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). 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]].

== Interval Base ==

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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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== 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 ==

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[[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.

== 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.

[[Category:Just intonation]]

[[Category:5-limit]]

[[Category:23-tone]]

[[Category:Ergotonic]]

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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).  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]].
 == Interval Base ==
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 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 240: / Line 242: @@
 == 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 ==
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 [[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.
+== 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.
 [[Category:Just intonation]]
 [[Category:5-limit]]
 [[Category:23-tone]]
 [[Category:Ergotonic]]