Bozuji tuning

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Summary

Bozuji tuning is a 5-limit just intonation tuning set with specified intervals proposed by Bostjan Zupancic (Bostjan Zupancic Just Intonation), which are closely related to the tones available in meantone temperament. 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, 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 Ptolemy's Intense Diatonic Scale.

Interval Base

The basis for the tuning is the 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 (#).

The imperfect degrees are the second, third, sixth, and seventh. In this approach, four varieties are allowed: diminished (d), minor (m), major (M), and augmented (A). Diminished is indicated with a double flat accidental sign (𝄫) or two lowercase letter b's without a space in between them (bb), minor is indicated by a regular flat accidental sign (♭) or single lowercase letter b (b), major is indicated without an accidental sign, and augmented is indicated with a sharp accidental sign (♯) or a number sign (#).

Table 1: Scale Degree/Interval Framework for Bozuji Approach
Degree Type Allowed flavors Formal accidentals Informal accidentals
1 reference P
2 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
3 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
4 perfect d, P, A ♭, ♯ b, #
5 perfect d, P, A ♭, ♯ b, #
6 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #
7 imperfect d, m, M, A 𝄫,♭, ♯ bb, b, #

A scale with seven degrees is called "heptatonic." This system is based on a seven note framework, but it's slightly more specified than general heptatonic scales. A general heptatonic scale would define an arbitrary number of tones in a tuning set, then arbitrarily choose seven notes from that set, and call it a scale. This system has some additional rules, but is more loosely defined than the rules used to make diatonic scales. It is, however, in the American sense of the term "functional." Each scale degree plays a certain role with a select few options. Breaking down "diatonic" into Greek, it means "across tones." "Heptatonic" means "seven tones." Since the Greek word for function (in this sense) is "ergon," and the scale degrees are dictated by musical function, a term for this sort of approach may be "ergotonic" ("functional tones").

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).

Table 2: List of Steps for Bozuji Approach
Step name Abbreviation Ratio Cents
Lesser quarter q 2048:2025 19.55
Greater quarter Q 128:125 41.06
Lesser half h 16:15 111.73
Greater half H 27:25 133.24
Lesser whole w 10:9 182.40
Greater whole W 9:8 203.91
Lesser grown g 125:108 253.08
Greater grown G 75:64 274.58
Lesser expanded x 625:512 345.25
Greater expanded X 10125:8192 366.76

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), 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 approach".

Generating the Intervals from Both Step Sizes and From Simplest Ratios

With the interval base of how many intervals need to be generated and the step sizes dictating the ratios between adjacent steps, there are still several choices for which interval is which ratio, but, given any set of possibilities, the proper interval is determined to be the one with the smallest integers in the ratio.

Intervals

Table 3: List of All Intervals from Bozuji Tuning
Interval

(abbreviated)

Ratio Cents
P1 1:1 0
d2 128:125 41.06
m2 16:15 111.73
M2 9:8 203.91
A2 75:64 274.58
d3 144:125 244.97
m3 6:5 315.64
M3 5:4 386.31
A3 675:512 478.49
d4 32:25 427.37
p4 4:3 498.05
A4 45:32 590.22
d5 64:45 609.78
p5 3:2 701.96
A5 25:16 772.63
d6 192:125 743.01
m6 8:5 813.69
M6 5:3 884.36
A6 225:128 976.54
d7 128:75 925.42
m7 9:5 1017.60
M7 15:8 1088.27
A7 125:64 1158.94

Use

The intervals given in Table 3 can be used to generate a seven tone scale, but there are some steps between adjacent scale degrees with conflicting flavours (i.e., diminished and augmented) that are too big or too small to match any of the step sizes listed. For example, augmented second to diminished third is a step of 9216:9375 (about -29.6 cents), which is less than 1:1 (!). Since there are no steps with negative cent values, this step is generally discouraged. Another example: diminished second to augmented third is a step of 84375:65536 (about 437.4 cents), which is greater than our largest defined step size.

However, the tuning does offer any combination of major and minor intervals possible with a twelve note system, as well as hundreds of alternative scales and experimental scales that still fit within the wider framework of generalized western music theory, all with a reasonably manageable number of note choices (23 per octave) and a limited number of step sizes between adjacent degrees (ten), which offers some familiarity to musicians more comfortable with equal temperaments.

Still, the system is by no means perfect. Because it is a just intonation system, key modulation is nontrivial. Twenty-three notes is a lot to only really use seven at a time, yet the 5-limit nature of the tuning causes it to miss out on the neutral intervals (e.g. from maqam) and spicier-sounding sixths (e.g. from some more advanced raga) found in non-western music.

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) 12-EDO, some subsets of this system are represented rather well by it, as are most meantone temperaments.

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.

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.