User:Ganaram inukshuk/TAMNAMS
| The following is a draft for a proposed rewrite of the following page: TAMNAMS
The primary changes are as follows:
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TAMNAMS (read "tame names"; from Temperament-Agnostic Mos NAMing System), devised by the XA Discord in 2021, is a system of temperament-agnostic names for scales – primarily octave-equivalent moment of symmetry scales – as well as their their intervals, their associated generator ranges, and the ratios describing the proportions of large and small steps.
The goal of TAMNAMS is to allow musicians and theorists to discuss moment-of-symmetry scales, or mosses, independent of the language of regular temperament theory. For example, the names flattone[7], meantone[7], pythagorean[7], and superpyth[7] all describe the same step pattern of 5L 2s, with different proportions of large and small steps. Under TAMNAMS parlance, these names can be described broadly as soft 5L 2s (for flattone and meantone) and hard 5L 2s (for pythagorean and superpyth). For discussions of the step pattern itself, the name 5L 2s or, in this example, diatonic, is used.
This article outlines TAMNAMS conventions as it applies to octave-equivalent moment of symmetry scales, or such scales with tempered octaves.
Credits
This page and its associated pages were mainly written by User:Godtone, User:SupahstarSaga, User:Inthar, and User:Ganaram inukshuk.
Step ratio spectrum
Simple step ratios
TAMNAMS provides names for nine specific simple step ratios. These correspond to the simplest edos that have the mos scale, and can be used in place of their respective step ratio.
| TAMNAMS Name | Ratio | Hardness | Diatonic example |
|---|---|---|---|
| Equalized | L:s = 1:1 | 1.000 | 7edo |
| Supersoft | L:s = 4:3 | 1.333 | 26edo |
| Soft (or monosoft) | L:s = 3:2 | 1.500 | 19edo |
| Semisoft | L:s = 5:3 | 1.667 | 31edo |
| Basic | L:s = 2:1 | 2.000 | 12edo |
| Semihard | L:s = 5:2 | 2.500 | 29edo |
| Hard (or monohard) | L:s = 3:1 | 3.000 | 17edo |
| Superhard | L:s = 4:1 | 4.000 | 22edo |
| Collapsed | L:s = 1:0 | ∞ (infinity) | 5edo |
For example, the 5L 2s (diatonic) scale of 19edo has a step ratio of 3:2, which is soft, and is thus called soft diatonic. Tunings of a mos with L:s larger than that ratio are harder, and tunings with L:s smaller than that are softer.
The two extremes, equalized and collapsed, are degenerate cases and define the boundaries for valid tuning ranges. An equalized mos has large and small steps be the same size (L=s), so the mos pattern is no longer apparent. A collapsed mos has small steps shrunken down to zero (s=0), merging adjacent tones s apart into a single tone. In both cases, the mos structure is no longer valid.
Step ratio ranges
In between the nine specific ratios there are eight named intermediate ranges of step ratios. These terms are used for classifying mos tunings which don't match any of the nine simple step ratios.
There are also two additional terms for broader ranges: the term hyposoft describes step ratios that are soft-of-basic but not as soft as 3:2; similarly, the term hypohard describes step ratios that are hard-of-basic but not as hard as 3:1.
By default, all ranges include their endpoints. For example, a hard tuning is considered a quasihard tuning. To exclude endpoints, the modifier strict can be used, for example strict hyposoft.
| TAMNAMS Name | Ratio range | Hardness |
|---|---|---|
| Hyposoft | 3:2 ≤ L:s ≤ 2:1 | 1.500 ≤ L/s ≤ 2.000 |
| Ultrasoft | 1:1 ≤ L:s ≤ 4:3 | 1.000 ≤ L/s ≤ 1.333 |
| Parasoft | 4:3 ≤ L:s ≤ 3:2 | 1.333 ≤ L/s ≤ 1.500 |
| Quasisoft | 3:2 ≤ L:s ≤ 5:3 | 1.500 ≤ L/s ≤ 1.667 |
| Minisoft | 5:3 ≤ L:s ≤ 2:1 | 1.667 ≤ L/s ≤ 2.000 |
| Minihard | 2:1 ≤ L:s ≤ 5:2 | 2.000 ≤ L/s ≤ 2.500 |
| Quasihard | 5:2 ≤ L:s ≤ 3:1 | 2.500 ≤ L/s ≤ 3.000 |
| Parahard | 3:1 ≤ L:s ≤ 4:1 | 3.000 ≤ L/s ≤ 4.000 |
| Ultrahard | 4:1 ≤ L:s ≤ 1:0 | 4.000 ≤ L/s ≤ ∞ |
| Hypohard | 2:1 ≤ L:s ≤ 3:1 | 2.000 ≤ L/s ≤ 3.000 |
Central spectrum
| Step ratio ranges | Specific step ratios | Notes | ||
|---|---|---|---|---|
| 1:1 (equalized) | Trivial/pathological | |||
| 1:1 to 2:1 (soft-of-basic) | 1:1 to 4:3 (ultrasoft) | Step ratios especially close to 1:1 may be called pseudoequalized | ||
| 4:3 (supersoft) | ||||
| 4:3 to 3:2 (parasoft) | ||||
| 3:2 (soft) | Also called monosoft | |||
| 3:2 to 2:1 (hyposoft) | 3:2 to 5:3 (quasisoft) | |||
| 5:3 (semisoft) | ||||
| 5:3 to 2:1 (minisoft) | ||||
| 2:1 (basic) | ||||
| 2:1 to 1:0 (hard-of-basic) | 2:1 to 3:1 (hypohard) | 2:1 to 5:2 (minihard) | ||
| 5:2 (semihard) | ||||
| 5:2 to 3:1 (quasihard) | ||||
| 3:1 (hard) | Also called monohard | |||
| 3:1 to 4:1 (parahard) | ||||
| 4:1 (superhard) | ||||
| 4:1 to 1:0 (ultrahard) | Step ratios especially close to 1:0 may be called pseudocollapsed | |||
| 1:0 (collapsed) | Trivial/pathological | |||
Expanded spectrum and other terminology
For a derivation of these ratio ranges, see <link>.
Naming mos intervals
Mos intervals, the gap between any two tones in the scale, are named after the number of steps (large or small) between. An interval that spans k mossteps is called a k-mosstep interval, or simply a k-mosstep (abbreviated kms). This can be further shortened to k-step if context allows.
Mossteps are zero-indexed, counting the number of steps subtended rather than the number of scale degrees, meaning that the unison is called a 0-mosstep, since a unison has zero steps. A mosstep that reaches the octave can simply be called the octave.
Generic mos intervals only denote how many mossteps an interval subtends. Specific mos intervals denote the sizes, or varieties, an interval has. Per the definition of a moment of symmetry scale (that is, maximum variety 2), every interval, except for the root and multiples of the period, has two sizes: large and small. The terms major, minor, augmented, perfect, and diminished are added before the phrase k-mosstep using the following rules:
- Multiples of the period such as the root and octave are perfect, as they only have one size each.
- The generators use the terms augmented, perfect, and diminished. Note that there are two generators (bright and dark) whose perfect varieties can be used to create the scale. Thus:
- The large size of the bright generator is perfect, and the small size is diminished.
- The large size of the dark generator is augmented, and the small size is perfect.
- For all other intervals, the large size is major and the small size is minor.
There is one exception to the above rules: the designations of augmented, perfect, and diminished don't apply for the generators for nL ns mosses. Instead, major and minor is used, so as to prevent ambiguity over calling every interval perfect.
Mosstep intervals can exceed the octave as they do in standard music theory (eg, a diatonic 9th is a diatonic 2nd raised one octave). For a single-period mos, any interval that is raised by an octave will be the same interval quality that it was before raising. Likewise, for a multi-period mos, any interval raised by the period, where the period is some fraction of the octave, will be the same interval quality that it was before raising.
Examples using 5L 2s and 4L 4s interval names are provided below. Note that 5L 2s interval names are identical to that of standard music theory, apart from the 0-indexed interval names. To differentiate intervals of a specific mos, the mos's corresponding prefix can be used in place of "mos-", outlined <link>. For a detailed derivation of these intervals, see <link>.
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Alterations by a chroma
The terms augmented and diminished are also used to describe intervals that are further lowered or raised by an interval called a moschroma (or simply chroma if context allows), a generalized sharp or flat. The rules for alteration are the same as with conventional music theory.
- Raising a minor interval by a chroma makes it minor.
- Lowering a major interval by a chroma makes it major.
- Raising a major interval by a chroma makes it augmented.
- Lowering a minor interval by a chroma makes it diminished.
- Raising an augmented interval by a chroma makes it doubly augmented.
- Lowering a diminished interval by a chroma makes it doubly diminished.
- Raising or lowering a perfect interval makes it augmented or diminished, respectively.
The terms augmented and diminished can be abbreviated using the letters A (capitalized A) and d (lowercase d). Repetition of "A" or "d" is used to denote repeatedly augmented/diminished intervals, and is sufficient in most cases. It's typically uncommon to alter an interval more than three times, and superscript numbers or alternate notation is advised for such cases. The table below shows how such intervals can be notated.
| Chromas | Perfectable intervals | Non-perfectable intervals | ||
|---|---|---|---|---|
| Interval quality | Abbrev. | Interval quality | Abbrev. | |
| +4 | Quadruply-augmented | A4 or A^4 | Quadruply-augmented | A4 or A^4 |
| +3 | Triply-augmented | AAA, A3, or A^3 | Triply-augmented | AAA, A3, or A^3 |
| +2 | Doubly-augmented | AA | Doubly-augmented | AA |
| +1 | Augmented | A | Augmented | A |
| 0 | Perfect | P | Major | M |
| Minor | m | |||
| -1 | Diminished | d | Diminished | d |
| -2 | Doubly-diminished | dd | Doubly-diminished | dd |
| -3 | Triply-diminished | ddd, d3, or d^3 | Triply-diminished | ddd, d3, or d^3 |
| -4 | Quadruply-diminished | d4 or d^4 | Quadruply-diminished | d4 or d^4 |
Intervals smaller than a chroma
| Interval name | Absolute value of a... |
|---|---|
| Moschroma (generalized chroma, provided for reference) | Large step minus a small step |
| Mosdiesis (generalized diesis) | Large step minus two small steps |
| Moskleisma (generalized kleisma) | Mosdiesis minus a moschroma |
| Mosgothma (generalized gothma) | Mosdiesis minus a small step |
Other terminology and intervals
Intervals that have a perfect variety (the unison, period intervals, and generators) are called perfectable intervals, whereas intervals that do not have a perfect variety are called non-perfectable intervals. Intervals corresponding to the generators may be called imperfect intervals since, unlike the period and unison, they have two varieties instead of one.
A discussion of neutral and interordinal intervals, which fall between major and minor, can be found at <link>.
Naming mos degrees
The pitches of a mos are called k-mosdegrees (abbreviated kmd), and follow the same rules as that with mosstep intervals. Mosdegrees are 0-indexed and are enumerated starting at the 0-mosdegree, representing the root or tonic of the scale. For example, if you go up a major k-mosstep up from the root, then the mos degree reached this way is a major k-mosdegree.
The phrase k-mosdegree may also be shortened to k-degree, if context allows. When the modifiers major, minor, augmented, perfect, and diminished are omitted, they are assumed to be the unmodified degrees of a particular mode.
Naming mos chords
To denote a chord or a mode on a given degree, write the notes of the chord separated by spaces or commas, or the mode, in parentheses after the degree symbol. The most explicit option is to write out the chord in cents, edosteps or mossteps (e.g. in 13edo 5L 3s, the (0 369 646) chord can be written (0 4 7)\13, (P0ms M2ms M4ms) or 7|0 (0 2 4ms) and to write the mode. To save space, you can use whatever names or abbreviations for the chord or mode you have defined for the reader. For example, in the LsLLsLLs mode of 5L 3s, we have m2md(0 369 646), or the chord (0 369 646) on the 2-mosdegree which is a minor 2-mosstep. The LsLLsLLs mode also has m2md(7|0), meaning that we have the 7| (LLsLLsLs) mode on the 2-mosdegree which is a minor 2-mosstep in LsLLsLLs (see below for the convention we have used to name the mode).
To analyze a chord as an inversion of another chord (i.e. when the bass is not seen as the root), the following strategies can be used:
- One can write the root degree first: (6s, 0s, 2s, 7s). The first degree is assumed to be the tonic unless the following method is used:
- One can write "T" to the left of the tonic: (0s, 2s, T6s, 7s).
- One can use 0 for the root, using negative numbers for notes below the root. For example, to analyze (0s, 2s, 6s, 7s) on the 7-degree of the LsLLsLLs mode as being rooted on its 6s (thus on the 5-degree of LsLLsLLs), we write 5d(0s, -6s, -4s, 1s). The "5d" here is essential for avoiding confusion with the previous notation.
- If clarity is desired as to what the root position chord is, slash notation can be used as in conventional notation. Thus the above chord can be written 5d(0s 1s 2s 4s)/7d.
Naming mos modes
TAMNAMS uses Modal UDP notation to name modes. For example, the names of modes for 5L 3s are the names of the mos followed by the UDP of that mode.
For modes with altered scale degrees, the abbreviations for the scale degrees are listed after the UDP for the mode.
Notation, such as diamond-mos, can be used instead of the abbreviation of a mosdegree. For example, LsLsLLLs can be written "5L 3s 5|2 m4md". "5L 3s 5|2 @4d".
| UDP | Cyclic order |
Step pattern |
Scale degree (oneirodegree) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
| 7|0 | 1 | LLsLLsLs | Perf. | Maj. | Maj. | Perf. | Maj. | Aug. | Maj. | Maj. | Perf. |
| 6|1 | 4 | LLsLsLLs | Perf. | Maj. | Maj. | Perf. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 5|2 | 7 | LsLLsLLs | Perf. | Maj. | Min. | Perf. | Maj. | Perf. | Maj. | Maj. | Perf. |
| 4|3 | 2 | LsLLsLsL | Perf. | Maj. | Min. | Perf. | Maj. | Perf. | Maj. | Min. | Perf. |
| 3|4 | 5 | LsLsLLsL | Perf. | Maj. | Min. | Perf. | Min. | Perf. | Maj. | Min. | Perf. |
| 2|5 | 8 | sLLsLLsL | Perf. | Min. | Min. | Perf. | Min. | Perf. | Maj. | Min. | Perf. |
| 1|6 | 3 | sLLsLsLL | Perf. | Min. | Min. | Perf. | Min. | Perf. | Min. | Min. | Perf. |
| 0|7 | 6 | sLsLLsLL | Perf. | Min. | Min. | Dim. | Min. | Perf. | Min. | Min. | Perf. |
| UDP and alterations |
Cyclic order |
Step pattern |
Scale degree (oneirodegree) | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||
| 5|2 m4md 3|4 M7md |
1 | LsLsLLLs | Perf. | Maj. | Min. | Perf. | Min. | Perf. | Maj. | Maj. | Perf. |
| 2|5 d3md 0|7 M6md |
2 | sLsLLLsL | Perf. | Min. | Min. | Dim. | Min. | Perf. | Maj. | Min. | Perf. |
| 7|0 m2md 5|2 A5md |
3 | LsLLLsLs | Perf. | Maj. | Min. | Perf. | Maj. | Aug. | Maj. | Maj. | Perf. |
| 4|3 m1md 2|5 M4md |
4 | sLLLsLsL | Perf. | Min. | Min. | Perf. | Maj. | Perf. | Maj. | Min. | Perf. |
| 7|0 A3md | 5 | LLLsLsLs | Perf. | Maj. | Maj. | Aug. | Maj. | Aug. | Maj. | Maj. | Perf. |
| 6|1 m7md 4|3 M2md |
6 | LLsLsLsL | Perf. | Maj. | Maj. | Perf. | Maj. | Perf. | Maj. | Min. | Perf. |
| 3|4 m6md 1|6 M1md |
7 | LsLsLsLL | Perf. | Maj. | Min. | Perf. | Min. | Perf. | Min. | Min. | Perf. |
| 0|7 d5md | 8 | sLsLsLLL | Perf. | Min. | Min. | Dim. | Min. | Dim. | Min. | Min. | Perf. |
Mos pattern names
This section contains unapproved namechanges. They are provided for reference/completeness and, unless approved, should not be included in the main-namespace rewrite.
TAMNAMS primarily uses the following names for octave-equivalent (or tempered-octave) mosses with step counts between 6 and 10. These names are optional, and conventional xL ys names can be used instead in discussions regarding mosses, its intervals, scale degrees, and modes.
Prefixes and abbreviations for each name are also provided, and can used in place of the prefix mos- and its abbreviation of m-, as seen in mos-related terms, such as mosstep and mosdegree, and their abbreviations of ms and md, respectively. For example, discussion of the intervals and scale degrees of oneirotonic uses the terms oneirosteps and oneirodegrees, abbreviated as oneis and oneid, respectively.
This list is maintained by User:Inthar and User:Godtone.
| 6-note mosses | ||||
|---|---|---|---|---|
| Pattern | Name | Prefix | Abbr. | Etymology |
| 1L 5s | selenite | sel- | sel | References luna temperament (selenite is named after the moon); also called antimachinoid[1]. |
| 2L 4s | malic | mal- | mal | Sister mos of 4L 2s; apples have concave ends, whereas lemons/limes have convex ends. |
| 3L 3s | triwood | triwd- | tw | Blackwood[10] and whitewood[14] generalized to 3 periods. |
| 4L 2s | citric | citro- | cit | Parent (or subset) mos of 4L 6s and 6L 4s. |
| 5L 1s | machinoid | mech- | mk | From machine temperament. |
| 7-note mosses | ||||
| Pattern | Name | Prefix | Abbr. | Etymology |
| 1L 6s | onyx | on- | on | Sounds like "one-six" depending on one's pronunciation; also called anti-archeotonic[1]. |
| 2L 5s | pelotonic | pel- | pel | From pelog; also called antidiatonic[1], a common name. |
| 3L 4s | mosh | mosh- | mosh | From "mohajira-ish", a name from Graham Breed's naming scheme. |
| 4L 3s | smitonic | smi- | smi | From "sharp minor third". |
| 5L 2s | diatonic | dia- | dia | |
| 6L 1s | archaeotonic | arch- | arc | Originally a name for 13edo's 6L 1s scale; also called archæotonic/archeotonic[2]. |
| 8-note mosses | ||||
| Pattern | Name | Prefix | Abbr. | Etymology |
| 1L 7s | spinel | spin- | sp | Contains the string "pine", referencing its sister mos; also called antipine[1]. |
| 2L 6s | subaric | subar- | sb | Parent (or subset) mos of 2L 8s and 8L 2s. |
| 3L 5s | checkertonic | check- | chk | From the Kite guitar checkerboard scale. |
| 4L 4s | tetrawood | tetrawd- | ttw | Blackwood[10] and whitewood[14] generalized to 4 periods; also called diminished[3]. |
| 5L 3s | oneirotonic | oneiro- | or | Originally a name for 13edo's 5L 3s scale; also called oneiro[4]. |
| 6L 2s | ekic | ek- | ek | From echidna and hedgehog temperaments. |
| 7L 1s | pine | pine- | p | From porcupine temperament. |
| 9-note mosses | ||||
| Pattern | Name | Prefix | Abbr. | Etymology |
| 1L 8s | agate | ag- | ag | Rhymes with "eight", depending on one's pronunciation; also called antisubneutralic[1]. |
| 2L 7s | balzano | bal- | bz | Originally a name for 20edo's 2L 7s (and 2L 11) scales; bal- is pronounced /bæl/. |
| 3L 6s | tcherepnin | cher- | ch | In reference to Tcherepnin's 9-note scale in 12edo. |
| 4L 5s | gramitonic | gram- | gm | From "grave minor third". |
| 5L 4s | semiquartal | cthon- | ct | From "half fourth"; cthon- is from "chthonic". |
| 6L 3s | hyrulic | hyru- | hy | References triforce temperament. |
| 7L 2s | armotonic | arm- | am | From Armodue theory; also called superdiatonic[3]. |
| 8L 1s | subneutralic | blu- | bl | Derived from the generator being between supraminor and neutral quality; blu- is from bleu temperament. |
| 10-note mosses | ||||
| Pattern | Name | Prefix | Abbr. | Etymology |
| 1L 9s | olivnie | oli- | ol | Rhymes with "nine", depending on one's pronunciation; also called antisinatonic[1]. |
| 2L 8s | jaric | jara- | ja | From pajara, injera, and diaschismic temperaments. |
| 3L 7s | sephiroid | seph- | sp | From sephiroth temperament. |
| 4L 6s | lime | lime- | lm | Sister mos of 6L 4s; limes are smaller than lemons, as are 4L 6s's step sizes compared to 6L 4s. |
| 5L 5s | pentawood | pentawd- | pw | Blackwood[10] and whitewood[14] generalized to 5 periods. |
| 6L 4s | lemon | lem- | le | From lemba temperament. Also sister mos of 4L 6s. |
| 7L 3s | dicoid | dico- | di | From dichotic and dicot (dicoid) exotemperaments; pronounced /'daɪˌkɔɪd/. |
| 8L 2s | taric | tara- | ta | Sister mos of 2L 8s; based off of Hindi word for 18 (aṭhārah), since 18edo contains basic 8L 2s. |
| 9L 1s | sinatonic | sina- | si | Derived from the generator being within the range of a sinaic. |
Names for smaller mosses
In addition to the names listed above are names for smaller mosses, provided for completeness. These names, with the exception of monowood and biwood, are meant to be as general as possible so as to avoid flavor and to allow for valid reuse for non-octave mosses.
| 2-note mosses | |||||
|---|---|---|---|---|---|
| Pattern | Name | Prefix | Abbr. | Must be 2/1-equivalent? | Etymology |
| 1L 1s | trivial | triv- | tw | No | The simplest valid mos pattern. |
| monowood | monowd- | w | Yes | Blackwood[10] and whitewood[14] generalized to 1 period. | |
| 3-note mosses | |||||
| Pattern | Name | Prefix | Abbr. | Must be 2/1-equivalent? | Etymology |
| 1L 2s | antrial | atri- | at | No | Opposite pattern of 2L 1s, with broader range. Shortening of anti-trial. |
| 2L 1s | trial | tri- | t | No | From tri- for 3. |
| 4-note mosses | |||||
| Pattern | Name | Prefix | Abbr. | Must be 2/1-equivalent? | Etymology |
| 1L 3s | antetric | atetra- | att | No | Opposite pattern of 3L 1s, with broader range. Shortening of anti-tetric. |
| 2L 2s | biwood | biwd- | bw | Yes | Blackwood[10] and whitewood[14] generalized to 2 periods. |
| 3L 1s | tetric | tetra- | tt | No | From tetra- for 4. |
| 5-note mosses | |||||
| Pattern | Name | Prefix | Abbr. | Must be 2/1-equivalent? | Etymology |
| 1L 4s | pedal | ped- | pd | No | From Latin ped, for foot; one big toe and four small toes. |
| 2L 3s | pentic | pent- | pt | No | Common pentatonic; from penta- for 5. |
| 3L 2s | antipentic | apent- | apt | No | Opposite pattern of 2L 3s. |
| 4L 1s | manual | manu- | mn | No | From Latin manus, for hand; one thumb and four longer fingers. |
Names for larger mosses
For a discussion of names for mosses with more than 10 steps, see <link>.
Generalizations to non-mos scales
Non-octave mosses
The terminology for intervals and scale degrees can be applied to scales with arbitrary equivalence intervals, replacing the term octave with the term equave.
The mos names provided for step counts 6-10 do not apply for non-octave mos patterns, unless the equave in question is seen as a tempered octave.
Intervals for MV3 scales
Scales with maximum-variety 3, such as diasem, have at most 3 varieties for each interval class, called large k-step, medium k-step, and small k-step. Interval classes with only two varieties are given the phrases large k-step and small k-step, and interval classes with only one variety are given the phrase perfect k-step.
Step ratios for ternary scales
Ternary scales, i.e. those with three step sizes L > M > S, including MV3 scales, can also be defined by their L:M:S ratios. Here TAMNAMS names the L/M ratio and then the M/S ratio as if these were mos step ratios: for example, 21edo diasem (5L 2M 2s, LMLSLMLSL or its inverse) has a step ratio of L:M:S = 3:2:1, so we name it soft-basic diasem. If the ratios are the same, repetition may optionally be omitted, so that 26edo diasem, 4:2:1, may optionally be called "basic diasem" instead of "basic-basic diasem". Not to be confused with step ratios where one ratio is unspecified; for that, use:
- x:y:z (where x:y is known but y:z is not) is called (hardness term for x/y)-any. x:x:1 is called equalized-any or LM-equalized (where x >= 1 represents a free variable).
- x:y:z (where y:z is known but x:y is not) is called any-(hardness term for y/z). x:1:1 is called any-equalized or MS-equalized (where x >= 1 represents a free variable).
- x:y:z (where x:z is known but x:y and y:z are not) is called outer-(hardness term for x/z)-any. x:1:x is called outer-equalized-any or LS-equalized. (where x >= 0 represents a free variable).
Arbitrary scales
Zero-indexed interval and degrees can be used for arbitrary scales. However, instead of using the terms k-mosstep and k-mosdegree, the terms k-scalestep and k-scaledegree are used. As with octave-equivalent mosses, these terms can be further shortened to k-step and k-degree, if context allows.
Frequently asked questions
Do I need to use this system over temperament names?
Why are intervals zero-indexed?
What's the difference between mosdegrees and mosintervals?