Path-based goodness - Revision history

Hkm at 18:56, 6 September 2025

2025-09-06T18:56:40Z

← Older revision		Revision as of 18:56, 6 September 2025
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	#* <code>inaccuracy_fondness</code> = 0.98,		#* <code>inaccuracy_fondness</code> = 0.98,
	#* <code>target_score</code> = 0.002.		#* <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 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.		# 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.

Hkm at 00:29, 26 August 2025

2025-08-26T00:29:25Z

← Older revision		Revision as of 00:29, 26 August 2025
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	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. ~~brb going to put that on the path-based goodness page~~		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).		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).

Hkm at 20:17, 17 August 2025

2025-08-17T20:17:09Z

← Older revision		Revision as of 20:17, 17 August 2025
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	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. brb going to put that on the path-based goodness page		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. brb going to put that on the path-based goodness page

	The parameters are calibrated according to what ~~I believe~~ best reflects the preferences of composers (how much they use simple intervals, and how much ~~unjustness~~ 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 ~~inaccuracy fondness~~. If one is interested in essentially tempered chords more than long comma pumps, one may want to lower target_score. hkm has found that changing target_score 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 inaccuracy_fondness, 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, because it is not entirely agreed upon whether they do.		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 ==		== Method of computation ==

Hkm at 20:11, 17 August 2025

2025-08-17T20:11:07Z

← Older revision		Revision as of 20:11, 17 August 2025
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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:		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. brb going to put that on the path-based goodness page

			The parameters are calibrated according to what I believe best reflects the preferences of composers (how much they use simple intervals, and how much unjustness 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 inaccuracy fondness. If one is interested in essentially tempered chords more than long comma pumps, one may want to lower target_score. hkm has found that changing target_score 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 inaccuracy_fondness, 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, because it is not entirely agreed upon whether they do.

			== 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:		# 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:
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	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 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, ideal GPVs, 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;"		{\| class="wikitable sortable" style="text-align: right;"
	\|+		\|+

Hkm at 06:09, 17 August 2025

2025-08-17T06:09:51Z

← Older revision		Revision as of 06:09, 17 August 2025
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	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 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;"		{\| class="wikitable sortable" style="text-align: right;"
	\|+		\|+

Hkm: accidentally removed this category earlier

2025-08-11T18:39:23Z

accidentally removed this category earlier

← Older revision		Revision as of 18:39, 11 August 2025
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	[[Category:Math]]		[[Category:Math]]
			[[Category:Badness]]

BudjarnLambeth: Uncategorised page

2025-08-10T23:43:01Z

Uncategorised page

← Older revision		Revision as of 23:43, 10 August 2025
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	\| 140 \|\| 140.00 \|\| 43.6 \|\| +0.09		\| 140 \|\| 140.00 \|\| 43.6 \|\| +0.09
	\|}		\|}

			[[Category:Math]]

Hkm at 23:06, 6 August 2025

2025-08-06T23:06:50Z

← Older revision		Revision as of 23:06, 6 August 2025
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	=== Path-based goodnesses and octave stretches for the equal temperaments 1-140: ===		=== Path-based goodnesses and octave stretches for the equal temperaments 1-140: ===
	{\| class="wikitable sortable" style="text-align: right;"		{\| class="wikitable sortable" style="text-align: right;"
	\|+ ~~Temperament Rankings~~		\|+
	! style="width: 25%;" \| EDO		! style="width: 25%;" \| EDO
	! style="width: 25%;" \| GPV		! style="width: 25%;" \| GPV

Hkm at 23:05, 6 August 2025

2025-08-06T23:05:58Z

← Older revision		Revision as of 23:05, 6 August 2025
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	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 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 1-90: ===		=== Path-based goodnesses and octave stretches for the equal temperaments 1-140: ===
	{\| class="wikitable sortable" style="text-align: right;"		{\| class="wikitable sortable" style="text-align: right;"
	\|+ Temperament Rankings		\|+ Temperament Rankings

Hkm at 22:57, 6 August 2025

2025-08-06T22:57:29Z

@@ Line 206: / Line 206: @@
 |-
 | 90 || 90.06 || 39.6 ||  -1.06
 |}