User:Zastari

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JI scales on the Harmonic Series

/2^n Scales of Interest

A /2^n scale is a JI scale with the following properties:

  • All denominators of all notes in the scale are of the form /2^n
  • The scale has constant structure

Impropriety is not important; many of these scales will be improper.

3-Limit

Pyth (2.3)

Pyth is a line in 2.3 space. It can be traversed only by stacking 4/3. It forms constant structure at the same sizes that it forms a MOS.

Pyth frame: AGS(4/3)

  • [5]: 9/8 81/64 3/2 27/16 2/1
  • [7]: 9/8 81/64 729/512 3/2 27/16 243/128 2/1

5-Limit

Duodene (2.3.5)

Duodene is a 4x3 rectangular prism in 2.3.5 space. It can be traversed along prime axes in any order yielding 2 possible AGS recipes that generate it.

  • Augmented frame (5->3): AGS(8/5, 8/5, 25/24):
    • [6]: 75/64 5/4 3/2 25/16 15/8 2/1
    • [9]: 9/8 75/64 5/4 45/32 3/2 25/16 225/128 15/8 2/1
    • [12]: 135/128 9/8 75/64 5/4 675/512 45/32 3/2 25/16 27/16 225/128 15/8 2/1
  • Zarlino frame (3->5) AGS(4/3, 4/3, 4/3, 27/20):
    • [5]: 9/8 5/4 3/2 27/16 2/1
    • [7]: 9/8 5/4 45/32 3/2 27/16 15/8 2/1 (aka Zarlino[7])
    • [12]: 135/128 9/8 75/64 5/4 675/512 45/32 3/2 25/16 27/16 225/128 15/8 2/1 (aka Duodene[12])

7-Limit

Tas (2.3.7)

AGS(7/6, 8/7)

  • [5]: 9/8 21/16 3/2 7/4 2/1
  • [9]: 9/8 81/64 (or 5/4) 21/16 189/128 3/2 27/16 7/4 63/32 2/1
Detempered Marveldene (2.3.5.7)
  • [7] Mixolydian: 9/8 5/4 21/16 3/2 27/16 7/4 2/1
  • [7] Lydian (aka Zarlino[7]): 9/8 5/4 45/32 3/2 27/16 15/8 2/1
  • [7] Locrian: 135/125 75/64 21/16 45/32 25/16 7/4 2/1
  • [12]: 135/128 9/8 75/64 5/4 21/16 45/32 3/2 25/16 27/16 7/4 15/8 2/1
Zil (2.3.5.7)

Note: Zil is sometimes defined as a chiral version that contains 405/256 and 2835/2048 in place of 25/16 and 175/128 which yields the AGS (8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6). The version discussed below uses 25/16 and 175/128 yielding the truncated AGS (8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 189/160).

Zil is a 4x3x2 rectangular prism in 2.3.5.7 space. It can be traversed along the prime axes in any order yielding 6 possible AGS recipes that generate it.

  • Tas frame (7->3->5) : AGS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160)
    • [5]: 9/8 21/16 3/2 7/4 2/1
    • [9]: 9/8 5/4 21/16 189/128 3/2 27/16 7/4 63/32 2/1
    • [14]: 35/32 9/8 315/256 5/4 21/16 45/32 189/128 3/2 105/64 27/16 7/4 15/8 63/32 2/1
    • [19]: 135/128 35/32 9/8 75/64 315/256 5/4 21/16 175/128 45/32 189/128 3/2 25/16 105/64 27/16 7/4 945/512 15/8 63/32 2/1
  • Hexatonic septimal-pental frame (7->5->3): AGS(8/7, 7/5, 8/7, 7/5, 8/7, 175/96)
    • [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (aka Hexatonic pental-septimal[6])
    • [12]: 525/512 35/32 75/64 5/4 21/16 175/128 3/2 25/16 105/64 7/4 15/8 2/1 (aka Hexatonic pental-septimal[6])
  • Hexatonic pental-septimal frame (5->7->3): AGS(8/5, 8/5, 25/14, 8/5, 8/5, 175/96)
    • [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (aka Hexatonic septimal-pental[6])
    • [9]: 35/32 75/64 5/4 175/128 3/2 25/16 7/4 15/8 2/1
    • [12]: 525/512 35/32 75/64 5/4 21/16 175/128 3/2 25/16 105/64 7/4 15/8 2/1 (aka Hexatonic septimal-pental[12])
    • [15]: 525/512 35/32 9/8 75/64 5/4 21/16 175/128 45/32 3/2 25/16 105/64 7/4 225/128 15/8 2/1
  • Duodene Augmented frame (5->3->7): AGS(8/5, 8/5, 25/24, 8/5, 8/5, 25/24, 8/5, 8/5, 25/24, 8/5, 8/5, 675/448)
    • [6], [9], and [12] are the same as the Duodene Augmented frame in 2.3.5.
  • Duodene Zarlino frame (3->5->7) AGS(4/3, 4/3, 4/3, 40/27, 4/3, 4/3, 4/3, 40/27, 4/3, 4/3, 4/3, 896/675)
    • [5], [7], and [12] are the same as Duodene Zarlino frame in 2.3.5.

Transitional Modes

  • Unknown mode (2.3.7.25): 525/512 75/64 21/16 175/128 25/16 7/4 2/1
  • Unknown mode (2.3.5.7): 35/32 5/4 21/16 3/2 105/64 7/4 2/1
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