Scala analysis: rodpoole
Rod Poole's 13-limit scale
Generated by Scala: https://www.huygens-fokker.org/scala/
SHOW
0: 1/1 0.000000 unison, perfect prime 1: 33/32 53.272943 undecimal comma, al-Farabi's 1/4-tone 2: 13/12 138.572661 tridecimal 2/3-tone 3: 9/8 203.910002 major whole tone 4: 7/6 266.870906 septimal minor third 5: 11/9 347.407941 undecimal neutral third 6: 14/11 417.507964 undecimal diminished fourth or major third 7: 4/3 498.044999 perfect fourth 8: 11/8 551.317942 undecimal semi-augmented fourth 9: 13/9 636.617660 tridecimal diminished fifth 10: 3/2 701.955001 perfect fifth 11: 14/9 764.915905 septimal minor sixth 12: 44/27 845.452940 neutral sixth 13: 27/16 905.865003 Pythagorean major sixth 14: 7/4 968.825906 harmonic seventh 15: 11/6 1049.362941 undecimal neutral seventh, 21/4-tone 16: 21/11 1119.462965 undecimal major seventh 17: 2/1 1200.000000 octave
SHOW/INTERVAL
0: 22/21 80.5370 undecimal minor semitone 1: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 2: 104/99 85.2997 3: 27/26 65.3373 tridecimal comma 4: 28/27 62.9609 Archytas' 1/3-tone 5: 22/21 80.5370 undecimal minor semitone 6: 126/121 70.1000 7: 22/21 80.5370 undecimal minor semitone 8: 33/32 53.2729 undecimal comma, al-Farabi's 1/4-tone 9: 104/99 85.2997 10: 27/26 65.3373 tridecimal comma 11: 28/27 62.9609 Archytas' 1/3-tone 12: 22/21 80.5370 undecimal minor semitone 13: 729/704 60.4121 undecimal major diesis 14: 28/27 62.9609 Archytas' 1/3-tone 15: 22/21 80.5370 undecimal minor semitone 16: 126/121 70.1000 17: 22/21 80.5370 undecimal minor semitone
SHOW INTERVALS
Interval class, Number of incidences, Size: 1: 2 33/32 53.273 cents undecimal comma, al-Farabi's 1/4-tone 1: 1 729/704 60.412 cents undecimal major diesis 1: 3 28/27 62.961 cents Archytas' 1/3-tone 1: 2 27/26 65.337 cents tridecimal comma 1: 2 126/121 70.100 cents 1: 5 22/21 80.537 cents undecimal minor semitone 1: 2 104/99 85.300 cents 2: 1 189/176 123.373 cents 2: 2 14/13 128.298 cents 2/3-tone 2: 2 121/112 133.810 cents 2: 2 13/12 138.573 cents tridecimal 2/3-tone 2: 1 243/224 140.949 cents Archytas' 2/3-tone 2: 3 88/81 143.498 cents 2nd undecimal neutral second 2: 6 12/11 150.637 cents undecimal neutral second, 3/4-tone 3: 7 9/8 203.910 cents major whole tone 3: 2 44/39 208.835 cents 3: 4 112/99 213.598 cents 3: 2 143/126 219.110 cents 3: 2 8/7 231.174 cents septimal whole tone 4: 3 7/6 266.871 cents septimal minor third 4: 1 243/208 269.247 cents 4: 1 567/484 274.010 cents 4: 1 168/143 278.935 cents 4: 5 33/28 284.447 cents undecimal minor third 4: 2 13/11 289.210 cents tridecimal minor third 4: 4 32/27 294.135 cents Pythagorean minor third 5: 1 63/52 332.208 cents 5: 7 11/9 347.408 cents undecimal neutral third 5: 5 27/22 354.547 cents neutral third, Zalzal wosta of al-Farabi 5: 1 16/13 359.472 cents tridecimal neutral third 5: 1 448/363 364.235 cents 5: 2 26/21 369.747 cents 6: 2 81/64 407.820 cents Pythagorean major third 6: 2 33/26 412.745 cents tridecimal major third 6: 5 14/11 417.508 cents undecimal diminished fourth or major third 6: 2 242/189 427.945 cents 6: 2 104/81 432.708 cents 6: 3 9/7 435.084 cents septimal major third, BP third 6: 1 128/99 444.772 cents 7: 1 21/16 470.781 cents narrow fourth 7: 1 189/143 482.845 cents 7: 2 297/224 488.357 cents 7: 1 117/88 493.120 cents tridecimal gentle fourth 7: 12 4/3 498.045 cents perfect fourth 8: 4 11/8 551.318 cents undecimal semi-augmented fourth 8: 2 243/176 558.457 cents 8: 2 112/81 561.006 cents 8: 2 18/13 563.382 cents tridecimal augmented fourth 8: 2 168/121 568.145 cents 8: 1 39/28 573.657 cents 8: 3 88/63 578.582 cents 8: 1 416/297 583.345 cents 9: 1 297/208 616.655 cents 9: 3 63/44 621.418 cents 9: 1 56/39 626.343 cents 9: 2 121/84 631.855 cents 9: 2 13/9 636.618 cents tridecimal diminished fifth 9: 2 81/56 638.994 cents 9: 2 352/243 641.543 cents 9: 4 16/11 648.682 cents undecimal semi-diminished fifth 10: 12 3/2 701.955 cents perfect fifth 10: 1 176/117 706.880 cents 10: 2 448/297 711.643 cents 10: 1 286/189 717.155 cents 10: 1 32/21 729.219 cents wide fifth 11: 1 99/64 755.228 cents 11: 3 14/9 764.916 cents septimal minor sixth 11: 2 81/52 767.292 cents 11: 2 189/121 772.055 cents 11: 5 11/7 782.492 cents undecimal augmented fifth 11: 2 52/33 787.255 cents tridecimal minor sixth 11: 2 128/81 792.180 cents Pythagorean minor sixth 12: 2 21/13 830.253 cents 12: 1 363/224 835.765 cents 12: 1 13/8 840.528 cents tridecimal neutral sixth 12: 5 44/27 845.453 cents neutral sixth 12: 7 18/11 852.592 cents undecimal neutral sixth 12: 1 104/63 867.792 cents 13: 4 27/16 905.865 cents Pythagorean major sixth 13: 2 22/13 910.790 cents tridecimal major sixth 13: 5 56/33 915.553 cents 13: 1 143/84 921.065 cents 13: 1 968/567 925.990 cents 13: 1 416/243 930.753 cents 13: 3 12/7 933.129 cents septimal major sixth 14: 2 7/4 968.826 cents harmonic seventh 14: 2 252/143 980.890 cents 14: 4 99/56 986.402 cents 14: 2 39/22 991.165 cents 14: 7 16/9 996.090 cents Pythagorean minor seventh 15: 6 11/6 1049.363 cents undecimal neutral seventh, 21/4-tone 15: 3 81/44 1056.502 cents 2nd undecimal neutral seventh 15: 1 448/243 1059.051 cents 15: 2 24/13 1061.427 cents tridecimal neutral seventh 15: 2 224/121 1066.190 cents 15: 2 13/7 1071.702 cents 16/3-tone 15: 1 352/189 1076.627 cents 16: 2 99/52 1114.700 cents 16: 5 21/11 1119.463 cents undecimal major seventh 16: 2 121/63 1129.900 cents 16: 2 52/27 1134.663 cents 16: 3 27/14 1137.039 cents septimal major seventh 16: 1 1408/729 1139.588 cents 16: 2 64/33 1146.727 cents 33rd subharmonic Highest number of different intervals for one interval class: 8 Average number of different intervals per interval class: 6.50000 = 13/2
SHOW/LINE/CENTS INTERVALS
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.0 : 53.3 138.6 203.9 266.9 347.4 417.5 498.0 551.3 636.6 702.0 764.9 845.5 905.9 968.8 1049.4 1119.5 1200.0 53.3 : 85.3 150.6 213.6 294.1 364.2 444.8 498.0 583.3 648.7 711.6 792.2 852.6 915.6 996.1 1066.2 1146.7 1200.0 138.6 : 65.3 128.3 208.8 278.9 359.5 412.7 498.0 563.4 626.3 706.9 767.3 830.3 910.8 980.9 1061.4 1114.7 1200.0 203.9 : 63.0 143.5 213.6 294.1 347.4 432.7 498.0 561.0 641.5 702.0 764.9 845.5 915.6 996.1 1049.4 1134.7 1200.0 266.9 : 80.5 150.6 231.2 284.4 369.7 435.1 498.0 578.6 639.0 702.0 782.5 852.6 933.1 986.4 1071.7 1137.0 1200.0 347.4 : 70.1 150.6 203.9 289.2 354.5 417.5 498.0 558.5 621.4 702.0 772.1 852.6 905.9 991.2 1056.5 1119.5 1200.0 417.5 : 80.5 133.8 219.1 284.4 347.4 427.9 488.4 551.3 631.9 702.0 782.5 835.8 921.1 986.4 1049.4 1129.9 1200.0 498.0 : 53.3 138.6 203.9 266.9 347.4 407.8 470.8 551.3 621.4 702.0 755.2 840.5 905.9 968.8 1049.4 1119.5 1200.0 551.3 : 85.3 150.6 213.6 294.1 354.5 417.5 498.0 568.1 648.7 702.0 787.3 852.6 915.6 996.1 1066.2 1146.7 1200.0 636.6 : 65.3 128.3 208.8 269.2 332.2 412.7 482.8 563.4 616.7 702.0 767.3 830.3 910.8 980.9 1061.4 1114.7 1200.0 702.0 : 63.0 143.5 203.9 266.9 347.4 417.5 498.0 551.3 636.6 702.0 764.9 845.5 915.6 996.1 1049.4 1134.7 1200.0 764.9 : 80.5 140.9 203.9 284.4 354.5 435.1 488.4 573.7 639.0 702.0 782.5 852.6 933.1 986.4 1071.7 1137.0 1200.0 845.5 : 60.4 123.4 203.9 274.0 354.5 407.8 493.1 558.5 621.4 702.0 772.1 852.6 905.9 991.2 1056.5 1119.5 1200.0 905.9 : 63.0 143.5 213.6 294.1 347.4 432.7 498.0 561.0 641.5 711.6 792.2 845.5 930.8 996.1 1059.1 1139.6 1200.0 968.8 : 80.5 150.6 231.2 284.4 369.7 435.1 498.0 578.6 648.7 729.2 782.5 867.8 933.1 996.1 1076.6 1137.0 1200.0 1049.4: 70.1 150.6 203.9 289.2 354.5 417.5 498.0 568.1 648.7 702.0 787.3 852.6 915.6 996.1 1056.5 1119.5 1200.0 1119.5: 80.5 133.8 219.1 284.4 347.4 427.9 498.0 578.6 631.9 717.2 782.5 845.5 926.0 986.4 1049.4 1129.9 1200.0 1200.0
SHOW/SPAN INTERVALS
Interval class, Interval span, Span size, Gap to prev. class: 1: 53.2729 .. 85.2997 cents 3328/3267, 32.0268 cents 33/32, 53.2729 cents 2: 123.3730 .. 150.6371 cents 64/63, 27.2641 cents 1701/1664, 38.0732 cents 3: 203.9100 .. 231.1741 cents 64/63, 27.2641 cents 33/32, 53.2729 cents 4: 266.8709 .. 294.1350 cents 64/63, 27.2641 cents 49/48, 35.6968 cents 5: 332.2082 .. 369.7468 cents 1352/1323, 37.5385 cents 1701/1664, 38.0732 cents 6: 407.8200 .. 444.7721 cents 8192/8019, 36.9521 cents 1701/1664, 38.0732 cents 7: 470.7809 .. 498.0450 cents 64/63, 27.2641 cents 2079/2048, 26.0089 cents 8: 551.3179 .. 583.3447 cents 3328/3267, 32.0268 cents 33/32, 53.2729 cents 9: 616.6553 .. 648.6821 cents 3328/3267, 32.0268 cents 88209/86528, 33.3106 cents 10: 701.9550 .. 729.2191 cents 64/63, 27.2641 cents 33/32, 53.2729 cents 11: 755.2279 .. 792.1800 cents 8192/8019, 36.9521 cents 2079/2048, 26.0089 cents 12: 830.2532 .. 867.7918 cents 1352/1323, 37.5385 cents 1701/1664, 38.0732 cents 13: 905.8650 .. 933.1291 cents 64/63, 27.2641 cents 1701/1664, 38.0732 cents 14: 968.8259 .. 996.0900 cents 64/63, 27.2641 cents 49/48, 35.6968 cents 15: 1049.3629 .. 1076.6270 cents 64/63, 27.2641 cents 33/32, 53.2729 cents 16: 1114.7003 .. 1146.7271 cents 3328/3267, 32.0268 cents 1701/1664, 38.0732 cents
SHOW DATA
Number of notes : 17 -- Interval properties -- Smallest interval : 33/32, 53.2729 cents, class 1 Average step (divided formal octave): 70.5882 cents Largest one step interval : 104/99, 85.2997 cents Average / Smallest step : 1.325030 Largest / Average step : 1.208413 Largest / Smallest step : 1.601183 Median interval of one step : 126/121, 70.1000 cents, amount: 2 Most common interval of one step : 22/21, 80.5370 cents, amount: 5 Least squares average step : 69.94334 cents, oct.: 1189.03673 cents Scale is strictly proper Step pattern alph. order: ABCDEFEABCDEGDEFE Step pattern size order : AGDCFEFAGDCFBCFEF Scale is a Constant Structure, by a margin of 26.00885 cents Scale diversity : 1.320071 Rothenberg stability : 1.000000 = 1 Lumma stability : 0.587333 Number of different interval sizes : 104 = 6.50000 / class Number of one step interval sizes : 7 Highest interval variety : 8 Mean interval variety : 6.50000 = 13/2 Median interval variety : 5 Lowest interval variety : 5 Smallest interval difference : 729/728, 2.3764 cents Most common intervals : 4/3, 498.0450 cents & inv., amount: 12 Most common triad is 0.0 498.045 701.955 cents, amount: 7 Number of recognisable fifths : 16, average 704.4238 cents Number of appreciable fifths : 15, average 703.5751 cents Best fifths form a closed circle Scale contains a complete diamond : 1 3 Length of longest harmonic part : 2 at degree 2 Length of longest subharmonic part : 1 at degree 0 Formal octave complements present : 3 = 17.6471% Scale is n-1 differentially coherent in interval classes 1 and 2 combined -- Rational properties -- Prime limit : 13 Odd number limit : 729 (O: 729 U: 729) Highest odd numerator or denominator: 33 Scale harmonicity : 0.010893 Average absolute harmonicity : 0.132528 Specific harmonicity : 0.076993 Fundamental : 1/9504, -13.2143 octaves, 0.0275 Hz. Guide tone : 108108, 16.7221 octaves, 28283816.614 Hz. Exponens Consonantiae : 1.027458E+09, 29.93643 octaves Euler's gradus suavitatis : 58 Sum of Mann's harmonic distance : 209.0, average 12.29412 Mersenne's string divisions : too high to compute Sum of van Prooijen's expressibility: 16.64984, average 0.97940 Sum of Tenney's harmonic distance : 32.49254, average 1.91133 Vogel's harmonic complexity : 28.76471 Wille's k value : 4900 Wilson's harmonic complexity : 60 Rectangular lattice diameter : 10 Triangular lattice diameter : 6 Lattice compactness : 521.44853, average 3.40816 Lattice compactness (without 2's) : 339.53060, average 2.21915 Number of different primes : 5 Prime exponents' range, average, count, tones@limit: 2: -5 .. 2 -0.88235 29 1 3: -3 .. 3 -0.29412 21 4 7: 0 .. 1 0.29412 5 3 11: -1 .. 1 0.17647 7 7 13: 0 .. 1 0.11765 2 2 Average exponent except 2's : 5 / 17 = 0.29412 Average absolute exponent except 2's: 35 / 17 = 2.05882 Scale is JI-epimorphic with val: <17 27 48 59 63| = patent
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: 10.0786 c. M:-19.4094 c. 31: 1 3 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 P D3 S SD: 9.4074 c. M: 17.2628 c. 32: 1 3 1 2 2 2 2 2 2 2 1 3 1 2 2 2 2 P D3 S SD: 10.1631 c. M:-17.0471 c. 37: 2 2 2 2 3 2 2 2 3 2 2 2 2 2 2 3 2 P D3 SD: 9.2588 c. M:-15.6722 c. 38: 2 2 2 2 3 2 3 1 3 2 2 3 2 2 2 2 3 P D3 SD: 9.7107 c. M: 14.4758 c. 41: 2 3 2 2 3 2 3 2 3 2 2 3 2 2 3 2 3 SP D3 S SD: 4.6066 c. M:-7.7688 c. 58: 3 4 3 3 4 3 4 3 4 3 3 4 3 3 4 3 4 SP D3 S SD: 4.3565 c. M:-8.7960 c. 69: 3 5 4 3 5 4 5 3 5 3 4 5 3 4 4 4 5 SP SD: 4.6242 c. M:-6.8606 c. 72: 3 5 4 4 5 4 5 3 5 4 4 5 3 4 5 4 5 SP SD: 2.9913 c. M: 5.8650 c. 89: 4 6 5 5 6 5 6 4 6 5 5 6 4 5 6 5 6 SP SD: 2.3242 c. M:-3.9853 c. 94: 4 7 5 5 6 6 6 4 7 5 5 6 5 5 6 6 6 SP SD: 2.0842 c. M:-3.9413 c. 111: 5 8 6 6 7 7 7 5 8 6 6 7 6 6 7 7 7 SP SD: 2.4038 c. M:-4.8614 c. 113: 5 8 6 6 8 6 8 5 8 6 6 8 5 6 8 6 8 SP SD: 2.2657 c. M: 4.4187 c. 118: 5 8 7 6 8 7 8 5 8 7 6 8 6 6 8 7 8 SP SD: 2.6242 c. M: 6.3693 c. 121: 5 9 7 6 8 7 8 6 8 7 6 8 6 7 8 7 8 SP SD: 2.3971 c. M:-4.3545 c. 130: 6 9 7 7 9 7 9 6 9 7 7 9 6 7 9 7 9 SP SD: 1.8963 c. M:-3.7778 c. 135: 6 10 7 7 9 8 9 6 10 7 7 9 7 7 9 8 9 SP SD: 1.2902 c. M:-3.6496 c. 152: 7 11 8 8 10 9 10 7 11 8 8 10 8 8 10 9 10 SP SD: 1.6377 c. M:-3.5326 c. 158: 7 11 9 8 11 9 11 7 11 8 9 10 8 9 10 9 11 SP SD: 2.1502 c. M:-3.3260 c. 159: 7 11 9 8 11 9 11 7 11 9 8 11 8 8 11 9 11 SP SD: 1.7020 c. M: 2.7882 c. 166: 7 12 9 9 11 10 11 7 12 9 9 11 8 9 11 10 11 SP SD: 1.3062 c. M: 2.6705 c. 176: 8 12 10 9 12 10 12 8 12 10 9 12 9 9 12 10 12 SP SD: 1.1630 c. M: 2.5268 c. 183: 8 13 10 10 12 11 12 8 13 10 10 12 9 10 12 11 12 SP SD: 1.1831 c. M:-2.2972 c. 207: 9 15 11 11 14 12 14 9 15 11 11 14 10 11 14 12 14 SP SD: 0.7240 c. M: 1.5172 c. 224: 10 16 12 12 15 13 15 10 16 12 12 15 11 12 15 13 15 SP SD: 0.6491 c. M:-1.1555 c. 270: 12 19 15 14 18 16 18 12 19 15 14 18 14 14 18 16 18 SP SD: 0.5582 c. M: 1.0621 c. 311: 14 22 17 16 21 18 21 14 22 17 16 21 16 16 21 18 21 SP SD: 0.5235 c. M: 0.9288 c. 383: 17 27 21 20 26 22 26 17 27 21 20 26 19 20 26 22 26 SP SD: 0.4850 c. M: 0.9251 c. 400: 18 28 22 21 27 23 27 18 28 22 21 27 20 21 27 23 27 SP SD: 0.4416 c. M:-0.7271 c. 477: 21 34 26 25 32 28 32 21 34 26 25 32 24 25 32 28 32 SP SD: 0.2162 c. M: 0.4428 c. 494: 22 35 27 26 33 29 33 22 35 27 26 33 25 26 33 29 33 SP SD: 0.2102 c. M:-0.4049 c. 701: 31 50 38 37 47 41 47 31 50 38 37 47 35 37 47 41 47 SP SD: 0.1623 c. M: 0.3015 c. 877: 39 62 48 46 59 51 59 39 62 48 46 59 44 46 59 51 59 SP SD: 0.1588 c. M: 0.3743 c. 971: 43 69 53 51 65 57 65 43 69 53 51 65 49 51 65 57 65 SP SD: 0.1195 c. M:-0.2075 c. 1012: 45 72 55 53 68 59 68 45 72 55 53 68 51 53 68 59 68 SP SD: 0.0793 c. M:-0.1625 c. 1489: 66 106 81 78 100 87 100 66 106 81 78 100 75 78 100 87 100 SP SD: 0.0659 c. M: 0.1221 c. 1506: 67 107 82 79 101 88 101 67 107 82 79 101 76 79 101 88 101 SP SD: 0.0626 c. M:-0.1135 c. 1983: 88 141 108 104 133 116 133 88 141 108 104 133 100 104 133 116 133 SP SD: 0.0324 c. M: 0.0672 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: 7.7986 cents 5 .. 10 match: 5 H SD: 12.6498 cents 6 .. 12 match: 6 H SD: 3.5844 cents 7 .. 14 match: 7 H SD: 7.7847 cents 8 .. 16 match: 8 H SD: 5.5487 cents 9 .. 18 match: 9 H SD: 4.9784 cents 10 .. 20 match: 10 H SD: 7.9233 cents 11 .. 22 match: 11 H SD: 3.9556 cents 12 .. 24 match: 12 H SD: 4.9663 cents 13 .. 26 match: 13 H SD: 4.6103 cents 14 .. 28 match: 14 H SD: 5.3941 cents 15 .. 30 match: 14 S SD: 6.1651 cents 16 .. 32 match: 15 SD: 4.7066 cents 17 .. 34 match: 14 SD: 6.0304 cents 18 .. 36 match: 16 S SD: 4.1341 cents 19 .. 38 match: 16 SD: 4.9264 cents 20 .. 40 match: 16 SD: 5.1281 cents 21 .. 42 match: 17 SD: 4.7759 cents 22 .. 44 match: 17 SD: 3.3381 cents 24 .. 48 match: 17 SD: 2.8735 cents 30 .. 60 match: 17 SD: 2.7246 cents 32 .. 64 match: 17 SD: 1.9607 cents 36 .. 72 match: 17 SD: 1.8868 cents 40 .. 80 match: 17 SD: 1.8458 cents 48 .. 96 match: 17 SD: 1.4480 cents 54 .. 108 match: 17 SD: 1.3594 cents 58 .. 116 match: 17 SD: 1.3405 cents 59 .. 118 match: 17 SD: 1.2678 cents 67 .. 134 match: 17 SD: 1.1898 cents 70 .. 140 match: 17 SD: 1.1165 cents 72 .. 144 match: 17 SD: 0.8083 cents 95 .. 190 match: 17 SD: 0.7243 cents 96 .. 192 match: 17 SD: 0.5705 cents 120 .. 240 match: 17 SD: 0.5642 cents 121 .. 242 match: 17 SD: 0.5569 cents 144 .. 288 match: 17 SD: 0.4027 cents 168 .. 336 match: 17 SD: 0.3734 cents 192 .. 384 match: 17 SD: 0.2984 cents 216 .. 432 match: 17 SD: 0.2043 cents 288 .. 576 match: 17 SD: 0.1493 cents 360 .. 720 match: 17 SD: 0.1354 cents 384 .. 768 match: 17 SD: 0.1300 cents 432 .. 864 match: 17 SD: 0.1238 cents 480 .. 960 match: 17 SD: 0.1025 cents 576 .. 1152 match: 17 SD: 0.0510 cents