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 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.

The reference degree is the unison. In this approach, since the scale degree of 1 references the (movable) key of the scale, it is considered to be unaltered and only come in one flavor: 1.

There will, typically, also be a second reference pitch defined as a recursion of the first reference pitch. In conventional theory, this is usually the octave (2:1). This could be generalized to be anything (for instance 3:1), though, so long as there is a ratio that does not translate our pitch concepts (the way that 2:1 of any pitch, for example, C# is still given the same name, i.e. C#). The second reference pitch also acts as a stopping point for our scale.

The perfect degrees are the fourth and the fifth. In this approach, three varieties are allowed: diminished (d), perfect (P), and augmented (A). Diminished is indicated with a flat accidental sign (♭) or lowercase letter b (b), perfect without an accidental sign, and augmented with a sharp accidental sign ('''♯''') or number sign (#).

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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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== Examples ==

The most trivial example of this tuning used to generate a seven note scale is Ptolemy's Intense Diatonic Scale: with degrees 1 2 3 4 5 6 7, or P1 M2 M3 P4 P5 M6 M7, with interval ratios relative to the root 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, and 15:8, and step sizes W w h W w W h.

=== Classical Modes ===

The seven classical modes are represented in JI by this theory by using only four of the step sizes from Table 2 (Two whole steps: W and w, and two half steps: H and h):

Ionian: W w h W w W h = M2 M3 P4 P5 M6 M7

Aeolian: W h w W h W w = M2 m3 P4 P5 m6 m7

Mixolydian: W w h W w H w = M2 M3 P4 P5 M6 m7

Dorian: W h w W w H w = M2 m3 P4 P5 M6 m7

Lydian: W w W h w W h = M2 M3 A4 P5 M6 M7

Phrygian: h W w W h W w = m2 m3 P4 P5 m6 m7

Locrian: h W w h W W w = m2 m3 P4 d5 m6 m7

=== The Overtone Scale Family ===

The so-called "overtone scale" also consists of the same four of the step sizes. If modal scale theory is extrapolated and applied to this scale, a set of seven "modes" results, which includes the ascending melodic minor scale:

Melodic Minor Ascending: W h w W w W h = M2 m3 P4 P5 M6 M7

Hindu/Acoustic: W w h W h W w = M2 M3 P4 P5 m6 m7

Lydian Dominant/Overtone: W w W h w H w = M2 M3 A4 P5 M6 m7

Locrian Major Second: W h w h W W w = M2 m3 P4 d5 m6 m7

Javanese: h W w W w H w = m2 m3 P4 P5 M6 m7

Lydian Augmented: W w W w h W h = M2 M3 A4 A5 M6 M7

Altered/"Super Locrian": h W h w W W w = m2 m3 d4 d5 m6 m7

=== Harmonic Major ===

Even adding one potential step from Table 2 (greater grown step: G, essentially a step and a half) opens up a plethora of new possibilities:

Harmonic Major: W w h W h G h = M2 M3 P4 P5 m6 M7

Lydian Minor Third: W h G h w W h = M2 m3 A4 P5 M6 M7

Dorian Diminished Fifth: W h w h G H w = M2 m3 P4 d5 M6 m7

Dominant Minor Second: h G h W w H w = m2 M3 P4 P5 M6 m7

Augmented Major Sixth: G h W w h W h = A2 M3 A4 A5 M6 M7

Diminished Perfect Fourth: h W w h W h G = m2 m3 P4 d5 m6 d7

Altered Perfect Fifth: h W h G h W w = m2 m3 d4 P5 m6 m7

=== Harmonic Minor ===

Harmonic Minor: W h w W h G h = M2 m3 P4 P5 m6 M7

Spanish Gypsy: h G h W h W w = m2 M3 P4 P5 m6 m7

Romanian Minor: W h G h w H w = M2 m3 A4 P5 M6 m7

Locrian Major Sixth: h W w h G H w = m2 m3 P4 d5 M6 m7

Ionian Augmented Fifth: W w h G h W h = M2 M3 P4 A5 M6 M7

Lydian Augmented Second: G h W h w W h = A2 M3 A4 P5 M6 M7

Diminished: h W h w W h G = m2 m3 d4 d5 m6 d7

=== Hungarian Minor ===

We will have to add one more step from table 2 to get the next family of "modes" worked out. This set involves one obscure use of the lesser grown step (g):

Hungarian Minor: W h G h h G h = M2 m3 A4 P5 m6 M7

Oriental Major: h G h h G H w = m2 M3 P4 d5 M6 m7

Byzantine/Double Harmonic: h G h W h G h = m2 M3 P4 P5 m6 M7

Major Augmented: G h h G h W h = A2 M3 P4 A5 M6 M7

Undiminished: h W h G h h G = m2 m3 d4 P5 m6 d7

Unaugmented: G h W h G h h = A2 M3 A4 P5 A6 M7

"12-7-55-96": h H g h W h G = m2 d3 P4 d5 m6 d7

With those six step sizes involved, any ergotonic 12edo scale can be translated into a set of just intervals. But some limitations arise; for example, the diminished second and augmented seventh are allowed intervals from our general theory, but no such intervals exist in 12edo. To remedy the situation, the pair of quarter step ratios are necessary. To get from those intervals to other conventional intervals, the pair of extended steps are necessary.

== 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 ==

This approach ignores neutral intervals (neutral second, neutral third, neutral sixth, and neutral seventh). These intervals are widely understood, although, like most just intervals, there is some debate as to their exact ratio definitions. Such intervals have existed in non-Western music theory for hundreds of years.

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 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.
 The reference degree is the unison.  In this approach, since the scale degree of 1 references the (movable) key of the scale, it is considered to be unaltered and only come in one flavor: 1.
+There will, typically, also be a second reference pitch defined as a recursion of the first reference pitch.  In conventional theory, this is usually the octave (2:1).  This could be generalized to be anything (for instance 3:1), though, so long as there is a ratio that does not translate our pitch concepts (the way that 2:1 of any pitch, for example, C# is still given the same name, i.e. C#).  The second reference pitch also acts as a stopping point for our scale.
 The perfect degrees are the fourth and the fifth.  In this approach, three varieties are allowed: diminished (d), perfect (P), and augmented (A).  Diminished is indicated with a flat accidental sign (♭) or lowercase letter b (b), perfect without an accidental sign, and augmented with a sharp accidental sign ('''♯''') or number sign (#).
@@ Line 66: / 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 126: / 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 148: / Line 156: @@
 |m2
 |16:15
-|11.73
+|111.73
 |-
 |M2
@@ Line 240: / Line 248: @@
 == Examples ==
 The most trivial example of this tuning used to generate a seven note scale is Ptolemy's Intense Diatonic Scale: with degrees 1 2 3 4 5 6 7, or P1 M2 M3 P4 P5 M6 M7, with interval ratios relative to the root 1:1, 9:8, 5:4, 4:3, 3:2, 5:3, and 15:8, and step sizes W w h W w W h.
+=== Classical Modes ===
+The seven classical modes are represented in JI by this theory by using only four of the step sizes from Table 2 (Two whole steps: W and w, and two half steps: H and h):
+Ionian: W w h W w W h = M2 M3 P4 P5 M6 M7
+Aeolian: W h w W h W w = M2 m3 P4 P5 m6 m7
+Mixolydian: W w h W w H w = M2 M3 P4 P5 M6 m7
+Dorian: W h w W w H w = M2 m3 P4 P5 M6 m7
+Lydian: W w W h w W h = M2 M3 A4 P5 M6 M7
+Phrygian: h W w W h W w = m2 m3 P4 P5 m6 m7
+Locrian: h W w h W W w = m2 m3 P4 d5 m6 m7
+=== The Overtone Scale Family ===
+The so-called "overtone scale" also consists of the same four of the step sizes.  If modal scale theory is extrapolated and applied to this scale, a set of seven "modes" results, which includes the ascending melodic minor scale:
+Melodic Minor Ascending: W h w W w W h = M2 m3 P4 P5 M6 M7
+Hindu/Acoustic: W w h W h W w = M2 M3 P4 P5 m6 m7
+Lydian Dominant/Overtone: W w W h w H w = M2 M3 A4 P5 M6 m7
+Locrian Major Second: W h w h W W w = M2 m3 P4 d5 m6 m7
+Javanese: h W w W w H w = m2 m3 P4 P5 M6 m7
+Lydian Augmented: W w W w h W h = M2 M3 A4 A5 M6 M7
+Altered/"Super Locrian": h W h w W W w = m2 m3 d4 d5 m6 m7
+=== Harmonic Major ===
+Even adding one potential step from Table 2 (greater grown step: G, essentially a step and a half) opens up a plethora of new possibilities:
+Harmonic Major: W w h W h G h = M2 M3 P4 P5 m6 M7
+Lydian Minor Third: W h G h w W h = M2 m3 A4 P5 M6 M7
+Dorian Diminished Fifth: W h w h G H w = M2 m3 P4 d5 M6 m7
+Dominant Minor Second: h G h W w H w = m2 M3 P4 P5 M6 m7
+Augmented Major Sixth: G h W w h W h = A2 M3 A4 A5 M6 M7
+Diminished Perfect Fourth: h W w h W h G = m2 m3 P4 d5 m6 d7
+Altered Perfect Fifth: h W h G h W w = m2 m3 d4 P5 m6 m7
+=== Harmonic Minor ===
+Harmonic Minor: W h w W h G h = M2 m3 P4 P5 m6 M7
+Spanish Gypsy: h G h W h W w = m2 M3 P4 P5 m6 m7
+Romanian Minor: W h G h w H w = M2 m3 A4 P5 M6 m7
+Locrian Major Sixth: h W w h G H w = m2 m3 P4 d5 M6 m7
+Ionian Augmented Fifth: W w h G h W h = M2 M3 P4 A5 M6 M7
+Lydian Augmented Second: G h W h w W h = A2 M3 A4 P5 M6 M7
+Diminished: h W h w W h G = m2 m3 d4 d5 m6 d7
+=== Hungarian Minor ===
+We will have to add one more step from table 2 to get the next family of "modes" worked out.  This set involves one obscure use of the lesser grown step (g):
+Hungarian Minor: W h G h h G h = M2 m3 A4 P5 m6 M7
+Oriental Major: h G h h G H w = m2 M3 P4 d5 M6 m7
+Byzantine/Double Harmonic: h G h W h G h = m2 M3 P4 P5 m6 M7
+Major Augmented: G h h G h W h = A2 M3 P4 A5 M6 M7
+Undiminished: h W h G h h G = m2 m3 d4 P5 m6 d7
+Unaugmented: G h W h G h h = A2 M3 A4 P5 A6 M7
+"12-7-55-96": h H g h W h G = m2 d3 P4 d5 m6 d7
+With those six step sizes involved, any ergotonic 12edo scale can be translated into a set of just intervals.  But some limitations arise; for example, the diminished second and augmented seventh are allowed intervals from our general theory, but no such intervals exist in 12edo.  To remedy the situation, the pair of quarter step ratios are necessary.  To get from those intervals to other conventional intervals, the pair of extended steps are necessary.
 == 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 ==
+This approach ignores neutral intervals (neutral second, neutral third, neutral sixth, and neutral seventh).  These intervals are widely understood, although, like most just intervals, there is some debate as to their exact ratio definitions.  Such intervals have existed in non-Western music theory for hundreds of years.
+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]]