User:Jbcristian/The Average Tuning System - Revision history

Jbcristian at 00:34, 17 January 2024

2024-01-17T00:34:55Z

← Older revision		Revision as of 00:34, 17 January 2024
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	The final generated system consists of 13 notes, or 14 when including the square root of 2. This selection exhibits near-complete coverage of the tuning space. Graphically, their common-tone aggregate resembles the added tones for the entire collection, which is interesting. The intervals that were left out from the average 17 due to their proximity haven't disappeared entirely; they remain popular, even surpassing those included, although the major whole-tone was removed from the main key, it still exists in some of the others.		The final generated system consists of 13 notes, or 14 when including the square root of 2. This selection exhibits near-complete coverage of the tuning space. Graphically, their common-tone aggregate resembles the added tones for the entire collection, which is interesting. The intervals that were left out from the average 17 due to their proximity haven't disappeared entirely; they remain popular, even surpassing those included, although the major whole-tone was removed from the main key, it still exists in some of the others.
			[[File:Data-comparision2.jpg\|none\|thumb\|The first image corresponds to the analysis of the full archive's interval matrix, showcasing the 17 most popular intervals. The second image depicts the same graphic process, computing the interval matrix and accumulating the repetitions vertically, but on the newly generated tuning system. The general contour of both is similar, which is intriguing since this type of tuning analysis typically provides the fingerprint for a tuning. This means the 14-note system generates a similar fingerprint to the entire database of 2.5 million notes.]]

	The system does not match any of the existing files, further exploration across matrices might uncover resemblances or close parallels at specific keys.		The system does not match any of the existing files, further exploration across matrices might uncover resemblances or close parallels at specific keys.

Jbcristian at 22:07, 7 January 2024

2024-01-07T22:07:56Z

← Older revision		Revision as of 22:07, 7 January 2024
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	The system does not match any of the existing files, further exploration across matrices might uncover resemblances or close parallels at specific keys.		The system does not match any of the existing files, further exploration across matrices might uncover resemblances or close parallels at specific keys.

			Analysis using subsets of the archive—half or a third selected randomly—still yielded the same most frequent intervals. However, for a more accurate representation of an average world tuning system, it's essential to curate the data better. This would involve handpicking the most well-known tunings that are or were actually in use, rather than relying on the full Scala archive, which contains numerous modern tunings seldom used.

Jbcristian at 21:10, 7 January 2024

2024-01-07T21:10:34Z

← Older revision		Revision as of 21:10, 7 January 2024
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	====== Statistics and tuning construction: ======		====== Statistics and tuning construction: ======
	~~TODO add all images~~

	Out of the 5,176 files, the range of system sizes extends from 2 to 579. The average system size is 17, with a median of 12. The mode is also 12, appearing 1,546 times, followed by 7-note size tunings with 715 occurrences. This signifies a diverse collection, albeit with a notable concentration of systems hovering around the 12-note mark.		Out of the 5,176 files, the range of system sizes extends from 2 to 579. The average system size is 17, with a median of 12. The mode is also 12, appearing 1,546 times, followed by 7-note size tunings with 715 occurrences. This signifies a diverse collection, albeit with a notable concentration of systems hovering around the 12-note mark.
	{\| class="wikitable"		{\| class="wikitable"

Jbcristian at 20:36, 7 January 2024

2024-01-07T20:36:34Z

← Older revision		Revision as of 20:36, 7 January 2024
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	The [[perfect fifth]] emerges as the second most popular interval, succeeded by the [[perfect fourth]] and major third.		The [[perfect fifth]] emerges as the second most popular interval, succeeded by the [[perfect fourth]] and major third.
	[[File:Scala archive intervals.jpg\|thumb\|none\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.\|~~381x381px~~]]		[[File:Scala archive intervals.jpg\|thumb\|none\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.\|alt=]]
	{\| class="wikitable"		{\| class="wikitable"
	\|+Top 5 Intervals		\|+Top 5 Intervals
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	When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.		When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.
	[[File:Intervals scan comparision.jpg\|thumb\|none\|The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.\|~~376x376px~~]]		[[File:Intervals scan comparision.jpg\|thumb\|none\|The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.\|alt=]]

	''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''		''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''
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	The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.		The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.
	[[File:Truncation comparision.jpg\|thumb\|none\|Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.\|~~409x409px~~]]		[[File:Truncation comparision.jpg\|thumb\|none\|Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.\|alt=]]

	Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.		Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.

	Additionally, the graph experiences intrinsic truncation due to its fixed resolution, significantly lower than the data range. Consequently, different notes are depicted on the same pixel, this is used to add a third dimension to the graph, highlighting note concentrations, which are always very close to some of the already favored intervals. For example, the perfect fifth has a concentration of notes next to it, hinting at systems like 12-tone equal temperament, where the fifth is 700 cents. However, without altering the graphical scale, these clusters won't even be apparent.		Additionally, the graph experiences intrinsic truncation due to its fixed resolution, significantly lower than the data range. Consequently, different notes are depicted on the same pixel, this is used to add a third dimension to the graph, highlighting note concentrations, which are always very close to some of the already favored intervals. For example, the perfect fifth has a concentration of notes next to it, hinting at systems like 12-tone equal temperament, where the fifth is 700 cents. However, without altering the graphical scale, these clusters won't even be apparent.
			[[File:Key Analysis Comparison.jpg\|none\|thumb\|Both graphics represent the analysis of the initial keys, displaying the same dataset. However, the first graphic features a vertical logarithmic scale, while the second employs a linear scale. Presented as a heat map, red areas denote note concentrations (which are not visible in the linear view), while blue indicates fewer notes.]]

	The generated systems employing the 17 most frequent intervals, are symmetric in both cases, reflecting a mirror image via the square root of 2. They comprise half superparticular intervals and half their reduced inversions, the perfect fourth and fifth, major third and minor sixth, minor third and major sixth, etc.		The generated systems employing the 17 most frequent intervals, are symmetric in both cases, reflecting a mirror image via the square root of 2. They comprise half superparticular intervals and half their reduced inversions, the perfect fourth and fifth, major third and minor sixth, minor third and major sixth, etc.

Jbcristian at 20:23, 7 January 2024

2024-01-07T20:23:52Z

← Older revision		Revision as of 20:23, 7 January 2024
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	\|}		\|}
	While some files span multiple octaves or include [[Subharmonic\|non-reduced intervals below the unison]], these instances are relatively rare. Most are periodic tunings in alignment with the [[octave]], the archive's most common interval. (Note: rather than relatively rare, some are intentionally wrong, since scala file definition specifies the omission of the 1, and conclude with the equave, implementations may totally ignore those values)		While some files span multiple octaves or include [[Subharmonic\|non-reduced intervals below the unison]], these instances are relatively rare. Most are periodic tunings in alignment with the [[octave]], the archive's most common interval. (Note: rather than relatively rare, some are intentionally wrong, since scala file definition specifies the omission of the 1, and conclude with the equave, implementations may totally ignore those values)
	[[File:Scala archive intervals.jpg\|thumb\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.]]

	In a direct analysis of the files, the first key from each tuning, totaling 87,558 notes, reveals the octave as the most common, appearing with its exact representation in 4,481 total files and with close variations in practically all tunings.		In a direct analysis of the files, the first key from each tuning, totaling 87,558 notes, reveals the octave as the most common, appearing with its exact representation in 4,481 total files and with close variations in practically all tunings.

	The [[perfect fifth]] emerges as the second most popular interval, succeeded by the [[perfect fourth]] and major third.		The [[perfect fifth]] emerges as the second most popular interval, succeeded by the [[perfect fourth]] and major third.
			[[File:Scala archive intervals.jpg\|thumb\|none\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.\|381x381px]]
	{\| class="wikitable"		{\| class="wikitable"
	\|+Top 5 Intervals		\|+Top 5 Intervals
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	When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.		When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.
	[[File:Intervals scan comparision.jpg\|thumb\|The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.]]		[[File:Intervals scan comparision.jpg\|thumb\|none\|The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.\|376x376px]]

	''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''		''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''
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	The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.		The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.
	[[File:Truncation comparision.jpg\|thumb\|Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.]]		[[File:Truncation comparision.jpg\|thumb\|none\|Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.\|409x409px]]

	Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.		Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.
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	The first ~500 most frequent intervals comprise just, rational, and integer ratio intervals before cent-defined intervals like the octave at 1200 cents appear.		The first ~500 most frequent intervals comprise just, rational, and integer ratio intervals before cent-defined intervals like the octave at 1200 cents appear.

	====== How to: ======		======How to:======
	The program developed for this analysis is open-source and available at [LINK]. It's designed for straightforward usage—simply load any .scl file or files, and it will promptly conduct and showcase statistics on them. The analysis comes in two modes: 'direct' examines files as they are, focusing on the first key, while 'full' generates interval matrices for all files. Notably, the 'full' analysis uses a fixed equave of 2:1, a setting implemented after discovering that 95% of the database concludes with a 2:1 equave. This equave parameter can be adjusted within the code for further exploration and customization."		The program developed for this analysis is open-source and available at [LINK]. It's designed for straightforward usage—simply load any .scl file or files, and it will promptly conduct and showcase statistics on them. The analysis comes in two modes: 'direct' examines files as they are, focusing on the first key, while 'full' generates interval matrices for all files. Notably, the 'full' analysis uses a fixed equave of 2:1, a setting implemented after discovering that 95% of the database concludes with a 2:1 equave. This equave parameter can be adjusted within the code for further exploration and customization."

Jbcristian at 20:14, 7 January 2024

2024-01-07T20:14:12Z

← Older revision		Revision as of 20:14, 7 January 2024
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	You can attempt to correct this by equally limiting the number of digits, which would effectively reduce the number of individual distinct intervals. However, since truncation occurs in their decimal format, an uneven definition loss of musical notes is observed due to their original distribution, which is nonlinear (without repetitions).		You can attempt to correct this by equally limiting the number of digits, which would effectively reduce the number of individual distinct intervals. However, since truncation occurs in their decimal format, an uneven definition loss of musical notes is observed due to their original distribution, which is nonlinear (without repetitions).

	The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically. ~~The octave~~, the ~~highest peak~~, which ~~appears 4~~,~~379 times~~ in the ~~direct analysis and 72~~,~~146 times after computing all matrices, can be ignored for better displaying~~ the ~~rest~~.		The graph represents the tuning space horizontally and accumulates identical exact repetitions vertically.
			[[File:Truncation comparision.jpg\|thumb\|Both graphics portray identical data, but the second one illustrates the data after truncation (with a maximum error of approximately 0.2 cents). Both visuals display the top 17 intervals, which remained consistent even after truncation. This reduction resulted in 242,538 unique intervals being compressed to just 9,997. The logarithmic view in the graphic also highlights the uneven definition loss of musical notes post-truncation, which was executed on the decimal data.]]

	Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.		Progressively truncating the notes in this way, doesn't significantly alter popularity, even a 2-cent error proved insufficient to dislodge any peak prominence.

Jbcristian at 19:57, 7 January 2024

2024-01-07T19:57:34Z

← Older revision		Revision as of 19:57, 7 January 2024
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	There are precision issues affecting interval categorization, resulting from the conversion of fractions and cents, the dual languages of scala files, to a common decimal representation. This inherits machine number problems. When calculating the complete matrix of equal division systems, where a size of any given number should imply the same diversity, the precision nuances in floating-point arithmetic may lead to some being counted as different.		There are precision issues affecting interval categorization, resulting from the conversion of fractions and cents, the dual languages of scala files, to a common decimal representation. This inherits machine number problems. When calculating the complete matrix of equal division systems, where a size of any given number should imply the same diversity, the precision nuances in floating-point arithmetic may lead to some being counted as different.

	Another problem arises in categorizing cent tunings. Some files may refer to the same note, but due to differences in the amount of digits in their definitions, no program will consider them equal.		Another problem arises in categorizing cent tunings. Some files may refer to the same note, but due to differences in the amount of digits in their definitions, no program will consider them equal. ''(701.955 != 701.95)''

	You can attempt to correct this by equally limiting the number of digits, which would effectively reduce the number of individual distinct intervals. However, since truncation occurs in their decimal format, an uneven definition loss of musical notes is observed due to their original distribution, which is nonlinear (without repetitions).		You can attempt to correct this by equally limiting the number of digits, which would effectively reduce the number of individual distinct intervals. However, since truncation occurs in their decimal format, an uneven definition loss of musical notes is observed due to their original distribution, which is nonlinear (without repetitions).

Jbcristian at 19:51, 7 January 2024

2024-01-07T19:51:36Z

← Older revision		Revision as of 19:51, 7 January 2024
Line 108:		Line 108:

	When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.		When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.
			[[File:Intervals scan comparision.jpg\|thumb\|The two graphics present distinct datasets. The first graphic represents the scan of the initial key in each file, while the second illustrates the scan subsequent to computing all matrices. Both graphics showcase the top 17 intervals, which exhibit remarkable similarity. Each graph encompasses a single octave, with both vertical and horizontal axes set to a logarithmic scale.]]

	''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''		''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''

Jbcristian at 19:35, 7 January 2024

2024-01-07T19:35:19Z

← Older revision		Revision as of 19:35, 7 January 2024
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	\|206		\|206
	\|}		\|}
	While some files span multiple octaves or include [[Subharmonic\|non-reduced intervals below the unison]], these instances are relatively rare. Most are periodic tunings in alignment with the [[octave]], the archive's most common interval. (Note: rather than relatively rare, some are ~~purposefully~~ wrong, since scala file definition ~~asks for omit~~ the 1, and ~~end~~ with the equave)		While some files span multiple octaves or include [[Subharmonic\|non-reduced intervals below the unison]], these instances are relatively rare. Most are periodic tunings in alignment with the [[octave]], the archive's most common interval. (Note: rather than relatively rare, some are intentionally wrong, since scala file definition specifies the omission of the 1, and conclude with the equave, implementations may totally ignore those values)
	[[File:Scala archive intervals.jpg\|thumb\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.]]		[[File:Scala archive intervals.jpg\|thumb\|Distribution of intervals. The two graphics depict identical data. The first graphic displays both vertical and horizontal axes on a linear scale, while the second utilizes a logarithmic scale for the vertical axis. This logarithmic scale highlights intervals that occur only once, significantly beyond the octave, as well as those appearing below a value of 1.]]

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	When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.		When calculating all added tones, the complete interval matrix only for the octave-ending tunings yields a total of 2,641,310 intervals, and the list of the most frequent remains largely unchanged.

			''(Why is it important to calculate the interval matrix and added tones to determine the most common intervals?''

			''Take this periodic tuning, for example: 16/15 6/5 8/5 9/5 2/1.''

			''If you're not very familiar with intervals, simply seeing the initial key doesn't tell you anything. However, upon computing the matrix for this 5-note periodic tuning, it reveals 14 unique intervals. Among these, the most common intervals are the fifth (3/2), the fourth (4/3), the major whole tone (9/8), and the Pythagorean minor seventh (16/9) – all of which aren't explicitly mentioned in the first key.)''

	There are precision issues affecting interval categorization, resulting from the conversion of fractions and cents, the dual languages of scala files, to a common decimal representation. This inherits machine number problems. When calculating the complete matrix of equal division systems, where a size of any given number should imply the same diversity, the precision nuances in floating-point arithmetic may lead to some being counted as different.		There are precision issues affecting interval categorization, resulting from the conversion of fractions and cents, the dual languages of scala files, to a common decimal representation. This inherits machine number problems. When calculating the complete matrix of equal division systems, where a size of any given number should imply the same diversity, the precision nuances in floating-point arithmetic may lead to some being counted as different.

Jbcristian at 19:09, 7 January 2024

2024-01-07T19:09:19Z

@@ Line 74: / Line 74: @@
 |}
 While some files span multiple octaves or include [[Subharmonic|non-reduced intervals below the unison]], these instances are relatively rare. Most are periodic tunings in alignment with the [[octave]], the archive's most common interval. (Note: rather than relatively rare, some are purposefully wrong, since scala file definition asks for omit the 1, and end with the equave)
 In a direct analysis of the files, the first key from each tuning, totaling 87,558 notes, reveals the octave as the most common, appearing with its exact representation in 4,481 total files and with close variations in practically all tunings.
 The [[perfect fifth]] emerges as the second most popular interval, succeeded by the [[perfect fourth]] and major third.
+{| class="wikitable"
 Assuming all tunings are periodic, cyclical pitch sets, the octave is identified as the [[interval of equivalence]] in 4,379 tuning files. The next most common [[equave]] is the [[Tritave|twelfth]], with only 93 files.