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.