User:Zastari: Difference between revisions

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== /2^n JI Scales of Interest ==
== JI scales on the Harmonic Series ==


=== 3-Limit ===
=== /2^n Scales of Interest ===
 
==== 3-Limit ====


====== Pyth (2.3) ======
====== Pyth (2.3) ======


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


=== 5-Limit ===
==== 5-Limit ====


====== [[Zarlino|Zarlino (2.3.5)]] ======
=====[[Zarlino|Zarlino (2.3.5)]]=====


* [7]: 9/8 5/4 45/32 3/2 27/16 15/8 2/1
* [7] Lydian: 9/8 5/4 45/32 3/2 27/16 15/8 2/1  


'''Duodene'''
===== Duodene (2.3.5) =====


AGS: (8/5, 8/5, 25/24)
* [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


* [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
==== 7-Limit ====


==='''7-Limit'''===
===== Tas (2.3.7) =====


==== Tas (2.3.7) ====
* [5]: 9/8 21/16 3/2 7/4 2/1
* [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


* [9]: 9/8 81/64 21/16 189/128 3/2 27/16 7/4 63/32 2/1
===== Detempered Marveldene (2.3.5.7) =====


==== Detempered Marveldene (2.3.5.7) ====
* [7] Mixolydian: 9/8 5/4 21/16 3/2 27/16 7/4 2/1
* [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 (aka a Marveldene detempering)


* [7] (Lydian): 9/8 5/4 45/32 3/2 27/16 15/8 2/1 (aka Zarlino[7])
===== 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).


* [7] (Locrian): 135/125 75/64 21/16 45/32 25/16 7/4 2/1
Chromatic version: 135/128 9/8 75/64 5/4 21/16 45/32 3/2 25/16 27/16 7/4 15/8 2/1 (aka a Marveldene detempering)
==== Zil (2.3.5.7) ====
Zil is a 4x3x2 rectangle in the (3, 5, 7) space. It can be traversed along the prime axes in any order yielding 6 possible AGS recipes that generate it.
Zil is a 4x3x2 rectangle in the (3, 5, 7) space. It can be traversed along the prime axes in any order yielding 6 possible AGS recipes that generate it.
AGS generators (all operations are downwards starting at 1/1 and generate zil[24] at the 24 note size):


* Tas frame (7->3->5) : AGS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160)
* Tas frame (7->3->5) : AGS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160)
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* Hexatonic septimal-pental frame (7->5->3): AGS(8/7, 7/5, 8/7, 7/5, 8/7, 175/96)
* 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 (same as Hexatonic pental-septimal)
** [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (same as Hexatonic pental-septimal)
** [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 (same as Hexatonic pental-septimal)
** [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
* Hexatonic pental-septimal frame (5->7->3): AGS(8/5, 8/5, 25/14, 8/5, 8/5, 175/96)
* Hexatonic pental-septimal frame (5->7->3):
** [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (same as Hexatonic septimal-pental)
** [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (same as Hexatonic septimal-pental)
** [9]: 35/32 75/64 5/4 175/128 3/2 25/16 7/4 15/8 2/1
** [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 (same as Hexatonic septimal-pental)
** [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 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)
* Duodene 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]: 75/64 5/4 3/2 25/16 15/8 2/1
** [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
** [9]: 9/8 75/64 5/4 45/32 3/2 25/16 225/128 15/8 2/1
Unknown hexatonic: 35/32 5/4 175/128 25/16 7/4 2/1
Chromatic Extension (3/2 offset): 525/512 35/32 75/64 5/4 21/16 175/128 3/2 25/16 105/64 7/4 15/8 2/1
* Offsetting this scale by 9/8 yields zil[24]
Contains tas[14] with 81/64 warped to 5/4 instead.
Diatonic-like Modes:
All Marveldene[12] 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

Revision as of 03:54, 6 December 2023

JI scales on the Harmonic Series

/2^n Scales of Interest

3-Limit

Pyth (2.3)
  • [7] Lydian: 9/8 81/64 729/512 3/2 27/16 243/128 2/1

5-Limit

Zarlino (2.3.5)
  • [7] Lydian: 9/8 5/4 45/32 3/2 27/16 15/8 2/1
Duodene (2.3.5)
  • [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

7-Limit

Tas (2.3.7)
  • [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 (aka a Marveldene detempering)
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 rectangle in the (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
  • 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 (same as Hexatonic pental-septimal)
    • [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
  • Hexatonic pental-septimal frame (5->7->3):
    • [6]: 35/32 5/4 175/128 25/16 7/4 2/1 (same as Hexatonic septimal-pental)
    • [9]: 35/32 75/64 5/4 175/128 3/2 25/16 7/4 15/8 2/1
  • Duodene 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]: 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

Unknown hexatonic: 35/32 5/4 175/128 25/16 7/4 2/1

Chromatic Extension (3/2 offset): 525/512 35/32 75/64 5/4 21/16 175/128 3/2 25/16 105/64 7/4 15/8 2/1

  • Offsetting this scale by 9/8 yields zil[24]

Contains tas[14] with 81/64 warped to 5/4 instead.

Diatonic-like Modes:

All Marveldene[12] 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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