Nicetone

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Nicetone (also known as the Zarlino pattern or Ptolemaic diatonic) is a 7-note maximum-variety-3 scale with the step signature 3L 2M 2s. Nicetone is a chiral scale with left-handed (LH, step pattern LMLsMLs) and right-handed (RH, step pattern LMLsLMs) variants that are rotationally non-equivalent. 15edo is the first equal division that supports nicetone.

Nicetone has the same pattern of the 5-limit Zarlino scale, though it encompasses the whole range of 3L 2M 2s. It's also a subset of the 5L 2m 3s blackdye scale.

Nicetone is intermediate between the 5L 2s diatonic scale and the 3L 4s neutral scale.

Nicetone can be tuned as a 5-limit JI scale or a tempered version thereof, where L represents 9/8, M represents 10/9, and s represents 16/15.


Comparison of diatonic scales in JI, Pythagorean tuning, and meantone
Comparison with mosh and antipentic in 33edo
Name Structure Step Sizes Graphical Representation
Mosh 3L 4s 7\33, 3\33 ├──┼──────┼──┼──────┼──┼──┼──────┤
Nicetone 3L 2M 2s 7\33, 4\33, 2\33 ├───┼──────┼─┼──────┼───┼─┼──────┤
Antipentic 3L 2s 7\33, 6\33 ├─────┼──────┼──────┼─────┼──────┤

Intervals

The following is a table of nicetone intervals and their abstract sizes in terms of L, M, and s. Given concrete sizes of L, M, and s in EDO steps or cents, you can compute the concrete size of any interval in nicetone using the following expressions.

Interval sizes in nicetone
Interval class Sizes 5-limit JI 15edo
(L:M:s = 3:2:1) 		41edo
(L:M:s = 7:6:4)
Second
(1-step) 		Small s 16/15, 111.73¢ 1\15, 80.00¢ 4\41, 117.07¢
Medium M 10/9, 182.40¢ 2\15, 160.00¢ 6\41, 175.61¢
Large L 9/8, 203.91¢ 3\15, 240.00¢ 7\41, 204.88¢
Third
(2-step) 		Small M + s 32/27, 294.13¢ 3\15, 240.00¢ 10\41, 292.68¢
Medium L + s 6/5, 315.64¢ 4\15, 320.00¢ 11\41, 321.95¢
Large L + M 5/4, 386.31¢ 5\15, 400.00¢ 13\41, 380.49¢
Fourth
(3-step) 		Small L + M + s 4/3, 498.04¢ 6\15, 480.00¢ 17\41, 497.56¢
Medium 2L + s 27/20, 519.55¢ 7\15, 560.00¢ 18\41, 526.83¢
Large 2L + M 45/32, 590.22¢ 8\15, 640.00¢ 20\41, 585.37¢
Fifth
(4-step) 		Small L + M + 2s 64/45, 609.78¢ 7\15, 560.00¢ 21\41, 614.63¢
Medium L + 2M + s 40/27, 680.45¢ 8\15, 640.00¢ 23\41, 673.17¢
Large 2L + M + s 3/2, 701.96¢ 9\15, 720.00¢ 24\41, 702.44¢
Sixth
(5-step) 		Small 2L + M + 2s 8/5, 813.69¢ 10\15, 800.00¢ 28\41, 819.51¢
Medium 2L + 2M + s 5/3, 884.36¢ 11\15, 880.00¢ 30\41, 878.05¢
Large 3L + M + s 27/16, 905.87¢ 12\15, 960.00¢ 31\41, 907.32¢
Seventh
(6-step) 		Small 2L + 2M + 2s 16/9, 996.09¢ 12\15, 960.00¢ 34\41, 995.12¢
Medium 3L + M + 2s 9/5, 1017.60¢ 13\15, 1040.00¢ 35\41, 1024.39¢
Large 3L + 2M + s 15/8, 1088.27¢ 14\15, 1120.00¢ 37\41, 1082.93¢

Modes

Nicetone has 14 modes total, with 7 LH and 7 RH modes. The names are based on their diatonic (5L 2s) counterparts.

The modes are arranged by brightest to darkest.

Nicetone modes
Left handed Right handed
LMLsMLs
LH Nice-Lydian 		LMLsLMs
RH Nice-Lydian
MLsLMLs
LH Nice-Ionian 		LMsLMLs
RH Nice-Ionian
MLsMLsL
LH Nice-Mixolydian 		MLsLMsL
RH Nice-Mixolydian
LsLMLsM
LH Nice-Dorian 		MsLMLsL
RH Nice-Dorian
LsMLsLM
LH Nice-Aeolian 		LsLMsLM
RH Nice-Aeolian
sLMLsML
LH Nice-Phrygian 		sLMLsLM
RH Nice-Phrygian
sMLsLML
LH Nice-Locrian 		sLMsLML
RH Nice-Locrian

Tunings

3L2m2s GO scale region.png

Tuning range of nicetone
Tuning range (in octaves)
Outer generator
(G1 = 2L + M + s) 		[math]\displaystyle \frac{4}{7} < G_\text{1} < \frac{2}{3}[/math]
RH inner generator
(G2R = L + M) 		[math]\displaystyle \frac{1}{2} G_\text{1} < G_\text{2R} < 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} < G_\text{1} ≤ \frac{3}{5}[/math]
[math]\displaystyle \frac{1}{2} G_\text{1} < G_\text{2R} < 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} ≤ G_\text{1} < \frac{2}{3}[/math]
LH inner generator
(G2L = L + s) 		[math]\displaystyle 2 - 3 G_\text{1} < G_\text{2L} < \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} < G_\text{1} ≤ \frac{3}{5}[/math]
[math]\displaystyle 2 G_\text{1} - 1 < G_\text{2L} < \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} ≤ G_\text{1} < \frac{2}{3}[/math]
Large step
(L = 2G1 − 1) 		[math]\displaystyle \frac{1}{7} < L < \frac{1}{3}[/math]
Middle step
(M = 1 − G1 − G2L) 		[math]\displaystyle \frac{1}{4} (1 - 3 L) < M < L \,\text{ for }\, \frac{1}{7} < L ≤ \frac{1}{5}[/math]
[math]\displaystyle \frac{1}{4} (1 - 3 L) < M < \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} ≤ L < \frac{1}{3}[/math]
Small step
(s = 1 − G1 − G2R) 		[math]\displaystyle \frac{1}{2} (1 - 5 L) < s < \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} < L ≤ \frac{1}{5}[/math]
[math]\displaystyle 0 < s < \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} ≤ L < \frac{1}{3}[/math]
Common Nicetone tunings
Tuning L:M:s Size of step (¢) Inner generator Outer generator
(2L + M + s)		 Comments
L M s LH (L + s) RH (L + M)
5-limit JI 203.910 182.404 111.731 315.641 386.314 701.955 L = 9/8, M = 10/9, s = 16/15
15edo 3:2:1 240.000 160.000 80.000 320.000 400.000 720.000 5-limit patent val
18edo 4:2:1 266.667 133.333 66.667 333.333 400.000 733.333 5-limit patent val
20edo 4:3:1 240.000 180.000 60.000 300.000 420.000 720.000
21edo 5:2:1 285.714 114.286 57.143 342.857 400.000 742.857
22edo 4:3:2 218.182 163.636 109.091 327.273 381.818 709.091 5-limit patent val
23edo 5:3:1 260.870 156.522 52.174 313.043 417.391 730.435
24edo 6:2:1 300.000 100.000 50.000 350.000 400.000 750.000
25edo 5:3:2 240.000 144.000 96.000 336.000 384.000 720.000 5-limit patent val
5:4:1 240.000 192.000 48.000 288.000 432.000 720.000
26edo 6:3:1 276.923 138.462 46.154 323.077 415.385 738.462
27edo 5:4:2 222.222 177.778 88.889 311.111 400.000 711.111 5-limit patent val
7:2:1 311.111 88.889 44.444 355.556 400.000 755.556
28edo 6:3:2 257.143 128.571 85.714 342.857 385.714 728.571
6:4:1 257.143 171.429 42.857 300.000 428.571 728.571
29edo 5:4:3 206.897 165.517 124.138 331.034 372.414 703.448 5-limit patent val
7:3:1 289.655 124.138 41.379 331.034 413.793 744.828
30edo 6:5:1 240.000 200.000 40.000 280.000 440.000 720.000
8:2:1 320.000 80.000 40.000 360.000 400.000 760.000
31edo 7:3:2 270.968 116.129 77.419 348.387 387.097 735.484
7:4:1 270.968 154.839 38.710 309.677 425.806 735.484
32edo 6:4:3 225.000 150.000 112.500 337.500 375.000 712.500 5-limit patent val
6:5:2 225.000 187.500 75.000 300.000 412.500 712.500
8:3:1 300.000 112.500 37.500 337.500 412.500 750.000
33edo 7:4:2 254.545 145.455 72.727 327.273 400.000 727.273
7:5:1 254.545 181.818 36.364 290.909 436.364 727.273
9:2:1 327.273 72.727 36.364 363.636 400.000 763.636
34edo 6:5:3 211.765 176.471 105.882 317.647 388.235 705.882 5-limit patent val
8:3:2 282.353 105.882 70.588 352.941 388.235 741.176
8:4:1 282.353 141.176 35.294 317.647 423.529 741.176
35edo 7:4:3 240.000 137.143 102.857 342.857 377.143 720.000
7:5:2 240.000 171.429 68.571 308.571 411.429 720.000
7:6:1 240.000 205.714 34.286 274.286 445.714 720.000
9:3:1 308.571 102.857 34.286 342.857 411.429 754.286
36edo 6:5:4 200.000 166.667 133.333 333.333 366.667 700.000
8:5:1 266.667 166.667 33.333 300.000 433.333 733.333
10:2:1 333.333 66.667 33.333 366.667 400.000 766.667
37edo 7:5:3 227.027 162.162 97.297 324.324 389.189 713.514 5-limit patent val
7:6:2 227.027 194.595 64.865 291.892 421.622 713.514
9:3:2 291.892 97.297 64.865 356.757 389.189 745.946
9:4:1 291.892 129.730 32.432 324.324 421.622 745.946
38edo 8:4:3 252.632 126.316 94.737 347.368 378.947 726.316
8:5:2 252.632 157.895 63.158 315.789 410.526 726.316
8:6:1 252.632 189.474 31.579 284.211 442.105 726.316
10:3:1 315.789 94.737 31.579 347.368 410.526 757.895
39edo 7:5:4 215.385 153.846 123.077 338.462 369.231 707.692
7:6:3 215.385 184.615 92.308 307.692 400.000 707.692 5-limit patent val
9:4:2 276.923 123.077 61.538 338.462 400.000 738.462
9:5:1 276.923 153.846 30.769 307.692 430.769 738.462
11:2:1 338.462 61.538 30.769 369.231 400.000 769.231
40edo 8:5:3 240.000 150.000 90.000 330.000 390.000 720.000
8:7:1 240.000 210.000 30.000 270.000 450.000 720.000
10:3:2 300.000 90.000 60.000 360.000 390.000 750.000
10:4:1 300.000 120.000 30.000 330.000 420.000 750.000
41edo 7:6:4 204.878 175.610 117.073 321.951 380.488 702.439 5-limit patent val
9:4:3 263.415 117.073 87.805 351.220 380.488 731.707
9:5:2 263.415 146.341 58.537 321.951 409.756 731.707
9:6:1 263.415 175.610 29.268 292.683 439.024 731.707
11:3:1 321.951 87.805 29.268 351.220 409.756 760.976
42edo 8:5:4 228.571 142.857 114.286 342.857 371.429 714.286
8:6:3 228.571 171.429 85.714 314.286 400.000 714.286 5-limit patent val
8:7:2 228.571 200.000 57.143 285.714 428.571 714.286
10:5:1 285.714 142.857 28.571 314.286 428.571 742.857
12:2:1 342.857 57.143 28.571 371.429 400.000 771.429
43edo 7:6:5 195.349 167.442 139.535 334.884 362.791 697.674
9:5:3 251.163 139.535 83.721 334.884 390.698 725.581
9:6:2 251.163 167.442 55.814 306.977 418.605 725.581
9:7:1 251.163 195.349 27.907 279.070 446.512 725.581
11:3:2 306.977 83.721 55.814 362.791 390.698 753.488
11:4:1 306.977 111.628 27.907 334.884 418.605 753.488
44edo 8:7:3 218.182 190.909 81.818 300.000 409.091 709.091
10:4:3 272.727 109.091 81.818 354.545 381.818 736.364
10:5:2 272.727 136.364 54.545 327.273 409.091 736.364
10:6:1 272.727 163.636 27.273 300.000 436.364 736.364
12:3:1 327.273 81.818 27.273 354.545 409.091 763.636
45edo 9:5:4 240.000 133.333 106.667 346.667 373.333 720.000
9:7:2 240.000 186.667 53.333 293.333 426.667 720.000
9:8:1 240.000 213.333 26.667 266.667 453.333 720.000
11:4:2 293.333 106.667 53.333 346.667 400.000 746.667
11:5:1 293.333 133.333 26.667 320.000 426.667 746.667
13:2:1 346.667 53.333 26.667 373.333 400.000 773.333
46edo 8:6:5 208.696 156.522 130.435 339.130 365.217 704.348
8:7:4 208.696 182.609 104.348 313.043 391.304 704.348 5-limit patent val
10:5:3 260.870 130.435 78.261 339.130 391.304 730.435
10:7:1 260.870 182.609 26.087 286.957 443.478 730.435
12:3:2 313.043 78.261 52.174 365.217 391.304 756.522
12:4:1 313.043 104.348 26.087 339.130 417.391 756.522
47edo 9:6:4 229.787 153.191 102.128 331.915 382.979 714.894
9:7:3 229.787 178.723 76.596 306.383 408.511 714.894
9:8:2 229.787 204.255 51.064 280.851 434.043 714.894
11:4:3 280.851 102.128 76.596 357.447 382.979 740.426
11:5:2 280.851 127.660 51.064 331.915 408.511 740.426
11:6:1 280.851 153.191 25.532 306.383 434.043 740.426
13:3:1 331.915 76.596 25.532 357.447 408.511 765.957
48edo 8:7:5 200.000 175.000 125.000 325.000 375.000 700.000 5-limit patent val
10:5:4 250.000 125.000 100.000 350.000 375.000 725.000
10:6:3 250.000 150.000 75.000 325.000 400.000 725.000
10:7:2 250.000 175.000 50.000 300.000 425.000 725.000
10:8:1 250.000 200.000 25.000 275.000 450.000 725.000
12:5:1 300.000 125.000 25.000 325.000 425.000 750.000
14:2:1 350.000 50.000 25.000 375.000 400.000 775.000
49edo 9:6:5 220.408 146.939 122.449 342.857 367.347 710.204
9:7:4 220.408 171.429 97.959 318.367 391.837 710.204 5-limit patent val
9:8:3 220.408 195.918 73.469 293.878 416.327 710.204
11:5:3 269.388 122.449 73.469 342.857 391.837 734.694
11:6:2 269.388 146.939 48.980 318.367 416.327 734.694
11:7:1 269.388 171.429 24.490 293.878 440.816 734.694
13:3:2 318.367 73.469 48.980 367.347 391.837 759.184
13:4:1 318.367 97.959 24.490 342.857 416.327 759.184
50edo 8:7:6 192.000 168.000 144.000 336.000 360.000 696.000
10:7:3 240.000 168.000 72.000 312.000 408.000 720.000
10:9:1 240.000 216.000 24.000 264.000 456.000 720.000
12:4:3 288.000 96.000 72.000 360.000 384.000 744.000
12:5:2 288.000 120.000 48.000 336.000 408.000 744.000
12:6:1 288.000 144.000 24.000 312.000 432.000 744.000
14:3:1 336.000 72.000 24.000 360.000 408.000 768.000
51edo 9:7:5 211.765 164.706 117.647 329.412 376.471 705.882 5-limit patent val
9:8:4 211.765 188.235 94.118 305.882 400.000 705.882
11:5:4 258.824 117.647 94.118 352.941 376.471 729.412
11:6:3 258.824 141.176 70.588 329.412 400.000 729.412
11:7:2 258.824 164.706 47.059 305.882 423.529 729.412
11:8:1 258.824 188.235 23.529 282.353 447.059 729.412
13:4:2 305.882 94.118 47.059 352.941 400.000 752.941
13:5:1 305.882 117.647 23.529 329.412 423.529 752.941
15:2:1 352.941 47.059 23.529 376.471 400.000 776.471
52edo 10:6:5 230.769 138.462 115.385 346.154 369.231 715.385
10:7:4 230.769 161.538 92.308 323.077 392.308 715.385
10:8:3 230.769 184.615 69.231 300.000 415.385 715.385
10:9:2 230.769 207.692 46.154 276.923 438.462 715.385
12:5:3 276.923 115.385 69.231 346.154 392.308 738.462
12:7:1 276.923 161.538 23.077 300.000 438.462 738.462
14:3:2 323.077 69.231 46.154 369.231 392.308 761.538
14:4:1 323.077 92.308 23.077 346.154 415.385 761.538
53edo 9:7:6 203.774 158.491 135.849 339.623 362.264 701.887
9:8:5 203.774 181.132 113.208 316.981 384.906 701.887 5-limit patent val
11:6:4 249.057 135.849 90.566 339.623 384.906 724.528
11:7:3 249.057 158.491 67.925 316.981 407.547 724.528
11:8:2 249.057 181.132 45.283 294.340 430.189 724.528
11:9:1 249.057 203.774 22.642 271.698 452.830 724.528
13:4:3 294.340 90.566 67.925 362.264 384.906 747.170
13:5:2 294.340 113.208 45.283 339.623 407.547 747.170
13:6:1 294.340 135.849 22.642 316.981 430.189 747.170
15:3:1 339.623 67.925 22.642 362.264 407.547 769.811
54edo 10:7:5 222.222 155.556 111.111 333.333 377.778 711.111 5-limit patent val
10:9:3 222.222 200.000 66.667 288.889 422.222 711.111
12:5:4 266.667 111.111 88.889 355.556 377.778 733.333
12:7:2 266.667 155.556 44.444 311.111 422.222 733.333
12:8:1 266.667 177.778 22.222 288.889 444.444 733.333
14:5:1 311.111 111.111 22.222 333.333 422.222 755.556
16:2:1 355.556 44.444 22.222 377.778 400.000 777.778

Supersets

Regions of MV3 supersets of 3L2m2s.png

Nicetone has following generator-offset MV3 supersets:

  • Sephipechroid: 13-note 3L 5M 5s scale (LMsMLsMsLMsMs and LMsMLsMsMLsMs)
  • Interoneichro: 13-note 5L 3M 5s scale (LMsLsLMsLsMLs and LMsLsMLsLsMLs)
  • Sephimechroid: 13-note 5L 5M 3s scale (LMLMsLMLsMLMs and LMLsMLMLsMLMs)
  • Beatloid: 17-note 5L 5M 7s scale (LMsLsMLsMsLMsLsMs and LMsLsMLsMsLsMLsMs)
  • Enharoid: 17-note 5L 7M 5s scale (LMsLMsMLMsLMsMLsM and LMsLMsMLsMLMsMLsM)
  • Moharoid: 17-note 7L 5M 5s scale (LMLsLMsLMLsMLsLMs and LMLsMLsLMLsMLsLMs)

Remarkable non-MV3 generator-offset supersets include blackdye (10-note, LmLsLmLsLs).

See also

  • Blackdye – A 10-note scale that is an extension to nicetone.
  • Zarlino – A 5-limit JI scale with the same pattern.
  • Omnidiatonic­­ ­– Sister 2L 3M 2s scale
  • Antinicetone – Sister 2L 2M 3s scale
  • 5L 2s – LM-equalized version of nicetone
  • 3L 4s – MS-equalized version of nicetone
  • 3L 2s – Collapsed version of nicetone
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