TAMNAMS: Difference between revisions

'''TAMNAMS''' (read "tame names"; from '''''T'''emperament-'''A'''gnostic '''M'''os '''NAM'''ing '''S'''ystem''), devised by the XA Discord, is a system of temperament-agnostic names for octave-[[mos]] scales and their associated generator ranges, taking into account the relative sizes of large and small steps.

{| class="wikitable"

|Paucitonic (from "few tones")

For example, the 5L2s (diatonic) scale of 19edo has a step ratio of 3:2, which is "soft". We call the 19edo diatonic scale "soft diatonic". Tunings of a mos with L:s larger are "harder", and tunings with L:s smaller are "softer".

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".

::: ('''Ultrasoft''' range here, may also be called "pseudoequalized" if especially close to equalized.)

::: ('''Ultrahard''' range here, may also be called "pseudopaucitonic" if especially close to paucitonic.)

To denote interval classes within the mos, TAMNAMS uses the generic prefix ''mos-'', or the specific prefixes and abbreviations listed under "mos pattern names". One might be tempted to  

generalize diatonic 1-indexed ordinal names: "In 31edo's ultrasoft [[mosh]] scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth."

The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named "(n+1)th". This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.

For this reason the originally suggested 1-indexed interval names such as "mos-kth" are deprecated for non-diatonic mosses. TAMNAMS now prefers a 0-indexed name system for non-diatonic mos intervals: First, use the term "mosstep" for steps of the mos, large or small. From there, an interval which is k mossteps wide is a "k-mosstep", short for "k-mosstep interval". Major, minor, perfect, etc would apply as established. The names "mosoctave" (or "mosequave" for nonoctave mosses) and "mosunison" could still be used, interchangeably with "n-mosstep" (for an n-tone mos) and "0-mosstep" respectively. This change makes the arithmetic needed to understand mos intervals much smoother.

In contexts where it doesn't cause ambiguity, "k-mosstep" can be shortened to "k-step". "k-step" is also generalizable to non-mos scale types such as 3-step-size scales; see below for naming in scales with 3 step sizes.

* For nL ns scales, there's an exception to the above two rules. Only multiples of the period (1\n) are called "perfect". Other intervals are called major or minor, despite being period-equivalent to a generator. The reason for this exception is that otherwise all intervals would be called "perfect", leading to ambiguity.

** When modifying unisons or octave multiples, "mosdiminished" and "mosaugmented" could be used (e.g. "mosdiminished octave" instead of "diminished mosoctave"), because the unison and the octave don't change depending on the mos pattern, but the meanings of "augmented" and "diminished".

The following names are suggested for certain octave-period mosses of sizes between 6 and 10. These names are optional; interval size names and step ratio names can be combined with conventional "xL ys" names. For example: ''21edo is the soft [[5L 3s]] tuning and its major mosthird is a neutral third of size 342.9 cents.'' (Pattern names are the least important part of TAMNAMS.)

Some of these come from temperament-agnostic mos names coined by [[Igliashon Jones]] and others, as well as some of the names (such as "mosh") from [[Graham Breed]]'s [[mos naming#Graham_Breed.27s_naming_scheme|mos names]]. Some are named by taking an arbitrary temperament that generates the scale (preferably in the mos's [[proper]] range) and suffixing ''-oid''. These names have been coined so that mosses can be discussed more independently of RTT temperaments (while drawing on an established RTT tradition in the xen community which may help make the names more meaningful to more people).

! Pattern !! Name !! Interval prefix<ref name="prefix">used in interval names, e.g. "perfect 3-oneirostep"</ref> !! Abbreviation<ref name="abbr">used in abbreviations of interval names, e.g. "P3ons"</ref> !! Notes

! Pattern !! Name !! Interval prefix<ref name="prefix"/>!! Abbreviation<ref name="abbr"/> !! Notes

| [[3L 4s]] || mosh || mosh- || mosh || Graham Breed's name, from "[[mohajira]]-ish".

! Pattern !! Name !! Interval prefix<ref name="prefix"/> !! Abbreviation<ref name="abbr"/> !! Notes

! Pattern !! Name !! Interval prefix<ref name="prefix"/> !! Abbreviation<ref name="abbr"/> !! Notes

| [[5L 4s]] || semiquartal || sequar- || seq || From "half-fourth".

| [[8L 1s]] || subneutralic || blu- || blu || From "subneutral 2nd" generator. ''blu'' comes from [[bleu]] temperament.

! Pattern !! Name !! Interval prefix<ref name="prefix"/> !! Abbreviation<ref name="abbr"/> !! Notes

Analogously to 2-step-size scales including mosses, scales 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".

[[MV3]] scales, such as [[diasem]], have at most 3 sizes for each interval class. For every interval class that occurs in exactly 3 sizes, we use "large", "medium" and "small k-step". For every interval class that occurs in 2 sizes, we use "large k-step" and "small k-step".  If an interval class only has one size, then we call it "perfect k-step".

* Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the "quintessential" ("quintess." or "essential" for short) or "basic" tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)

* As L/s = 1/1 represents L and s being equal in size, it is called "equalized".

* As L/s = 1/0 represents s = 0, it is called "paucitonic", meaning "few tones", as the resulting scale is also equalized but with fewer tones per period than expected.

* The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest [[Odd limit#Relationship_to_other_limits|integer limit]]) ratio to represent such a property, it is simply called the "soft" tuning.

* Analogously, the mediant of 2/1 and 1/0, 3/1, is called the "hard" tuning. Thus you can say that a step ratio tuning is "hard of" or "soft of" another step ratio tuning.

* To get something between soft and basic we take the mediant again and get 5/3 for "semisoft", and analogously 5/2 for "semihard". To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for "superhard" and analogously 4/3 for "supersoft".

* Anything softer than supersoft is "ultrasoft," and anything harder than superhard is "ultrahard". Something between soft and supersoft is "parasoft", as "para-" means both "beyond" and "next to". Something between hard and superhard is "parahard".

* Something between soft and basic is "hyposoft" as it is less soft than soft. Something between hard and basic is "hypohard" for the same reason. Between semisoft and quintessential is "minisoft" and between semihard and quintessential is "minihard".

* Finally, between soft and semisoft is "quasisoft" as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix "quasi-", and between hard and semihard is "quasihard" for the same reason.

The reasoning for the "para- super- ultra-" progression (note that "super-" is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because "parapythagorean" is between "pythagorean" and "superpythagorean".

This results in the "central spectrum" - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an [[EPD|equal pitch division]] low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum.

Extending the spectrum builds on the central spectrum and relies on a few key observations. Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of "simple" tunings for the step ratio for a given mos pattern and period, the notion of "simple" used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s. The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the "hard" end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, ''are not'' temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.

The most obvious adjustment to the edges is to draw a distinction between "ultrasoft" and "pseudoequalized" by adding a step ratio corresponding to "semiequalized", and between "ultrahard" and "pseudopaucitonic" by adding a step ratio corresponding to "semipaucitonic". Thus:

Then all that's left is to decide what the step ratios for semipaucitonic and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, '''semiequalized''' should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for '''semipaucitonic''' we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as '''extrahard'''. Although L/s = 10/1 for '''semipaucitonic''' may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most "equally spaced" continuing on from 6/1 for the same reason that L/s = 6/5 is the most "equally spaced". Note that while the range from '''superhard''' to '''semipaucitonic''' is '''ultrahard''', the region may be split into two sub-ranges:

A ratio of L/s = k/1 can be called ''k-hard'' and a ratio of L/s = k/(k-1) can analogously be called ''k-soft'', so the simplest ultrasoft tuning is 5-soft or "pentasoft", the simplest hyperhard tuning is 5-hard or "pentahard", the simplest clustered tuning is 7-hard or "heptahard", 8-hard is "octahard", 9-hard is "nonahard", and finally, the characteristic simple ultrahard tuning is 6-hard or "extrahard", as previously discussed, which can be seen to be similar to "hexahard" - hopefully helping with memorisation.

A perhaps useful (or otherwise mildly amusing) mnemonic is "2-soft is too soft to be hard and 2-hard is too hard to be soft", representing that 2-soft = 2-hard = 2/1 = '''basic'''.