User:Ganaram inukshuk/Sandbox

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This is a sandbox page for me (Ganaram) to test out a few things before deploying things. (Expect some mess.) 

{{subst:User:Ganaram inukshuk/JI ratios|Int Limit=50|Prime Limit=7|Equave=2/1}}

produces

1/1, 50/49, 49/48, 36/35, 28/27, 25/24, 21/20, 16/15, 15/14, 27/25, 49/45, 35/32, 10/9, 28/25, 9/8, 8/7, 7/6, 32/27, 25/21, 6/5, 49/40, 5/4, 32/25, 9/7, 35/27, 21/16, 4/3, 27/20, 49/36, 48/35, 25/18, 7/5, 45/32, 10/7, 36/25, 35/24, 40/27, 3/2, 32/21, 49/32, 14/9, 25/16, 8/5, 45/28, 49/30, 5/3, 42/25, 27/16, 12/7, 7/4, 16/9, 25/14, 9/5, 49/27, 50/27, 28/15, 15/8, 40/21, 48/25, 27/14, 35/18, 49/25, 2/1

MOS scalesig

5L 2s

5L 2s

4L 5s (3/1-equivalent)

4L 5s⟨3/1⟩

4L 5s (3/1-equivalent)

4L 5s⟨3/1⟩

4L 5s (3/1-equivalent)

MOS tuning spectrum (AKA, scale tree)

Scale tree and tuning spectrum of 1L 1s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\2 600.000 600.000 1:1 1.000 Equalized 1L 1s
13\25 624.000 576.000 13:12 1.083
12\23 626.087 573.913 12:11 1.091
11\21 628.571 571.429 11:10 1.100
10\19 631.579 568.421 10:9 1.111
9\17 635.294 564.706 9:8 1.125
8\15 640.000 560.000 8:7 1.143
7\13 646.154 553.846 7:6 1.167
13\24 650.000 550.000 13:11 1.182
6\11 654.545 545.455 6:5 1.200
11\20 660.000 540.000 11:9 1.222
5\9 666.667 533.333 5:4 1.250
9\16 675.000 525.000 9:7 1.286
13\23 678.261 521.739 13:10 1.300
4\7 685.714 514.286 4:3 1.333 Supersoft 1L 1s
11\19 694.737 505.263 11:8 1.375
7\12 700.000 500.000 7:5 1.400
10\17 705.882 494.118 10:7 1.429
13\22 709.091 490.909 13:9 1.444
3\5 720.000 480.000 3:2 1.500 Soft 1L 1s
11\18 733.333 466.667 11:7 1.571
8\13 738.462 461.538 8:5 1.600
13\21 742.857 457.143 13:8 1.625
5\8 750.000 450.000 5:3 1.667 Semisoft 1L 1s
12\19 757.895 442.105 12:7 1.714
7\11 763.636 436.364 7:4 1.750
9\14 771.429 428.571 9:5 1.800
11\17 776.471 423.529 11:6 1.833
13\20 780.000 420.000 13:7 1.857
2\3 800.000 400.000 2:1 2.000 Basic 1L 1s
13\19 821.053 378.947 13:6 2.167
11\16 825.000 375.000 11:5 2.200
9\13 830.769 369.231 9:4 2.250
7\10 840.000 360.000 7:3 2.333
12\17 847.059 352.941 12:5 2.400
5\7 857.143 342.857 5:2 2.500 Semihard 1L 1s
13\18 866.667 333.333 13:5 2.600
8\11 872.727 327.273 8:3 2.667
11\15 880.000 320.000 11:4 2.750
3\4 900.000 300.000 3:1 3.000 Hard 1L 1s
13\17 917.647 282.353 13:4 3.250
10\13 923.077 276.923 10:3 3.333
7\9 933.333 266.667 7:2 3.500
11\14 942.857 257.143 11:3 3.667
4\5 960.000 240.000 4:1 4.000 Superhard 1L 1s
13\16 975.000 225.000 13:3 4.333
9\11 981.818 218.182 9:2 4.500
5\6 1000.000 200.000 5:1 5.000
11\13 1015.385 184.615 11:2 5.500
6\7 1028.571 171.429 6:1 6.000
13\15 1040.000 160.000 13:2 6.500
7\8 1050.000 150.000 7:1 7.000
8\9 1066.667 133.333 8:1 8.000
9\10 1080.000 120.000 9:1 9.000
10\11 1090.909 109.091 10:1 10.000
11\12 1100.000 100.000 11:1 11.000
12\13 1107.692 92.308 12:1 12.000
13\14 1114.286 85.714 13:1 13.000
1\1 1200.000 0.000 1:0 → ∞ Collapsed 1L 1s


Scale tree and tuning spectrum of 3L 4s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\7 342.857 857.143 1:1 1.000 Equalized 3L 4s
39\136 344.118 855.882 20:19 1.053
37\129 344.186 855.814 19:18 1.056
35\122 344.262 855.738 18:17 1.059
33\115 344.348 855.652 17:16 1.062
31\108 344.444 855.556 16:15 1.067
29\101 344.554 855.446 15:14 1.071
27\94 344.681 855.319 14:13 1.077
25\87 344.828 855.172 13:12 1.083
23\80 345.000 855.000 12:11 1.091
21\73 345.205 854.795 11:10 1.100
19\66 345.455 854.545 10:9 1.111
36\125 345.600 854.400 19:17 1.118
17\59 345.763 854.237 9:8 1.125
32\111 345.946 854.054 17:15 1.133
15\52 346.154 853.846 8:7 1.143
28\97 346.392 853.608 15:13 1.154
13\45 346.667 853.333 7:6 1.167
37\128 346.875 853.125 20:17 1.176
24\83 346.988 853.012 13:11 1.182
35\121 347.107 852.893 19:16 1.188
11\38 347.368 852.632 6:5 1.200 Mohaha / ptolemy↑
31\107 347.664 852.336 17:14 1.214
20\69 347.826 852.174 11:9 1.222
29\100 348.000 852.000 16:13 1.231
9\31 348.387 851.613 5:4 1.250 Mohaha / migration / mohajira
34\117 348.718 851.282 19:15 1.267
25\86 348.837 851.163 14:11 1.273
16\55 349.091 850.909 9:7 1.286
23\79 349.367 850.633 13:10 1.300
30\103 349.515 850.485 17:13 1.308
7\24 350.000 850.000 4:3 1.333 Supersoft 3L 4s
33\113 350.442 849.558 19:14 1.357
26\89 350.562 849.438 15:11 1.364
19\65 350.769 849.231 11:8 1.375 Mohaha / mohamaq
31\106 350.943 849.057 18:13 1.385
12\41 351.220 848.780 7:5 1.400 Mohaha / neutrominant
29\99 351.515 848.485 17:12 1.417
17\58 351.724 848.276 10:7 1.429 Hemif / hemififths
22\75 352.000 848.000 13:9 1.444
27\92 352.174 847.826 16:11 1.455
32\109 352.294 847.706 19:13 1.462
5\17 352.941 847.059 3:2 1.500 Soft 3L 4s
33\112 353.571 846.429 20:13 1.538
28\95 353.684 846.316 17:11 1.545
23\78 353.846 846.154 14:9 1.556
18\61 354.098 845.902 11:7 1.571 Suhajira
31\105 354.286 845.714 19:12 1.583
13\44 354.545 845.455 8:5 1.600
21\71 354.930 845.070 13:8 1.625 Golden suhajira (354.8232¢)
29\98 355.102 844.898 18:11 1.636
8\27 355.556 844.444 5:3 1.667 Semisoft 3L 4s
Suhajira / ringo
27\91 356.044 843.956 17:10 1.700
19\64 356.250 843.750 12:7 1.714 Beatles
30\101 356.436 843.564 19:11 1.727
11\37 356.757 843.243 7:4 1.750
25\84 357.143 842.857 16:9 1.778
14\47 357.447 842.553 9:5 1.800
31\104 357.692 842.308 20:11 1.818
17\57 357.895 842.105 11:6 1.833
20\67 358.209 841.791 13:7 1.857
23\77 358.442 841.558 15:8 1.875
26\87 358.621 841.379 17:9 1.889
29\97 358.763 841.237 19:10 1.900
3\10 360.000 840.000 2:1 2.000 Basic 3L 4s
Scales with tunings softer than this are proper
28\93 361.290 838.710 19:9 2.111
25\83 361.446 838.554 17:8 2.125
22\73 361.644 838.356 15:7 2.143
19\63 361.905 838.095 13:6 2.167
16\53 362.264 837.736 11:5 2.200
29\96 362.500 837.500 20:9 2.222
13\43 362.791 837.209 9:4 2.250
23\76 363.158 836.842 16:7 2.286
10\33 363.636 836.364 7:3 2.333
27\89 364.045 835.955 19:8 2.375
17\56 364.286 835.714 12:5 2.400
24\79 364.557 835.443 17:7 2.429
7\23 365.217 834.783 5:2 2.500 Semihard 3L 4s
25\82 365.854 834.146 18:7 2.571
18\59 366.102 833.898 13:5 2.600 Unnamed golden tuning (366.2564¢)
11\36 366.667 833.333 8:3 2.667
26\85 367.059 832.941 19:7 2.714
15\49 367.347 832.653 11:4 2.750
19\62 367.742 832.258 14:5 2.800
23\75 368.000 832.000 17:6 2.833
27\88 368.182 831.818 20:7 2.857
4\13 369.231 830.769 3:1 3.000 Hard 3L 4s
25\81 370.370 829.630 19:6 3.167
21\68 370.588 829.412 16:5 3.200
17\55 370.909 829.091 13:4 3.250
13\42 371.429 828.571 10:3 3.333
22\71 371.831 828.169 17:5 3.400
9\29 372.414 827.586 7:2 3.500 Sephiroth
23\74 372.973 827.027 18:5 3.600
14\45 373.333 826.667 11:3 3.667
19\61 373.770 826.230 15:4 3.750
24\77 374.026 825.974 19:5 3.800
5\16 375.000 825.000 4:1 4.000 Superhard 3L 4s
21\67 376.119 823.881 17:4 4.250
16\51 376.471 823.529 13:3 4.333
11\35 377.143 822.857 9:2 4.500 Muggles
17\54 377.778 822.222 14:3 4.667
23\73 378.082 821.918 19:4 4.750
6\19 378.947 821.053 5:1 5.000 Magic
19\60 380.000 820.000 16:3 5.333
13\41 380.488 819.512 11:2 5.500
20\63 380.952 819.048 17:3 5.667
7\22 381.818 818.182 6:1 6.000 Würschmidt↓
22\69 382.609 817.391 19:3 6.333
15\47 382.979 817.021 13:2 6.500
23\72 383.333 816.667 20:3 6.667
8\25 384.000 816.000 7:1 7.000
17\53 384.906 815.094 15:2 7.500
9\28 385.714 814.286 8:1 8.000
19\59 386.441 813.559 17:2 8.500
10\31 387.097 812.903 9:1 9.000
21\65 387.692 812.308 19:2 9.500
11\34 388.235 811.765 10:1 10.000
12\37 389.189 810.811 11:1 11.000
13\40 390.000 810.000 12:1 12.000
14\43 390.698 809.302 13:1 13.000
15\46 391.304 808.696 14:1 14.000
16\49 391.837 808.163 15:1 15.000
17\52 392.308 807.692 16:1 16.000
18\55 392.727 807.273 17:1 17.000
19\58 393.103 806.897 18:1 18.000
20\61 393.443 806.557 19:1 19.000
21\64 393.750 806.250 20:1 20.000
1\3 400.000 800.000 1:0 → ∞ Collapsed 3L 4s


Scale tree and tuning spectrum of 3L 4s⟨3/2⟩
Generator(edf) Cents Step ratio Comments
Bright Dark L:s Hardness
2\7 200.559 501.396 1:1 1.000 Equalized 3L 4s⟨3/2⟩
7\24 204.737 497.218 4:3 1.333 Supersoft 3L 4s⟨3/2⟩
5\17 206.457 495.498 3:2 1.500 Soft 3L 4s⟨3/2⟩
8\27 207.987 493.968 5:3 1.667 Semisoft 3L 4s⟨3/2⟩
3\10 210.587 491.369 2:1 2.000 Basic 3L 4s⟨3/2⟩
Scales with tunings softer than this are proper
7\23 213.638 488.317 5:2 2.500 Semihard 3L 4s⟨3/2⟩
4\13 215.986 485.969 3:1 3.000 Hard 3L 4s⟨3/2⟩
5\16 219.361 482.594 4:1 4.000 Superhard 3L 4s⟨3/2⟩
1\3 233.985 467.970 1:0 → ∞ Collapsed 3L 4s⟨3/2⟩

MOS intro

First sentence:

  • Single-period 2/1-equivalent: xL ys (TAMNAMS name tamnams-name), also called other-name, is an octave-repeating moment of symmetry scale that divides the octave (2/1) into x large and y small steps.
  • Multi-period 2/1-equivalent: nxL nys (TAMNAMS name tamnams-name), also called other-name, is an octave-repeating moment of symmetry scale that divides the octave (2/1) into nx large steps and ny small steps, with n periods of c cents containing x large and y small steps each.
  • Single-period 3/1-equivalent: 3/1-equivalent xL ys, also called other-name, is a twelfth-repeating moment of symmetry scale that divides the tritave or perfect 12th (3/1, c cents) into x large and y small steps.
  • Multi-period 3/1-equivalent: 3/1-equivalent nxL nys, also called other-name, is a twelfth-repeating moment of symmetry scale that divides the tritave or perfect 12th (3/1, nc cents) into nx large steps and ny small steps, with n periods of c cents containing x large and y small steps each.
  • Single-period 3/2-equivalent: 3/2-equivalent xL ys, also called other-name, is a fifth-repeating moment of symmetry scale that divides the perfect 5th (3/2, c cents) into x large and y small steps.
  • Multi-period 3/2-equivalent: 3/2-equivalent nxL nys, also called other-name, is a fifth-repeating moment of symmetry scale that divides the perfect 5th (3/2, nc cents) into nx large steps and ny small steps, with n periods of c cents containing x large and y small steps each.

Second sentence:

  • Generators that produce this scale range from g1 cents to g2 cents, or from d1 cents to d2 cents.

Octave-equivalent relational info:

  • Parents of mosses with 6-10 steps: xL ys is the parent scale of both child-soft and child-hard.
  • Children of mosses with 6-10 steps: xL ys expands parent-scale by adding step-count-difference tones.

Rothenprop:

  • Single-period: Scales of this form are always proper because there is only one small step.
  • Multi-period: Scales of this form, where every period is the same, are proper because there is only one small step per period.

Sandbox for proposed templates

Cent ruler

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MOS characteristics

NOTE: not suitable for displaying intervals or scale degrees. Repurpose for other content.

Scale degrees of the modes of 5L 2s
UDP Cyclic
order		 Step
pattern		 Scale degree (diadegree)
0 1 2 3 4 5 6 7
6|0 1 LLLsLLs Perf. Maj. Maj. Aug. Perf. Maj. Maj. Perf.
5|1 5 LLsLLLs Perf. Maj. Maj. Perf. Perf. Maj. Maj. Perf.
4|2 2 LLsLLsL Perf. Maj. Maj. Perf. Perf. Maj. Min. Perf.
3|3 6 LsLLLsL Perf. Maj. Min. Perf. Perf. Maj. Min. Perf.
2|4 3 LsLLsLL Perf. Maj. Min. Perf. Perf. Min. Min. Perf.
1|5 7 sLLLsLL Perf. Min. Min. Perf. Perf. Min. Min. Perf.
0|6 4 sLLsLLL Perf. Min. Min. Perf. Dim. Min. Min. Perf.
Intervals of 5L 2s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-diastep Perfect 0-diastep P0dias 0 0.0 ¢
1-diastep Minor 1-diastep m1dias s 0.0 ¢ to 171.4 ¢
Major 1-diastep M1dias L 171.4 ¢ to 240.0 ¢
2-diastep Minor 2-diastep m2dias L + s 240.0 ¢ to 342.9 ¢
Major 2-diastep M2dias 2L 342.9 ¢ to 480.0 ¢
3-diastep Perfect 3-diastep P3dias 2L + s 480.0 ¢ to 514.3 ¢
Augmented 3-diastep A3dias 3L 514.3 ¢ to 720.0 ¢
4-diastep Diminished 4-diastep d4dias 2L + 2s 480.0 ¢ to 685.7 ¢
Perfect 4-diastep P4dias 3L + s 685.7 ¢ to 720.0 ¢
5-diastep Minor 5-diastep m5dias 3L + 2s 720.0 ¢ to 857.1 ¢
Major 5-diastep M5dias 4L + s 857.1 ¢ to 960.0 ¢
6-diastep Minor 6-diastep m6dias 4L + 2s 960.0 ¢ to 1028.6 ¢
Major 6-diastep M6dias 5L + s 1028.6 ¢ to 1200.0 ¢
7-diastep Perfect 7-diastep P7dias 5L + 2s 1200.0 ¢
Tamnams suggests the name NAME for this scale, which comes from ORIGIN. Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
4
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MOS intervals (using large/small instead of MmAPd)

Intervals of 5L 2s
Interval Size(s) Steps Range in cents Abbrev.
0-diastep (root) Perfect 0-diastep 0 0.0¢ P0ms
1-diastep Small 1-diastep s 0.0¢ to 171.4¢ s1ms
Large 1-diastep L 171.4¢ to 240.0¢ L1ms
2-diastep Small 2-diastep L + s 240.0¢ to 342.9¢ s2ms
Large 2-diastep 2L 342.9¢ to 480.0¢ L2ms
3-diastep Small 3-diastep 2L + s 480.0¢ to 514.3¢ s3ms
Large 3-diastep 3L 514.3¢ to 720.0¢ L3ms
4-diastep Small 4-diastep 2L + 2s 480.0¢ to 685.7¢ s4ms
Large 4-diastep 3L + s 685.7¢ to 720.0¢ L4ms
5-diastep Small 5-diastep 3L + 2s 720.0¢ to 857.1¢ s5ms
Large 5-diastep 4L + s 857.1¢ to 960.0¢ L5ms
6-diastep Small 6-diastep 4L + 2s 960.0¢ to 1028.6¢ s6ms
Large 6-diastep 5L + s 1028.6¢ to 1200.0¢ L6ms
7-diastep (octave) Perfect 7-diastep 5L + 2s 1200.0¢ P7ms

MOS mode degrees (using large/small instead of MmAPd)

Scale degree qualities of 5L 2s modes
Mode names Ordering Step pattern Scale degree
Default Names Bri. Rot. 0 1 2 3 4 5 6 7
5L 2s 6|0 Lydian 1 1 LLLsLLs Perf. Lg. Lg. Lg. Lg. Lg. Lg. Perf.
5L 2s 5|1 Ionian (major) 2 5 LLsLLLs Perf. Lg. Lg. Sm. Lg. Lg. Lg. Perf.
5L 2s 4|2 Mixolydian 3 2 LLsLLsL Perf. Lg. Lg. Sm. Lg. Lg. Sm. Perf.
5L 2s 3|3 Dorian 4 6 LsLLLsL Perf. Lg. Sm. Sm. Lg. Lg. Sm. Perf.
5L 2s 2|4 Aeolian (minor) 5 3 LsLLsLL Perf. Lg. Sm. Sm. Lg. Sm. Sm. Perf.
5L 2s 1|5 Phrygian 6 7 sLLLsLL Perf. Sm. Sm. Sm. Lg. Sm. Sm. Perf.
5L 2s 0|6 Locrian 7 4 sLLsLLL Perf. Sm. Sm. Sm. Sm. Sm. Sm. Perf.

KB vis

Type Visualization Individual steps Notes
Start Large step Small step End
Small vis 
┌╥╥╥┬╥╥┬┐
│║║║│║║││
│││││││││
└┴┴┴┴┴┴┴┘
 
┌
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└
 
╥
║
│
┴
 
┬
│
│
┴
 
┐
│
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┘
 Not enough room for note names.
Large vis 
┌──┬─┬─┬─┬─┬─┬──┬──┬─┬─┬─┬──┬───┐
│░░│▒│░│▒│░│▒│░░│░░│▒│░│▒│░░│░░░│
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└───┴───┴───┴───┴───┴───┴───┴───┘
 
┌──
│  
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─┘
 Black squares indicate notes one equave apart.

Contains shading characters, meant for spacing.

Type Visualization Individual steps Notes
Start Size 1 Size 2 Size 3 Size 4 Size 5 End
Multisize vis (large) 
┌────┬───┬──┬───┬──┬─┬─┬────┬────┬─┬─┬──┬─┬─┬────┬──────┐
│░░░░│▒▒▒│░░│▒▒▒│░░│▒│▒│░░░░│░░░░│▒│▒│░░│▒│▒│░░░░│░░░░░░│
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X's are placeholders for note names.

Naturals only, as there is not enough room for accidentals.

May not display correctly on some devices.

Testing with unintrusive filler characters

TAMNAMS use

This article assumes TAMNAMS conventions for naming scale degrees, intervals, and step ratios.

Names for the scale degrees of xL ys, the position of the scales tones, are called mosdegrees, or prefixdegrees. Its intervals, the pitch difference between any two tones, are based on the number of large and small steps between them and are called mossteps, or prefixsteps. Both mosdegrees and mossteps use 0-indexed numbering, as opposed to using 1-indexed ordinals, such as mos-1st instead of 0-mosstep. The use of 1-indexed ordinal names is discouraged for nondiatonic MOS scales.

JI ratio intro

For general ratios: m/n, also called interval-name, is a p-limit just intonation ratio of exactly/about r¢.

For harmonics: m/1, also called interval-name, is a just intonation ration that represents the mth harmonic of exactly/about r¢.

MOS step sizes

3L 4s step sizes
Interval Basic 3L 4s

(10edo, L:s = 2:1)

Hard 3L 4s

(13edo, L:s = 3:1)

Soft 3L 4s

(17edo, L:s = 3:2)

Approx. JI ratios
Steps Cents Steps Cents Steps Cents
Large step 2 240¢ 3 276.9¢ 3 211.8¢ Hide column if no ratios given
Small step 1 120¢ 1 92.3¢ 2 141.2¢
Bright generator 3 360¢ 4 369.2¢ 5 355.6¢

Notes:

  • Allow option to show the bright generator, dark generator, or no generator.
  • JI ratios column only shows if there are any ratios to show

Mos ancestors and descendants

2nd ancestor 1st ancestor Mos 1st descendants 2nd descendants
uL vs zL ws xL ys xL (x+y)s xL (2x+y)s
(2x+y)L xs
(x+y)L xs (2x+y)L (x+y)s
(x+y)L (2x+y)s


Encoding scheme for module:mos

Mossteps as a vector of L's and s's

For an arbitrary step sequence consisting of L's and s's, the sum of the quantities of L's and s's denotes what mosstep it is. EG, "LLLsL" is a 5-mosstep since it has 5 L's and s's total. This can be expressed as a vector denoting how many L's and s's there are. EG, "LLLsL" becomes { 4, 1 }, denoting 4 large steps and 1 small step.

Alterations by adding a chroma always adds one L and subtracts one s (or subtracts one L and adds one s, if lowering by a chroma), so the sum of L's and s's, even if one of the quantities is negative, will always denote what k-mosstep that interval is. EG, raising "LLLsL" by a chroma produces the vector { 5, 0 }, and raising it by another chroma produces the vector { 6, -1 }.

Through this, the "original size" of the interval can always be deduced.

EG, the vector { 6, -2 } is given, assuming a mos of 5L 2s. Adding 6 and -2 shows that the interval is a 4-mosstep. Taking the brightest mode of 5L 2s (LLLsLLs) and truncating it to the first 4 steps (LLLs), the corresponding vector is { 3, 1 }. This is the vector to compare to. Subtracting the given vector from the comparison vector ( as { 6-3, -2-1 }) produces the vector { 3, -3 }, meaning that { 6, -2 } is the large 4-mosstep raised by 3 chromas. (A shortcut can be employed by simply subtracting only the L-values.) The decoding scheme below shows how the "large 4-mosstep plus 3 chromas" can be decoded into more familiar terms. In this example, since the large 4-mosstep is the perfect bright generator, adding 3 chromas makes it triply augmented.

Encoding scheme
Value Encoded Decoded
Intervals with 2 sizes Intervals with 1 size Nonperfectable intervals Bright gen Dark gen Period intervals
2 Large plus 2 chromas Perfect plus 2 chromas 2× Augmented 2× Augmented 3× Augmented 2× Augmented
1 Large plus 1 chroma Perfect plus 1 chroma Augmented Augmented 2× Augmented Augmented
0 Large Perfect Major Perfect Augmented Perfect
-1 Small Perfect minus 1 chroma Minor Diminished Perfect Diminished
-2 Small minus 1 chroma Perfect minus 2 chromas Diminished 2× Diminished Diminished 2× Diminished
-3 Small minus 2 chromas Perfect minus 3 chromas 2× Diminished 3× Diminished 2× Diminished 3× Diminished

Rationale:

  • Vectors of L's and s's can always be translated back to the original k-mosstep, no matter how many chromas were added. The "unmodified" vector (the large k-mosstep, or perfect k-mosstep for period intervals) can be compared with the mosstep vector to produce the number of chromas.
    • Alterations by entire large steps or small steps is considered interval arithmetic.
  • Easy to translate values to number of chromas for mos notation. Best done with notation assigned to the brightest mode, but can be adapted for arbitrary notations by adjusting the approprite chroma offsets.

Examples of encodings for 5L 2s

Interval encodings for 5L 2s
Interval in mossteps Encoding Decoding Standard notation in the key of F
Mossteps Chroma
0 0 0 Perfect 0-diastep F
s 1 -1 Minor 1-diastep Gb
L 1 0 Major 1-diastep G
L + s 2 -1 Minor 2-diastep Ab
2L 2 0 Major 2-diastep A
2L + s 3 -1 Perfect 3-diastep Bb
3L 3 0 Augmented 3-diastep B
2L + 2s 4 -1 Diminished 4-diastep Cb
3L + s 4 0 Perfect 4-diastep C
3L + 2s 5 -1 Minor 5-diastep Db
4L + s 5 0 Major 5-diastep D
4L + 2s 6 -1 Minor 6-diastep Eb
5L + s 6 0 Major 6-diastep E
5L + 2s 7 0 Perfect 7-diastep F
Mode names Ordering Step pattern Scale degree (encoded)
Default Names Bri. Rot. 0 1 2 3 4 5 6 7
5L 2s 6|0 Lydian 1 1 LLLsLLs 0 0 0 0 0 0 0 0
5L 2s 5|1 Ionian (major) 2 5 LLsLLLs 0 0 0 -1 0 0 0 0
5L 2s 4|2 Mixolydian 3 2 LLsLLsL 0 0 1 -1 0 0 -1 0
5L 2s 3|3 Dorian 4 6 LsLLLsL 0 0 -1 -1 0 0 -1 0
5L 2s 2|4 Aeolian (minor) 5 3 LsLLsLL 0 0 -1 -1 0 -1 -1 0
5L 2s 1|5 Phrygian 6 7 sLLLsLL 0 -1 -1 -1 0 -1 -1 0
5L 2s 0|6 Locrian 7 4 sLLsLLL 0 -1 -1 -1 -1 -1 -1 0
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