Reply to Justin White

Justin -- let me give you an example of how a periodicity block approach
might help you.

Let's say you want to stay with strict JI, no tempering. Let's also say you
decide that an interval smaller than 15 cents is going to be too small to
have two adjacent frets on your guitar for both of its notes. So as you
expand in the lattice, you need to locate the first three intervals you find
that are smaller than 15 cents. Since you said you wanted 27s and 21s in
your chords, I take it you're going to expand furthest along the 3-axis,
less far along the 7-axis, and least far along the 5-axis. So the three
unison vectors you're likely to find first are

224:225
243:245
1024:1029

Using these as unison vectors, you end up with the following 41-tone system:

tone # cents numerator denominator

0 0 1 1
1 35.697 49 48
2 62.961 28 27
3 84.467 21 20
4 111.73 16 15
5 155.14 35 32
6 182.4 10 9
7 203.91 9 8
8 231.17 8 7
9 266.87 7 6
10 294.13 32 27
11 315.64 6 5
12 342.91 128 105
13 386.31 5 4
14 422.01 245 192
15 435.08 9 7
16 470.78 21 16
17 498.04 4 3
18 533.74 49 36
19 546.82 48 35
20 582.51 7 5
21 617.49 10 7
22 653.18 35 24
23 666.26 72 49
24 701.96 3 2
25 729.22 32 21
26 764.92 14 9
27 777.99 384 245
28 813.69 8 5
29 857.09 105 64
30 884.36 5 3
31 905.87 27 16
32 933.13 12 7
33 968.83 7 4
34 996.09 16 9
35 1017.6 9 5
36 1044.9 64 35
37 1088.3 15 8
38 1115.5 40 21
39 1137 27 14
40 1164.3 96 49
41 1200 2 1

Let's lattice this out:

245/192
,' / `.
35/24/----35/32-----105/64
,'/|\/. ,'/ \`. ,'/
10/9-------5/3-/-|/\-5/4-/---\15/8 /
/|\`49/36----/49/48\/|\/ \/|\/
/ | ,40/21.-|/\10/7 /\|/\ /\|/\
28/27-----14/9-------7/6-------7/4------21/16\
,' `. /,' `.\|/,' \`.\|/.'/ \`.\|/,'/ `.\
32/27-----16/9-------4/3-----\-1/1-/---\-3/2-/-----9/8------27/16
\`. /,'/ \`.\ /,\/|\/.\ /,\/|\/. ,'/ `. ,'
\32/21/---\-8/7-/\|/\12/7-/\|/\-9/7-/----27/14
\/| / \ |`/ 7/5------21/20\'| /
/\|/ \|/,' *-`.\|/,'-@ `.\|/
/16/15------8/5-------6/5-------9/5
/,' `. ,' `. ,'
128/105----64/35-----48/35
`. /,'
384/245
legend:

*=96/49
@=72/49

Maybe Robert Walker could do this in VRML so that it would be easier to see.

Anyway, you can immediately see one complete 7-limit diamond and (if you
don't miss the 5/3-40/12-10/7-4/3-1/1-8/7 hexany) six complete 7-limit
hexanies, along with plenty of fragments thereof.

You can modify the periodicity block by transposing one or more of the
"outer" tones by a unison vector (you'll preserve all the properties of the
periodicity block in doing so). For example, making use of the 1024:1029
unison vector, 384/245 becomes 63/40, and 245/192 becomes 80/63, and, if you
wish, 96/49 becomes 63/32, and, if you wish, 49/48 becomes 64/43. . . .

That last ratio should have been 64/63 . . .

Hi Justin,

I wrote,

>Let's say you want to stay with strict JI, no tempering.
>. . .
>224:225
>243:245
>1024:1029
>Using these as unison vectors
>. . .
>You can modify the periodicity block by transposing one or more of the
>"outer" tones by a unison vector (you'll preserve all the properties of
>the periodicity block in doing so). For example, making use of the
>1024:1029 unison vector, 384/245 becomes 63/40, and 245/192 becomes >80/63,
and, if you wish, 96/49 becomes 63/32, and, if you wish, 49/48 >becomes
64/43. . . .

In fact it would be _necessary_ to do this for some ratios, since 245/192 is
only 13 cents away from 9/7, 384/245 is only 13 cents away from 14/9 . . .
Let's see what we can do . . .

using 1024:1024
384/245 becomes 63/40
64/35 becomes 147/80
48/35 becomes 441/320
245/192 becomes 80/63
128/105 becomes 49/40
32/21 becomes 49/32
96/49 becomes 63/32
72/49 becomes 189/128
105/64 becomes

using 245:243
9/5 becomes 49/27

using 224:225
105/64 becomes 49/30
16/15 becomes 15/14

tone # cents numerator denominator

0 0 1 1
1 35.697 49 48
2 62.961 28 27
3 84.467 21 20
4 119.44 15 14
5 155.14 35 32
6 182.4 10 9
7 203.91 9 8
8 231.17 8 7
9 266.87 7 6
10 294.13 32 27
11 315.64 6 5
12 351.34 49 40
13 386.31 5 4
14 413.58 80 63
15 435.08 9 7
16 470.78 21 16
17 498.04 4 3
18 533.74 49 36
19 555.25 441 320
20 582.51 7 5
21 617.49 10 7
22 653.18 35 24
23 674.69 189 128
24 701.96 3 2
25 737.65 49 32
26 764.92 14 9
27 786.42 63 40
28 813.69 8 5
29 849.39 49 30
30 884.36 5 3
31 905.87 27 16
32 933.13 12 7
33 968.83 7 4
34 996.09 16 9
35 1031.8 49 27
36 1053.3 147 80
37 1088.3 15 8
38 1115.5 40 21
39 1137 27 14
40 1172.7 63 32
41 1200 2 1

Let's lattice this out:

35/24-----35/32
,'/|\`. ,'/|\`.
10/9-------5/3-/-|-\-5/4-/-|-\15/8
49/27-----49/36-----49/48-----49/32\/|\
,' *-`.-|-,'-#-`.-|/\'/$\`/\|/,'/@\`.\| \
28/27-----14/9-------7/6-/---\-7/4-/---\21/16-----63/32----189/128
,' `. |/,' `.\|/,' |\/.\|/.\/|\/.\|/,\/|\`.\ ,'/| `. ,'
32/27-----16/9-------4/3---|/\-1/1-/\|/\-3/2-/\|-\-9/8-/----27/16
\`49/30-----49/40----147/80----441/320
\ 8/7-/\|/,'-%-`.\|/,'9/7`.\|/,'-&
\ |`/ 7/5------21/20-----63/40
\|/,' `.\|/,'
8/5-------6/5

@=15/14
*=80/63
#=40/21
$=10/7
%=12/7
&=27/14

There, now the smallest step is the syntonic comma, 21.5 cents.

You'll notice that the structure is perfectly symmetrical about the point
halfway between 1/1 and 7/4, except for the 15/8, whose reflection would be
28/15, only a 225:224 (one of the unison vectors) away.

This new structure has _two_ complete 7-limit diamonds and _seven_ complete
hexanies, while maintaining the "JI major scale", etc.