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Hello Paul,

you 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
>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.

I have just read part 1 of your gentle intro to periodicity blocks and
understand all you have written in that. I don't yet understand how you
got the above system. But I will tell you what I think when I have fully
read the Gentle Intro...[i'm sure any confusions will be cleared up once I
have fully read the article]
Thank you for your interest [these tonal arrays look great]

PS. Despite my mentioning of the 21st and 27th identies I have ear tested
them and prefer the 11th and 13th harmonics in these circumstances. So I
will probably venture further north and south and further north east and
south west in my periodicity blocks [ie more 5 and 7 less 3]

Justin White

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