Path-based goodness

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Path-based goodness is a metric for judging the accuracy and simplicity of temperaments. To compute path-based goodness, we can apply the following steps:

  1. Choose values for the parameters error_power, complexity_fondness, inaccuracy_fondness, and target_score. The canonical values are:
    • error_power = 1.5,
    • complexity_fondness = 0.9,
    • inaccuracy_fondness = 0.98,
    • target_score = 0.002.
  2. Let g be a variable. Assign to every JI interval (except 1/1) n/d with error e (in cents) a score equal to inaccuracy_fondness^(e^error_power) * complexity_fondness^(n+d) / g.
  3. Assign to every ordered list of JI intervals a score equal to the product of the scores of all of its steps.
  4. Assign to every temperament with tuning a score equal to the sum of the scores of all the ordered lists that yield a comma when the intervals are stacked.
  5. Assign to every temperament with tuning a goodness equal to the value g such that the score is equal to the target_score.
  6. Assign to every temperament a goodness equal to the largest goodness of that temperament across all generator tunings. In this way, path-based goodness also provides optimal tunings for temperaments.

Path-based goodness observes several properties that make it easier to compare across subgroups than Smith or Dirichlet badness.

  1. If we have two temperaments A and B which are equal except that in temperament A, some prime is free, while in temperament B, the prime is tuned arbitrarily close to temperament A's optimal tuning for that prime but related to the other primes by an arbitrarily complex comma, A and B have the same goodness.
  2. If we have two temperaments A and B which are equal except that B gives a mapping to a prime that A does not, but the tuned value for that prime is arbitrarily inaccurate, then A and B have the same goodness.
  3. Just intonation has a defined goodness, and (at least under the canonical parameters) some tempered systems have larger goodness.

Path-based goodness can also be calculated for scales without JI interpretations; we assign to each note and just interval a tempered interval rooted on that note, and compute the goodness of the best assignment.

Path-based goodnesses and octave stretches for the equal temperaments 7-41:

(these use target_score 0.02 instead of 0.002; the order remains mostly unchanged, but all the values here are smaller than they would be at the correct target_score.) 

 1. 41et: goodness 30.8. 2:  0.10. score <3, <6:  3.9%, 11.5%
 2. 31et: goodness 28.2. 2:  0.33. score <3, <6:  4.2%, 13.0%
 3. 34et: goodness 26.0. 2: -1.06. score <3, <6:  4.6%, 13.9%
 4. 36et: goodness 26.0. 2:  0.44. score <3, <6:  4.5%, 13.2%
 5. 37et: goodness 24.3. 2: -0.68. score <3, <6:  4.6%, 13.0%
 6. 38et: goodness 23.3. 2:  2.26. score <3, <6:  4.8%, 13.6% which is likely easy to navigate due to sharing features with 19edo
 7. 24et: goodness 22.8. 2:  0.03. score <3, <6:  4.9%, 15.3%
 8. 29et: goodness 22.8. 2:  2.45. score <3, <6:  4.6%, 14.0% Note that 29et is identical to 58et on the 2.3.7/5.11/5.13/5 subgroup
 9. 40et: goodness 22.7. 2:  1.51. score <3, <6:  4.4%, 12.4% which is likely easy to navigate due to supporting orwell and diminished
10. 22et: goodness 22.5. 2: -0.51. score <3, <6:  4.8%, 15.1% the best temperament below 29
11. 26et: goodness 22.4. 2:  2.57. score <3, <6:  4.5%, 15.0%
12. 27et: goodness 22.4. 2: -2.56. score <3, <6:  5.0%, 14.4%
13. 19et: goodness 21.5. 2:  3.75. score <3, <6:  5.5%, 17.7% the best temperament below 22
14. 32et: goodness 20.2. 2: -0.45. score <3, <6:  4.9%, 14.3% which is likely easy to navigate due to supporting pajara
15. 39et: goodness 20.1. 2: -1.63. score <3, <6:  5.6%, 14.4%
16. 33et: goodness 19.2. 2: -0.02. score <3, <6:  4.9%, 13.3% which is likely placed so high due to higher-limit accuracy
17. 35et: goodness 18.7. 2:  1.97. score <3, <6:  5.2%, 14.7%
18. 17et: goodness 17.4. 2: -2.42. score <3, <6:  7.1%, 21.4%
19. 30et: goodness 16.9. 2: -2.47. score <3, <6:  5.4%, 14.8% which shares many features with 15et and lands close to it in badness
20. 28et: goodness 16.7. 2: -0.05. score <3, <6:  6.5%, 16.8%
21. 21et: goodness 16.5. 2: -0.21. score <3, <6:  5.8%, 17.6%
22. 12et: goodness 16.3. 2:  0.35. score <3, <6:  8.7%, 28.2% octave-stretched for an even worse 5/1 because it was never that good anyway
23. 25et: goodness 15.9. 2:  0.90. score <3, <6:  7.2%, 19.0%
24. 15et: goodness 15.8. 2: -3.91. score <3, <6:  5.9%, 19.6%
25. 20et: goodness 14.7. 2:  1.79. score <3, <6:  6.3%, 16.2%
26. 16et: goodness 14.6. 2:  4.64. score <3, <6:  6.3%, 18.1%
27. 14et: goodness 12.6. 2:  9.46. score <3, <6:  8.2%, 21.3% near 22edt even though we're using the wrong mapping for prime 5, thus ignoring it
28. 10et: goodness 12.6. 2:  3.35. score <3, <6:  6.7%, 15.6%
29. 23et: goodness 12.0. 2:  0.03. score <3, <6:  9.2%, 21.7%
30. 18et: goodness 11.4. 2:  1.48. score <3, <6:  8.4%, 18.0%
31. 13et: goodness  9.8. 2:  4.95. score <3, <6:  9.2%, 26.7%
32.  9et: goodness  9.7. 2: -0.26. score <3, <6: 13.1%, 38.8%
33.  7et: goodness  8.6. 2: 10.80. score <3, <6: 12.1%, 38.2% near 11edt
34.  8et: goodness  6.9. 2: -0.07. score <3, <6: 21.9%, 71.7%
35. 11et: goodness  6.9. 2: -0.28. score <3, <6: 21.9%, 83.5%
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