Path-based goodness: Difference between revisions

(4 intermediate revisions by the same user not shown)

Line 1:

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:

Path-based goodness is a measure of your ability as a composer to create music recognizable as tempered JI. we assume a composer wants to return to previously written pitches; otherwise, they would just use JI. so their music contains multiple paths from a tone to the same tone. They are likelier to use a path if the intervals within it are simple and close to just. So path-based goodness is a measure of how much total ''option'' (which I use as a mass noun here because there is an infinite number of options and some paths are more of a compositional option for them than others) a composer has when writing music.

The parameters are calibrated according to what best reflects the preferences of composers (how much they use simple intervals, and how much error they're willing to tolerate). If a composer works under different aesthetic criteria, they may want to use different parameters. For example, if one uses timbres less sensitive to mistuning, one may want to increase <code>inaccuracy_fondness</code>. If one is interested in essentially tempered chords more than long comma pumps, one may want to lower <code>target_score</code>. hkm has found that changing <code>target_score</code> by an order of magnitude affects rankings minimally; changing the other parameters changes the rankings in a predictable and small way. (For example, if we increase <code>inaccuracy_fondness</code>, all scores go up, but scores for less accurate systems increase more.) We do not treat some JI intervals as having more room for error than others, both because it is not entirely agreed upon whether they do, and because it should have little change in the results (since simple intervals can be stacked upon other intervals to become complex intervals).

== Method of computation ==

To compute path-based goodness, we can apply the following steps:

# Choose values for the parameters <code>error_power</code>, <code>complexity_fondness</code>, <code>inaccuracy_fondness</code>, and <code>target_score</code>. The canonical values are:

Line 6:

Line 11:

#* <code>inaccuracy_fondness</code> = 0.98,

#* <code>target_score</code> = 0.002.

# Let ''g'' be a variable. Assign to every JI interval (except 1/1) n/d with error e (in cents) a score equal to <code>inaccuracy_fondness</code>^(e^<code>error_power</code></code>) * <code>complexity_fondness</code>^(n+d) / ''g''.

# Let ''g'' be a variable. Assign to every JI interval (except 1/1) n/d with error r (in cents) a score equal to <code>inaccuracy_fondness</code>^(r^<code>error_power</code></code>) * <code>complexity_fondness</code>^(n+d) / ''g''.

# Assign to every ordered list of JI intervals a score equal to the product of the scores of all of its steps.

# 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.

Line 19:

Line 24:

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: ===

== Path-based goodnesses, ideal GPVs, and octave stretches for the equal temperaments 1-140: ==

{| class="wikitable sortable" style="text-align: right;"

|+

@@ Line 1: / Line 1: @@
-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:
+Path-based goodness is a measure of your ability as a composer to create music recognizable as tempered JI. we assume a composer wants to return to previously written pitches; otherwise, they would just use JI. so their music contains multiple paths from a tone to the same tone. They are likelier to use a path if the intervals within it are simple and close to just. So path-based goodness is a measure of how much total ''option'' (which I use as a mass noun here because there is an infinite number of options and some paths are more of a compositional option for them than others) a composer has when writing music.
+The parameters are calibrated according to what best reflects the preferences of composers (how much they use simple intervals, and how much error they're willing to tolerate). If a composer works under different aesthetic criteria, they may want to use different parameters. For example, if one uses timbres less sensitive to mistuning, one may want to increase <code>inaccuracy_fondness</code>. If one is interested in essentially tempered chords more than long comma pumps, one may want to lower <code>target_score</code>. hkm has found that changing <code>target_score</code> by an order of magnitude affects rankings minimally; changing the other parameters changes the rankings in a predictable and small way. (For example, if we increase <code>inaccuracy_fondness</code>, all scores go up, but scores for less accurate systems increase more.) We do not treat some JI intervals as having more room for error than others, both because it is not entirely agreed upon whether they do, and because it should have little change in the results (since simple intervals can be stacked upon other intervals to become complex intervals).
+== Method of computation ==
+To compute path-based goodness, we can apply the following steps:
 # Choose values for the parameters <code>error_power</code>, <code>complexity_fondness</code>, <code>inaccuracy_fondness</code>, and <code>target_score</code>. The canonical values are:
@@ Line 6: / Line 11: @@
 #* <code>inaccuracy_fondness</code> = 0.98,
 #* <code>target_score</code> = 0.002.
-# Let ''g'' be a variable. Assign to every JI interval (except 1/1) n/d with error e (in cents) a score equal to <code>inaccuracy_fondness</code>^(e^<code>error_power</code></code>) * <code>complexity_fondness</code>^(n+d) / ''g''.
+# Let ''g'' be a variable. Assign to every JI interval (except 1/1) n/d with error r (in cents) a score equal to <code>inaccuracy_fondness</code>^(r^<code>error_power</code></code>) * <code>complexity_fondness</code>^(n+d) / ''g''.
 # Assign to every ordered list of JI intervals a score equal to the product of the scores of all of its steps.
 # 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.
@@ Line 19: / Line 24: @@
 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: ===
+== Path-based goodnesses, ideal GPVs, and octave stretches for the equal temperaments 1-140: ==
 {| class="wikitable sortable" style="text-align: right;"
 |+