User:Ganaram inukshuk/TAMNAMS: Difference between revisions
No edit summary |
No edit summary |
||
| Line 212: | Line 212: | ||
==Naming mos intervals== | ==Naming mos intervals== | ||
Mos intervals are | Mos intervals are generally named after the number of steps (large or small) they subtend. | ||
The term "mosstep" can be further shortened to "step" if context allows. | The term "mosstep" can be further shortened to "step" if context allows. | ||
| Line 220: | Line 220: | ||
===Specific mos intervals=== | ===Specific mos intervals=== | ||
Specific mos intervals denote the specific sizes 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 a large and small size. | |||
The phrase ''k-mosstep'' by itself does not specify the exact size of an interval. To refer to specific intervals, the familiar modifiers of ''major'', ''minor'', ''augmented'', ''diminished'' and ''perfect'' are used. As mosses have [[maximum variety]] 2, every interval (except for the [[1/1|unison]] and multiples of the [[period]] which is usually the [[2/1|octave]]) will be in no more than two sizes. | The phrase ''k-mosstep'' by itself does not specify the exact size of an interval. To refer to specific intervals, the familiar modifiers of ''major'', ''minor'', ''augmented'', ''diminished'' and ''perfect'' are used. As mosses have [[maximum variety]] 2, every interval (except for the [[1/1|unison]] and multiples of the [[period]] which is usually the [[2/1|octave]]) will be in no more than two sizes. | ||
Revision as of 06:44, 9 June 2024
| The following is a draft for a proposed rewrite of the following page: TAMNAMS
The primary changes are as follows:
The original page can be compared with this page here. |
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 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 L:s ratios. These correspond to the simplest edos that have the mos scale, and can be usedd 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 names are useful 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.
Note that mosses with soft-of-basic step ratios always exhibit Rothenberg propriety, or are proper, whereas mosses with hard-of-basic step ratios do not, or are not proper, with one exception: mosses with only one small step per period are always proper, regardless of the step ratio.
| 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
Naming mos intervals
Mos intervals are generally named after the number of steps (large or small) they subtend.
The term "mosstep" can be further shortened to "step" if context allows.
Generic mos intervals
Generic mos intervals only denote how many mossteps an interval subtends.
Specific mos intervals
Specific mos intervals denote the specific sizes 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 a large and small size.
The phrase k-mosstep by itself does not specify the exact size of an interval. To refer to specific intervals, the familiar modifiers of major, minor, augmented, diminished and perfect are used. As mosses have maximum variety 2, every interval (except for the unison and multiples of the period which is usually the octave) will be in no more than two sizes.
The modifiers of major, minor, augmented, perfect, and diminished (abbreviated as M, m, A, P, and d respectively) are given as such:
- Integer multiples of the period, such as the unison and (often but not always) the octave, are perfect because they only have one size each.
- The generating intervals, or generators, are referred to as perfect. Note that a mos actually has two generators - a bright and dark generator - and both generators have two sizes each, specifically, the only time the less common size appears is at the end of the generator chain. For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep (the following subsection explains how to find these). Referring to a mos's generating intervals usually implies its perfect form (a.k.a the common form); specifically:
- 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.
- For k-mossteps where k is greater than the number of pitches in the mos, those intervals have the same modifiers as an octave-reduced interval. Similarly, multiples of the octave are perfect, as are generators raised by some multiple of the octave.
For multi-period mosses, note that both the bright and dark generators appear in every period, not just every octave, as what it means for a mos to be multi-period is that there is multiple periods per octave so that some number of periods is (intended to be interpreted to) equal the octave. Therefore, generators that are raised or lowered by some integer multiple of the mos's period are also perfect. There is an important exception in interval naming for nL ns mosses, in which the generators are major and minor (for the bright and dark generator respectively) rather than augmented, perfect and diminished, and all other intervals (the octave, unison and multiples of the period) are perfect as would be expected. This is to prevent ambiguity over calling every interval present perfect.
| Interval classes | Specific intervals | Interval size | Abbreviation | Gens up |
|---|---|---|---|---|
| 0-mosstep (unison) | Perfect unison | 0 | P0ms | 0 |
| 1-mosstep | Minor mosstep (or small mosstep) | s | m1ms | -3 |
| Major mosstep (or large mosstep) | L | M1ms | 4 | |
| 2-mosstep | Diminished 2-mosstep | 2s | d2ms | -6 |
| Perfect 2-mosstep | L+s | P2ms | 1 | |
| 3-mosstep | Minor 3-mosstep | 1L+2s | m3ms | -2 |
| Major 3-mosstep | 2L+s | M3ms | 5 | |
| 4-mosstep | Minor 4-mosstep | 1L+3s | m4ms | -5 |
| Major 4-mosstep | 2L+2s | M4ms | 2 | |
| 5-mosstep | Perfect 5-mosstep | 2L+3s | P5ms | -1 |
| Augmented 5-mosstep | 3L+2s | A5ms | 6 | |
| 6-mosstep | Minor 6-mosstep | 2L+4s | m6ms | -4 |
| Major 6-mosstep | 3L+3s | M6ms | 3 | |
| 7-mosstep (octave) | Perfect octave | 3L+4s | P7ms | 0 |
Alterations by a chroma
TAMNAMS also uses the modifiers of augmented and diminished to refer to alterations of a mos interval, much like with using sharps and flats in standard notation. Mos intervals are altered by raising or lowering it by a moschroma (or simply chroma, if context allows), a generalized sharp/flat that is the difference between a large step and a small step. Raising a minor mos interval by a chroma makes it major; the reverse is true. Raising a major or perfect mos interval repeatedly makes an augmented, doubly-augmented, and a triply-augmented mos interval. Likewise, lowering a minor or perfect mos interval repeatedly makes a diminished, doubly-diminished, and a triply-diminished mos interval. A unison, period or equave that is itself augmented or diminished may also be referred to a mosaugmented or mosdiminished unison, period or equave, respectively. Here, the meaning of unison and octave does not change depending on the mos pattern, but the meanings of augmented and diminished do.
Repetition of "A" or "d" is used to denote repeatedly augmented/diminished mos intervals, and is sufficient in most cases. It's typically uncommon to alter an interval more than three times, such as with a quadruply-augmented and quadruply-diminished interval; in such cases, it's preferable to use a shorthand such as A^n and d^n, or to use alternate notation or terminology.
| Number of chromas | Perfect intervals | Major/minor intervals |
|---|---|---|
| +3 chromas | Triply-augmented (AAA, A³, or A^3) | Triply-augmented (AAA, A³, or A^3) |
| +2 chromas | Doubly-augmented (AA) | Doubly-augmented (AA) |
| +1 chroma | Augmented (A) | Augmented (A) |
| 0 chromas (unaltered) | Perfect (P) | Major (M) |
| Minor (m) | ||
| -1 chroma | Diminished (d) | Diminished (d) |
| -2 chromas | Doubly-diminished (dd) | Doubly-diminished (dd) |
| -3 chromas | Triply-diminished (ddd, d³, or d^3) | Triply-diminished (ddd, d³, or d^3) |
Naming neutral and interordinal intervals
For a discussion of semi-moschroma-altered versions of mos intervals, see Neutral and interordinal k-mossteps.
Other terminology
The tonic (unison), the period, the generator and the period-complement of the generator make up all the intervals in any given mos scale that might be labelled "perfect". With the exception of the tonic and the period, they may also be "imperfect". Therefore, the degrees of a mos scale which come in a "perfect" variety are called perfectable degrees and the degrees of a mos scale which do not come in a "perfect" variety are called non-perfectable degrees.
Naming mos degrees
Individual mos degrees, (that is, specific notes of a mos scale,) or k-mosdegrees (abbreviated kmd), are based on the modifiers given to intervals using the process for naming mos intervals and alterations. Mosdegrees are 0-indexed and are enumerated starting at the 0-mosdegree, the tonic/root 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. Much like mossteps, the prefix of mos- may also be replaced with the mos's prefix. If context allows, k-mosdegree may also be shortened to k-degree to allow generalization to non-mos scales. When the modifiers major/minor or augmented/perfect/diminished are omitted, they are assumed to be the unmodified degrees of the current 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 uses the following names for octave-equivalent (or tempered-octave) mosses with step counts between 6 and 10, called the named range. 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].
(Provided for lack of a better name) |
| 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- | mech | 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- | arch | 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- | onei | 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- | pine | 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- | bal | Originally a name for 20edo's 2L 7s (and 2L 11) scales; bal- is pronounced /bæl/. |
| 3L 6s | tcheretonic | cher- | ch | In reference to Tcherepnin's 9-note scale in 12edo. Also called cheretonic[2]. |
| 4L 5s | gramitonic | gram- | gram | From "grave minor third". |
| 5L 4s | semiquartal | cthon- | cth | From "half fourth"; cthon- is from "chthonic". |
| 6L 3s | hyrulic | hyru- | hy | References triforce temperament. |
| 7L 2s | armotonic | arm- | arm | From Armodue theory; also called superdiatonic[3]. |
| 8L 1s | subneutralic | blu- | blu | 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- | oli | Rhymes with "nine", depending on one's pronunciation; also called antisinatonic[1]. |
| 2L 8s | jaric | jara- | jar | From pajara, injera, and diaschismic temperaments. |
| 3L 7s | sephiroid | seph- | seph | From sephiroth temperament. |
| 4L 6s | lime | lime- | lim | 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- | lem | From lemba temperament. |
| 7L 3s | dicoid | dico- | dico | From dichotic and dicot (dicoid) exotemperaments; pronounced /'daɪˌkɔɪd/. |
| 8L 2s | taric | tara- | tar | 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. |
Extending the named range
For a discussion of names for mosses with fewer than 6 steps, see <link>. For a discussion of names for mosses with more than 10 steps, see <link>.