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Home , Diminished seventh, Just intonation, Major sixth

SPRING 2002 1 1 Volume 11, Number 1 THE JOURNAL OF THE NETWORK A PROJECT OF OTHER , INC. A Whole Different Serving of Tapioca: Exploring Just Intonation Dissonance

by Kyle Gann wrote a song, “Jargon,” completely in seconds, sevenths, Just Intonation dissonance, said in one of and , and justified them with a satirical text (“Let tuning theory’s more ringing phrases, is “a whole dif- horrid Jargon split the air/And rive the Nerves asunder”). ferent serving of tapioca” from equally tempered dis- Somewhat analogously, I found a text and allowed pro- sonance. I had no idea what that meant when I first heard grammatic considerations to lead me into several differ- it. I’ve now used enough Just Intonation dissonance in ent kinds and degrees of just dissonance. my music to learn how right he was. My text was taken from My Life on the Plains, the My early Just Intonation works were rather fanatically autobiographical memoir of General George Armstrong consonant. Having expended years of time and energy Custer. 1 If ever a text cried out for dissonance, it was to achieve pure consonance, I was at a loss as to how to this, bristling with unacknowledged contradictions. One justify returning to dissonance, and as to what strategy or moment, Custer would admit that if he were an Indian, he theory to use as a guideline. After all, if you start working would ignore U.S. Government mandates and go live on in Just Intonation to get rid of beats in your sonic surface, the plains; a few pages later he would damn all Indians under what theoretical aegis do you bring beats back in? as brutes only governable by force. And so I was in danger of replicating the fate of one of Overall, the book seemed schizophrenic, so for our American forebears, William Billings (1746 –1800), “Custer: ‘If I Were an Indian…’ ” — the first movement who was criticized because his hymns made almost no of my music-theater piece Custer and Sitting Bull 2 — use of dissonance. And for exactly parallel reasons: Bill- I came up with a schizophrenic . Half of the scale ings had no theoretical sanction for breaking the rules was a normal major/minor Just Intonation scale on C that he had learned at such cost. In response, Billings (Table 1a). (text continued on page 4)

Table 1a Pitch CDEb EFGAb ABb B 1 9 6 5 4 3 8 5 9 15 /1 /8 /5 /4 /3 /2 /5 /3 /5 /8 Cents 0 203.9 315.6 386.3 498.0 702.0 813.7 884.4 1017.6 1088.3 Table 1b Pitch CiY Cl+Di EiY FiY Fl+Gi AiY BR Bi

Ratio 2673 4455 297 2673 891 1485 99 891 297 495 2560 4096 256 2048 640 1024 64 512 160 256 Cents 74.8 145.5 257.2 461.1 572.8 643.5 755.2 959.1 1070.9 1141.5

Volume“Just Intonation 11, Number is the 1 Best Intonation”1/1 ——Lou Harrison1 Example 1 4 3 3 3 / 4 ä j ÎÎä Voice À À À À. À À À À À À À.. À À À À À À À À À À À À À À 4 In- di - an can ne - ver be per- mit - ted ú. to view the ques - tion in this de - li - be - rate way. ú When the English & 4iYú Rú bú. Ï iYú ú Horn iY iY 4 j j & 4iYÏ Ï ÏÏä Ï Ï Ï ÎäÏ Ï Ï iY Ï Ï Ï Ï Ï Ï Ï Ï Marimba iYiÏ Ï Ï Ï Ï bÏ Ï Ï Ï iiY Ï Ï Ï Ï Ï Ï Ï Ï 4 À À À À À À À À À À À À À À À À À À À À À À À À Drum / 4 ú Rú w ú ú ? 4iY ú bw iY ú Bass 4 i i

(Tapioca… continued from page 1) think of eleven as the number of mediation, providing as The other half was this same scale transposed it does so many midpoints between close 5- and 7-limit upward 257.2 cents to the 297 : that is, as a intervals, so 11 times 27 promised to mediate among quick calculation will show, to the harmonic of my other pitches. the Pythagorean . Stated in terms of C, the Among the paragraphs I chose from Custer’s book other scale’s numbers looked more impressive (given were evident many different degrees and levels of hypoc- in Table 1b in the specific Just Intonation pitch notation risy, a real connoisseur’s cornucopia of mendacity and devised by my teacher ). rationalization. At times, Custer was capable of very Keep in mind that although I have listed this second reasonably portraying the Indian people as victims of iY scale starting from C so it can be seen in conjunction white imperialist aggression. At these moments, I had i with the just scale on C, its actual tonic is D . Recalcu- the music move fluidly back and forth between the two 297 1 late the with /256 as /1, and you’ll find the same scales, as in Example 1. When he began justifying the ratios as in the C major/ in Table 1a. Washita massacre, however, I used intrusions in FiY to Thus, the entire movement operated within a bitonal, interrupt the Eb major statement of “Garry Owen,” the twenty-pitch just pitch collection, with one tonic on C 7th Cavalry’s theme song, as in Example 2. When Custer i i and the other on D , the 297th harmonic of C. I chose D reached the height of hypocrisy, accusing the Indians explicitly for its approximation of quarter-tone values. of depredations that were actually the direct result of Although I would never compose in equally tempered white actions, I undercut his sanctimonious speech in quarter tones, considering them merely a convenient Eb major with a bass line and a soprano countermelody addition to an equally tempered twelve-pitch scale that in the Di scale as in Example 3. And finally, after his was a historical mistake in the first place, I admit I’ve plea for mercy at his 1868 court martial (“Judge me not always found quarter tones rather raucously titillating to by what is known now, but in the light of what I knew the ear, and I wanted to build in as much incommensu- when these events transpired”), I climaxed with a final rability to Custer’s scale as possible. Also, I’ve come to chord in both keys at once (Example 4).

Example 2 114 2 Å j 6 Voice / 8 À. 8 À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. À. My firm con- vic - tion based on a-ly na --of sis the cha -rac - ter traits of the In -andi - bÏ Ï . bÏ Ï Ï . bÏ Ï & 82 86 Ï bÏ bÏ Ï RÏ.iY Ï Ï Ï Ï bÏ bÏ Ï RÏ.i Ï Ï Ï Flute Ï bÏ i Ï bÏ iY Ï.i Ï Ï 2 6 English & 8 äÅr 8 Å r Å r Å r Å r Horn iY Ï bÏ. Ï iY Ï bÏ. Ï iY Ï RÏ. Ï iY Ï RÏ. Ï iY Ï ú. i ú. ú. iY ú. ? 2 · 6 bú. ú. bú. iY ú. Brass 8 8 i

4 1/1 Volume 11, Number 1 Example 3 154 3 3 3 4 Îä. r ä î j Å r Voice / 4 À À À À À À À À À À À À À À À À À À À À À À À À À À À À À À

Al- though ne- ver claimed as an ex - po - nent of the peace po- li - cy, Yet I en - ter- tained the most peace- a - ble sen - ti - ments iY Ï Ï Y Ï Ï RÏ Ï. iY ÏRÏ Ï Ï Ï Ï Ï RÏ Ï Ï Ï Ï Ï RÏ Ï Ï Ï Ï 4 îäJ J ä Y RÏ äÎ ä Y Ï Ï Ï Ï ÏYÏ Ï Flute & 4 J R Y 4 Ï Ï ÏbÏ Ï Ï Ï bÏ Ï Ï Ï Ï Ï Ï Ï Ï bÏ Trumpet & 4 Î bÏ Ï Ï Ï Ï ä J bú b in B Ï bÏ bbÏ ú ú bú Ï bbÏ ú ú bÏ Ï ? 4 bÏ Ï ú ú b ú Ï ú Ï Bassoon 4b Ï bÏ ? 4 Ï Double 4iY ú ú RÏ Bass iY Ï lYú ú iY ú Ï iY Ï iY i iY Ï

I felt that these varied deployments of bitonal just century headaches because, occurring between D and A scales — successive, simultaneous, layered, fused — of- in a justly tuned scale based on C, it made the ii chord fered me a nuanced way to comment on Custer’s words, unavailable in a twelve-pitch just tuning. Besides, it as though the C scale were his conscious mind and the Di seemed appropriate that a Native American sun dance scale his unconscious poking its head out at inopportune would be pervaded by the spirit of the wolf. (for Custer) moments. Hypocrisy, however, is a rather I set up a series of drones (Table 2) that would move superficial emotional state, and this was still a superfi- gradually from the interval of a fourth to one of a , 40 cial approach to Just Intonation dissonance. Similar, if with /27 as the penultimate interval before reaching less subtle, effects could have been achieved in . The movement of Custer and Sit- Table 2 ting Bull demanded a deeper treatment. This was “Sun Pitch Ratio Cents Dance/Battle of the Greasy Grass River,” a depiction of I 21 the sun dance that Sitting Bull led before the battle of the B /16 470.8 Z 4 Little Bighorn (the Sioux name for the river translates as B /3 498.0 27 “Greasy Grass River”), along with the subsequent battle. B /20 519.6 During the sun dance, Sitting Bull cut a hundred notches BiZ 11/ 551.3 of flesh out of his arms and legs, letting the blood run 8 I 7 down until he had a vision. This sounded uncomfortable. C /5 582.5 LZ 10 Dissonance seemed appropriate. B /7 617.5 ∑ 16 To symbolize Sitting Bull’s trance and vision, I C /11 648.7 wanted intervals that would pulsate with acoustic mZ 40 40 C /27 680.5 beats. In particular, I was curious to work with /27, m 3 the notorious “wolf fifth” that caused so many sixteenth- C /2 702.0

Example 4 232 À. À À À ÀÎ À. À À À ÀÎ Voice /

Ask the Sax -on race! Ask theSax -on race! Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï & iY Ï Ï Ï Ï äÎ iY Ï Ï Ï Ï äÎ Marimba bÏiYiÏ Ï Ï Ï Ï Ï Ï JÏ bÏiYiÏ Ï Ï Ï Ï Ï Ï JÏ bú Ï bú Ï bú Ï bú Ï & J äÎ J äÎ ú Ï ú Ï Winds iYiú Ï iYiú Ï ?iY úú ÏÏ äÎiY úú ÏÏ äÎ J J

Volume 11, Number 1 1/1 5 3 4 the consonant goal /2. I began, though, not with /3, but contrast to work with. 21 ∑ 16 /16, an interval I had come to enjoy from La Monte C /11 is the eleventh undertone; in other words, the 1 Young’s use of it in The Well-Tuned . Over a tonic pitch of which /1 is the eleventh harmonic. I harmonized m 12 4 F , the series of nine drone intervals turned out as follows it with its third harmonic, /11, and also with /3, which 12 (pitches once again with Johnston’s accidentals). made a /11 interval with it. mZ 40 This list includes four , two septimal (seven- C /27 was the crucial interval, the wolf, the penul- m 3 based) and two undecimal (eleven-based). Not included, timate pitch before the climactic resolution into C /2. 45 m mZ interestingly (to me), was /32, the somewhat historically The ratio implies that the pitches involved, F and C , standard European of 590 cents. Only eight cents are the second and sixth scale degrees of a scale on EZ 7 16 16 16 away, /5 was so close as to render it superfluous. Then /9. Thus, I used /9, and also the of /9, 10 also, it is a more complex ratio than any of the tritones /9, as mediating pitches. m 3 I already have. C /2 was the sun dance’s consonant goal, coinciding My approach to tuning has generally aimed at render- with Sitting Bull’s vision: white soldiers and cavalry ing complex intervals as intelligible as possible, and I falling down, as a voice said, “I give you these because continued that strategy with this series. Each interval they have no ears.” This drone I was able to harmonize implied a certain and according to with all of the small-number ratios I had available. where it fell in a harmonic series. This information is summarized in Example 5, which I 21 B /16, the septimal fourth, is, of course, the shows the drone interval, the root of the harmonic series 3 m harmonic of a /2. With reference to the tonic F , the pitch implied, and the scales I spun my melodies around to m 3 implied a dominant seventh on C /2. make each drone interval intelligible. Z 4 B /3 is the traditional . Its conso- An even more dissonant passage than the sun dance nance with the tonic was unproblematic, and many was required by the aftermath of the battle scene. Here, pitches could be used with both for a fair amount of as the 200-plus dead soldiers lay on the battlefield consonance. beginning to swell and stink, I wanted no mediating 27 B /20 was more interesting. 27 and consonances, just an unintelligible sonority to match 20 imply an interval between the Pythagorean major the unfathomability of the scene. 27 5 20 sixth ( /16) and a five-limit major third ( /4, or /16). In the bass I used the drones that had run throughout Z 8 Therefore D /5 was an important note to include, as the sun dance. Above those, quite intuitively, I chose m 1 27 being a fundamental of which F /1 and B /20 are both pitches from the ones I had available that I suspected harmonics. would be particularly incommensurate. Specifically, for iZ 11 B /8 was again a simpler case, since it is the a drone whose ratio had a 7 in the numerator, I would eleventh harmonic of Fm. I harmonized it with other pick a pitch with a 7 in the denominator, for example, m 8 21 7 low-numbered of F . /7 above /16 and again above /5. The seven chords in I 7 LZ 10 C /5 and B /7, the midpoints between the per- the battle aftermath are shown in Table 3. fect fourth and , are an interesting pair. The The most complex of these are the chords that last numbers suggest that CI is the seventh harmonic in a harmonic series of which Fm is the fifth harmonic; that Table 3 Z 8 m 1 harmonic series has D /5 as its fundamental. Likewise, Ratios to F /1 10 LZ 7 15 25 9 45 7 25 /7 suggests that B is the tenth (or fifth) harmonic in a /4 /8 /14 /5 /28 /4 /14 series in which Fm is the seventh; that series is founded 12 8 8 7 10 12 27 /11 /7 /7 /6 /9 /11 /20 LZ 8 on G /7. Thus, every time in the music that the drone 1 1 1 1 1 1 1 I 7 LZ /1 /1 /1 /1 /1 /1 /1 pitch shifted from C /5 to B (only 35 cents, after all) 27 21 7 11 16 10 40 the tonality tended to flip-flop a tritone, from DZ to GLZ /20 /16 /5 /8 /11 /7 /27 or back. Both in classical music, where they represent Cents above Fm Neapolitan or German sixth progressions, and in jazz 968.8 1088.3 1003.8 1017.6 821.4 968.8 1003.8 where the tritone substitution chord is involved, tritone 150.6 231.2 231.2 266.9 182.4 150.6 519.6 root movements between seventh chords tend to guar- antee interesting and effective . Having 0 000000 7 10 that /5 – /7 progression right in the middle was a juicy 519.6 470.8 582.5 551.3 648.7 617.5 680.5

6 1/1 Volume 11, Number 1 Example 5

1 7 21 3 7 1 10 4 3 16 9 27 6 9 1 1 9 11 3 1 6 7 8 9 1 6 16 2 4 1 9 3 2 9 5 20 5 8 1 1 8 8 2 1 5 5 5 5 #Ï IÏ #Ï ZÏ Ï #Ï Ï ZÏ Ï & #Ï IÏ IÏ #Ï #ZÏ ZÏ #Ï #Ï Ï Ï #Ï #Ï iZÏ #Ï Ï I ? #w #w #w #w #w I w Zw w iZ w I w

Ratio: Cents: 4/3 498.0 27/20 519.6 11/8 551.3 7/5 582.5 ?21/16 470.8 Zw #w Zw #w Zw Implied Fundamental: 1/1 0.0 4/3 498.0 8/5 813.7 1/1 0.0 8/5 813.7

1 8 10 12 1 12 4 12 1 10 4 40 16 1 9 7 6 5 4 3 8 5 7 1 7 7 7 1 11 3 11 1 9 3 27 Ï9 1 8 6 5 4 3 2 5 3 4Ï ÏLZ Ï ZÏ ∑Ï ZÏ#ZÏ Z ZÏ #Ï ZÏ #Ï I & #ÏLZ ÏLZ #Ï ∑Ï #Ï#ZÏ #Ï #Ï I Ï nÏ #Ï ? #w #w #w #w LZ w ∑ w #Zw # w 16/11 648.7 40/27 680.5 3/2 702.0 ? 10/7 617.5 LZ w Zw #w ∑w 8/7 231.2 16/11 648.7 32/27 294.1 1/1 0.0 the longest, the third from last — at ratios of 1386:3080: will mimic Just Intonation chords at first, looking ini- 4032:4455 — and the last, at ratios of 1890:5103:5600: tially regular in their patterns, but then flattening out as 6750. The chords themselves contain no close pitches or their ratios go out of sync. These just chords, however, tiny intervals; in spacing, they approximate the seventh look weirder as they begin, but one can find large-scale chords of tonal music, revolving roughly around a half- periodicities recurring at different rates, for a truly mul- diminished , spelled upward as Cm, Fm, A, tidimensional sonority. Dm. The half- chord is a traditional I am a long way from coming to any general theory of device for creating musical tension. Tuned in incom- Just Intonation dissonance. But I am already impressed mensurate just intervals, it takes on a whole new aura at how vastly expanded the range of consonance and dis- of dissonance. sonance is in Just Intonation, from beatless note fusions With a good sound system, and played at high that blend into a single to wild assemblages that volume, these seven chords growl and pulsate and on several different levels at once. It’s great news throb in unpredictable acoustic patterns, presenting for someone who can’t stand serving an audience the just the image I wanted: the stillness of death (since the same old tapioca they’ve had before. fundamentals of these pitches remain motionless), but seething with emotion and ominous implications. Notes: Looking at the waveforms of these chords on a sine 1. University of Oklahoma Press. Norman, OK: 1962. wave chart or oscilloscope, their difference from equal 2. On the CD Custer’s Ghost. Monroe Street MSM60104. tempered dissonance is visible. Equally tempered chords Available from the Just Intonation Store.

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