About the Tuning

This tuning was originally devised by Lou Harrison and Bill Slye for the National Reso-Phonic Guitar, used in Scenes from Nek Chand, Lou’s last completed composition before his death in 2003. I originally composed these three movements for that same guitar. The guitar tuning is in D, but I have transposed it to G for these keyboard arrangements. The tuning comprises a twelve-tone-per- scale in 11-limit Just Intonation. In particular, it features two harmonic series segments (1–3–5–7–9–11)—what Harry Partch termed otonality hexads—on fundamentals separated by a perfect (3:2), C and G. Two additional tones needed to complete a twelve-tone scale, A and E , are not part of either harmonic series, but were added by chaining fifths below B , the b b b harmonic of C. Although the harmonics of G are themselves harmonics of C (3–9–15–21–27–33), I have named G 1/1 in the diagrams on this and the following page because of the primary role it plays in these pieces. (Those interested in a detailed explanation of the Just Intonation terminology used here should consult my Just Intonation Primer.) Although the scale has twelve tones per octave, it bears little resemblance Cents deviation from 12-equal to an equally tempered chromatic scale. Step sizes between adjacent tones vary from as little as 53 cents (slightly larger than a tempered quartertone) G 1/1 = 0 Normalized to A to as much as 151 cents (more than a tone and one-half), with many other G ±0.00 G –3.91 sizes in between. The scale provides a great variety of modal possibilities, A –37.04 A –40.95 both historical and novel, but if one desires consonant harmony, one is limited b b A +3.91 A ±0.00 mainly to structures centered on C and G. Most of the intervals in the scale can be easily tuned by ear by an experienced B b –33.13 Bb –37.04 tuner, though the seventh and especially the eleventh harmonic may prove B –13.69 B –17.6 difficult for those unfamiliar with these intervals. The harmonic seventh, C –1.96 C –5.87 7:4, can be tuned by zero beating. It is 27 cents flatter than a Pythagorean . Probably the best procedure is to start from the Pythagorean C # –48.68 C # –52.59 interval and then lower the pitch until the desired beat-free position is found. D +1.95 D –1.96 The harmonic , 11:8, is harder to tune by ear, as the eleventh harmonic E b –35.08 E b –38.99 may be too weak for its beating to be easily detected. The 11:8 tritone has a E –15.64 E –19.55 width of approximately 551 cents, almost exactly the midpoint between the 12-equal and augmented fourth. It is probably best to use a F –31.17 F –35.08 programmable electronic tuner to set this interval until it becomes familiar to F +49.36 F +45.45 the ear. The composer will provide a CD with reference tones on request. G 0.00 G –3.91 The absolute pitch of the tuning is not crucial; the scale may be tuned based on A 440 or on any other historical A that your instrument may be accustomed to.

The Tuning Represented as a Lattice

5 5 E 3 B 4

Major / 28 14 7 7 Ab 27 Eb 9 Bb 6 F 4

C 4 G 1 D 3 A 9 3 1 2 8

Harmonic Seventh/ Perfect Supermajor Second FB 11 11 Fifth/Fourth 6 CB 8 Harmonic Augmented Fourth/ Diminished Fifth Tuning Schema

Keyboard Suite Tuning

cents from 0 63 204 267 386 498 551 702 765 884 969 1049 1200 G 1/1: 1 28 9 7 5 4 11 3 14 5 7 11 2 1 27 8 6 4 3 8 2 9 3 4 6 1 ww# w bww n ww# w &w28 bww 243 n 28 bww 15 n 16 33 12 28 15 21 22 12 27 224 27 14 15 32 11 27 14 20 21 11 cents between 63 141 63 119 112 53 151 63 119 84 81 151 degrees:

Tuning Sequence Harmonic 7th Harmonic Tritone (7:4) (11:8) Pure Fifths/Fourths Pure Fifth Pure Fifths/Fourths Pure Fifth w bw #w & w w w w bw bw w #w w Purew Majorw Third

Checks (Consonant Chords)

w w w w &ww w w w bw bw nw w Major Minorw bw Subminorw n wDominant (very narrow , (maximally consonant but quite consonant) harmonic pentad)

As Harmonic Series Degrees

4567911 4567911 Pure Fourth/Fifth w #w #w w w bw & w bw w w w bw bw w w

(1) ? (1) W W