Tuning for Sun Dance / Battle of the Greasy Grass River
The basis of Sun Dance / Battle of the Greasy Grass River is a series of drones over and under the tonic F#, moving slowly from a low B (21/16, 471 cents) up through several variously dissonant tritones to C# (3/2, 702 cents):
| Drone Pitches: | B7 | B- | B | B^- | C7 | B#L- | C#v | C#- | C# |
|---|---|---|---|---|---|---|---|---|---|
| Ratio to F#: | 21/16 | 4/3 | 27/20 | 11/8 | 7/5 | 10/7 | 16/11 | 40/27 | 3/2 |
| Cents above F#: | 471 | 498 | 520 | 551 | 583 | 617 | 649 | 680 | 702 |
Harmonizing these drones requires a scale, in the first half of the movement, of the following 25 pitches:
| Pitch: | F# | G#v | G#- | G# | G#L- | A7 | A | A# | B7 | B- | B | B^- | C7 | B#L- | C#v | C#- |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio: | 1/1 | 12/11 | 10/9 | 9/8 | 8/7 | 7/6 | 6/5 | 5/4 | 21/16 | 4/3 | 27/20 | 11/8 | 7/5 | 10/7 | 16/11 | 40/27 |
| Cents: | 0 | 151 | 182 | 204 | 231 | 267 | 316 | 386 | 471 | 498 | 520 | 551 | 583 | 617 | 649 | 680 |
| C# | D- | D#v | D# | D#L- | E7 | E- | E | E# |
| 3/2 | 8/5 | 18/11 | 5/3 | 12/7 | 7/4 | 16/9 | 9/5 | 15/8 |
| 702 | 814 | 853 | 884 | 933 | 969 | 996 | 1018 | 1088 |
The effect is a slow movement from the subdominant B to the tonic F# through a series of tritones, a gradual coming into focus from wildly pulsating dissonance to pure consonance, hopefully suggesting the painful discipline of the Sun Dance and the slowly emerging, ecstatic clarity at the end as Sitting Bull has his vision.
In the second half of the movement, the Battle scene, Custer's army comes blundering in in a tonality a tritone away, B#L- (10/7), and adding the pitches CxL (45/28), DxL- (25/14), E#L- (40/21), FxL (15/14), and GxL- (25/21), for a total overall scale of 30 pitches:
| Pitch: | F# | FxL | G#v | G#- | G# | G#L- | A7 | GxL- | A | A# | B7 | B- | B | B^- | C7 | B#L- |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio: | 1/1 | 15/14 | 12/11 | 10/9 | 9/8 | 8/7 | 7/6 | 25/21 | 6/5 | 5/4 | 21/16 | 4/3 | 27/20 | 11/8 | 7/5 | 10/7 |
| Cents: | 0 | 119 | 151 | 182 | 204 | 231 | 267 | 302 | 316 | 386 | 471 | 498 | 520 | 551 | 583 | 617 |
| C#v | C#- | C# | D- | CxL | D#v | D# | D#L- | E7 | E- | DxL- | E | E# | E#L- |
| 16/11 | 40/27 | 3/2 | 8/5 | 45/28 | 18/11 | 5/3 | 12/7 | 7/4 | 16/9 | 25/14 | 9/5 | 15/8 | 40/21 |
| 649 | 680 | 702 | 814 | 821 | 853 | 884 | 933 | 969 | 996 | 1004 | 1018 | 1088 | 1116 |
Following the battle comes a series of sustained chords, the most dissonant chords I could find within the scale. As Harry Partch says, just-intonation dissonance is "a whole different serving of tapioca" from equal-tempered dissonance. Just as the deceptive serenity of the dead on the battlefield belies the tremendous cultural ramifications of the battle, the static, unmoving chords are contradicted by the wild, multi-tempoed pulsations of the mutually and complexly dissonant overtones.
I have notated these scales (as closely as html will allow) in Ben Johnston's excellent microtonal notation, in which accidentals make the following alterations:
| Sign: | Ratio: | Effect | |
|---|---|---|---|
| + | (plus) | 81/80 | raises a pitch by 21.5 cents |
| - | (minus) | 80/81 | lowers a pitch by -21.5 cents |
| # | (sharp) | 25/24 | raises a pitch by 71 cents |
| b | (flat) | 24/25 | lowers a pitch by -71 cents |
| 7 | (seven) | 35/36 | lowers a pitch by -49 cents to get a seventh harmonic |
| L | (upside-down 7) | 36/35 | raises a pitch by 49 cents |
| ^ | (upward arrow) | 33/32 | raises a pitch by 53 cents |
| v | (downward arrow) | 32/33 | lowers a pitch by -53 cents |
In addition, F-A-C, C-E-G, and G-B-D are all perfectly tuned 4:5:6 major triads.
(If you don't have enough experience with just intonation to make sense of these charts, try reading the step-by-step Just Intonation Explained section.)
Kyle Gann
Photo of the Little Bighorn Battlefield taken by Kent LaCombe, courtesy of his excellent Custer Web Page.
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