Scala analysis: 415_D8

  0:          1/1               0.000000 unison, perfect prime
  1:         88/81            143.497939 2nd undecimal neutral second
  2:         88/75            276.735514
  3:          4/3             498.044999 perfect fourth

  0:         25/22            221.3095 undecimal acute whole tone
  1:         88/81            143.4979 2nd undecimal neutral second
  2:         27/25            133.2376 large limma, BP small semitone, Zarlino semitone
  3:         25/22            221.3095 undecimal acute whole tone

Interval class, Number of incidences, Size:
  1:   1 27/25               133.238 cents  large limma, BP small semitone, Zarlino semitone
  1:   1 88/81               143.498 cents  2nd undecimal neutral second
  1:   1 25/22               221.309 cents  undecimal acute whole tone
  2:   1 88/75               276.736 cents  
  2:   1 27/22               354.547 cents  neutral third, Zalzal wosta of al-Farabi
  2:   1 100/81              364.807 cents  grave major third
Highest number of different intervals for one interval class: 3
Average number of different intervals per interval class: 3.00000 = 3

        1     2     3    
 0.0  : 143.5 276.7 498.0
 143.5: 133.2 354.5 498.0
 276.7: 221.3 364.8 498.0
 498.0

Interval class, Interval span, Span size, Gap to prev. class:
  1:  133.2376  .. 221.3095 cents      625/594, 88.0719 cents             27/25, 133.2376 cents
  2:  276.7355  .. 364.8074 cents      625/594, 88.0719 cents             1936/1875, 55.4260 cents

Number of notes                     : 3
-- Interval properties --
Smallest interval                   : 27/25, 133.2376 cents, class 1
Average step (divided formal octave): 166.0150 cents
Largest one step interval           : 25/22, 221.3095 cents
Average / Smallest step             : 1.246007
Largest / Average step              : 1.333069
Largest / Smallest step             : 1.661014
Median interval of one step         : 88/81, 143.4979 cents, amount: 1
Least squares average step          : 156.50743 cents, oct.:  469.52228 cents
Scale is strictly proper
Scale has trivalence property
Scale is a mode of a 2381-tone equal temperament with octave  4/3
  degrees: 686 1323 2381
Least number of segments generator  : 1191 of 249.127 cents and inv.
  number of contiguous generator circle segments: 1
Shortest superset generator         : 2332 of 487.795 cents and inv.
  generated superset size: 28 = 25 more = 933.333%
Number of contiguous 1-step segments: 0
Step pattern alph. order: ABC
Step pattern size order : MSL
Scale is sum-free (all different intervals)
Scale is a Constant Structure, by a margin of 55.42603 cents
Scale diversity                     : 1.320553
Rothenberg stability                : 1.000000 = 1
Lumma stability                     : 0.646330
Rothenberg efficiency               : 0.666667   redundancy: 0.333333
Efficiency x scale size             : 2.000000
Number of different interval sizes  : 6 = 3.00000 / class
Number of one step interval sizes   : 3
Highest interval variety            : 3
Mean interval variety               : 3.00000 = 3
Median interval variety             : 3
Lowest interval variety             : 3
Smallest interval difference        : 2200/2187, 10.2604 cents
Number of recognisable fifths       : 0
Number of recognisable fourths      : 0
Best fifths form a closed circle
Best major thirds form a closed circle
Best minor thirds form a closed circle
Formal octave complements present   : 1 = 33.3333%
2/1 octave complements present      : 0 = 0.0000%
Limited inverse transpositions with margin 88.0719 cents:
 2
-- Rational properties --
Prime limit                         : 11
Odd number limit                    : 81 (O: 27 U: 81)
Highest odd numerator or denominator: 81
Scale harmonicity                   : 0.022397
Average absolute harmonicity        : 0.090990
Specific harmonicity                : 0.104488
Fundamental                         : 1/2025, -10.9837 octaves, 0.1292 Hz.
Guide tone                          : 88,  6.4594 octaves, 23023.050 Hz.
Exponens Consonantiae               : 1.782000E+05, 17.44314 octaves
Euler's gradus suavitatis           : 30
Sum of Mann's harmonic distance     : 169.5, average 56.50000
Mersenne's string divisions         : 192829
Sum of van Prooijen's expressibility: 4.26067, average 1.42022
Sum of Tenney's harmonic distance   : 8.75169, average 2.91723
Vogel's harmonic complexity         : 22.66667
Wille's k value                     : 337
Wilson's harmonic complexity        : 33
Rectangular lattice diameter        : 7
Triangular lattice diameter         : 3
Lattice compactness                 : 20.57974, average 3.42996
Lattice compactness (without 2's)   : 16.57649, average 2.76275
Number of different primes          : 4
Prime exponents' range, average, count, tones@limit:
  2:   2 .. 3         2.66667   8
  3:  -4 ..-1        -2.00000   6     1
  5:  -2 .. 0        -0.66667   2
 11:   0 .. 1         0.66667   2     2
Average exponent except 2's         :-6 / 3 =-2.00000
Average absolute exponent except 2's: 10 / 3 = 3.33333
Scale is weakly epimorphic with val: <6 10 16 25|
Scale is weakly epimorphic with val: <6 10 19 25|
Scale is weakly epimorphic with val: <7 10 16 24|
Scale is JI-epimorphic with val: <7 11 16 24|
Scale is weakly epimorphic with val: <7 11 16 27|
Scale is weakly epimorphic with val: <7 11 17 25| = patent
Scale is weakly epimorphic with val: <7 11 19 24|
Scale is weakly epimorphic with val: <7 11 19 27|
Scale is weakly epimorphic with val: <8 11 16 23|
Scale is weakly epimorphic with val: <8 11 17 24|
Scale is weakly epimorphic with val: <8 11 19 23|
Scale is weakly epimorphic with val: <8 12 16 23|
Scale is weakly epimorphic with val: <8 12 17 27|

  3:  1 1 1  SP B ME I SD: 34.4698 c. M:-55.2945 c. 
  4:  1 1 2   P M ME S SD: 19.3951 c. M: 27.7130 c. 
  6:  2 1 3   P   T3   SD: 20.6158 c. M: 27.7130 c. 
  7:  2 2 3  SP M ME S SD: 4.5914 c. M:-7.8616 c. 
 11:  3 3 5  SP M DE S SD: 5.3085 c. M: 7.6675 c. 
 18:  5 5 8  SP M DE S SD: 2.9747 c. M: 5.1521 c. 
 25:  7 7 11  SP M DE S SD: 2.6503 c. M: 4.0453 c. 
 27:  8 7 12  SP   T3   SD: 2.3505 c. M:-4.0709 c. 
 31:  9 8 14  SP   T3   SD: 2.1804 c. M: 3.6141 c. 
 32:  9 9 14  SP M DE S SD: 2.7914 c. M: 3.4228 c. 
 34:  10 9 15  SP   T3   SD: 1.9514 c. M:-2.9859 c. 
 38:  11 10 17  SP   T3   SD: 0.9493 c. M: 1.5001 c. 
 45:  13 12 20  SP   T3   SD: 0.2218 c. M:-0.3817 c. 
 142: 41 38 63  SP   T3   SD: 0.2658 c. M:-0.3459 c. 
 153: 44 41 68  SP   T3   SD: 0.1575 c. M: 0.2693 c. 
 198: 57 53 88  SP   T3   SD: 0.0745 c. M: 0.1213 c. 
 243: 70 65 108  SP   T3   SD: 0.0301 c. M: 0.0438 c. 
 288: 83 77 128  SP   T3   SD: 0.0327 c. M: 0.0438 c. 
 531: 153 142 236  SP   T3   SD: 0.0256 c. M: 0.0438 c. 
 736: 212 197 327  SP   T3   SD: 0.0290 c. M: 0.0393 c. 
 774: 223 207 344  SP   T3   SD: 0.0254 c. M: 0.0438 c. 
 781: 225 209 347  SP   T3   SD: 0.0179 c. M:-0.0270 c. 
 826: 238 221 367  SP   T3   SD: 0.0139 c. M:-0.0231 c. 
 1024: 295 274 455  SP   T3   SD: 0.0120 c. M: 0.0182 c. 
 1069: 308 286 475  SP   T3   SD: 0.0046 c. M:-0.0079 c. 
 1312: 378 351 583  SP   T3   SD: 0.0038 c. M: 0.0063 c. 
 2381: 686 637 1058  SP   T3   SD: 0.0028 c. M: 0.0041 c. 

 1 x x 2  S SD: 701.9550 cents
 2 x x 3  S SD: 203.9100 cents
 3 x x 4  S SD: 0.0000 cents
 4 x 5 x  S SD: 109.5782 cents
 5 x 6:7  S SD: 46.4983 cents
 6 x 7:8  S SD: 4.9323 cents
 7 x 8:9  S SD: 38.8585 cents
 8:9 x 11    SD: 40.2729 cents
 9:10:11:12  H SD: 26.8912 cents
 10:11:12:13  H SD: 20.8096 cents
 11:12:13:15    SD: 13.8252 cents
 12:13:14:16  S SD: 3.6753 cents
 45:49:53:60    SD: 2.5449 cents
 57:62:67:76    SD: 1.2434 cents
 69:75:81:92    SD: 0.4031 cents
 81:88:95:108    SD: 0.2429 cents
 150:163:176:200    SD: 0.1311 cents
 231:251:271:308    SD: 0.1204 cents
 381:414:447:508    SD: 0.1155 cents