Superparticular-Nonoctave-MOS

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Superparticular-Nonoctave-MOS

NOTE: I haven't completed the list of scales on this page. Consider that part under construction. You can check the intro & the few scales I have in the meantime, though!

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A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on the nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:

  1. Superparticular - meaning that the steps of the scale represent the intervals between adjacent notes in the harmonic series. You can identify these intervals easily, because they appear in the form n/n-1. Examples: 5:4, 7:6, 13:12, 41:40, etc.
  2. Nonoctave - meaning that the scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, & so on. These intervals can sound very harsh, but they can also sound incredibly rich & dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.
  3. Moment of Symmetry - meaning that the scale contains exactly two step sizes, spaced out as evenly as possible within the scale. Normally, you build MOS scales by continuously adding notes a given interval, called the generator, away from one another in one long chain until the resulting scale has only two step sizes. Pythagorean scales use 3/2 (the perfect fifth) as the generator. In 12edo, we can identify the standard pentatonic scale & the various diatonic scales as MOS scales because you can build them using a chain of fifths.

Jacky Ligon's scale meets these three criteria. I will use it as an example:

Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11

  1. It is superparticular because its intervals, 9/8 & 12/11, both fit the form n/n-1.
  2. It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.
  3. It is MOS because it contains exactly two step sizes, spaced out as evenly as possible within the scale.

Inspired by the peculiar musical qualities of this scale, I set about looking for others, & found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type within these (admittedly arbitrary) limits:

  1. Superparticular Limit: smallest interval: 41/40 = 42.8 cents.
  2. Nonoctave Limit: greatest deviation from octave allowed = 25 cents.
  3. Moment of Symmetry Limit: greatest number of notes in a scale = 10.

Even with these limits in place, this produces a multitude of fascinating scales for our enjoyment & fascination. I invite you to play & share your results!

Pentatonic (5-note) Scales:

MOS 2+3 : sLsLs

SNM230513 : 13:12, 5:4, 13:12, 5:4, 13:12 = 1188.3 cents

SNM230610 : 10:9, 6:5, 10:9, 6:5, 10:9 = 1178.5 cents

MOS 3+2 : LsLsL

SNM320614 : 6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents

SNM320615 : 6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents

Heptatonic (7-note) Scales:

MOS 2+5 : sLsssLs

SNM250520 : 20:19, 5:4, 20:19, 20:19, 20:19, 5:4, 20:19 = 1216.6 cents

SNM250521 : 21:20, 5:4, 21:20, 21:20, 21:20, 5:4, 21:20 = 1195.0 cents (nickname: Mercury Sand)

SNM250616 : 16:15, 6:5, 16:15, 16:15, 16:15, 6:5, 16:15 = 1189.9 cents

MOS 3+4 : sLsLsLs

MOS 4+3 : LsLsLsL

MOS 5+2 : LsLLLsL

Octatonic (8-note) Scales:

MOS 3+5 : sLssLsLs

MOS 5+3 : LsLLsLsL

Nonatonic (9-note) Scales:

MOS 2+7 : ssLsssLss

SNM270528 : 28:27, 28:27, 5:4, 28:27, 28:27, 28:27, 5:4, 28:27, 28:27 = 1213.4 cents

SNM270529 : 29:28, 29:28, 5:4, 29:28, 29:28, 29:28, 5:4, 29:28, 29:28 = 1197.9 cents

SNM270530 : 30:29, 30:29, 5:4, 30:29, 30:29, 30:29, 5:4, 30:29, 30:29 = 1183.5 cents

SNM270622 : 22:21, 22:21, 6:5, 22:21, 22:21, 22:21, 6:5, 22:21, 22:21 = 1195.0 cents

MOS 4+5 : LsLsLsLsL

MOS 5+4 : sLsLsLsLs

MOS 7+2 : LLsLLLsLL

Dekatonic (10-note) Scales:

MOS 3+7 : sLsssLssLs

MOS 7+3 : LsLLLsLLsL