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

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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 ~~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.]]

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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.

''(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.

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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 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.]]
@@ 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.
+''(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.