Path-based goodness

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 1-140:

EDO	GPV	Goodness	Octave stretch
1	1.06	10.8	+0.00
2	2.03	10.6	+0.00
3	3.12	13.5	-0.03
4	4.03	12.1	+0.01
5	5.06	13.7	+0.11
6	6.03	12.6	-0.47
7	7.03	14.1	+10.58
8	8.12	14.4	-0.09
9	8.97	15.9	-0.10
10	10.12	17.3	+0.19
11	11.03	17.4	-0.17
12	11.97	24.9	+0.49
13	13.03	15.2	+0.00
14	13.91	21.1	+9.93
15	15.12	22.6	-3.43
16	16.00	20.2	+4.52
17	17.12	26.2	-2.36
18	18.09	19.7	-8.70
19	18.97	29.5	+3.64
20	19.94	19.9	+2.01
21	21.00	23.1	-0.71
22	22.06	30.3	-1.04
23	22.91	21.5	+9.07
24	24.00	30.7	+0.13
25	25.06	23.7	+0.87
26	25.94	30.6	+3.03
27	27.12	32.8	-4.21
28	27.94	24.3	+0.04
29	28.94	31.8	+3.28
30	30.03	24.0	-2.72
31	31.00	37.4	+0.36
32	32.06	28.1	-2.29
33	32.88	26.1	+4.29
34	34.06	36.7	-1.67
35	34.94	28.8	+2.80
36	36.00	34.4	+0.45
37	37.06	32.8	-0.89
38	37.91	35.0	+3.82
39	39.09	35.1	-4.02
40	39.97	30.2	+1.10
41	41.00	40.4	+0.12
42	42.12	30.9	-4.47
43	43.00	37.2	-1.03
44	44.00	33.3	-0.71
45	44.88	35.4	+4.03
46	45.94	40.7	+0.14
47	46.91	30.2	+1.23
48	48.00	35.6	+0.37
49	49.12	37.4	-3.50
50	49.94	39.4	+1.33
51	51.09	34.4	-2.03
52	51.97	31.2	+0.21
53	53.00	42.6	+0.11
54	54.12	33.9	-2.76
55	54.91	35.9	+2.81
56	56.00	38.6	-0.10
57	56.94	36.7	+0.62
58	58.06	42.2	-1.51
59	59.09	32.8	-1.89
60	59.94	40.6	+1.56
61	61.09	35.8	-2.59
62	61.88	40.1	+1.39
63	63.00	40.2	-0.32
64	63.88	36.4	+3.48
65	65.09	41.5	-0.50
66	66.12	35.4	-3.41
67	67.09	37.8	-0.39
68	68.00	42.0	-0.71
69	68.91	36.8	+1.93
70	70.06	38.4	-0.59
71	71.09	36.3	-2.01
72	71.94	44.0	+0.93
73	73.12	38.5	-2.34
74	74.00	38.0	-0.00
75	75.12	40.1	-1.46
76	75.91	36.9	+2.05
77	77.00	42.9	+0.21
78	78.09	38.9	-0.77
79	78.88	39.7	+1.98
80	80.06	42.7	-1.03
81	80.88	39.4	+1.23
82	81.97	40.9	+0.56
83	83.12	37.1	-2.07
84	84.03	42.5	-0.03
85	85.12	38.8	-1.90
86	85.88	39.4	+1.99
87	87.06	43.6	-0.26
88	87.88	38.3	+1.80
89	89.00	42.0	-0.24
90	90.06	39.6	-1.06
91	90.88	41.3	+2.00
92	92.12	40.2	-1.97
93	93.06	39.2	+0.17
94	94.00	43.6	+0.14
95	95.09	40.7	-1.55
96	95.94	42.0	+0.68
97	97.00	39.4	-0.01
98	97.94	39.9	+1.21
99	99.09	43.5	-0.69
100	99.91	38.8	+1.43
101	100.88	40.5	+1.51
102	102.12	40.6	-1.34
103	102.97	43.4	+0.73
104	104.06	41.9	-0.72
105	104.97	38.5	+0.37
106	106.03	42.3	+0.03
107	107.12	39.5	-1.89
108	108.09	40.9	-0.40
109	109.03	41.9	-0.04
110	109.88	40.2	+1.53
111	111.06	43.6	-0.68
112	111.91	39.5	+1.42
113	113.00	43.2	+0.41
114	114.06	41.3	-0.86
115	114.91	41.4	+0.38
116	115.97	40.7	+0.00
117	117.12	38.7	-1.48
118	118.03	43.6	+0.17
119	119.12	39.2	-1.37
120	120.00	41.3	-0.00
121	121.09	43.3	-0.76
122	121.88	41.6	+1.17
123	122.97	41.1	+0.15
124	124.12	40.1	-0.38
125	124.91	43.0	+0.54
126	126.12	41.4	-1.39
127	126.91	41.0	+0.64
128	127.97	42.3	-0.04
129	128.88	40.1	+1.17
130	130.00	43.8	-0.01
131	131.06	40.4	-0.57
132	131.88	42.2	+1.28
133	133.06	41.9	-0.50
134	133.88	41.3	+1.05
135	135.03	42.0	-0.36
136	136.09	41.3	-0.95
137	137.03	42.7	-0.05
138	138.12	41.7	-1.41
139	138.97	41.0	+0.36
140	140.00	43.6	+0.09