Gann 8x8
The 8x8 scale, or Gann 8x8, is a 13-limit just intonation scale with 33 pitches per octave, developed by Kyle Gann and used in his cycle Hyperchromatica for three player pianos.
The scale is formed by taking the first eight odd harmonics (1, 3, 5, 7, 9, 11, 13, and 15), and then taking the first eight odd harmonics of each of those pitches. In the following table, the resulting numerators are shown, with the power-of-two denominators omitted. The red cells are duplicates of their mirrors above the diagonal (3×7 = 7×3 = 21, for example), and the yellow cells are duplicates due to having other factorizations (3×15 = 5×9 = 45, for example).
| 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 |
|---|---|---|---|---|---|---|---|
| 3 | 9 | 15 | 21 | 27 | 33 | 39 | 45 |
| 5 | 15 | 25 | 35 | 45 | 55 | 65 | 75 |
| 7 | 21 | 35 | 49 | 63 | 77 | 91 | 105 |
| 9 | 27 | 45 | 63 | 81 | 99 | 117 | 135 |
| 11 | 33 | 55 | 77 | 99 | 121 | 143 | 165 |
| 13 | 39 | 65 | 91 | 117 | 143 | 169 | 195 |
| 15 | 45 | 75 | 105 | 135 | 165 | 195 | 225 |
Properties of the scale
The scale is in 13-limit just intonation, and all the denominators are powers of two. The largest numerator is 225 = 15×15. The spacing between consecutive pitches is highly uneven, ranging from 7.71¢ between 225/128 and 7/4 (a 225/224 septimal kleisma) to 92.6¢ between 91/64 and 3/2 (a 96/91 ratio).
The ratios in the scale, as well as the resulting notes, in Ben Johnston's notation, when building the scale on E-flat (assigned to 1/1 in the two works by Gann using this scale) and C:
| Ratio | Cents | Note built on Eb | Note built on C |
|---|---|---|---|
| 1/1 | 0 | Eb | C |
| 65/64 | 26.8 | Eb13 | C13 |
| 33/32 | 53.3 | Eb^ | C^ |
| 135/128 | 92.2 | E+ | C#+ |
| 35/32 | 155.1 | F7+ | D7 |
| 143/128 | 191.8 | Fb13^ | Db13^- |
| 9/8 | 203.9 | F+ | D |
| 75/64 | 274.6 | F#+ | D# |
| 77/64 | 320.1 | Gb7^ | Eb7^ |
| 39/32 | 342.5 | Gb13 | Eb13 |
| 5/4 | 386.3 | G | E |
| 81/64 | 407.8 | G+ | E+ |
| 165/128 | 439.6 | G^ | E^ |
| 21/16 | 470.8 | Ab7+ | F7+ |
| 169/128 | 481.1 | Abb1313 | Fb1313 |
| 11/8 | 551.3 | Ab^ | F^ |
| 45/32 | 590.2 | A+ | F#+ |
| 91/64 | 609.4 | Bbb713 | Gb713 |
| 3/2 | 702.0 | Bb | G |
| 195/128 | 728.8 | Bb13 | G13 |
| 49/32 | 737.7 | Cb77+ | Ab77+ |
| 99/64 | 755.2 | Bb^ | G^ |
| 25/16 | 772.6 | B | G# |
| 13/8 | 840.5 | Cb13 | Ab13 |
| 105/64 | 857.1 | C7+ | A7+ |
| 27/16 | 905.9 | C+ | A+ |
| 55/32 | 937.6 | C^ | A^ |
| 7/4 | 968.8 | Db7 | Bb7 |
| 225/128 | 976.5 | C#+ | A#+ |
| 117/64 | 1044.4 | Db13 | Bb13 |
| 15/8 | 1088.3 | D | B |
| 121/64 | 1102.6 | Db^^- | Bb^^- |
| 63/32 | 1172.7 | Eb7+ | C7+ |
The most versatile pitches are 15/8 and 45/32 (D and A+ in Hyperchromatica), each appearing in four different harmonic series, which is the reason Andromeda Memories, the first movement of Hyperchromatica, builds its jazz harmonies on D.[1]
On top of 5/4 (G in Hyperchromatica), it is possible to build minor (6/5), major (5/4), and neutral thirds, using 49/40 for the neutral third (49/32 relative to 1/1) instead of the more common 11/9. This minor-neutral-major third sequence, as Bb—Cb77+—B, is the basis for the movement Busted Grooves, and appears as an ostinato throughout. However, 9/8 (F+) comes close, with relative 11/9 (absolute 11/8, Ab^) for the neutral third, but 7/6 (absolute 21/16, Ab7+) for the minor third.
Works using this scale
- Kyle Gann - Nursery Tunes for Weird Children (2012/2015), on E-flat.[2]
- Kyle Gann - Hyperchromatica (2015-2017/2021), on E-flat.[3]
Scl file
! Gann_8x8.scl
!
! Harmonics 8-16, reproduced on harmonics 8-16 (or 15-odd-limit order-2 otonal heterodyne)
!
Kyle Gann, 8x8 Scale ("Nursery Tunes for Weird Children", "Hyperchromatica")
33
!
65/64
33/32
135/128
35/32
143/128
9/8
75/64
77/64
39/32
5/4
81/64
165/128
21/16
169/128
11/8
45/32
91/64
3/2
195/128
49/32
99/64
25/16
13/8
105/64
27/16
55/32
7/4
225/128
117/64
15/8
121/64
63/32
2/1