truncating a lattice by a periodicity other than the octave

Truncating a lattice by a periodicity other than the octave can lead
to some interesting results.

For instance, if you were to take a condition where n amount of major
thirds and n+1 amount of minor thirds coverage on a given P other than
the octave you could alter the triadic lattice according to that P and
get a result somewhat like that of the Balzano's "12-Fold scales".

Let's say the periodicity is a tenth rather than an octave, this would
result in a 9-tone lattice with a 350/243 comma:

25/18---25/12----5/4----15/8-----9/8
\ / \ / \ / \ / \
\ / \ / \ / \ / \
\ / \ / \ / \ / \
5/3-----1/1-----3/2-----9/4----27/20

This could then be tempered out as ((LOG(N))-LOG(D))*(20/LOG(2)), as
this would agree with an interpretation of n=4 that assumes a
convergence on a tenth; and as with the Balzano's 12-Fold scales,
4+5=9 and 4*5=20.

This could be recast in 15-tET by spelling the lattice in 20 and
reading it in 15:

7--16---5--14---3
\ / \ / \ / \ / \
11---0---9--18---7

Making the periodicity a twelfth rather than an octave would result in
an 11-tone lattice with the more familiar 81/80 comma:

10/9-----5/3-----5/2-----5/4----15/8----45/16
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
4/3-----2/1-----1/1-----3/2-----9/4-----9/8

This could then be tempered out as ((LOG(N))-LOG(D))*(30/LOG(2)). This
agrees with an interpretation of n=5 that assumes a convergence on the
twelfth where 5+6=11 and 5*6=30.

This could be recast in 19-tET by spelling the lattice in 30 and
reading it in 19:

3--14--25---6--17--28
\ / \ / \ / \ / \ / \
8--19---0--11--22---3

--Dan Stearns

I wrote,

say the periodicity is a tenth rather than an octave, this would
result in a 9-tone lattice with a 350/243 comma:

25/18---25/12----5/4----15/8-----9/8
\ / \ / \ / \ / \
\ / \ / \ / \ / \
\ / \ / \ / \ / \
5/3-----1/1-----3/2-----9/4----27/20

This could then be tempered out as ((LOG(N))-LOG(D))*(20/LOG(2)). This
would agree with an interpretation of n=4 that assumes a convergence
on the tenth; and as with the Balzano's 12-Fold scales, 4+5=9 and
4*5=20.

This could be recast in 15-tET by spelling the lattice in 20 and
reading it in 15:

7--16---5--14---3
\ / \ / \ / \ / \
11---0---9--18---7

Here's the corresponding examples that I for some reason forgot to
include... first the strict, 1/1 9/8 5/4 27/20 3/2 5/3 15/8 25/12 9/4
5/2, JI interpretation:

0 204 386 520 702 884 1088 1271 1404 1586
0 182 316 498 680 884 1067 1200 1382 1586
0 133 316 498 702 884 1018 1200 1404 1586
0 182 365 569 751 884 1067 1271 1453 1586
0 182 386 569 702 884 1088 1271 1404 1586
0 204 386 520 702 906 1088 1222 1404 1586
0 182 316 498 702 884 1018 1200 1382 1586
0 133 316 520 702 835 1018 1200 1404 1586
0 182 386 569 702 884 1067 1271 1453 1586

Then the non-octave ET, i.e., 0 3 5 7 9 11 14 16 18 20 as
1:(5/2)^(n/20):

0 238 397 555 714 872 1110 1269 1427 1586
0 159 317 476 634 872 1031 1190 1348 1586
0 159 317 476 714 872 1031 1190 1427 1586
0 159 397 555 714 872 1031 1269 1427 1586
0 159 397 555 714 872 1110 1269 1427 1586
0 238 397 555 714 872 1110 1190 1427 1586
0 159 317 476 714 872 1031 1190 1348 1586
0 159 317 555 714 872 1031 1190 1427 1586
0 159 397 555 714 872 1031 1269 1427 1586

Then the last example, the octave ET, i.e., 0 3 5 7 9 11 14 16 18 20
as 1:(2/1)^(n/15):

0 240 400 560 720 880 1120 1280 1440 1600
0 160 320 480 640 880 1040 1200 1360 1600
0 160 320 480 720 880 1040 1200 1440 1600
0 160 320 480 720 800 1040 1200 1440 1600
0 160 400 560 720 880 1120 1280 1440 1600
0 240 400 480 720 880 1120 1280 1440 1600
0 160 320 480 720 880 1040 1200 1360 1600
0 160 320 480 720 880 1040 1200 1440 1600
0 160 400 560 720 880 1040 1280 1440 1600

Here's the other bit and it's missing examples...

Making the periodicity a twelfth rather than an octave would result in
an 11-tone lattice with the more familiar 81/80 comma:

10/9-----5/3-----5/2-----5/4----15/8----45/16
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
\ / \ / \ / \ / \ / \
4/3-----2/1-----1/1-----3/2-----9/4-----9/8

This could then be tempered out as ((LOG(N))-LOG(D))*(30/LOG(2)). This
agrees with an interpretation of n=5 that assumes a convergence on the
twelfth where 5+6=11 and 5*6=30.

This could be recast in 19-tET by spelling the lattice in 30 and
reading it in 19:

3--14--25---6--17--28
\ / \ / \ / \ / \ / \
8--19---0--11--22---3

Here's the strict, 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 9/4 5/2 45/16 3/1,
JI interpretation:

0 204 386 498 702 884 1088 1200 1404 1586 1790 1902
0 182 294 498 680 884 996 1200 1382 1586 1698 1902
0 112 316 498 702 814 1018 1200 1404 1516 1720 1902
0 204 386 590 702 906 1088 1292 1404 1608 1790 1902
0 182 386 498 702 884 1088 1200 1404 1586 1698 1902
0 204 316 520 702 906 1018 1222 1404 1516 1720 1902
0 112 316 498 702 814 1018 1200 1312 1516 1698 1902
0 204 386 590 702 906 1088 1200 1404 1586 1790 1902
0 182 386 498 702 884 996 1200 1382 1586 1698 1902
0 204 316 520 702 814 1018 1200 1404 1516 1720 1902
0 112 316 498 610 814 996 1200 1312 1516 1698 1902

Then the non-octave ET, i.e., 0 3 6 8 11 14 17 19 22 25 28 30, as
1:(3/1)^(n/30):

0 190 380 507 697 888 1078 1205 1395 1585 1775 1902
0 190 317 507 697 888 1014 1205 1395 1585 1712 1902
0 127 317 507 697 824 1014 1205 1395 1522 1712 1902
0 190 380 571 697 888 1078 1268 1395 1585 1775 1902
0 190 380 507 697 888 1078 1205 1395 1585 1712 1902
0 190 317 507 697 888 1014 1205 1395 1522 1712 1902
0 127 317 507 697 824 1014 1205 1331 1522 1712 1902
0 190 380 571 697 888 1078 1205 1395 1585 1775 1902
0 190 380 507 697 888 1014 1205 1395 1585 1712 1902
0 190 317 507 697 824 1014 1205 1395 1522 1712 1902
0 127 317 507 634 824 1014 1205 1331 1522 1712 1902

Then the last example, 0 3 6 8 11 14 17 19 22 25 28 30 as
1:(3/1)^(n/19):

0 189 379 505 695 884 1074 1200 1389 1579 1768 1895
0 189 316 505 695 884 1011 1200 1389 1579 1705 1895
0 126 316 505 695 821 1011 1200 1389 1516 1705 1895
0 189 379 568 695 884 1074 1263 1389 1579 1768 1895
0 189 379 505 695 884 1074 1200 1389 1579 1705 1895
0 189 316 505 695 884 1011 1200 1389 1516 1705 1895
0 126 316 505 695 821 1011 1200 1326 1516 1705 1895
0 189 379 568 695 884 1074 1200 1389 1579 1768 1895
0 189 379 505 695 884 1011 1200 1389 1579 1705 1895
0 189 316 505 695 821 1011 1200 1389 1516 1705 1895
0 126 316 505 632 821 1011 1200 1326 1516 1705 1895

perhaps this will help flesh out the process a bit,

-- Dan Stearns