Bozuji tuning: Difference between revisions

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== Summary ==

Bozuji tuning is a [[5-limit|5-limit just intonation]] tuning set with specified intervals proposed by [[Bostjan Zupancic]] ('''Bo'''stjan '''Zu'''pancic '''J'''ust '''I'''ntonation). The approach to generating the intervals is somewhat unique, as all intervals were generated by choosing adaptive step sizes (which have been shown to work with software keyboards) and stepping through scales with different tonalities. The tuning contains 23 intervals per [[octave]], and it is intended to be an expansion of [[wikipedia:Ptolemy's_intense_diatonic_scale|Ptolemy's Intense Diatonic Scale]].

Bozuji tuning is a [[5-limit|5-limit just intonation]] tuning set with specified intervals proposed by [[Bostjan Zupancic]] ('''Bo'''stjan '''Zu'''pancic '''J'''ust '''I'''ntonation), which are closely related to the tones available in meantone temperament. The [[Bathomotonic|approach]] to generating the intervals is somewhat unique, as all intervals were generated by choosing adaptive step sizes (which have been shown to work with software keyboards, see AdaptiveJI) and stepping through scales with different tonalities. The tuning contains 23 intervals per [[octave]], and it is intended to be an expansion of [[wikipedia:Ptolemy's_intense_diatonic_scale|Ptolemy's Intense Diatonic Scale]].

== Interval Base ==

The basis for the tuning is the [[~~Diatonic, Chromatic, Enharmonic, Subchromatic~~|diatonic scale]]. Using seven notes as a rough framework to step through a scale and ultimately get to a perfect octave. The rough intervals are simply those given by classical western music theory scale degrees: unison (1), the second (2), the third (3), the fourth (4), the fifth (5), the sixth (6), and the seventh (7). The octave is taken for granted as exactly double the frequency of unison, and then the scale repeats the same intervals from there, such that the ninth is equivalent to the second, the tenth is equivalent to the third, and so on.

The basis for the tuning is the [[diatonic|diatonic scale]]. Using seven notes as a rough framework to step through a scale and ultimately get to a perfect octave. The rough intervals are simply those given by classical western music theory scale degrees: unison (1), the second (2), the third (3), the fourth (4), the fifth (5), the sixth (6), and the seventh (7). The octave is taken for granted as exactly double the frequency of unison, and then the scale repeats the same intervals from there, such that the ninth is equivalent to the second, the tenth is equivalent to the third, and so on.

This approach considers three general kinds of scale degrees: reference, perfect, and imperfect.

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== Step Sizes ==

Ptolemy's work generated a scale with seven degrees and used three step sizes between adjacent intervals. Zarlino later expanded Ptolemy's scale into a more generalized 5-limit just intonation tuning with four step sizes. This approach proposes a matrix of two kinds each of five types of step sizes.

The "type" of the step is determined by the musical context and the desired tonality of the interval after the step versus the tonality of the proceeding step. For example, going from the perfect interval to a major interval is achieved by using a whole step, whilst a minor interval is achieved by using a half step, a diminished interval by a quarter step, and an augmented interval by a grown step. The "kind" of step is determined by the positions of the intervals within the scale, in order to minimize the creation of new intervals by accounting for commas and such.

The two kinds of step sizes are lesser steps and greater steps. The lesser steps being indicated with a lowercase letter and the greater steps indicated with an uppercase letter. The five types of step sizes are: quarter (q), half (h), whole (w), grown (g), and expanded (x).

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In Ptolemy's system, whole and half steps were used to step through the major diatonic scale. In this system, the steps are used to generate scales based on the choice of which step to use. If you think of diminished, minor, major/perfect, then augmented scale degrees as decreasing in "flatness," or increasing in "sharpness," relative to one another, then, broadly speaking, quarter steps are used to increase flatness, half steps are used to either increase flatness or maintain flatness, whole steps are used to generally maintain flatness (might change by one unit of flatness in either direction, though), grown steps are used to decrease flatness, and expanded steps are used to greatly decrease flatness.

Such a general set of steps is difficult to notate using the more familiar (to those more steeped in xenharmonic music theory) symbols akin to "L" and "s" (see [[MOS scale|MOS]]), simply because there are necessarily multiple kinds and multiple types of steps.

The general approach of defining step sizes and generating scales based off of those sizes is called the [[Bathomotonic|"bathomotonic approach"]].

== Generating the Intervals from Both Step Sizes and From Simplest Ratios ==

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|m2

|16:15

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|111.73

|-

|M2

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== Approximation by Equal Temperaments ==

As Zarlino's system of tuning ended up being pretty well approximated by the (later developed) [[12edo|12-EDO]], some subsets of this system are represented rather well by it.

As Zarlino's system of tuning ended up being pretty well approximated by the (later developed) [[12edo|12-EDO]], some subsets of this system are represented rather well by it, as are most meantone temperaments.

[[19edo|19-EDO]] is also representative of Bozuji with the limitation of adjacent diminished and augmented imperfect tones being enharmonically equivalent to one another. Since scales with combinations of those are discouraged by the limitations of step sizes, though, that may not be a significant concern. With that in mind, 19-EDO is basically analogous to this tuning as much as 12-EDO is to Zarlino's system.

[[31edo|31-EDO]] is an excellent representative of every interval in Bozuji. 31-EDO also has some additional intervals to represent neutral seconds, thirds, sixths, and sevenths, and a few additional intervals.

== Limitations and into the Future ==

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Adapting this approach to include more intervals should simply be a matter of choosing the best ratio to represent their relationships to unison, and then number-crunching, but it is not a trivial task.

Scales and other higher rank temperaments that sound more xenharmonically exotic (for example [[Orwell]]), are poorly represented; however, the bulk of the potential applications of augmented and diminished constructions within the tuning are quite unusual to most listeners with little experience outside of music composed outside of Western Classical Music Theory, in spite of the fact that the tones are constructed strictly within the guidelines of that music theory.

[[Category:Just intonation]]

[[Category:5-limit]]

[[Category:23-tone]]

[[Category:23-tone scales]]

[[Category:Ergotonic]]

[[Category:Bathomotonic]]

@@ Line 1: / Line 1: @@
 == Summary ==
-Bozuji tuning is a [[5-limit|5-limit just intonation]] tuning set with specified intervals proposed by [[Bostjan Zupancic]] ('''Bo'''stjan '''Zu'''pancic '''J'''ust '''I'''ntonation).  The approach to generating the intervals is somewhat unique, as all intervals were generated by choosing adaptive step sizes (which have been shown to work with software keyboards) and stepping through scales with different tonalities.  The tuning contains 23 intervals per [[octave]], and it is intended to be an expansion of [[wikipedia:Ptolemy's_intense_diatonic_scale|Ptolemy's Intense Diatonic Scale]].
+Bozuji tuning is a [[5-limit|5-limit just intonation]] tuning set with specified intervals proposed by [[Bostjan Zupancic]] ('''Bo'''stjan '''Zu'''pancic '''J'''ust '''I'''ntonation), which are closely related to the tones available in meantone temperament. The [[Bathomotonic|approach]] to generating the intervals is somewhat unique, as all intervals were generated by choosing adaptive step sizes (which have been shown to work with software keyboards, see AdaptiveJI) and stepping through scales with different tonalities.  The tuning contains 23 intervals per [[octave]], and it is intended to be an expansion of [[wikipedia:Ptolemy's_intense_diatonic_scale|Ptolemy's Intense Diatonic Scale]].
 == Interval Base ==
-The basis for the tuning is the [[Diatonic, Chromatic, Enharmonic, Subchromatic|diatonic scale]].  Using seven notes as a rough framework to step through a scale and ultimately get to a perfect octave.  The rough intervals are simply those given by classical western music theory scale degrees: unison (1), the second (2), the third (3), the fourth (4), the fifth (5), the sixth (6), and the seventh (7).  The octave is taken for granted as exactly double the frequency of unison, and then the scale repeats the same intervals from there, such that the ninth is equivalent to the second, the tenth is equivalent to the third, and so on.
+The basis for the tuning is the [[diatonic|diatonic scale]].  Using seven notes as a rough framework to step through a scale and ultimately get to a perfect octave.  The rough intervals are simply those given by classical western music theory scale degrees: unison (1), the second (2), the third (3), the fourth (4), the fifth (5), the sixth (6), and the seventh (7).  The octave is taken for granted as exactly double the frequency of unison, and then the scale repeats the same intervals from there, such that the ninth is equivalent to the second, the tenth is equivalent to the third, and so on.
 This approach considers three general kinds of scale degrees: reference, perfect, and imperfect.
@@ Line 68: / Line 68: @@
 == Step Sizes ==
 Ptolemy's work generated a scale with seven degrees and used three step sizes between adjacent intervals.  Zarlino later expanded Ptolemy's scale into a more generalized 5-limit just intonation tuning with four step sizes.  This approach proposes a matrix of two kinds each of five types of step sizes.
+The "type" of the step is determined by the musical context and the desired tonality of the interval after the step versus the tonality of the proceeding step.  For example, going from the perfect interval to a major interval is achieved by using a whole step, whilst a minor interval is achieved by using a half step, a diminished interval by a quarter step, and an augmented interval by a grown step.  The "kind" of step is determined by the positions of the intervals within the scale, in order to minimize the creation of new intervals by accounting for commas and such.
 The two kinds of step sizes are lesser steps and greater steps.  The lesser steps being indicated with a lowercase letter and the greater steps indicated with an uppercase letter.  The five types of step sizes are: quarter (q), half (h), whole (w), grown (g), and expanded (x).
@@ Line 128: / Line 130: @@
 |}
 In Ptolemy's system, whole and half steps were used to step through the major diatonic scale.  In this system, the steps are used to generate scales based on the choice of which step to use.  If you think of diminished, minor, major/perfect, then augmented scale degrees as decreasing in "flatness," or increasing in "sharpness," relative to one another, then, broadly speaking, quarter steps are used to increase flatness, half steps are used to either increase flatness or maintain flatness, whole steps are used to generally maintain flatness (might change by one unit of flatness in either direction, though), grown steps are used to decrease flatness, and expanded steps are used to greatly decrease flatness.
+Such a general set of steps is difficult to notate using the more familiar (to those more steeped in xenharmonic music theory) symbols akin to "L" and "s" (see [[MOS scale|MOS]]), simply because there are necessarily multiple kinds and multiple types of steps.
+The general approach of defining step sizes and generating scales based off of those sizes is called the [[Bathomotonic|"bathomotonic approach"]].
 == Generating the Intervals from Both Step Sizes and From Simplest Ratios ==
@@ Line 150: / Line 156: @@
 |m2
 |16:15
-|11.73
+|111.73
 |-
 |M2
@@ Line 329: / Line 335: @@
 == Approximation by Equal Temperaments ==
-As Zarlino's system of tuning ended up being pretty well approximated by the (later developed) [[12edo|12-EDO]], some subsets of this system are represented rather well by it.
+As Zarlino's system of tuning ended up being pretty well approximated by the (later developed) [[12edo|12-EDO]], some subsets of this system are represented rather well by it, as are most meantone temperaments.
 [[19edo|19-EDO]] is also representative of Bozuji with the limitation of adjacent diminished and augmented imperfect tones being enharmonically equivalent to one another.  Since scales with combinations of those are discouraged by the limitations of step sizes, though, that may not be a significant concern.  With that in mind, 19-EDO is basically analogous to this tuning as much as 12-EDO is to Zarlino's system.
+[[31edo|31-EDO]] is an excellent representative of every interval in Bozuji. 31-EDO also has some additional intervals to represent neutral seconds, thirds, sixths, and sevenths, and a few additional intervals.
 == Limitations and into the Future ==
@@ Line 337: / Line 345: @@
 Adapting this approach to include more intervals should simply be a matter of choosing the best ratio to represent their relationships to unison, and then number-crunching, but it is not a trivial task.
+Scales and other higher rank temperaments that sound more xenharmonically exotic (for example [[Orwell]]), are poorly represented; however, the bulk of the potential applications of augmented and diminished constructions within the tuning are quite unusual to most listeners with little experience outside of music composed outside of Western Classical Music Theory, in spite of the fact that the tones are constructed strictly within the guidelines of that music theory.
 [[Category:Just intonation]]
 [[Category:5-limit]]
-[[Category:23-tone]]
+[[Category:23-tone scales]]
 [[Category:Ergotonic]]
+[[Category:Bathomotonic]]