Final recommendations for Justin White's scale (was: RE: [tuning] Re: New Scala version)

OK -- I did it myself (sort of). Since this is a CS scale, I could just look
at the intervals formed by fixed numbers of scale steps.

+24 +13 +33 +7 +19
+29
root
1/1 3/2 5/4 7/4 9/8 11/8
13/8
65/64 98/65 56/45 1024/585 512/455 448/325
64/39
33/32 448/297 96/77 96/55 260/231 320/231
18/11
21/20 3/2 5/4 110/63 55/49 25/18
80/49
15/14 112/75 56/45 26/15 28/25 539/390
49/30
35/32 52/35 26/21 26/15 28/25 48/35
512/315
10/9 3/2 99/80 693/400 9/8 441/320
81/50
9/8 3/2 56/45 110/63 91/81 112/81
44/27
8/7 3/2 5/4 7/4 9/8 441/320
13/8
65/56 98/65 49/39 7/4 147/130 448/325
49/30
33/28 448/297 49/39 7/4 112/99 91/66
49/30
6/5 3/2 5/4 7/4 325/288 25/18
275/168
49/40 220/147 5/4 600/343 55/49 135/98
80/49
5/4 52/35 56/45 7/4 28/25 48/35
13/8
91/72 3/2 81/65 160/91 720/637 18/13
297/182
9/7 539/360 56/45 7/4 245/216 112/81
49/30
21/16 220/147 26/21 256/147 44/39 48/35
80/49
4/3 3/2 5/4 195/112 9/8 11/8
105/64
65/48 3/2 81/65 792/455 147/130 48/35
64/39
11/8 3/2 96/77 96/55 112/99 91/66
18/11
7/5 3/2 5/4 7/4 9/8 11/8
80/49
10/7 3/2 56/45 7/4 28/25 11/8
13/8
35/24 3/2 216/175 26/15 39/35 48/35
396/245
77/52 1040/693 26/21 936/539 260/231 845/616
624/385
3/2 3/2 26/21 7/4 9/8 11/8
49/30
49/32 512/343 26/21 256/147 384/343 48/35
80/49
14/9 585/392 99/80 195/112 9/8 135/98
13/8
63/40 220/147 550/441 110/63 640/567 25/18
80/49
8/5 3/2 5/4 7/4 9/8 25/18
105/64
13/8 98/65 5/4 160/91 44/39 18/13
64/39
5/3 3/2 99/80 7/4 39/35 48/35
13/8
27/16 364/243 56/45 616/351 91/81 260/189
44/27
12/7 3/2 5/4 7/4 539/480 11/8
49/30
7/4 3/2 5/4 7/4 55/49 48/35
80/49
16/9 3/2 5/4 7/4 9/8 441/320
105/64
9/5 325/216 5/4 7/4 325/288 25/18
385/234
11/6 3/2 96/77 96/55 9/8 91/66
18/11
13/7 98/65 5/4 7/4 147/130 18/13
343/208
91/48 960/637 792/637 160/91 720/637 18/13
64/39
77/40 50/33 96/77 135/77 25/22 320/231
18/11
55/28 98/65 343/275 96/55 112/99 91/66
448/275

13-limit otonalities with four or more identities:

1/1 has a complete 1-3-5-7-9-11-13.

7/5 has 1-3-5-7-9-11.
8/7 has 1-3-5-7-9-13.

10/7 has 1-3-7-11-13.
3/2 has 1-3-7-9-11.
4/3 has 1-3-5-9-11.
12/7 has 1-3-5-7-11.
8/5 has 1-3-5-7-9.
16/9 has 1-3-5-7-9.

6/5 has 1-3-5-7.
5/3 has 1-3-7-13.
7/4 has 1-3-5-7.

The only notes (of the 41) _not_ involved in any of these chords are #1
65/64 (which is the 7 identity of 65/32), #14 91/72, # 23 77/52, and #38
91/48. There is no way, using unison vectors, of reassigning any of these
notes so that they fall into the empty spaces in the chords above.

Continuing . . . three identities . . .

21/20 has 1-3-5.
10/9 has 1-3-9.
5/4 has 1-7-13 -- 65/64 is the 13.
14/9 has 1-9-13.
9/5 has 1-5-7 -- 77/52 could be changed to 117/80 to add the 13 -- but then
it would be only 5 cents above 35/24 :(
11/6 has 1-3-9 -- 91/72 could be changed to 121/96 to add the 11 -- but then
it would be only 14 cents above 5/4 :(
13/7 has 1-5-7

Down to two identities . . .

9/8 has 1-3 -- 91/72 could be changed to 81/64 to add the 9, using the
unison vectors 385:384, 196:195, 100:99, 105:104, and 105:104 again.
65/56 has 1-7 -- 65/64 is the 7; 91/48 could be changed to 845/448 to add
the 13, using the unison vector 196:195.
33/28 has 1-7 -- 77/52 should be changed to 165/112 to add the 5, using the
unison vector 196:195.
49/40 has 1-5
91/72 has 1-3 with 91/48, two notes under investigation . . .
9/7 has 1-7
65/48 has 1-3 -- 65/64 is the 3.
11/8 has 1-3 -- 91/48 could be changed to 121/64 to add the 11, using the
unison vectors 275:273 and 100:99.
35/24 has 1-3 -- 91/72 could be changed to 245/192 to add the 7 -- but then
it would be only 13 cents below the 9/7 :(
13/8 has 1-5 -- 65/64 is the 5.

So . . . 65/64 needs to stay where it is; 77/52 should become 165/112; 91/72
should become 81/64; and 91/48 should change to either 121/64 or 845/448 --
I vote for the former since adding the 11 to a 1-3 is more audibly consonant
than adding the 13 to a 1-7.

So here's the scale in its final version:

note # ratio cents
0 1 0
1 65/64 27
2 33/32 53
3 21/20 84
4 15/14 119
5 35/32 155
6 10/9 182
7 9/8 204
8 8/7 231
9 65/56 258
10 33/28 284
11 6/5 316
12 49/40 351
13 5/4 386
14 81/64 408
15 9/7 435
16 21/16 471
17 4/3 498
18 65/48 525
19 11/8 551
20 7/5 583
21 10/7 617
22 35/24 653
23 165/112 671
24 3/2 702
25 49/32 738
26 14/9 765
27 63/40 786
28 8/5 814
29 13/8 841
30 5/3 884
31 27/16 906
32 12/7 933
33 7/4 969
34 16/9 996
35 9/5 1018
36 11/6 1049
37 13/7 1072
38 121/64 1103
39 77/40 1134
40 55/28 1169

Good luck with the guitar and let me know how it turns out!

As you may have noticed by now, 165/112 was not a good choice, as it
falls too close to 35/24. Sorry for the slip-up! Fortunately, I have
a better recommendation. 40/27 would be the 1 of a 1-3-9 with 10/9
and 5/3. So here's the FINAL scale:

note # ratio cents
0 1 0
1 65/64 27
2 33/32 53
3 21/20 84
4 15/14 119
5 35/32 155
6 10/9 182
7 9/8 204
8 8/7 231
9 65/56 258
10 33/28 284
11 6/5 316
12 49/40 351
13 5/4 386
14 81/64 408
15 9/7 435
16 21/16 471
17 4/3 498
18 65/48 525
19 11/8 551
20 7/5 583
21 10/7 617
22 35/24 653
23 40/27 680
24 3/2 702
25 49/32 738
26 14/9 765
27 63/40 786
28 8/5 814
29 13/8 841
30 5/3 884
31 27/16 906
32 12/7 933
33 7/4 969
34 16/9 996
35 9/5 1018
36 11/6 1049
37 13/7 1072
38 121/64 1103
39 77/40 1134
40 55/28 1169

Enjoy!

Here are the step sizes in the scale, in cents:

pitch distance to previous pitch

26.84 26.84
53.27 26.43
84.47 31.19
119.44 34.98
155.14 35.70
182.40 27.26
203.91 21.51
231.17 27.26
258.02 26.84
284.45 26.43
315.64 31.19
351.34 35.70
386.31 34.98
407.82 21.51
435.08 27.26
470.78 35.70
498.04 27.26
524.89 26.84
551.32 26.43
582.51 31.19
617.49 34.98
653.18 35.70
680.45 27.26
701.96 21.51
737.65 35.70
764.92 27.26
786.42 21.51
813.69 27.26
840.53 26.84
884.36 43.83
905.87 21.51
933.13 27.26
968.83 35.70
996.09 27.26
1017.60 21.51
1049.36 31.77
1071.70 22.34
1102.64 30.93
1133.83 31.19
1168.81 34.98
1200.00 31.19

I sent that prematurely!
> Here are the step sizes in the scale, in cents and ratios:

pitch step step
(¢) cents ratio

26.84 26.84 64/65
53.27 26.43 65/66
84.47 31.19 55/56
119.44 34.98 49/50
155.14 35.70 48/49
182.40 27.26 63/64
203.91 21.51 80/81
231.17 27.26 63/64
258.02 26.84 64/65
284.45 26.43 65/66
315.64 31.19 55/56
351.34 35.70 48/49
386.31 34.98 49/50
407.82 21.51 80/81
435.08 27.26 63/64
470.78 35.70 48/49
498.04 27.26 63/64
524.89 26.84 64/65
551.32 26.43 65/66
582.51 31.19 55/56
617.49 34.98 49/50
653.18 35.70 48/49
680.45 27.26 63/64
701.96 21.51 80/81
737.65 35.70 48/49
764.92 27.26 63/64
786.42 21.51 80/81
813.69 27.26 63/64
840.53 26.84 64/65
884.36 43.83 39/40
905.87 21.51 80/81
933.13 27.26 63/64
968.83 35.70 48/49
996.09 27.26 63/64
1017.60 21.51 80/81
1049.36 31.77 54/55
1071.70 22.34 77/78
1102.64 30.93 832/847
1133.83 31.19 55/56
1168.81 34.98 49/50
1200.00 31.19 55/56

Kraig Grady would be proud -- almost. This is a 13-limit, 41-tone CS
scale, with all superparticular ratios for successive scale degrees,
_except for one_! Looks like that 121/64 was a poor choice after all
(if you're super particular). Returning to 91/48 fixes this numerical
blemish (anyone see any other possibilities?). So here, at last, is a
13-limit, 41-tone CS scale that even Erv Wilson himself would
probably approve of:

note # cents ratio step step
cents ratio

1 26.84 64/65 26.84 64/65
2 53.27 32/33 26.43 65/66
3 84.47 20/21 31.19 55/56
4 119.44 14/15 34.98 49/50
5 155.14 32/35 35.70 48/49
6 182.40 9/10 27.26 63/64
7 203.91 8/9 21.51 80/81
8 231.17 7/8 27.26 63/64
9 258.02 56/65 26.84 64/65
10 284.45 28/33 26.43 65/66
11 315.64 5/6 31.19 55/56
12 351.34 40/49 35.70 48/49
13 386.31 4/5 34.98 49/50
14 407.82 64/81 21.51 80/81
15 435.08 7/9 27.26 63/64
16 470.78 16/21 35.70 48/49
17 498.04 3/4 27.26 63/64
18 524.89 48/65 26.84 64/65
19 551.32 8/11 26.43 65/66
20 582.51 5/7 31.19 55/56
21 617.49 7/10 34.98 49/50
22 653.18 24/35 35.70 48/49
23 680.45 27/40 27.26 63/64
24 701.96 2/3 21.51 80/81
25 737.65 32/49 35.70 48/49
26 764.92 9/14 27.26 63/64
27 786.42 40/63 21.51 80/81
28 813.69 5/8 27.26 63/64
29 840.53 8/13 26.84 64/65
30 884.36 3/5 43.83 39/40
31 905.87 16/27 21.51 80/81
32 933.13 7/12 27.26 63/64
33 968.83 4/7 35.70 48/49
34 996.09 9/16 27.26 63/64
35 1017.60 5/9 21.51 80/81
36 1049.36 6/11 31.77 54/55
37 1071.70 7/13 22.34 77/78
38 1107.40 48/91 35.70 48/49
39 1133.83 40/77 26.43 65/66
40 1168.81 28/55 34.98 49/50
41 1200.00 1/2 31.19 55/56

I wrote,

> Returning to 91/48 fixes this numerical
> blemish (anyone see any other possibilities?)

66/35 is another possibility for this note that both produces
superparticulars above and below it, and is faithful to the
periodicity block formalism.

I guess my final recommendation was not so final!