Nicetone

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Nicetone (also known as the Zarlino pattern, simply Zarlino, 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. This pattern is a variety of diatonic, distinct from the mos pattern 5L 2s officially called diatonic and found as such in fifth-generated tunings.

Nicetone has the same pattern of the 5-limit Zarlino scale, where L represents 9/8, M represents 10/9, and s represents 16/15, though it encompasses the whole range of 3L 2M 2s. It's also a subset of the 5L 2m 3s blackdye scale. Note that "Zarlino" by itself can refer to both the JI scale and the scale pattern.

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

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 ├─────┼──────┼──────┼─────┼──────┤

5-limit Zarlino scale

Ptolemy's intense diatonic, 5-limit Zarlino or sometimes Zarlino for short is a heptatonic 5-limit JI scale with the nicetone step pattern. It consists of the intervals 1/1–9/8–5/4–4/3–3/2–5/3–15/8–2/1. It corresponds to the case where L represents 9/8, M represents 10/9, and s represents 16/15. See the dedicated page for Scala files and Fokker blocks information.

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

Tuning range of nicetone
Tuning range (in octaves)
Outer generator
(G1 = 2L + M + s) 		[math]\displaystyle{ \displaystyle \frac{4}{7} \lt G_\text{1} \lt \frac{2}{3} }[/math]
RH inner generator
(G2R = L + M) 		[math]\displaystyle{ \displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 4 G_\text{1} - 2 \,\text{ for }\, \frac{4}{7} \lt G_\text{1} \le \frac{3}{5} }[/math]
[math]\displaystyle{ \displaystyle \frac{1}{2} G_\text{1} \lt G_\text{2R} \lt 1 - G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3} }[/math]
LH inner generator
(G2L = L + s) 		[math]\displaystyle{ \displaystyle 2 - 3 G_\text{1} \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{4}{7} \lt G_\text{1} \le \frac{3}{5} }[/math]
[math]\displaystyle{ \displaystyle 2 G_\text{1} - 1 \lt G_\text{2L} \lt \frac{1}{2} G_\text{1} \,\text{ for }\,\frac{3}{5} \le G_\text{1} \lt \frac{2}{3} }[/math]
Large step
(L = 2G1 − 1) 		[math]\displaystyle{ \displaystyle \frac{1}{7} \lt L \lt \frac{1}{3} }[/math]
Middle step
(M = 1 − G1 − G2L) 		[math]\displaystyle{ \displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt L \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5} }[/math]
[math]\displaystyle{ \displaystyle \frac{1}{4} (1 - 3 L) \lt M \lt \frac{1}{2} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \frac{1}{3} }[/math]
Small step
(s = 1 − G1 − G2R) 		[math]\displaystyle{ \displaystyle \frac{1}{2} (1 - 5 L) \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{7} \lt L \le \frac{1}{5} }[/math]
[math]\displaystyle{ \displaystyle 0 \lt s \lt \frac{1}{4} (1 - 3 L) \,\text{ for }\, \frac{1}{5} \le L \lt \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

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

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