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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