13-limit tuning for Justin White

Starting with the unison vectors

100:99
105:104
196:195
275:273
385:384

the Fokker periodicity block is

note # cents numerator denominator

0 0 1 1
1 31.194 56 55
2 70.672 25 24
3 84.467 21 20
4 128.3 14 13
5 150.64 12 11
6 182.4 10 9
7 187.34 39 35
8 231.17 8 7
9 266.87 7 6
10 301.85 25 21
11 315.64 6 5
12 363.97 385 312
13 386.31 5 4
14 417.51 14 11
15 443.94 84 65
16 454.21 13 10
17 498.04 4 3
18 520.38 104 77
19 551.32 11 8
20 582.51 7 5
21 617.49 10 7
22 648.68 16 11
23 679.62 77 52
24 701.96 3 2
25 745.79 20 13
26 756.06 65 42
27 782.49 11 7
28 813.69 8 5
29 836.03 624 385
30 884.36 5 3
31 898.15 42 25
32 933.13 12 7
33 968.83 7 4
34 1012.7 70 39
35 1017.6 9 5
36 1049.4 11 6
37 1071.7 13 7
38 1115.5 40 21
39 1129.3 48 25
40 1168.8 55 28

Using the unison vector 385:384, we can change
624/385 to 13/8
12/11 to 35/32
104/77 to 65/48
16/11 to 35/24
48/25 to 77/40
49/48 to 55/56

Using the unison vector 105:104, we can change
39/35 to 9/8
70/39 to 16/9
13/10 to 21/16

Using the unison vector 196:195, we can change
14/13 to 15/14
84/65 to 9/7
65/42 to 14/9
7/6 to 65/56

Using the unison vector 100:99, we can change
25/21 to 33/28
25/24 to 33/32

Using the unison vector 275:273, we can change
385/312 to 49/40

Using _both_ 385:384 _and_ 275:273, we can change
20/13 to 49/32
40/21 to 91/48

Using _both_ 196:195 _and_ 105:104, we can change
42/25 to 27/16

Using _both_ 105:104 _and_ 275:273, we can change
11/7 to 63/40

Using _both_ 385:384 _and_ 196:195, we can change
56/55 to 65/64

Using _both_ 385:384 _and_ 105:104, we can change
14/11 to 91/72

0 0 1 1
1 26.841 65 64
2 53.273 33 32
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
9 258.02 65 56
10 284.45 33 28
11 315.64 6 5
12 351.34 49 40
13 386.31 5 4
14 405.44 91 72
15 435.08 9 7
16 470.78 21 16
17 498.04 4 3
18 524.89 65 48
19 551.32 11 8
20 582.51 7 5
21 617.49 10 7
22 653.18 35 24
23 679.62 77 52
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
29 840.53 13 8
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 1049.4 11 6
37 1071.7 13 7
38 1107.4 91 48
39 1133.8 77 40
40 1168.8 55 28

Perhaps Monz (or someone else) would like to produce a lattice? I've tried
my best to maximize the consonant intervals and chords while keeping the
smallest step to a syntonic comma -- but there may still be further
improvements I've missed . . . a lattice may point out some obvious
deficiencies.

This is a response to Paul's post of 3-11-2000 where he created
a 41-tone periodicity block for Justin White.

>Starting with the unison vectors
>100:99
>105:104
>196:195
>275:273
>385:384

I wrote a new routine for Scala to optimise periodicity blocks in
the way Paul had been doing by hand, shown in that post. The command
is APPROXIMATE/MOULD and it will appear in the next version.
One can choose the attribute function that represents the
optimality criterion for the ratios, whether it be harmonic entropy,
Barlow harmonicity, triangular lattice distance, Euler gradus, etc.
It was indeed able to improve on Paul's version. It has
(for the whole scale) an Euler gradus of 83 and Wilson complexity 86.
Using Tenney's measure as criterion this was improved to
Euler gradus 81 and Wilson complexity 84. All intervals are also
superparticular here although this required some manual intervention:

1: 65/64 26.841 13th-partial chroma
2: 33/32 53.273 undecimal comma
3: 21/20 84.467 minor semitone
4: 15/14 119.443 major diatonic semitone
5: 12/11 150.637 3/4-tone, undecimal neutral second
6: 10/9 182.404 minor whole tone
7: 9/8 203.910 major whole tone
8: 8/7 231.174 septimal whole tone
9: 7/6 266.871 septimal minor third
10: 13/11 289.210
11: 6/5 315.641 minor third
12: 11/9 347.408 undecimal neutral third
13: 5/4 386.314 major third
14: 14/11 417.508 undecimal diminished fourth
15: 9/7 435.084 septimal major third, BP third
16: 21/16 470.781 narrow fourth
17: 4/3 498.045 perfect fourth
18: 27/20 519.551 acute fourth
19: 11/8 551.318 undecimal semi-augmented fourth
20: 7/5 582.512 septimal or Huygens' tritone, BP fourth
21: 10/7 617.488 Euler's tritone
22: 16/11 648.682 undecimal semi-diminished fifth
23: 40/27 680.449 grave fifth
24: 3/2 701.955 perfect fifth
25: 49/32 737.652
26: 14/9 764.916 septimal minor sixth
27: 11/7 782.492 undecimal augmented fifth
28: 8/5 813.686 minor sixth
29: 13/8 840.528 tridecimal neutral sixth
30: 5/3 884.359 major sixth, BP sixth
31: 27/16 905.865 Pythagorean major sixth
32: 12/7 933.129 septimal major sixth
33: 7/4 968.826 harmonic seventh
34: 16/9 996.090 Pythagorean minor seventh
35: 9/5 1017.596 just minor seventh, BP seventh
36: 11/6 1049.363 21/4-tone, undecimal neutral seventh
37: 13/7 1071.702 16/3-tone
38: 91/48 1107.399
39: 77/40 1133.830
40: 55/28 1168.806
41: 2/1 1200.000 octave

Using Barlow harmonicity instead for the optimisation it could be
further improved to Euler gradus 75 and Wilson complexity 76. All
intervals are still superparticular, again requiring a little manual
intervention:

1: 64/63 27.264 septimal comma
2: 28/27 62.961 1/3-tone
3: 21/20 84.467 minor semitone
4: 16/15 111.731 minor diatonic semitone
5: 49/45 147.428 BP minor semitone
6: 10/9 182.404 minor whole tone
7: 9/8 203.910 major whole tone
8: 8/7 231.174 septimal whole tone
9: 7/6 266.871 septimal minor third
10: 32/27 294.135 Pythagorean minor third
11: 6/5 315.641 minor third
12: 11/9 347.408 undecimal neutral third
13: 5/4 386.314 major third
14: 81/64 407.820 Pythagorean major third
15: 9/7 435.084 septimal major third, BP third
16: 21/16 470.781 narrow fourth
17: 4/3 498.045 perfect fourth
18: 27/20 519.551 acute fourth
19: 11/8 551.318 undecimal semi-augmented fourth
20: 7/5 582.512 septimal or Huygens' tritone, BP fourth
21: 10/7 617.488 Euler's tritone
22: 35/24 653.185 septimal semi-diminished fifth
23: 40/27 680.449 grave fifth
24: 3/2 701.955 perfect fifth
25: 32/21 729.219 wide fifth
26: 14/9 764.916 septimal minor sixth
27: 63/40 786.422 narrow minor sixth
28: 8/5 813.686 minor sixth
29: 13/8 840.528 tridecimal neutral sixth
30: 5/3 884.359 major sixth, BP sixth
31: 27/16 905.865 Pythagorean major sixth
32: 12/7 933.129 septimal major sixth
33: 7/4 968.826 harmonic seventh
34: 16/9 996.090 Pythagorean minor seventh
35: 9/5 1017.596 just minor seventh, BP seventh
36: 11/6 1049.363 21/4-tone, undecimal neutral seventh
37: 15/8 1088.269 classic major seventh
38: 40/21 1115.533 acute major seventh
39: 27/14 1137.039 septimal major seventh
40: 63/32 1172.736 octave - septimal comma
41: 2/1 1200.000 octave

Manuel Op de Coul

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> This is a response to Paul's post of 3-11-2000 where he created
> a 41-tone periodicity block for Justin White.

Hi Manuel.

In case you didn't notice, I was also trying to maintain at least one
complete 13-limit otonality for Justin, as well as a good number of
incomplete 13-limit consonant chords. I haven't looked at how your
proposals break down in this respect.

Paul wrote:
>In case you didn't notice, I was also trying to maintain at least one
>complete 13-limit otonality for Justin, as well as a good number of
>incomplete 13-limit consonant chords. I haven't looked at how your
>proposals break down in this respect.

Ah yes, I forgot. So with more luck than wisdom it turns out that the
first scale even has two: on 0 and 8. Yours on 8 has a 3.93 cents flat
11/8. The second scale has only one complete one, on 0 like yours.
For the number of incomplete 13-limit consonant chords the first scale
also seems a slight improvement over yours where the second scale
contains less.

Manuel