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:
- 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.
- 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.
- 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
- It is superparticular because its intervals, 9/8 & 12/11, both fit the form n/n-1.
- It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.
- 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:
- Superparticular Limit: smallest interval: 41/40 = 42.8 cents.
- Nonoctave Limit: greatest deviation from octave allowed = 25 cents.
- 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