User:Jbcristian/The Average Tuning System: Difference between revisions

Line 78:

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"

|+Top 5 Intervals

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.

[[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?''

Line 123:

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.

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.

@@ Line 78: / Line 78: @@
 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"
 |+Top 5 Intervals
@@ 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.
-[[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?''
@@ Line 123: / Line 123: @@
 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.
 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.