Get the PD patch here. It’s simplified and permits manual control. It runs in PD Vanilla. The full album is here.

Here are 8 sawtooth oscillators gliding up and down in pitch in equal intervals. They start at unison and end in octaves, a tritone from the starting pitch. In between are a surprising number of audible consonances, even though the overall sound is generally very dissonant. To avoid extremes in pitch, each oscillator is actually a Shepard-Risset glissando that returns to the same pitch every two octaves.

The most frequently used model of consonance is harmonic concordance. For a simultaneity of multiple tones, consonance is heard when partials from one tone coincide with partials from another. For tones consisting of a harmonic series, this happens at integer ratios. For inharmonic tones, consonances occur at different intervals where the inharmonic partials coincide.

In the video you can see only a very small number of harmonics, but the effect is illustrated rather clearly. Consonances are audible when lines intersect and overlap. This isn’t the only way to represent it on a spectrogram. There’s a lot to explore by viewing it at different scales. It probably looks more interesting when zoomed out, but I think the video corresponds most closely to what’s actually audible.

Here the consonances occur both at small integer ratios, and because the same inteveral is stacked 8 times, also at roots of those ratios, up to the seventh root. The greatest consonances occur when fundamental frequencies (and consequently all harmonics) coincide at equal divisions of the octave, up to 7 EDO. At 8 EDO intervals (odd multiples of 150 cents), all tones are equally spaced and none overlap. In general, partials audibly coincide about every 2 cents.

Here’s a list of audibly consonant intervals. The columns represent the time where the interval occurs, the interval size in cents and the equivalent ratio. This isn’t a complete list; it includes only the relatively simple intervals. I made it by stepping through all possible intervals 0.1 cents at a time and noting the consonances. Then I found the ratios using an octave script. The output is edited somewhat.

00:00 0000.000 unison
00:25 0025.106 sixth root of 12/11
00:32 0032.075 fourth root of 14/13
00:32 0032.093 sixth root of 19/17
00:32 0032.985 cube root of 18/17
00:33 0033.001 fifth root of 11/10
00:33 0033.025 seventh root of 8/7
00:33 0033.985 sixth root of 9/8
00:36 0036.481 fifth root of 10/9
00:38 0038.512 fifth root of 19/17
00:38 0038.529 sixth root of 8/7
00:41 0041.251 fourth root of 11/10
00:42 0042.766 cube root of 14/13
00:44 0044.400 square root of 20/19
00:44 0044.478 sixth root of 7/6
00:45 0045.092 seventh root of 6/5
00:46 0046.191 cube root of 13/12
00:46 0046.235 fifth root of 8/7
00:50 0050.212 cube root of 12/11
00:52 0052.607 sixth root of 6/5
00:53 0053.374 fifth root of 7/6
00:55 0055.001 cube root of 11/10
00:55 0055.866 square root of 16/15
00:57 0057.756 cube root of 21/19
00:57 0057.777 seventh root of 24/19
00:57 0057.794 fourth root of 8/7
00:57 0057.842 fifth root of 13/11
01:00 0060.801 cube root of 10/9
01:03 0063.128 fifth root of 6/5
01:04 0064.386 sixth root of 5/4
01:06 0066.718 fourth root of 7/6
01:07 0067.970 cube root of 9/8
01:09 0069.286 square root of 13/12
01:11 0071.090 eighth root of 25/18
01:12 0072.514 sixth root of 9/7
01:15 0075.319 square root of 12/11
01:17 0077.058 cube root of 8/7
01:17 0077.186 eighth root of 10/7
01:18 0078.910 fourth root of 6/5
01:22 0082.502 square root of 11/10
01:23 0083.007 sixth root of 4/3
01:24 0084.467 21/20
01:24 0084.602 cube root of 22/19
01:26 0086.852 fourth root of 11/9
01:28 0088.801 20/19
01:28 0088.957 cube root of 7/6
01:29 0089.868 fourth root of 16/13
01:31 0091.202 square root of 10/9
01:33 0093.603 19/18
01:36 0096.578 fourth root of 5/4
01:39 0099.609 fifth root of 4/3
01:41 0101.955 square root of 9/8
01:45 0105.214 cube root of 6/5
01:48 0108.771 fourth root of 9/7
01:50 0110.254 cube root of 23/19
01:51 0111.591 fourth root of 22/17
01:51 0111.731 16/15
01:52 0112.043 cube root of 17/14
01:55 0115.587 square root of 8/7
01:56 0116.107 fourth root of 17/13
01:56 0116.993 sixth root of 3/2
02:04 0124.511 fourth root of 4/3
02:08 0128.697 eighth root of 29/16
02:13 0133.435 square root of 7/6
02:15 0135.614 sixth root of 8/5
02:18 0138.376 sixth root of 21/13
02:18 0138.404 seventh root of 7/4
02:18 0138.573 13/12
02:20 0140.391 fifth root of 3/2
02:27 0147.393 sixth root of 5/3
02:30 0150.000 one step in 8 EDO (all tones are equally spaced)
02:30 0150.637 12/11
02:30 0150.727 fifth root of 17/11
02:34 0154.372 fourth root of 10/7
02:37 0157.821 square root of 6/5
02:46 0166.015 cube root of 4/3
02:51 0171.429 one step in 7 EDO
02:55 0175.489 fourth root of 3/2
02:56 0176.872 fifth root of 5/3
02:56 0176.905 sixth root of 24/13
02:59 0179.736 square root of 16/13
03:02 0182.404 10/9
03:03 0183.728 fifth root of 17/10
03:06 0186.626 fifth root of 12/7
03:13 0193.157 square root of 5/4
03:15 0195.623 fourth root of 11/7
03:19 0199.218 fifth root of 16/9
03:20 0200.000 one step in 6 EDO
03:23 0203.422 fourth root of 8/5
03:23 0203.910 9/8
03:25 0205.829 cube root of 10/7
03:26 0206.999 fifth root of 20/11
03:28 0208.754 square root of 14/11
03:29 0209.873 fifth root of 11/6
03:32 0212.182 square root of 23/18
03:32 0212.206 cube root of 13/9
03:36 0216.687 17/15
03:37 0217.493 sixth root of 17/8
03:37 0217.542 square root of 9/7
03:41 0221.090 fourth root of 5/3
03:43 0223.095 sixth root of 13/6
03:43 0223.181 square root of 22/17
03:47 0227.107 square root of 13/10
03:51 0231.174 8/7
03:53 0233.985 cube root of 3/2
04:00 0240.000 one step in 5 EDO
04:02 0242.206 fourth root of 7/4
04:09 0249.022 square root of 4/3
04:10 0250.188 seventh root of 11/4
04:12 0252.607 sixth root of 12/5
04:13 0253.805 22/19
04:14 0254.972 cube root of 14/9
04:18 0258.688 seventh root of 37/13
04:18 0258.721 fifth root of 19/9
04:24 0264.344 square root of 19/14
04:24 0264.386 sixth root of 5/2
04:26 0266.871 7/6
04:28 0268.475 square root of 15/11
04:31 0271.229 cube root of 8/5
04:31 0271.708 seventh root of 3/1
04:33 0273.001 fifth root of 11/5
04:35 0275.659 square root of 11/8
04:35 0275.702 sixth root of 13/5
04:40 0280.176 cube root of 13/8
04:40 0280.782 fifth root of 9/4
04:43 0283.007 sixth root of 8/3
04:43 0283.025 fourth root of 25/13
04:44 0284.197 cube root of 18/11
04:49 0289.210 13/11
04:51 0291.256 square root of 7/5
04:54 0294.786 cube root of 5/3
05:00 0300.000 one step in 4 EDO
05:00 0300.004 seventh root of 37/11
05:01 0301.500 square root of 17/12
05:08 0308.744 square root of 10/7
05:11 0311.043 cube root of 12/7
05:15 0315.641 6/5
05:18 0318.309 square root of 13/9
05:22 0322.942 cube root of 7/4
05:24 0324.341 square root of 16/11
05:27 0327.017 fifth root of 18/7
05:32 0332.030 cube root of 16/9
05:32 0332.075 fourth root of 28/13
05:36 0336.130 17/14
05:39 0339.609 fifth root of 8/3
05:42 0342.857 two steps in 7 EDOs
05:44 0344.999 cube root of 20/11
05:45 0345.601 fourth root of 20/9
05:47 0347.390 fifth root of 30/11
05:47 0347.393 sixth root of 10/3
05:47 0347.408 11/9
05:50 0350.978 square root of 3/2
05:56 0356.502 fifth root of 14/5
05:57 0357.794 fourth root of 16/7
05:59 0359.472 16/13
06:03 0363.498 fifth root of 20/7
06:06 0366.718 fourth root of 7/3
06:07 0367.300 sixth root of 25/7
06:10 0370.400 cube root of 19/10
06:12 0372.893 square root of 20/13
06:16 0376.819 square root of 17/11
06:18 0378.910 fourth root of 12/5
06:20 0380.391 fifth root of 3/1
06:20 0380.436 cube root of 29/15
06:22 0382.458 square root of 14/9
06:26 0386.314 5/4
06:31 0391.246 square root of 11/7
06:36 0396.578 fourth root of 5/2
06:40 0400.000 one step in 3 EDO
06:44 0404.377 fourth root of 28/11
06:44 0404.442 24/19
06:46 0406.843 square root of 8/5
06:54 0414.006 seventh root of 16/3
06:57 0417.493 sixth root of 17/4
06:57 0417.508 14/11
06:57 0417.695 fourth root of 21/8
07:00 0420.264 square root of 13/8
07:04 0424.364 23/18
07:04 0424.511 fourth root of 8/3
07:13 0433.803 seventh root of 52/9
07:14 0434.985 cube root of 17/8
07:15 0435.084 9/7
07:22 0442.179 square root of 5/3
07:26 0446.191 cube root of 13/6
07:26 0446.363 22/17
07:26 0446.999 fifth root of 40/11
07:30 0450.000 three steps in 8 EDO
07:30 0450.212 cube root of 24/11
07:32 0452.607 sixth root of 24/5
07:34 0454.214 13/10
07:37 0457.901 sixth root of 44/9
07:39 0459.098 seventh root of 32/5
07:40 0460.801 cube root of 20/9
07:40 0460.816 fourth root of 29/10
07:44 0464.428 17/13
07:46 0466.565 square root of 12/7
07:47 0467.970 cube root of 9/4
07:50 0470.781 21/16
07:53 0473.098 square root of 19/11
07:55 0475.489 fourth root of 3/1
08:00 0480.000 two steps in 5 EDO
08:03 0483.007 sixth root of 16/3
08:04 0484.413 square root of 7/4
08:18 0498.045 4/3
08:25 0505.214 cube root of 12/5
08:33 0513.001 fifth root of 22/5
08:34 0514.286 three steps in 7 EDO
08:35 0515.803 cube root of 22/9
08:39 0519.824 cube root of 32/13
08:41 0521.090 fourth root of 10/3
08:44 0524.681 square root of 11/6
08:48 0528.687 19/14
08:48 0528.771 cube root of 5/2
08:50 0530.714 square root of 24/13
08:53 0533.282 fourth root of 24/7
09:00 0540.279 square root of 28/15
09:02 0542.206 fourth root of 7/2
09:03 0543.015 26/19
09:04 0544.134 square root of 15/8
09:05 0545.028 cube root of 18/7
09:11 0551.318 11/8
09:11 0551.405 cube root of 13/5
09:14 0554.399 fourth root of 18/5
09:19 0559.420 cube root of 29/11
09:23 0563.382 18/13
09:26 0566.015 cube root of 8/3
09:26 0566.050 square root of 25/13
09:29 0569.599 sixth root of 36/5
09:39 0579.609 fifth root of 16/3
09:42 0582.512 7/5
09:47 0587.794 cube root of 36/13
09:47 0587.807 fourth root of 35/9
09:54 0594.171 cube root of 14/5
10:00 0600.000 one step in 2 EDO
10:05 0605.829 cube root of 20/7
10:12 0612.206 cube root of 26/9
10:17 0617.488 10/7
10:30 0630.376 square root of 29/14
10:33 0633.985 cube root of 3/1
10:36 0636.498 fifth root of 44/7
10:36 0636.618 13/9
10:38 0638.478 square root of 23/11
10:45 0645.601 fourth root of 40/9
10:48 0648.682 16/11
10:55 0655.866 square root of 32/15
10:56 0656.985 19/13
10:57 0657.794 fourth root of 32/7
10:59 0659.721 square root of 15/7
11:09 0669.286 square root of 13/6
11:15 0675.222 cube root of 29/9
11:15 0675.319 square root of 24/11
11:18 0678.910 fourth root of 24/5
11:20 0680.176 cube root of 13/4
11:24 0684.197 cube root of 36/11
11:25 0685.714 four steps in 7 EDO
11:34 0694.786 cube root of 10/3
11:41 0701.955 3/2
11:55 0715.587 square root of 16/7
12:00 0720.000 three steps in 5 EDO
12:04 0724.511 fourth root of 16/3
12:13 0733.435 square root of 7/3
12:15 0735.572 26/17
12:19 0739.199 cube root of 18/5
12:20 0740.006 23/15
12:29 0749.788 cube root of 11/3
12:30 0750.000 five steps in 8 EDO
12:33 0753.001 fifth root of 44/5
12:33 0753.637 17/11
12:37 0757.821 square root of 12/5
12:44 0764.916 14/9
12:59 0779.736 square root of 32/13
13:02 0782.492 11/7
13:13 0793.157 square root of 5/2
13:15 0795.558 19/12
13:20 0800.000 two steps in 3 EDO
13:23 0803.400 seventh root of 103/4
13:33 0813.686 8/5
13:47 0827.107 square root of 13/5
13:50 0830.253 21/13
13:56 0836.502 fifth root of 56/5
14:00 0840.528 13/8
14:02 0842.206 fourth root of 7/1
14:09 0849.022 square root of 8/3
14:12 0852.592 18/11
14:14 0854.399 fourth root of 36/5
14:15 0855.001 cube root of 22/5
14:17 0857.143 five steps in 7 EDO
14:23 0863.870 28/17
14:35 0875.659 square root of 11/4
14:41 0881.691 square root of 36/13
14:44 0884.359 5/3
14:51 0891.256 square root of 14/5
14:56 0896.859 square root of 31/11
15:00 0900.000 three steps in 4 EDO
15:05 0905.214 cube root of 24/5
15:10 0910.790 22/13
15:15 0915.803 cube root of 44/9
15:24 0924.341 square root of 32/11
15:33 0933.129 12/7
15:46 0946.195 19/11
15:50 0950.978 square root of 3/1
15:57 0957.794 fourth root of 64/7
16:00 0960.000 four steps in 5 EDO
16:06 0966.015 cube root of 16/3
16:08 0968.826 7/4
16:27 0987.747 23/13
16:33 0993.001 fifth root of 88/5
16:36 0996.090 16/9
16:40 1000.000 five steps in 6 EDO
16:57 1017.596 9/5
17:00 1020.264 square root of 13/4
17:08 1028.571 six steps in 7 EDO
17:22 1042.179 square root of 10/3
17:29 1049.363 11/6
17:30 1050.000 seven steps in 8 EDO (all tones are equally spaced)
17:30 1050.013 square root of 37/11
17:41 1061.427 24/13
17:46 1066.565 square root of 24/7
18:04 1084.413 square root of 7/2
18:08 1088.269 15/8
18:28 1108.798 square root of 18/5
18:37 1117.498 square root of 40/11
18:44 1124.681 square root of 11/3
18:52 1132.100 25/13
18:59 1139.199 cube root of 36/5
19:04 1144.134 square root of 15/4
19:05 1145.036 31/16
20:00 1199.700 octave (slightly detuned)

One thing worth noting here is that many intervals are almost exactly equal divisions of other intervals. For example, the 11/9 neutral third (347.408 cents) is close to two equal divisions of 3/2 (350.978 cents), three equal divisions of 20/11 (344.999 cents), four equal divisions of 20/9 (345.601 cents), five equal divisions of 30/11 (347.390 cents) and six equal divisions of 10/3 (347.393 cents). This implies that interesting chords can be constructed by stacking these intervals.

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