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Recording (MIDI realization) here
PDF score here
CAUTION: Score is meaningless as to pitch without retuning, key given at bottom of this page
Triskaidekaphonia is a piece for virtual electronic piano, i.e., electric keyboard that sounds like a piano. Unlike my earlier microtonal works, this one can be performed live by a soloist, without additional electronics. The tuning is superlatively simple: it consists of merely all the ratios formed by the whole numbers from 1 to 13, of which there are 29:
13/12, 13/11, 13/10, 13/9, 13/8, 13/7 (13/6, 13/5, and so on, are merely octaves of those already mentioned)
12/11, 12/7 (12/10 is the same as 6/5, 12/9 = 4/3, and so on)
11/10, 11/9, 11/8, 11/7, 11/6
10/9, 10/7 (10/8 = 5/4, 10/6 = 5/3)
9/8, 9/7, 9/5
8/7, 8/5
7/6, 7/5, 7/4
6/5
5/4, 5/3
4/3
3/2
1/1
The resulting scale (given in Ben Johnston's notation) is as follows:
| Pitch: | D | E13b | Ev+ | E^b | E | E+ | EL | F7+ | F13v+ | F+ | F^ | F#+ | F#L+ | G13b | G |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ratio: | 1/1 | 13/12 | 12/11 | 11/10 | 10/9 | 9/8 | 8/7 | 7/6 | 13/11 | 6/5 | 11/9 | 5/4 | 9/7 | 13/10 | 4/3 |
| Cents: | 0 | 138.6 | 150.6 | 165 | 182.4 | 203.9 | 231.2 | 266.9 | 289.2 | 315.6 | 347.4 | 386.3 | 435.1 | 454.2 | 498 |
| G^ | A7b+ | G#L | A13b | A+ | A^L | Bb | B13b | B | BL | C7+ | C+ | C^ | C13L |
| 11/8 | 7/5 | 10/7 | 13/9 | 3/2 | 11/7 | 8/5 | 13/8 | 5/3 | 12/7 | 7/4 | 9/5 | 11/6 | 13/7 |
| 551.3 | 582.5 | 617.5 | 636.6 | 702 | 782.5 | 813.7 | 840.5 | 884.4 | 933.1 | 968.8 | 1017.6 | 1049.4 | 1071.7 |
(If you don't have enough experience with just intonation to make sense of this chart, try reading the step-by-step Just Intonation Explained section.) In Johnston's notation, + raises a pitch by 81/80, - lowers it by 80/81, # raises it by 25/24, b lowers it by 24/25, 7 lowers it by 35/36, L raises it by 36/35, ^ raises it by 33/32, v lowers it by 32/33, 13 raises it by 65/64, and F-A-C, C-E-G, and G-B-D are all perfectly tuned 4:5:6 major triads.
This is my first piece to go beyond 11-limit tuning to use the 13th harmonic. I figured out that I could make different scales within this network by taking all notes expressible by the form 13/X, or 11/X, or X/7, and the scales with the smallest numbers would be closest to simple tonality, while the larger-numbered scales will have a much more oblique relationship. Thus the scales that run through the piece are:
| 13/X: | 1/1 | 13/12 | 13/11 | 13/10 | 13/9 | 13/8 | 13/7 |
|---|---|---|---|---|---|---|---|
| 12/X: | 1/1 | 12/11 | 6/5 | 4/3 | 3/2 | 12/7 | |
| 11/X: | 1/1 | 11/10 | 11/9 | 11/8 | 11/7 | 11/6 | |
| 10/X: | 1/1 | 10/9 | 5/4 | 10/7 | 5/3 | ||
| 9/X: | 1/1 | 9/8 | 9/7 | 3/2 | 9/5 | ||
| 8/X: | 1/1 | 8/7 | 4/3 | 8/5 | |||
| 7/X: | 1/1 | 7/6 | 7/5 | 7/4 | |||
| X/9: | 1/1 | 10/9 | 11/9 | 4/3 | 13/9 | 5/3 | |
| X/7: | 1/1 | 8/7 | 9/7 | 10/7 | 11/7 | 12/7 | 13/7 |
| X/5: | 1/1 | 11/10 | 6/5 | 13/10 | 7/5 | 8/5 | 9/5 |
| X/3: | 1/1 | 13/12 | 7/6 | 4/3 | 3/2 | 5/3 | 11/6 |
| X/2: | 1/1 | 9/8 | 5/4 | 11/8 | 3/2 | 13/8 | 7/4 |
Of course, the scales with X in the numerator are overtone series', and those with X in the denominator are undertone series'. Since all the pitches have a simple relationship to 1/1 D, a rhythmicized drone on that pitch runs throughout. The piece, then, moves "in and out of focus" depending on which scale is used at a given moment.
The score (available here as a PDF) uses conventional notation that does not indicate pitch, but merely tells the keyboardist which keys to play. The keyboard is tuned thusly:
| Key: | Pitch: | Ratio: | Tuning (relative to normal 12tet): |
| C1 | D | 1/1 | D0 + 0 cents |
| D1 | D | 1/1 | D1 + 0 cents |
| E1 | A+ | 3/2 | A1 + 2 cents |
| F1 | C7+ | 7/4 | C2 - 31.2 cents |
| G1 | D | 1/1 | D2 + 0 cents |
| A1 | A+ | 3/2 | A2 + 2 cents |
| A#1 | A^L | 11/7 | A#2 - 17.5 cents |
| B1 | Bb | 8/5 | Bb2 + 13.7 cents |
| C2 | B13b | 13/8 | Bb2 + 40.5 cents |
| C#2 | B | 5/3 | B2 - 15.6 cents |
| D2 | BL | 12/7 | B2 + 33.1 cents |
| D#2 | C7+ | 7/4 | C3 - 31.2 cents |
| E2 | C+ | 9/5 | C3 + 17.6 cents |
| F2 | C^ | 11/6 | C3 + 49.4 cents |
| F#2 | C13L | 13/7 | C#3 - 18.3 cents |
| G2 | D | 1/1 | D3 + 0 cents |
| G#2 | E13b | 13/12 | Eb3 + 38.6 cents |
| A2 | Ev+ | 12/11 | E3 - 49.4 cents |
| A#2 | E^b | 11/10 | E3 - 35 cents |
| B2 | E | 10/9 | E3 - 17.6 cents |
| C3 | E+ | 9/8 | E3 + 3.9 cents |
| C#3 | EL | 8/7 | E3 + 31.2 cents |
| D3 | F7+ | 7/6 | F3 - 33.1 cents |
| D#3 | F13v+ | 13/11 | F3 - 10.8 cents |
| E3 | F+ | 6/5 | F3 + 15.6 cents |
| F3 | F^ | 11/9 | F3 + 47.4 cents |
| F#3 | F#+ | 5/4 | F#3 - 13.7 cents |
| G3 | F#L+ | 9/7 | F#3 + 35.1 cents |
| G#3 | G13b | 13/10 | G3 - 45.8 cents |
| A3 | G | 4/3 | G3 - 2 cents |
| A#3 | G^ | 11/8 | G#3 - 48.7 cents |
| B3 | A7b+ | 7/5 | Ab3 - 17.5 cents |
| C4 | G#L | 10/7 | G#3 + 17.5 cents |
| C#4 | A13b | 13/9 | Ab3 + 36.6 cents |
| D4 | A+ | 3/2 | A3 + 2 cents |
| D#4 | A^L | 11/7 | Ab3 - 17.5 cents |
| E4 | Bb | 8/5 | Bb3 + 13.7 cents |
| F4 | B13b | 13/8 | Bb3 + 40.5 cents |
| F#4 | B | 5/3 | B3 - 15.6 cents |
| G4 | BL | 12/7 | B3 + 33.1 cents |
| G#4 | C7+ | 7/4 | C4 - 31.2 cents |
| A4 | C+ | 9/5 | C4 + 17.6 cents |
| A#4 | C^ | 11/6 | C4 + 49.4 cents |
| B4 | C13L | 13/7 | C#4 - 28.3 cents |
| C5 | D | 1/1 | D4 + 0 cents |
| C#5 | E13b | 13/12 | Eb4 + 38.6 cents |
| D5 | Ev+ | 12/11 | E4 - 49.4 cents |
| D#5 | E^b | 11/10 | E4 - 35 cents |
| E5 | E | 10/9 | E4 -17.6 cents |
| F5 | E+ | 9/8 | E4 + 3.9 cents |
| F#5 | EL | 8/7 | E4 + 31.2 cents |
| G5 | F7+ | 7/6 | F4 - 33.1 cents |
| G#5 | F13v+ | 13/11 | F4 - 10/8 cents |
| A5 | F+ | 6/5 | F4 + 15.6 cents |
| A#5 | F^ | 11/9 | F4 + 47.4 cents |
| B5 | F#+ | 5/4 | F#4 - 13.7 cents |
| C6 | F#L+ | 9/7 | F#4 + 35.1 cents |
Enough information is given here to facilitate a performance - kindly contact the composer if this is contemplated. The piece is performable on a five-octave keyboard.
Kyle Gann
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