MOS
A MOS (or mos, or moment of symmetry scale) is a member of a family of scales which generalize properties of the diatonic and pentatonic scales within 12 equal temperament. Microtonal MOS scales are often seen as some of the easier microtonal scales to work with, because the number of distinct intervals within the scale is quite low.
As in any scale, we can define "n-step interval" to refer to one of many intervals that arise from ascending by n steps of the scale. Formally, a MOS is a scale where, for all n, there are at most two distinct sizes of n-step intervals. Any multiple of the period (which is usually an octave or a fraction thereof) has only one size.[1] MOS scales are also known as MV2 (or maximum variety 2) scales.
MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.
MOSses can be named according to their number of large and small steps (for example, 5L 2s for the diatonic MOS), because there is exactly one step pattern that fits the MOS criteria with any given number of small and large steps. This form of name does not specify the tuning of the MOS or which scale degree is defined as the tonic.
Examples
The most widely used MOS scale is the 12edo diatonic scale, which has five equal large steps (major seconds) and two equal small steps (minor seconds) within the octave. It can thus be notated 5L 2s. In contrast, while the melodic minor scale (LsLLLLs) has only two step sizes, it is still not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.
A MOS exists for any whole number of large and small steps, for example 3L 4s (mosh), which functions as a "neutral" version of the diatonic scale, and 1L 6s (onyx), which has 1 large step and thus a very wide range of tunings.
The equave of a MOS is denoted using angle brackets: for example, 3L2s⟨3/2⟩ denotes the 3L 2s MOS pattern but using 3/2 as the interval of equivalence rather than 2/1.
Periods and generators
Every MOS scale can be generated by stacking a certain interval called the generator a number of times, then moving each note by multiples of the period (which is usually the octave) to fit within the span of one period.[2] The latter step is called period reduction. For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s, is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produce a MOS, because there are more than 2 sizes of some interval classes.
The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...
A 2/1-equivalent MOS scale aLbs always has 1\gcd(a, b) as its period. For example, 5L2s has period 1\1; 5L5s has period 1\5; 2L8s has period 1\2.
Hardness
One way to specify the tuning of a given MOS pattern (with a given equave) is hardness, which refers to the logarithmic ratio between the size of the L step versus the size of the s step. A tuning of a given MOS that has a higher hardness is harder, and one with a lower hardness is softer.
The hardness of a MOS scale is mostly associated with its melodic shape. Softer tunings (with more similar step sizes) may sound melodically smoother, softer, or more mellow. In contrast, harder tunings of the same MOS (with steps of very different sizes) may sound jagged, dramatic, or sparkly.
Hardness is usually given as L/s, and can fall anywhere between 1 and positive infinity. A hardness value of 1 is called equalized (since L = s), and positive infinity is called collapsed (since s = 0). We call hardness 2/1 the basic tuning of the MOS; the basic tuning is the smallest equal tuning that meaningfully supports the MOS scale. More generally, the basic tuning of a MOS aLbs is always (2a+b)-edo.[3]
Examples for MOS diatonic:
- 12edo diatonic is 2221221, so it has hardness 2/1.
- 17edo diatonic is 3331331, so it has hardness 3/1.
- 19edo diatonic is 3332332, so it has hardness 3/2.
- The equalized tuning is 7edo (1111111).
- The collapsed tuning is 5edo (1110110).
Table of MOS scales
The following table lists common MOS scales grouped by scale size, along with their step patterns and associated temperaments. All temperaments mentioned in the table be found in the List of regular temperaments. All names in the first column do not specify anything about tuning or which scale degree is the tonic. Common MOSes are highlighted.
| Name | aLbs | Brightest mode | Equalized (softest) gen. | Collapsed (hardest) gen. | Period (1\1 (1200c) unless otherwise stated) |
Description |
|---|---|---|---|---|---|---|
| 5-note MOSses | ||||||
| pentic | 2L3s | LsLss | 2\5 (480c) | 1\2 (600c) | Called "pentatonic" in 12edo music theory. Five-note subset of both MOS diatonic and antidiatonic. | |
| antipentic | 3L2s | LLsLs | 2\5 (480c) | 1\3 (400c) | Five-note subset of both oneirotonic and checkertonic. | |
| manual | 4L1s | LLLLs | 1\5 (240c) | 1\4 (300c) | Five-note subset of both semiquartal and gramitonic. | |
| 6-note MOSses | ||||||
| machinoid | 5L1s | LLLLLs | 1\6 (200c) | 1\5 (240c) | Some temperament interpretations: Machine[6], Gorgo[6], Slendric[6]. | |
| 7-note MOSses | ||||||
| onyx | 1L6s | Lssssss | 1\7 (171.4c) | 0\6 (0c) | One temperament interpretation is Porcupine[7]. | |
| antidiatonic | 2L5s | LssLsss | 3\7 (685.7c) | 1\2 (600c) | When using a very flat fifth, reverses the interval qualities of diatonic. One temperament interpretation is Mabilic[7]. | |
| mosh | 3L4s | LsLsLss | 2\7 (342.9c) | 1\3 (400c) | Neutral thirds generate this MOS. | |
| smitonic | 4L3s | LLsLsLs | 2\7 (342.9c) | 1\4 (300c) | So named because the generator is a sharp minor third. Inthar finds that it sounds like a brighter, stretched version of the diatonic scale. In fact, it can be seen as a "warped" diatonic scale, because it can be found by replacing one large step of diatonic with a small step. One temperament interpretation is Orgone[7]. | |
| (MOS) diatonic | 5L2s | LLLsLLs | 4\7 (685.7c) | 3\5 (720c) | ||
| arch(a)eotonic | 6L1s | LLLLLLs | 1\7 (171.4c) | 1\6 (200c) | Some temperament interpretations: Tetracot[7], Didacus[7]. | |
| 8-note MOSses | ||||||
| checkertonic | 3L5s | LsLssLss | 3\8 (450c) | 1\3 (400c) | Somewhat like a stretched tcherepnin scale. Associated with Squares (when relatively hard) and Sensi (hardness 3/2 or softer). In softer tunings, it is the most consonant 8-tone MOS. | |
| tetrawood | 4L4s | LsLsLsLs | 1\8 (150c) | 1\4 (300c) | 1\4 (300c) | Exists in 12edo; often called the "diminished" or "octatonic" scale in 12edo theory.[4] |
| oneirotonic | 5L3s | LLsLLsLs | 3\8 (450c) | 2\5 (480c) | Sounds like a darker, compressed version of the diatonic scale according to Inthar. | |
| ekic | 6L2s | LLLsLLLs | 1\8 (150c) | 1\6 (200c) | 1\2 (600c) | The 22edo (step ratio L:s = 3:2) tuning is associated with Hedgehog temperament. Lumi Pakkanen has described the sound of this MOS as "dreamy, yet oppressing", especially when tuned to step ratio 3:1.[5] |
| pine | 7L1s | LLLLLLLs | 1\8 (150c) | 1\7 (171.4c) | ||
| 9-note MOSses | ||||||
| tcherepnin | 3L6s | LssLssLss | 1\9 (133.3c) | 1\3 (400c) | 1\3 (400c) | Exists in 12edo; associated with Augmented temperament, which collapses the circle of just (5/4) major thirds into one augmented chord formed from 400c major thirds. |
| gramitonic | 4L5s | LsLsLsLss | 2\9 (266.7c) | 1\4 (300c) | So named because the generator is a grave minor third. Associated with Orwell temperament. One of the more consonant 9-note MOSses, especially when tuned with a step ratio 4:3 < L:s < 3:2. | |
| semiquartal | 5L4s | LLsLsLsLs | 2\9 (266.7c) | 1\5 (240c) | So named because the generator is half a fourth. One of the more consonant 9-note MOSses. | |
| armotonic | 7L2s | LLLLsLLLs | 5\9 (666.7c) | 4\7 (685.7c) | Contains antidiatonic as a subset. One temperament interpretation is Mabilic[9], or it can be seen in lower complexity as Mavila[9], where three just perfect fourths stack to a just (5/2) major tenth. | |
| subneutralic | 8L1s | LLLLLLLLs | 1\9 (133.3c) | 1\8 (150c) | Has tunings that split the perfect fifth into 5 equal parts, e.g. in 17edo. | |
| 10-note MOSses | ||||||
| jaric | 2L8s | LssssLssss | 1\10 (120c) | 1\2 (600c) | 1\2 (600c) | So named because of Pajara[10] and Injera[10] interpretations. Interpreted as Diaschismic[10] in 34edo and 46edo. |
| pentawood | 5L5s | LsLsLsLsLs | 1\10 (120c) | 0\5 (0c) | 1\5 (240c) | One temperament interpretation is Blackwood[10]. Often called the Blackwood scale because composer Easley Blackwood was one of the first to use it. Contains either a major or minor triad on every note, and thus can be seen as having similar properties to MOSdiatonic in 5n-edos. |
| dicoid | 7L3s | LLLsLLsLLs | 3\10 (360c) | 2\7 (342.9c) | The 10-note MOS generated by neutral thirds. So named because of the exotemperament Dichotic. | |
| taric | 8L2s | LLLLsLLLLs | 1\10 (120c) | 1\8 (150c) | 1\2 (600c) | Generated by an oneirotonic generator (3\5 to 5\8). Named after Hindi for 18 (aṭhārah), because 18edo is the basic tuning. |
| sinatonic | 9L1s | LLLLLLLLLs | 1\10 (120c) | 1\9 (133.3c) | So named because of the "sinaic" generator (named after ibn Sina), which is 1/4 of a perfect fourth. | |
| Larger MOSses | ||||||
| slentonic | 5L6s | sLsLsLsLsLs | 2\11 (218.2c) | 1\5 (240c) | The 11-note MOS of Slendric. | |
| p-chro smitonic | 4L7s | LsLssLssLss | 3\11 (327.3c) | 1\4 (300c) | The 11-note MOS associated with Kleismic/Cata and Orgone, both accurate temperaments. Older material may call this scale "kleistonic." | |
| p-chromatic | 5L7s | LsLsLssLsLss | 7\12 (700c) | 3\5 (720c) | The chromatic scale generated by sharp-of-12edo fifths. Superpyth[12] is a particularly interesting interpretation. | |
| hexawood | 6L6s | LsLsLsLsLsLs | 1\12 (100c) | 1\6 (200c) | 1\6 (200c) | A "straddle-fifth chromatic scale", as it can be constructed by stacking alternating flat and sharp fifths as long as they stack to 1\6, in e.g. 18edo. |
| m-chromatic | 7L5s | LLsLsLLsLsLs | 7\12 (700c) | 4\7 (685.7c) | The chromatic scale generated by flat-of-12edo fifths. Used in many 17th-century keyboards and still used in some church organs. | |
| telluric | 10L2s | LLLLLsLLLLLs | 1\12 (100c) | 1\10 (120c) | 1\2 (600c) | Commonly interpreted as Pajara[12] or Diaschismic[12]. |
| heptawood | 7L7s | LsLsLsLsLsLsLs | 1\14 (85.7c) | 0\7 (0c) | 1\7 (171.4c) | Two offset rings of 7edo fifths; the 7edo analogue of the blackwood MOS. |
| 11L4s | LLLsLLLsLLLsLLs | 4\15 (320c) | 3\11 (327.3c) | The first MOS unambiguously interpreted as Orgone. | ||
| 15L4s | LLLLsLLLLsLLLLsLLLs | 7\19 (315.8c) | 4\15 (320c) | The first MOS unambiguously interpreted as Kleismic. | ||
