A fun experiment

Someone on the list once made a remark to the effect of, if Schoenberg
really wanted to avoid consonance, he would have been better off in
something like 13TET. Of course 13TET still has an octave in it, which is
highly consonant. I wondered, somewhat perversely, what would happen if I
tried to construct a scale in which dissonance, that is, the average
dissonance of each interval, is maximized. Sort of an anti-JI.

I'm afraid I haven't followed all of the noise and fury, Plomp and
circumstance of the discussion of dissonance (or "complexity") calculations
so ably discussed by other contributors. Nevertheless, I calculated my own
table of dissonance based on ratio complexity to the 13-limit, and a curve
following the critical band dissonance curve in between those ratios. Now
even I don't think that it's the absolute best approximation of dissonance
there is, but it will do for now. Then I tried step sizes from 10 to 300
cents and intervals based on those steps from one step to 2 octaves. (Since
most of them don't divide evenly into an octave, I could go on
indefinitely. I chose 2 octaves just because most adjacent harmonic
intervals in music are within that range.) I calculated the average
dissonance of the intervals in that scale and sorted the results.

Of course I would have gotten different results with different dissonance
curves. I encourage others to try.

And the most dissonant award goes to: 257 cents.
In the < 100 cents category, it goes to 93 cents.
By the way, the least dissonant is nearly a tie between 200 and 240 cents
(6TET and 5TET).
In the < 100 cents category, most consonant is 63 cents (about Carlos beta).

I hope the whole list isn't too long to include here (least to most
dissonant, cents per step followed by average dissonance rating):

200,19.834225
240,19.973856
201,20.593583
298,20.710084
241,20.944444
199,21.13369
239,21.428104
198,21.625669
300,21.689075
299,21.689076
171,21.825792
297,21.844538
202,21.917112
172,21.959276
238,22
237,22.277778
242,22.411765
173,22.794118
149,22.876471
243,22.986928
203,23.072192
197,23.213904
133,23.392733
150,23.447059
169,23.470588
151,23.639216
170,23.647059
296,23.676471
148,24.066667
134,24.280277
244,24.326797
63,24.398251
174,24.429864
152,24.592157
119,24.681114
126,24.745098
236,24.794118
100,24.836317
204,24.842246
267,24.904412
266,25.003676
132,25.150519
295,25.491597
168,25.626697
271,25.680147
175,25.71267
50,25.886108
265,25.897059
109,25.931373
99,25.94757
131,25.958478
205,25.973262
127,25.995098
65,26.008987
55,26.046512
15,26.076582
25,26.157895
268,26.194853
11,26.208051
77,26.214706
13,26.215473
10,26.221265
38,26.23482
37,26.247243
89,26.274887
118,26.281734
35,26.307093
27,26.319853
19,26.320471
26,26.333549
22,26.336057
18,26.337344
12,26.35974
23,26.376356
61,26.380805
14,26.382526
17,26.390966
33,26.399101
54,26.400137
20,26.408799
53,26.450535
147,26.458824
135,26.460208
235,26.473856
39,26.476374
36,26.484849
24,26.510398
272,26.522059
121,26.547988
103,26.550802
16,26.554283
69,26.568339
270,26.573529
29,26.591463
294,26.592437
31,26.602554
51,26.60422
21,26.652004
41,26.657201
196,26.665775
71,26.675579
88,26.705882
245,26.722222
46,26.732411
32,26.733704
70,26.742424
30,26.785555
34,26.785714
44,26.786492
42,26.790966
57,26.817073
120,26.826625
73,26.83364
47,26.847059
146,26.847059
125,26.867647
101,26.87468
114,26.875
194,26.879679
79,26.883367
48,26.883554
130,26.891004
234,26.915033
68,26.918685
74,26.931689
45,26.938914
76,26.961101
195,26.962567
28,27.011073
113,27.058824
56,27.102941
43,27.113369
78,27.121569
59,27.127941
293,27.138655
80,27.156187
52,27.161438
155,27.226891
49,27.249387
40,27.259222
95,27.264706
58,27.311029
264,27.319853
72,27.321691
67,27.330252
81,27.341785
218,27.347059
122,27.411765
108,27.415966
91,27.434118
96,27.443628
64,27.461049
66,27.469748
62,27.49226
167,27.531674
82,27.651261
83,27.701681
153,27.705882
87,27.75817
159,27.920168
75,27.925996
141,27.928309
219,27.944118
97,27.946078
162,27.997899
163,28.016807
269,28.029412
106,28.062834
105,28.073529
90,28.085973
189,28.107843
160,28.126051
142,28.141544
138,28.147059
253,28.172794
140,28.204044
117,28.207353
85,28.269063
104,28.283423
84,28.295168
233,28.303922
139,28.3125
112,28.326471
98,28.383632
116,28.385294
86,28.420479
110,28.429972
217,28.458824
107,28.467787
102,28.470588
60,28.514329
94,28.521177
176,28.554299
178,28.607843
136,28.622837
190,28.703431
154,28.747059
246,28.754902
115,28.773529
158,28.817227
252,28.833333
273,28.955882
137,28.958477
166,29.036199
193,29.122995
164,29.12395
92,29.124706
251,29.169935
292,29.180672
161,29.254202
123,29.270898
206,29.296791
143,29.409926
128,29.509804
183,29.556373
93,29.565882
232,29.640523
124,29.648693
254,29.661765
250,29.72549
111,29.77451
192,29.802139
231,29.836601
156,29.972689
220,30.035294
263,30.0625
182,30.151961
188,30.257353
184,30.259804
177,30.316742
185,30.360294
216,30.461765
179,30.480392
157,30.558823
221,30.661765
228,30.708824
129,30.723856
144,30.727941
209,30.738235
145,30.748162
213,30.805882
222,30.820588
230,31
282,31.095588
227,31.185294
191,31.186275
229,31.222222
186,31.22549
212,31.229412
262,31.272059
247,31.313725
274,31.4375
187,31.446078
283,31.457983
281,31.474265
249,31.477124
181,31.482843
286,31.516807
165,31.60084
223,31.658824
210,31.691177
285,31.743697
291,31.806723
255,31.944853
261,31.948529
211,31.970588
284,31.987395
207,32.160428
214,32.182353
287,32.197479
180,32.497549
224,32.514706
248,32.545752
260,32.591912
215,32.670588
208,32.860963
226,33.114706
225,33.176471
288,33.294118
259,33.569853
275,33.94853
280,34.018382
290,34.07563
289,34.668067
256,34.709559
258,35.654412
277,36.069853
279,36.172794
276,36.301471
278,36.551471
257,36.650735

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^