more Bohlen-Pierce

Scales that are interpretable as having a two-stepsize cardinality can
be indexed [A,B] where A is the amount of small steps and B is the
amount of large steps.

This is handy in many ways...

The Bohlen-Pierce scale can be generalized as a [5,4] scale where P =
1:3 (P being the periodicity). In the past I've posted quite a bit
about how an index can be used to define the range of the generator
that is able to create the given scale.

The [5,4] Bohlen-Pierce scale would have to have a generator that
falls inside the range of 1:3^(3/4) and 1:3^(7/9).

Suppose one wanted to generalize the Pythagorean scale as an (A+B)-1
chain of the simplest (smallest) rational that falls in a given [A,B]
generator range. Well, if this were the case then the Bohlen-Pierce
Pythagorean scale would be either a

1/1 2401/2187 5764801/4782969 343/243 823543/531441 49/27 117649/59049
7/3 16807/6561 3/1

or a

1/1 19683/16807 9/7 177147/117649 81/49 1594323/823543 729/343
14348907/5764801 6561/2401 3/1

where the simplest rational that falls in the [A,B] generator range is
a 7/3 or a 9/7:

0 162 435 597 870 1032 1305 1467 1740 1902
0 273 435 709 870 1144 1305 1579 1740 1902
0 162 435 597 870 1032 1305 1467 1628 1902
0 273 435 709 870 1144 1305 1467 1740 1902
0 162 435 597 870 1032 1193 1467 1628 1902
0 273 435 709 870 1032 1305 1467 1740 1902
0 162 435 597 758 1032 1193 1467 1628 1902
0 273 435 597 870 1032 1305 1467 1740 1902
0 162 323 597 758 1032 1193 1467 1628 1902

The scale I first posted in this thread takes this one-dimensional two
term interpretation to a two-dimensional three term interpretation by
using the 118098/117649 comma (which is analogous to taking the two
simplest rationals in a given range):

7/6-------3/2------27/14----243/98
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
1/1-------9/7------81/49----729/343--6561/2401

The Bohlen-Pierce 3:5:7, which is roughly analogous to the syntonic
diatonic's 4:5:6, also works quite nicely as a generator for an n*n
figurate number lattice

25/21
/ \
/ \
15/7---5/3
/ \ / \
/ \ / \
9/7---1/1---7/3
\ / \ /
\ / \ /
9/5---7/5
\ /
\ /
63/25

(Note how closely this resembles the actual Bohlen-Pierce.)

--Dan Stearns

> 25/21
> / \
> / \
> 15/7---5/3
> / \ / \
> / \ / \
>9/7---1/1---7/3
> \ / \ /
> \ / \ /
> 9/5---7/5
> \ /
> \ /
> 63/25

>(Note how closely this resembles the actual Bohlen-Pierce.)

Just change 63/25 to 25/9 to get Bohlen's Lambda scale.

Though I'm going to have to be cutting back on my tuning list
involvement, I wanted to add a little something to an earlier
Bohlen-Pierce thread that I had started.

I wrote that the [5,4] Bohlen-Pierce scale would have to have a
generator that falls inside the range of 1:3^(3/4) and 1:3^(7/9).

And that if one wanted to generalize the Pythagorean scale as an
(A+B)-1 chain of the simplest (smallest) rational that falls in a
given [A,B] generator range, then the Bohlen-Pierce "Pythagorean"
would be a scale where the simplest rational that falls in the [A,B]
generator range is either a 7/3 or a 9/7.

I also wrote that by taking this one-dimensional two term
interpretation to a two-dimensional three term interpretation (which
is analogous to taking the two simplest rationals in a given range)
one can create a 5-limit like Bohlen-Pierce "syntonic diatonic".

Taking this line of reasoning to the next logical step would result in
a QCM type analogue (where QCM is seen as tempering a chain so as to
make the simplest consonance in the chain pure).

This would be accomplished by using 1/6th of the 118098/117649 comma
to make the 177147/117649 in a chain of 9/7s pure.

1468 434
1034 868
600 1302
166 1736
1634 268
1200 702
766 1136
332 1570
1800 102
1366 536
932 970
498 1404

Here's the 1/6 comma Bohlen-Pierce meantone rotations:

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

Note the pure 1:2s, 2:3s and the perfect 1:2^(1/2)s.

Oddly enough this is nearly identical to a P/((5+phi*4))*(1+phi*1) = X
Golden Bohlen-Pierce. And both the 1/6th comma meantone and Golden
meantone are all but a strict two-dimensional JI interpretation!

49/18------7/6-------3/2------27/14
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
7/3-------1/1-------9/7------81/49----729/343

--Dan Stearns