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Added more info including how it's made and the resulting pitches on Eb and C
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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 '''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.<syntaxhighlight lang="text">
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).
{| class="wikitable"
|+
!1
!3
!5
!7
!9
!11
!13
!15
|-
|style="background-color:red"|3
|style="background-color:yellow"|9
|style="background-color:yellow"|15
|21
|27
|33
|39
|45
|-
|style="background-color:red"|5
|style="background-color:red"|15
|25
|35
|style="background-color:yellow"|45
|55
|65
|75
|-
|style="background-color:red"|7
|style="background-color:red"|21
|style="background-color:red"|35
|49
|63
|77
|91
|105
|-
|style="background-color:red"|9
|style="background-color:red"|27
|style="background-color:red"|45
|style="background-color:red"|63
|81
|99
|117
|135
|-
|style="background-color:red"|11
|style="background-color:red"|33
|style="background-color:red"|55
|style="background-color:red"|77
|style="background-color:red"|99
|121
|143
|165
|-
|style="background-color:red"|13
|style="background-color:red"|39
|style="background-color:red"|65
|style="background-color:red"|91
|style="background-color:red"|117
|style="background-color:red"|143
|169
|195
|-
|style="background-color:red"|15
|style="background-color:red"|45
|style="background-color:red"|75
|style="background-color:red"|105
|style="background-color:red"|135
|style="background-color:red"|165
|style="background-color:red"|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:
{| class="wikitable"
|+
!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+
|}
 
== Works using this scale ==
 
* Kyle Gann - ''Nursery Tunes for Weird Children'' (2012/2015), on E-flat.<ref>https://kylegann.com/NurseryTunes.html</ref>
* Kyle Gann - ''Hyperchromatica'' (2015-2017/2021), on E-flat.<ref>https://kylegann.com/Hyperchromatica.html</ref>
 
== Scl file ==
<syntaxhighlight lang="text">
! Gann_8x8.scl
! Gann_8x8.scl
!
!
! Harmonics 8-16, reproduced on harmonics 8-16 (or 15-odd-limit order-2 otonal heterodyne)
! Harmonics 8-16, reproduced on harmonics 8-16 (or 15-odd-limit order-2 otonal heterodyne)
!
!
Kyle Gann, 8x8 Scale ("Nursery Tunes for Demented Children", "Hyperchromatica")
Kyle Gann, 8x8 Scale ("Nursery Tunes for Weird Children", "Hyperchromatica")
33
33
!
!
Retrieved from "https://en.xen.wiki/w/Gann_8x8"