 268 434 702 868 1136 1302 1468 1736 1902
0 166 434 600 868 1034 1200 1468 1634 1902
0 268 434 702 868 1034 1302 1468 1736 1902
0 166 434 600 766 1034 1200 1468 1634 1902

the Equal Lambda scale

0 293 439 585 878 1024 1317 1463 1756 1902
0 146 293 585 732 1024 1170 1463 1609 1902
0 146 439 585 878 1024 1317 1463 1756 1902
0 293 439 732 878 1170 1317 1609 1756 1902
0 146 439 585 878 1024 1317 1463 1609 1902
0 293 439 732 878 1170 1317 1463 1756 1902
0 146 439 585 878 1024 1170 1463 1609 1902
0 293 439 732 878 1024 1317 1463 1756 1902
0 146 439 585 732 1024 1170 1463 1609 1902

the Silver Lambda scale

0 313 443 573 886 1016 1329 1459 1772 1902
0 130 260 573 703 1016 1146 1459 1589 1902
0 130 443 573 886 1016 1329 1459 1772 1902
0 313 443 756 886 1199 1329 1642 1772 1902
0 130 443 573 886 1016 1329 1459 1589 1902
0 313 443 756 886 1199 1329 1459 1772 1902
0 130 443 573 886 1016 1146 1459 1589 1902
0 313 443 756 886 1016 1329 1459 1772 1902
0 130 443 573 703 1016 1146 1459 1589 1902

Note that the strictly proper Golden Lambda scale has, for all intents
and purposes, just octaves and fifths! Though I'm not certain how this
sits within the BP paradigm, it certainly is interesting.

In terms of "tetrachordality", the BP site sees this scale as a "2112
1 2121, with the first group of numbers describing the first
pentachord, the last group describing the second pentachord, and the
single semitone step in the center indicating their separation".

Using the adjacent fraction series of the Ls index, which arranges
"tetrachords" according to generator size, you'd have either a
0/1 -1/0, 1/2 0/1, 2/3 1/2, 3/4 2/3, 4/5 3/4, at:

s Ls Ls Ls Ls

Or a -3/1 -2/1, 1/5 1/4, 5/9 4/7, at:

sL sLsLsLs

The first has the "tetrachord" occupying a 1:2^(3/13) and the
disjunction a 1:2^(1/13), thereby giving four instances of the
"tetrachord".

The second would have the 9-out-of-13 as a sort of "pentatonic"
analogue to a "diatonic" 16-out-of-23 where the "tetrachord" would
occupy a 1:2^(10/23) and the disjunction a 1:2^(3/23). If you were to
rotated this so that generator occupies the tonic you would have the
following m-out-of-n:

sLsLsLs sL sLsLsLs

This would seem to have little relevance here though, for the BP
paradigm which considers the generalized tetrachord to be a pentachord
which "has 5/3 as a frame interval". But I thought I'd point it out
just to show how if fits in with some of the approaches I've been
working with... rotating the 5s4L scale where P=1:3 so that the
generator occupies the tonic would give the following Ls arrangement:

sLssLsLsL
sL s sL sL sL

--d.stearns

🔗Kraig Grady <kraiggrady@anaphoria.com>

Dan!
One could take any of the horagrams and treat the octave as 3/1. The BP scale always
struck me as a form of triadic diamond repeated at the 3/1.

"D.Stearns" wrote:

> The Bohlen-Pierce scale structures site gives the following diatonic
> scale -- the "Lambda scale":
>
> 1/1 25/21 9/7 7/5 5/3 9/5 15/7 7/3 25/9 3/1
>
> <http://members.aol.com/bpsite/scales.html#anchor245013>
>
> Looking at this scale as a generalized Apical mapping you'd have a
> 5s4L scale where "P" = 1:3. Running this through the following
> generalized algorithm:
>
> X = P/((b+A*d))*(a+A*c)
> X = P/((d+A*b))*(c+A*a)
>
> Where:
>
> "P" = any given periodicity
>
> "A" = one of three Apical constants ('a bud which terminates a stem')
> taken from the following Golden, Equal, and Silver, series:
>
> 1/1, 1/2, 2/3, 3/5, 5/8, ...
>
> 1/1.5, 1.5/3.25, 3.25/6.375, 6.375/12.8125, 12.8125/25.59375, ...
>
> 1/2, 2/5, 5/12, 12/29, 29/70, ...
>
> "a"/"b", "c"/"d" and "c"/"d", "a"/"b" = the two adjacent fractions of
> a given Ls index
>
> "X" = the resulting generator
>
> would give the following three scales...
>
> the Golden Lambda scale
>
> 0 268 434 600 868 1034 1302 1468 1736 1902
> 0 166 332 600 766 1034 1200 1468 1634 1902
> 0 166 434 600 868 1034 1302 1468 1736 1902
> 0 268 434 702 868 1136 1302 1570 1736 1902
> 0 166 434 600 868 1034 1302 1468 1634 1902
> 0 268 434 702 868 1136 1302 1468 1736 1902
> 0 166 434 600 868 1034 1200 1468 1634 1902
> 0 268 434 702 868 1034 1302 1468 1736 1902
> 0 166 434 600 766 1034 1200 1468 1634 1902
>
> the Equal Lambda scale
>
> 0 293 439 585 878 1024 1317 1463 1756 1902
> 0 146 293 585 732 1024 1170 1463 1609 1902
> 0 146 439 585 878 1024 1317 1463 1756 1902
> 0 293 439 732 878 1170 1317 1609 1756 1902
> 0 146 439 585 878 1024 1317 1463 1609 1902
> 0 293 439 732 878 1170 1317 1463 1756 1902
> 0 146 439 585 878 1024 1170 1463 1609 1902
> 0 293 439 732 878 1024 1317 1463 1756 1902
> 0 146 439 585 732 1024 1170 1463 1609 1902
>
> the Silver Lambda scale
>
> 0 313 443 573 886 1016 1329 1459 1772 1902
> 0 130 260 573 703 1016 1146 1459 1589 1902
> 0 130 443 573 886 1016 1329 1459 1772 1902
> 0 313 443 756 886 1199 1329 1642 1772 1902
> 0 130 443 573 886 1016 1329 1459 1589 1902
> 0 313 443 756 886 1199 1329 1459 1772 1902
> 0 130 443 573 886 1016 1146 1459 1589 1902
> 0 313 443 756 886 1016 1329 1459 1772 1902
> 0 130 443 573 703 1016 1146 1459 1589 1902
>
> Note that the strictly proper Golden Lambda scale has, for all intents
> and purposes, just octaves and fifths! Though I'm not certain how this
> sits within the BP paradigm, it certainly is interesting.
>
> In terms of "tetrachordality", the BP site sees this scale as a "2112
> 1 2121, with the first group of numbers describing the first
> pentachord, the last group describing the second pentachord, and the
> single semitone step in the center indicating their separation".
>
> Using the adjacent fraction series of the Ls index, which arranges
> "tetrachords" according to generator size, you'd have either a
> 0/1 -1/0, 1/2 0/1, 2/3 1/2, 3/4 2/3, 4/5 3/4, at:
>
> s Ls Ls Ls Ls
>
> Or a -3/1 -2/1, 1/5 1/4, 5/9 4/7, at:
>
> sL sLsLsLs
>
> The first has the "tetrachord" occupying a 1:2^(3/13) and the
> disjunction a 1:2^(1/13), thereby giving four instances of the
> "tetrachord".
>
> The second would have the 9-out-of-13 as a sort of "pentatonic"
> analogue to a "diatonic" 16-out-of-23 where the "tetrachord" would
> occupy a 1:2^(10/23) and the disjunction a 1:2^(3/23). If you were to
> rotated this so that generator occupies the tonic you would have the
> following m-out-of-n:
>
> sLsLsLs sL sLsLsLs
>
> This would seem to have little relevance here though, for the BP
> paradigm which considers the generalized tetrachord to be a pentachord
> which "has 5/3 as a frame interval". But I thought I'd point it out
> just to show how if fits in with some of the approaches I've been
> working with... rotating the 5s4L scale where P=1:3 so that the
> generator occupies the tonic would give the following Ls arrangement:
>
> sLssLsLsL
> sL s sL sL sL
>
> --d.stearns

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

Dan!
I knew this structure was a just a diamond at the 3/1, but it took me till now to figure out
how to map it. By the same method one could produce all types of diamonds that repeat at other
intervals than the 2/1

http://www.anaphoria.com/images/BPdiamond.gif

"D.Stearns" wrote:

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Kraig Grady <kraiggrady@anaphoria.com>

Dan!
This shows this same scale as an intersection of the 3-5-7 and the 5-7-9 both which
generates the same pitches. It shows also that if you wanted to use a 3-5-7-9 diamond the 3
and the 9 being the same pitch you would end up with triads? how very uninteresting.

http://www.anaphoria.com/images/BPdia2.gif

Why the golden lambda generates fifth and octaves is strange (if not suspicious) and I don't
understand why.

"D.Stearns" wrote:

> Kraig Grady wrote,
>
> > I knew this structure was a just a diamond at the 3/1, but it took
> me till now to figure out how to map it. By the same method one could
> produce all types of diamonds that repeat at other intervals than the
> 2/1
> >
> > http://www.anaphoria.com/images/BPdiamond.gif
>
> Neat. Note the he actually uses an "odd-numbered tetraktys" to
> essentially do the same thing, see:
>
> <http://members.aol.com/bpsite/scales.html#anchor48667>
>
> What do you make of the "Golden Lambda scale" with its just octaves
> and fifths? I thought it was pretty interesting...
>
> --d.stearns

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

🔗D.Stearns <STEARNS@CAPECOD.NET>

Kraig Grady wrote,

> Why the golden lambda generates fifth and octaves is strange (if not
suspicious) and I don't understand why.

The Lambda scales have a generator that is a near 3:7, and the
fortuitous 2:3s and 1:2s are an interesting result/byproduct of the
weighting, or tempering, of the generator.

Of course if just 2:3s and 1:2s were a priority of sorts, one could
set either (as one is the compliment of the other here) as a JI target
on the stack of near 3:7s and 7:9s. A generator of
(log(162)-log(1))*(1200/log(2^6)) would be one way to do this:

434 1468
868 1034
1302 600
1736 166
268 1634
702 1200
1136 766
1570 332

This is mimicked remarkably well by the Golden Lambda scale.

The Silver Lambda scale bumps the near 2:3s and 1:2s up a place in the
chain, and as a JI target on a stack of near 3:7s and 7:9s the best
you'd get is either a generator of (log(365)-log(1))*(1200/log(2^7)):

443 1459
886 1016
1328 574
1771 131
312 1590
755 1147
1198 704
1640 262

Or (log(364)-log(1))*(1200/log(2^7)):

443 1458
887 1015
1330 572
1774 128
315 1587
759 1143
1202 700
1646 256

The first of these, log 365, is mimicked by the Silver Lambda scale.

Though I'm still not exactly sure how this all actually sits in the
overall BP scheme of things as neither the octave or the fifth figure
in the BP 3:5:7 identity, it certainly is interesting!

--Dan Stearns