Scala analysis: uruk
Jon Lyle Smith's "Uruk" scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 21/20 84.467193 minor semitone 2: 15/14 119.442808 major diatonic semitone 3: 9/8 203.910002 major whole tone 4: 189/160 288.377195 5: 135/112 323.352810 septimal wide minor third 6: 81/64 407.820003 Pythagorean major third 7: 4/3 498.044999 perfect fourth 8: 7/5 582.512193 septimal or Huygens' tritone, BP fourth 9: 10/7 617.487807 Euler's tritone 10: 3/2 701.955001 perfect fifth 11: 63/40 786.422194 narrow minor sixth 12: 45/28 821.397809 septimale wide minor sixth 13: 27/16 905.865003 Pythagorean major sixth 14: 567/320 990.332196 15: 405/224 1025.307811 16: 243/128 1109.775004 Pythagorean major seventh 17: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 256/243 90.2250 limma, Pythagorean minor second 1: 21/20 84.4672 minor semitone 2: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 3: 21/20 84.4672 minor semitone 4: 21/20 84.4672 minor semitone 5: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 6: 21/20 84.4672 minor semitone 7: 256/243 90.2250 limma, Pythagorean minor second 8: 21/20 84.4672 minor semitone 9: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 10: 21/20 84.4672 minor semitone 11: 21/20 84.4672 minor semitone 12: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 13: 21/20 84.4672 minor semitone 14: 21/20 84.4672 minor semitone 15: 50/49 34.9756 Erlich's decatonic comma, tritonic diesis 16: 21/20 84.4672 minor semitone 17: 256/243 90.2250 limma, Pythagorean minor second
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 5 50/49 34.976 cents Erlich's decatonic comma, tritonic diesis 1: 10 21/20 84.467 cents minor semitone 1: 2 256/243 90.225 cents limma, Pythagorean minor second 2: 10 15/14 119.443 cents major diatonic semitone 2: 3 441/400 168.934 cents two (minor semitone) 2: 4 448/405 174.692 cents 3: 11 9/8 203.910 cents major whole tone 3: 4 640/567 209.668 cents 3: 2 784/675 259.159 cents 4: 3 225/196 238.886 cents two (major diatonic semitone) 4: 6 189/160 288.377 cents 4: 8 32/27 294.135 cents Pythagorean minor third 5: 6 135/112 323.353 cents septimal wide minor third 5: 2 1600/1323 329.111 cents 5: 1 3969/3200 372.844 cents 5: 8 56/45 378.602 cents septimal narrow major third 6: 5 81/64 407.820 cents Pythagorean major third 6: 8 80/63 413.578 cents wide major third 6: 4 98/75 463.069 cents 7: 1 2025/1568 442.796 cents 7: 2 1701/1280 492.287 cents 7: 14 4/3 498.045 cents perfect fourth 8: 2 1215/896 527.263 cents 8: 4 200/147 533.021 cents 8: 10 7/5 582.512 cents septimal or Huygens' tritone, BP fourth 8: 1 1024/729 588.270 cents Pythagorean diminished fifth 9: 1 729/512 611.730 cents Pythagorean tritone 9: 10 10/7 617.488 cents Euler's tritone 9: 4 147/100 666.979 cents 9: 2 1792/1215 672.737 cents 10: 14 3/2 701.955 cents perfect fifth 10: 2 2560/1701 707.713 cents 10: 1 3136/2025 757.204 cents two (septimal narrow major third) 11: 4 75/49 736.931 cents BP fifth 11: 8 63/40 786.422 cents narrow minor sixth 11: 5 128/81 792.180 cents Pythagorean minor sixth 12: 8 45/28 821.398 cents septimale wide minor sixth 12: 1 6400/3969 827.156 cents two (wide major third) 12: 2 1323/800 870.889 cents 12: 6 224/135 876.647 cents narrow septimal major sixth 13: 8 27/16 905.865 cents Pythagorean major sixth 13: 6 320/189 911.623 cents 13: 3 392/225 961.114 cents 14: 2 675/392 940.841 cents 14: 4 567/320 990.332 cents 14: 11 16/9 996.090 cents Pythagorean minor seventh 15: 4 405/224 1025.308 cents 15: 3 800/441 1031.066 cents 15: 10 28/15 1080.557 cents grave major seventh 16: 2 243/128 1109.775 cents Pythagorean major seventh 16: 10 40/21 1115.533 cents acute major seventh 16: 5 49/25 1165.024 cents BP eighth Highest number of different intervals for one interval class: 4 Average number of different intervals per interval class: 3.25000 = 13/4
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.0 : 84.5 119.4 203.9 288.4 323.4 407.8 498.0 582.5 617.5 702.0 786.4 821.4 905.9 990.3 1025.3 1109.8 1200.0 84.5 : 35.0 119.4 203.9 238.9 323.4 413.6 498.0 533.0 617.5 702.0 736.9 821.4 905.9 940.8 1025.3 1115.5 1200.0 119.4 : 84.5 168.9 203.9 288.4 378.6 463.1 498.0 582.5 667.0 702.0 786.4 870.9 905.9 990.3 1080.6 1165.0 1200.0 203.9 : 84.5 119.4 203.9 294.1 378.6 413.6 498.0 582.5 617.5 702.0 786.4 821.4 905.9 996.1 1080.6 1115.5 1200.0 288.4 : 35.0 119.4 209.7 294.1 329.1 413.6 498.0 533.0 617.5 702.0 736.9 821.4 911.6 996.1 1031.1 1115.5 1200.0 323.4 : 84.5 174.7 259.2 294.1 378.6 463.1 498.0 582.5 667.0 702.0 786.4 876.6 961.1 996.1 1080.6 1165.0 1200.0 407.8 : 90.2 174.7 209.7 294.1 378.6 413.6 498.0 582.5 617.5 702.0 792.2 876.6 911.6 996.1 1080.6 1115.5 1200.0 498.0 : 84.5 119.4 203.9 288.4 323.4 407.8 492.3 527.3 611.7 702.0 786.4 821.4 905.9 990.3 1025.3 1109.8 1200.0 582.5 : 35.0 119.4 203.9 238.9 323.4 407.8 442.8 527.3 617.5 702.0 736.9 821.4 905.9 940.8 1025.3 1115.5 1200.0 617.5 : 84.5 168.9 203.9 288.4 372.8 407.8 492.3 582.5 667.0 702.0 786.4 870.9 905.9 990.3 1080.6 1165.0 1200.0 702.0 : 84.5 119.4 203.9 288.4 323.4 407.8 498.0 582.5 617.5 702.0 786.4 821.4 905.9 996.1 1080.6 1115.5 1200.0 786.4 : 35.0 119.4 203.9 238.9 323.4 413.6 498.0 533.0 617.5 702.0 736.9 821.4 911.6 996.1 1031.1 1115.5 1200.0 821.4 : 84.5 168.9 203.9 288.4 378.6 463.1 498.0 582.5 667.0 702.0 786.4 876.6 961.1 996.1 1080.6 1165.0 1200.0 905.9 : 84.5 119.4 203.9 294.1 378.6 413.6 498.0 582.5 617.5 702.0 792.2 876.6 911.6 996.1 1080.6 1115.5 1200.0 990.3 : 35.0 119.4 209.7 294.1 329.1 413.6 498.0 533.0 617.5 707.7 792.2 827.2 911.6 996.1 1031.1 1115.5 1200.0 1025.3: 84.5 174.7 259.2 294.1 378.6 463.1 498.0 582.5 672.7 757.2 792.2 876.6 961.1 996.1 1080.6 1165.0 1200.0 1109.8: 90.2 174.7 209.7 294.1 378.6 413.6 498.0 588.3 672.7 707.7 792.2 876.6 911.6 996.1 1080.6 1115.5 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 34.9756 .. 90.2250 cents 6272/6075, 55.2494 cents 50/49, 34.9756 cents 2: 119.4428 .. 174.6922 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 3: 203.9100 .. 259.1594 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 4: 238.8856 .. 294.1350 cents 6272/6075, 55.2494 cents 151875/153664,-20.2738 cents 5: 323.3528 .. 378.6022 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 6: 407.8200 .. 463.0694 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 7: 442.7956 .. 498.0450 cents 6272/6075, 55.2494 cents 151875/153664,-20.2738 cents 8: 527.2628 .. 588.2700 cents 917504/885735, 61.0072 cents 3645/3584, 29.2178 cents 9: 611.7300 .. 672.7372 cents 917504/885735, 61.0072 cents 531441/524288, 23.4600 cents 10: 701.9550 .. 757.2044 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 11: 736.9306 .. 792.1800 cents 6272/6075, 55.2494 cents 151875/153664,-20.2738 cents 12: 821.3978 .. 876.6472 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 13: 905.8650 .. 961.1144 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 14: 940.8406 .. 996.0900 cents 6272/6075, 55.2494 cents 151875/153664,-20.2738 cents 15: 1025.3078 .. 1080.5572 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents 16: 1109.7750 .. 1165.0244 cents 6272/6075, 55.2494 cents 3645/3584, 29.2178 cents
SHOW DATA
Number of notes : 17 -- Interval properties -- Smallest interval : 50/49, 34.9756 cents, class 1 Average step (divided formal octave): 70.5882 cents Largest one step interval : 256/243, 90.2250 cents Average / Smallest step : 2.018213 Largest / Average step : 1.278187 Largest / Smallest step : 2.579654 Median interval of one step : 21/20, 84.4672 cents, amount: 10 Most common interval of one step : 21/20, 84.4672 cents, amount: 10 Least squares average step : 69.78149 cents, oct.: 1186.28533 cents Scale is not proper Step pattern alph. order: ABAABACABAABAABAC Step pattern size order : MSMMSMLMSMMSMMSML Scale is a Constant Structure, by a margin of 20.27377 cents Scale diversity : 1.272026 Lumma stability : 0.321324 Lumma impropriety factor : 0.067579 Number of different interval sizes : 52 = 3.25000 / class Number of one step interval sizes : 3 Highest interval variety : 4 Mean interval variety : 3.25000 = 13/4 Median interval variety : 3 Lowest interval variety : 3 Smallest interval difference : 5120/5103, 5.7578 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 14 Most common triad is 0.0 498.045 701.955 cents, amount: 11 Number of recognisable fifths : 16, average 702.6747 cents Scale contains two identical disjunct "tetrachords" of 8 notes Scale contains a complete diamond : 5 7 Length of longest harmonic part : 1 at degree 0 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 5 = 29.4118% Limited inverse transpositions : 6 Inversional symmetry on degrees : 3 Inversional symmetry on intervals : 11-12 -- Rational properties -- Prime limit : 7 Odd number limit : 3969 (O: 3969 U: 3969) Highest odd numerator or denominator: 567 Scale harmonicity : 0.017132 Average absolute harmonicity : 0.130765 Specific harmonicity : 0.092832 Fundamental : 1/13440, -13.7142 octaves, 0.0195 Hz. Guide tone : 34020, 15.0541 octaves, 8900501.732 Hz. Exponens Consonantiae : 4.572288E+08, 28.76834 octaves Euler's gradus suavitatis : 42 Sum of Mann's harmonic distance : 1489.0, average 87.58824 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 25.04283, average 1.47311 Sum of Tenney's harmonic distance : 48.57164, average 2.85716 Vogel's harmonic complexity : 25.35294 Wille's k value : 12757 Wilson's harmonic complexity : 42 Rectangular lattice diameter : 10 Triangular lattice diameter : 6 Lattice compactness : 650.30203, average 4.25034 Lattice compactness (without 2's) : 406.02739, average 2.65377 Number of different primes : 4 Prime exponents' range, average, count, tones@limit: 2: -7 .. 2 -2.64706 53 1 3: -1 .. 5 2.00000 36 6 5: -1 .. 1 0.00000 10 7: -1 .. 1 0.00000 10 10 Average exponent except 2's : 34 / 17 = 2.00000 Average absolute exponent except 2's: 56 / 17 = 3.29412 Scale is weakly epimorphic with val: <17 25 36 50| Scale is weakly epimorphic with val: <17 25 37 51| Scale is weakly epimorphic with val: <17 25 42 45| Scale is weakly epimorphic with val: <17 26 36 50| Scale is weakly epimorphic with val: <17 26 37 51| Scale is weakly epimorphic with val: <17 26 42 45| Scale is weakly epimorphic with val: <17 27 36 45| Scale is JI-epimorphic with val: <17 27 37 45| Scale is weakly epimorphic with val: <17 27 37 46| Scale is JI-epimorphic with val: <17 27 38 46| Scale is weakly epimorphic with val: <17 27 38 47| Scale is JI-epimorphic with val: <17 27 39 47| Scale is weakly epimorphic with val: <17 27 39 48| = patent Scale is JI-epimorphic with val: <17 27 40 48| Scale is weakly epimorphic with val: <17 27 40 49| Scale is JI-epimorphic with val: <17 27 41 49| Scale is weakly epimorphic with val: <17 27 41 50| Scale is JI-epimorphic with val: <17 27 42 50| Scale is weakly epimorphic with val: <17 27 42 51| Scale is weakly epimorphic with val: <17 28 36 51| Scale is weakly epimorphic with val: <17 29 36 49| Scale is weakly epimorphic with val: <17 29 37 50| Scale is weakly epimorphic with val: <17 29 38 51| Scale is weakly epimorphic with val: <17 29 41 45| Scale is weakly epimorphic with val: <17 29 42 46| Scale is weakly epimorphic with val: <17 30 36 46| Scale is weakly epimorphic with val: <17 30 37 47| Scale is weakly epimorphic with val: <17 30 38 45| Scale is weakly epimorphic with val: <17 30 38 48| Scale is weakly epimorphic with val: <17 30 39 46| Scale is weakly epimorphic with val: <17 30 39 49| Scale is weakly epimorphic with val: <17 30 40 47| Scale is weakly epimorphic with val: <17 30 40 50| Scale is weakly epimorphic with val: <17 30 41 48| Scale is weakly epimorphic with val: <17 30 41 51| Scale is weakly epimorphic with val: <17 30 42 49|
FIT/MODE
17: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 SP B ME I SD: 17.0808 c. M:-33.5157 c. 20: 1 1 1 2 0 2 1 2 0 2 1 1 1 1 1 1 2 N D3 SD: 18.0176 c. M: 30.3322 c. 21: 1 1 2 1 1 1 2 1 1 1 2 0 2 1 1 1 2 N D3 SD: 15.8362 c. M: 27.3243 c. 23: 2 0 2 2 0 2 2 1 1 1 2 1 1 2 1 1 2 N D3 SD: 14.9604 c. M:-24.6663 c. 24: 2 0 2 2 0 2 2 2 0 2 2 0 2 2 1 1 2 N D3 S SD: 14.2886 c. M:-24.6922 c. 26: 2 1 1 2 1 2 2 2 0 2 2 1 2 1 1 2 2 N T3 SD: 12.6425 c. M: 21.1014 c. 27: 2 1 2 1 1 2 2 2 1 2 2 0 2 2 1 2 2 N T3 SD: 12.2283 c. M: 21.7105 c. 28: 2 1 2 2 1 2 2 2 0 2 2 1 2 2 1 2 2 N T3 SD: 12.4616 c. M:-20.7514 c. 29: 2 1 2 2 1 2 2 2 1 2 2 1 2 2 1 2 2 P M ME S SD: 4.6093 c. M:-9.1749 c. 41: 3 1 3 3 1 3 3 3 1 3 3 1 3 3 1 3 3 N M DE S SD: 2.5276 c. M:-4.7898 c. 70: 5 2 5 5 2 5 5 5 2 5 5 2 5 5 2 5 5 N M DE S SD: 2.3714 c. M:-4.5107 c. 111: 8 3 8 8 3 8 8 8 3 8 8 3 8 8 3 8 8 N M DE S SD: 2.1831 c. M:-4.2624 c. 118: 8 4 8 8 4 8 9 8 4 8 8 4 8 8 4 8 9 N S SD: 2.2899 c. M: 3.8915 c. 130: 9 4 9 9 4 9 10 9 4 9 9 4 9 9 4 9 10 N S SD: 1.3197 c. M: 2.6399 c. 142: 10 4 10 10 4 10 11 10 4 10 10 4 10 10 4 10 11 N S SD: 1.4969 c. M: 2.7726 c. 159: 11 5 11 11 5 11 12 11 5 11 11 5 11 11 5 11 12 N S SD: 1.0734 c. M: 1.6530 c. 171: 12 5 12 12 5 12 13 12 5 12 12 5 12 12 5 12 13 N S SD: 0.5258 c. M: 1.0031 c. 200: 14 6 14 14 6 14 15 14 6 14 14 6 14 14 6 14 15 N S SD: 0.4100 c. M:-0.6922 c. 241: 17 7 17 17 7 17 18 17 7 17 17 7 17 17 7 17 18 N S SD: 0.3161 c. M:-0.5984 c. 371: 26 11 26 26 11 26 28 26 11 26 26 11 26 26 11 26 28 N S SD: 0.2922 c. M: 0.5748 c. 412: 29 12 29 29 12 29 31 29 12 29 29 12 29 29 12 29 31 N S SD: 0.0358 c. M: 0.0663 c. 1477: 104 43 104 104 43 104 111 104 43 104 104 43 104 104 43 104 111 N S SD: 0.0268 c. M:-0.0537 c. 1889: 133 55 133 133 55 133 142 133 55 133 133 55 133 133 55 133 142 N S SD: 0.0170 c. M:-0.0331 c. 2301: 162 67 162 162 67 162 173 162 67 162 162 67 162 162 67 162 173 N S SD: 0.0133 c. M:-0.0198 c.
FIT/HARMONIC
1 .. 2 match: 1 H SD: 0.0000 cents 2 .. 4 match: 2 H SD: 0.0000 cents 3 .. 6 match: 3 H SD: 7.1688 cents 4 .. 8 match: 4 H SD: 7.6036 cents 5 .. 10 match: 5 H SD: 2.6713 cents 6 .. 12 match: 6 H SD: 6.4629 cents 7 .. 14 match: 7 H SD: 8.6838 cents 8 .. 16 match: 8 H SD: 6.5271 cents 9 .. 18 match: 9 H SD: 5.4902 cents 10 .. 20 match: 10 H SD: 6.1469 cents 11 .. 22 match: 11 H SD: 7.8290 cents 12 .. 24 match: 12 H SD: 4.8317 cents 13 .. 26 match: 13 H SD: 7.5581 cents 14 .. 28 match: 13 SD: 5.8179 cents 15 .. 30 match: 12 SD: 4.6898 cents 16 .. 32 match: 15 S SD: 4.2626 cents 17 .. 34 match: 13 SD: 5.3838 cents 18 .. 36 match: 16 S SD: 3.9767 cents 19 .. 38 match: 14 SD: 4.9239 cents 20 .. 40 match: 14 SD: 4.4283 cents 21 .. 42 match: 16 SD: 4.5108 cents 22 .. 44 match: 15 SD: 3.9113 cents 23 .. 46 match: 17 SD: 3.3143 cents 24 .. 48 match: 15 SD: 3.7308 cents 25 .. 50 match: 16 SD: 3.3417 cents 26 .. 52 match: 16 SD: 2.7304 cents 27 .. 54 match: 16 SD: 3.5382 cents 28 .. 56 match: 17 SD: 3.0774 cents 29 .. 58 match: 16 SD: 3.4716 cents 30 .. 60 match: 16 SD: 2.1419 cents 31 .. 62 match: 15 SD: 2.7444 cents 32 .. 64 match: 16 SD: 2.8092 cents 33 .. 66 match: 16 SD: 2.3322 cents 34 .. 68 match: 16 SD: 2.6777 cents 35 .. 70 match: 17 SD: 2.6546 cents 36 .. 72 match: 16 SD: 2.8571 cents 37 .. 74 match: 17 SD: 2.6071 cents 38 .. 76 match: 17 SD: 1.6436 cents 40 .. 80 match: 17 SD: 1.6327 cents 54 .. 108 match: 17 SD: 1.3512 cents 56 .. 112 match: 17 SD: 1.3402 cents 57 .. 114 match: 17 SD: 1.2821 cents 68 .. 136 match: 17 SD: 1.2249 cents 72 .. 144 match: 17 SD: 1.0135 cents 78 .. 156 match: 17 SD: 0.9366 cents 82 .. 164 match: 17 SD: 0.7703 cents 98 .. 196 match: 17 SD: 0.6895 cents 117 .. 234 match: 17 SD: 0.6831 cents 120 .. 240 match: 17 SD: 0.6248 cents 122 .. 244 match: 17 SD: 0.5833 cents 128 .. 256 match: 17 SD: 0.5660 cents 137 .. 274 match: 17 SD: 0.5408 cents 138 .. 276 match: 17 SD: 0.5112 cents 150 .. 300 match: 17 SD: 0.4940 cents 158 .. 316 match: 17 SD: 0.4906 cents 165 .. 330 match: 17 SD: 0.4653 cents 166 .. 332 match: 17 SD: 0.4075 cents 184 .. 368 match: 17 SD: 0.4036 cents 198 .. 396 match: 17 SD: 0.3010 cents 238 .. 476 match: 17 SD: 0.2760 cents 240 .. 480 match: 17 SD: 0.2663 cents 264 .. 528 match: 17 SD: 0.2428 cents 282 .. 564 match: 17 SD: 0.1752 cents 336 .. 672 match: 17 SD: 0.1660 cents 418 .. 836 match: 17 SD: 0.1652 cents 438 .. 876 match: 17 SD: 0.1575 cents 448 .. 896 match: 17 SD: 0.1444 cents 464 .. 928 match: 17 SD: 0.1337 cents 520 .. 1040 match: 17 SD: 0.1175 cents 558 .. 1116 match: 17 SD: 0.1160 cents 560 .. 1120 match: 17 SD: 0.1123 cents 600 .. 1200 match: 17 SD: 0.0971 cents