Path-based goodness: Difference between revisions

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# 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.

# 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.

# 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.

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 # 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.
 # 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.
+# 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.