TAMNAMS
TAMNAMS (read "tame names"; from Temperament-Agnostic Mos NAMing System), devised by the XA Discord, is a system of temperament-agnostic names for scales (designed especially with small octave-equivalent mosses in mind), their intervals and their associated generator ranges, taking into account the relative sizes of large and small steps.
Step ratio spectrum
Simple step ratios
The TAMNAMS system names nine specific simple L:s ratios. These correspond to the simplest edos that have the mos scale.
TAMNAMS Name | Ratio | Diatonic example |
---|---|---|
Equalized | L:s = 1:1 | 7edo |
Supersoft | L:s = 4:3 | 26edo |
Soft (or monosoft) | L:s = 3:2 | 19edo |
Semisoft | L:s = 5:3 | 31edo |
Basic (or quintessential) | L:s = 2:1 | 12edo |
Semihard | L:s = 5:2 | 29edo |
Hard (or monohard) | L:s = 3:1 | 17edo |
Superhard | L:s = 4:1 | 22edo |
Collapsed | L:s = 1:0 | 5edo |
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.
The two extremes, equalized and collapsed, are degenerate cases. An equalized mos has L equal to s, so the mos pattern is no longer apparent. A collapsed mos has 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 In both cases, the mos structure is no longer valid.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 ranges of ratios. Each range has a name. These names are useful for classifying mos tunings which don't match any of the nine simple step ratios. Hypohard could be used for tunings that are harder than basic but not as hard as the 3:1 tuning; similarly, hyposoft can be used for the range between soft and basic. Note that the soft-of-basic range is always strictly proper while the hard-of-basic range is often improper but is always proper in the case that there is 1 small step per period in the mos pattern.
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 | Range |
---|---|
Hyposoft | 3:2 ≤ L:s ≤ 2:1 |
Ultrasoft | 1:1 ≤ L:s ≤ 4:3 |
Parasoft | 4:3 ≤ L:s ≤ 3:2 |
Quasisoft | 3:2 ≤ L:s ≤ 5:3 |
Minisoft | 5:3 ≤ L:s ≤ 2:1 |
Minihard | 2:1 ≤ L:s ≤ 5:2 |
Quasihard | 5:2 ≤ L:s ≤ 3:1 |
Parahard | 3:1 ≤ L:s ≤ 4:1 |
Ultrahard | 4:1 ≤ L:s ≤ 1:0 |
Hypohard | 2:1 ≤ L:s ≤ 3:1 |
Central spectrum
Equalized: L/s = 1/1 (trivial/pathological)
- (Ultrasoft range here, may also be called pseudoequalized if especially close to equalized.)
- Supersoft: L/s = 4/3
- (Parasoft range here.)
- Soft: L/s = 3/2
- (Beginning of hyposoft range here.)
- (Quasisoft range here.)
- Semisoft: L/s = 5/3
- (Minisoft range here.)
- (End of hyposoft range here.)
Quintesssential: L/s = 2/1
- (Beginning of hypohard range here.)
- (Minihard range here.)
- Semihard: L/s = 5/2
- (Quasihard range here.)
- (End of hypohard range here.)
- Hard: L/s = 3/1
- (Parahard range here.)
- Superhard: L/s = 4/1
- (Ultrahard range here, may also be called pseudocollapsed if especially close to collapsed.)
Collapsed: L/s = 1/0 = infinity (trivial/pathological)
Naming mos intervals
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.
Thus TAMNAMS uses 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.
(The ordinal names could still be suggestive for e.g. (tunings of) heptatonic mosses where the ordinal names tend to match up well with diatonic ordinal categories.)
Interval name | Abbreviation | 10edo Size | Gens up |
---|---|---|---|
Perfect mosunison | P0ms | 0\10 | 0 |
Minor mosstep (or small mosstep) | m1ms | 1\10 | -3 |
Major mosstep (or large mosstep) | M1ms | 2\10 | 4 |
Diminished 2-mosstep | d2ms | 2\10 | -6 |
Perfect 2-mosstep | P2ms | 3\10 | 1 |
Minor 3-mosstep | m3ms | 4\10 | -2 |
Major 3-mosstep | M3ms | 5\10 | 5 |
Minor 4-mosstep | m4ms | 5\10 | -5 |
Major 4-mosstep | M4ms | 6\10 | 2 |
Perfect 5-mosstep | P5ms | 7\10 | -1 |
Augmented 5-mosstep | A5ms | 8\10 | 6 |
Minor 6-mosstep | m6ms | 8\10 | -4 |
Major 6-mosstep | M6ms | 9\10 | 3 |
Perfect mosoctave | P7ms | 10\10 | 0 |
TAMNAMS uses the following modifiers to denote different interval sizes within a mos interval class:
- For multiples of the period plus or minus 0 or 1 generators: perfect. (Diatonic examples: perfect mos4th (Pmos4th), perfect mos5th (Pmos5th), perfect mos8th (Pmos8th), perfect mos12th (Pmos12th), etc.)
- For generic interval classes with 2 specific sizes of intervals therein (which are therefore separated by a chroma of c = L - s), major and minor are used to distinguish the larger (L) and smaller (s) intervals. Note that the generator, its period-equivalents, and the generator's period-complement and its period-equivalents are the only intervals excluded from this rule due to their inclusion in the previous rule. Diatonic examples: major mos2nd (abbreviated Lmos2nd), minor mos3rd (abbreviated smos3rd), major mos3rd (Lmos3rd), etc.)
- 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.
- If you subtract a chroma from a perfect (Pmos) or minor (smos) interval, it becomes diminished (d; dmos). If you subtract two chromas instead, it becomes doubly diminished (dd; ddmos). (Diatonic examples: diminished mos3rd (dmos3rd), diminished mos4th (dmos4th), doubly diminished mos5th (ddmos5th), etc.)
- 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.
- If you add a chroma to a perfect (Pmos) or major (Lmos) interval, it becomes augmented (A; Amos). If you add two chromas instead, it becomes doubly augmented (AA; AAmos). (Diatonic examples: augmented mos2nd (Amos2nd), augmented mos4th (Amos4th), doubly augmented mos5th (AAmos5th).)
- The pattern continues, ddd for triply diminished and AAA for triply augmented. Note that applying this operation more than 3 times is an unlikely usecase, and a shorthand notaton of d^3 and A^3 or an alternative notation or terminology entirely would likely be preferable in such circumstances, hence repetition of the corresponding letter is a sufficient system.
Other interval names:
- moschroma or chroma: L − s
- mosdiesis: |chroma − s| = |L − 2s|
Naming mos degrees
To denote degrees in a given mos (or altered mos) mode, we use the term k-mosdegree, abbreviated as kmd (or k-degree and kd if context allows). Just like for mos interval names, we use 0-indexing, so the perfect 0-mosdegree is the tonic. The modifiers perfect, minor, major, augmented and diminished can be used just like for mos interval names: for example, minor 3-mosdegree (m3md) denotes the degree that lies a minor 3-mosstep above the tonic. Modifiers can be omitted when clear from context.
To denote a chord or a mode on a given degree, write the chord 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|), 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).
Mos pattern names
NOTE: Names are finalized, prefixes and abbreviations are still open to change.
TAMNAMS suggests the following names for selected small mosses. 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.
Some of the names come from older temperament-agnostic mos names, such as names (such as mosh) from Graham Breed's mos names. These names have been coined so that mosses can be discussed more independently of RTT temperaments. Sometimes the prefix has a different source than the scale name for euphonic reasons.
1L ns names are named with the an- prefix if they are generalised names and anti- prefix if the name for the corresponding nL1s scale assumed a period of an octave.
This list is maintained by User:Inthar and User:Godtone.
2-note mosses | |||||
---|---|---|---|---|---|
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Allows non-octave tunings?^{[3]} | Etymology |
1L 1s | trivial | triv- | trv | Yes; can have any period | the simplest valid mos pattern |
1L 1s | monowood | monowd- | wood | No; must have octave period | blackwood[10] & whitewood[14] generalized to n-wood for nL ns |
3-note mosses (non-octave^{[3]}) | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | (Non-octave periods allowed)^{[3]} | Etymology |
1L 2s | antrial | atri- | atri | Yes; can have any period | broader range than trial so named w.r.t. it (anti-trial; antial; antrial) |
2L 1s | trial | tri- | tri | Yes; can have any period | from tri- for 3 |
4-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Allows non-octave tunings?^{[3]} | Etymology |
1L 3s | antetric | atetra- | att | Yes; can have any period | broader range than tetric so named w.r.t. it (anti-tetric; antetric) |
2L 2s | biwood | biwd- | bw | No; two periods must be an octave | from 2-wood |
3L 1s | tetric | tetra- | tt | Yes; can have any period | from tetra- for 4 |
5-note mosses (non-octave^{[3]}) | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | (Non-octave periods allowed)^{[3]} | Etymology |
1L 4s | pedal | ped- | ped | one big toe and four small toes | |
2L 3s | pentic | pent- | pt | common pentatonic; from penta- for 5 | |
3L 2s | antipentic | apent- | apt | opposite pattern of common pentatonic mos | |
4L 1s | manual | manu- | manu | one thumb and four longer fingers | |
6-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Notes on tuning^{[3]} | Etymology |
1L 5s | antimachinoid | amech- | amech | opposite pattern of machinoid | |
2L 4s | malic | mal- | mal | antrial mos with 2 periods per octave | apples have two concave ends, lemons have two pointy ends. |
3L 3s | triwood | triwd- | trw | trivial mos with 3 periods per octave | from 3-wood |
4L 2s | citric | citro- | cit | trial mos with 2 periods per octave | parent mos of lemon and lime |
5L 1s | machinoid | mech- | mech | from machine temperament | |
7-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | (Octave periods only.)^{[3]} | Etymology |
1L 6s | onyx | on- | on | from a lot of naming puns | |
2L 5s | antidiatonic | pel- | pel | pel- is from pelog | |
3L 4s | mosh | mosh- | mosh | Graham Breed's name; from "mohajira-ish" | |
4L 3s | smitonic | smi- | smi | from "sharp minor third" | |
5L 2s | diatonic | none | none | ||
6L 1s | arch(a)eotonic | arch- | arch | originally a name for 13edo's 6L 1s | |
8-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Notes on tuning^{[3]} | Etymology |
1L 7s | antipine | apine- | apine | opposite pattern of pine | |
2L 6s | subaric | subar- | subar | antetric mos with 2 periods per octave | largest subset mos of jaric and taric |
3L 5s | checkertonic | check- | chk | from the Kite guitar checkerboard scale | |
4L 4s | tetrawood; diminished | tetwd- | ttw | trivial mos with 4 periods per octave | from 4-wood |
5L 3s | oneirotonic | oneiro- | onr | originally a name for 13edo's 5L 3s | |
6L 2s | ekic | ek- | ek | tetric mos with 2 periods per octave | from temperaments echidna and hedgehog |
7L 1s | pine | pine- | pine | from porcupine temperament | |
9-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Notes on tuning^{[3]} | Etymology |
1L 8s | antisubneutralic | ablu- | ablu | opposite pattern of subneutralic | |
2L 7s | balzano | bal- /bæl/ | bal | from Balzano scale in 20edo which is 2L 7s | |
3L 6s | tcherepnin | cher- | ch | antrial mos with 3 periods per octave | common name |
4L 5s | gramitonic | gram- | gram | from "grave minor third" | |
5L 4s | semiquartal | cthon- | cth | from "half fourth" and "chthonic" | |
6L 3s | hyrulic | hyru- | hyru | trial mos with 3 periods per octave | allusion to triforce temperament |
7L 2s | superdiatonic | arm- | arm | common name, arm- is from Armodue | |
8L 1s | subneutralic | blu- | blu | blu- is from bleu temperament | |
10-note mosses | |||||
Pattern | Name | Prefix^{[1]} | Abbreviation^{[2]} | Notes on tuning^{[3]} | Etymology |
1L 9s | antisinatonic | asina- | asi | opposite pattern of sinatonic | |
2L 8s | jaric | jara- | jar | pedal mos with 2 periods per octave | from temperaments pajara, injera and diaschismic |
3L 7s | sephiroid | seph- | seph | from sephiroth temperament | |
4L 6s | lime | lime- | lime | pentic mos with 2 periods per octave | limes/4L 6s's steps tend to be smaller than lemons/6L 4s's steps |
5L 5s | pentawood | penwd- | pw | trivial mos with 5 periods per octave | from 5-wood |
6L 4s | lemon | lem- | lem | anpentic mos with 2 periods per octave | from lemba temperament |
7L 3s | dicoid /'daɪkɔɪd/ | dico- | dico | from exotemperaments dichotic and dicot | |
8L 2s | taric | tara- | tar | manual mos with 2 periods per octave | from Hindi aṭhārah '18' |
9L 1s | sinatonic | sina- | si | from sinaic |
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} ^{1.6} ^{1.7} ^{1.8} used in interval, degree and mode names, e.g. perfect 3-oneirostep, perfect 3-oneirodegree, oneiro-3-up
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} ^{2.4} ^{2.5} ^{2.6} ^{2.7} ^{2.8} written abbreviations of prefixes, e.g. P3neiros, P3neirod, on-3|
- ↑ ^{3.00} ^{3.01} ^{3.02} ^{3.03} ^{3.04} ^{3.05} ^{3.06} ^{3.07} ^{3.08} ^{3.09} ^{3.10} whether the name can be used for mosses with no octaves; lightly tempered octaves are allowed;
names for mosses with more than 5 notes do not admit nonoctave tunings because the names are specific to the corresponding valid tuning range
Reasoning for the names
The goal of TAMNAMS mos names is to choose memorable but aesthetically neutral names.
All names ending in -oid refer to an exotemperament which, when including extreme tunings, covers the entire range of the corresponding octave-period mos, such that many edos with simple step ratios for that mos will correspond to valid tunings, if not by patent val, then with a small number of warts.
All names for mosses with five or less notes - excluding (mono)wood and biwood (which like all n-wood mosses are specific to octave tuning) - require that some small integer multiple of the period is equal to an octave, under the reasoning that mosses with five or less notes are common and broad in tuning enough that they are much more likely to find interest in non-octave contexts. Because of this, their names were chosen to be extremely general, both to avoid bias/being too flavorful and (correspondingly) so that the terms could validly be reused for any mos for which the period is not equal to a (potentially tempered) octave.
Any multiperiod mos with more than five notes was given a name that wasn't reliant on the name of a mos with five or less notes as such names were based on those mos names formerly requiring an octave tuning (which is to say some small integer multiple of their period must be equal to a (potentially slightly tempered) octave).
Former names like "orwelloid" and "sensoid" were abandoned because the names were too temperament-specific in the sense that even considering extreme tunings didnt cover the whole range of the mos. The remaining temperament-based names have been abstracted or altered heavily, namely "pine", "hyrulic", "jaric", "ekic" and "lemon".
The inclusion of mos names for "multiperiod" mosses was from a desire to have all ten-note-and-under mosses named for completeness, which is also what prompted some of the reconsiderations mentioned earlier. Similarly, the inclusion of mosses of the form 1L ns using the "anti-" prefix (or an- for less-than-six-note mosses) was also for a practical consideration; although the tuning range is very unhelpful for knowing what such a mos will sound, it is nonetheless useful for describing structure in situations where one does not want to use the mathematical name, especially given that in such situations the tuning will likely be specified somewhere already. Jaric and taric specifically were chosen over bipedal and bimanual because of this, and to a lesser extent, lemon and lime were chosen over antibipentic and bipentic respectively (and for consistency with that their parent MOSS, 4L2s, is named citric).
The distinction between using the prefixes "anti-" vs "an-" for reversing the number of large vs. small steps is also not as trivial as it may sound. In the case of mosses with six or more notes, as the period is always an octave, there is a very large tuning range for the 1L ns mosses (hence the original reason for omitting such mosses), but the "anti-" prefix shows that what is significant is that it has the opposite structure to the corresponding nL 1s mos while pointing out the resulting ambiguity of range. In the case of mosses with five or less notes, as the period is not known and therefore could be very small, this is not as much of a concern as fuller specification is likely required anyway, especially in the case of larger periods, so the name should not be tediously long as the name refers to a very simple mos pattern, and for related reasons, the name shouldn't give as much of a sense of one 'orientation' of the structure being more 'primary' than the other, while with mosses with more than five notes, this suggestion of sense is very much intended, because it will almost always make more sense to talk about the (n+1)L 1s child mos of whatever 1L ns mos you want to speak of.
Name-specific reasonings
Pedal (1L 4s)
Pedals are operated with feet, which have one large toe and four small toes. Also comes from words like "bipedal", where in TAMNAMS, "bipedal" would literally mean a pedal scale with a period equal to half of some chosen interval, although such a scale would have either two right feet or two left feet depending on orientation chosen. If you think "car"/"vehicle" when you think "pedal" and don't think (or want to think) much about feet then you can think about "beeping" (as beep is the 7-limit 4&5 exotemperament). Because this name relies so heavily and fundamentally on there being 1 large and 4 small steps per period, it is appropriate to generalise for any size of period you would want. In that regard, same goes for manual, pentic and anpentic.
Malic (2L 4s) and citric (4L 2s)
Malic derives from Latin malus 'apple'. An apple has two concave ends, and large steps in a scale with more small steps are hole-like, hence the two large steps in malic. Citric (4L 2s) is named after the child mosses of citric, namely lemon (6L 4s) and lime (4L 6s). Unlike apples, lemons have two convex pointy ends, and small steps in a scale with more large steps are pointy, hence the two small steps. Malic and citric acids are both ubiquitous in food and biology, thus justifying their use for fairly small mos scales.
Machinoid (5L 1s)
Machine is the 5&6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.
Its list of EDO tunings is with non-patent val EDOs in brackets but included for demonstrating virtual completeness of the tuning range and the unusually large 33edo tuning being to show 11edo's strength as a tuning.
Onyx (1L 6s)
"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L 1s (pine), 8L 1s (subneutralic) and 9L 1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).
Subaric (2L 6s)
This name alludes to the fact that 2L 6s is the largest proper subset mos of both jaric (2L 8s) and taric (8L 2s).
Jaric (2L 8s)
This name alludes to a few highly notable and generally inaccurate (with the exception of diaschismic) temperaments that exist in the tuning range of this MOSS. Specifically, notice how the letters and sound of "jaric" has (or is intended to have) a lot of overlap with pajara, diaschismic and injera (listed in order of increasingly sharp fourths; note that diatonic fourths and 4-jarasteps are equated in jaric, a notable property).
Taric (8L 2s)
Taric was named based on it being the only octave-tuned TAMNAMS pattern with a basic tuning of 18edo (because 5L 6s has more than 10 notes so is out of the scope of TAMNAMS, although not necessarily out of the scope of extensions) and it was also named based on rhyming with jaric (as they share the parent mos 2L 6s).
Sephiroid (3L 7s)
Sephiroth is the 3&10 temperament in the 2.5.7.11.13.17.21 subgroup with commas including 65/64, 85/84, 221/220, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.
Its list of patent val EDO tunings is 3, 10, 13, 16, 23, 26 with non-patent val tunings including 6eg, 7e*, 19eg, 20e, 29g, 32egq, 33ce, 36c.
- Extreme tunings even occasionally go outside of this range like with 7e, but this would never be considered a good tuning.
(Note that q in the above is a placeholder symbol meaning that the generator 21 is warted.)
Note therefore how practically a full range of tunings is covered both in breadth and depth.
Dicoid (7L 3s)
Dichotic is the 7&10 temerament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63 and is an extension of the 5-limit exotemperament dicot which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. To help justify using these temperament for inspiration for the name, note that:
Its list of patent val EDO tunings is 7, 10, 17 with non-patent val tunings including 14cd(=7+7), 20e(=10+10), 24cd(=17+7), 27ce(=17+10).
Note there are many more warted tunings than this with even more extreme tunings, which makes it reasonable to associate the exotemperament with the range of vaguely saner tunings.
On the term diatonic
In TAMNAMS, diatonic exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as Zarlino, blackdye and diasem, are called detempered (if the philosophy is RTT-based) or deregularized/detuned (RTT-agnostic) diatonic scales. The adjectives diatonic-like or diatonic-based may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.
(The choice of how to define diatonic isn't bound by history, since many other terms have different meanings depending on the historical musical system referred to, for example the enharmonic scale may refer to a chain-of-fifths-based scale with 12edo enharmonics not equated, or to an ancient Greek genus.)
Proposal: Naming mos modes
For any mos pattern, TAMNAMS allows one to use a modified version of UDP notation to name a given mos's mode. We use "xL ys k|" to denote the mode of the mos xL ys that has k generators above the tonic and n − 1 − k generators below the tonic. For example, "5L 3s 5|" for LsLLsLLs, read "5 ell 3 ess 5 pipe". We skip the number of chroma-positive generators down and the number of periods, which are redundant though these are required by standard UDP notation. The | symbol is read "pipe" instead of "up", because "up" here may be interpreted as part of the accidental alteration modifier (see below).
For an altered mode we can use any accidental alteration whose meaning is clear. For non-diatonic mosses, the degree modified is indicated using TAMNAMS's 0-indexing convention. For example, LsLsLLLs can be written "5L 3s 5| @4d" (read "5L 3s 5 pipe at-4-degree"), using the @ accidental from diamond-mos notation. Using this convention, any mode of any mos can be referred to without introducing new mode names to memorize.
For a mos pattern given a name in TAMNAMS, there is also the option of using the prefix for the pattern instead of saying "xL ys": the 5L 3s mode LsLLsLLs can be called "oneiro-5-pipe" and written "onr-5|".
Proposal: Extensions for Descendent MOSes
- See also: TAMNAMS Extension
- See also: User:Frostburn/TAMNAMS Extension
There is currently a proposal for a series of systematic extensions to this system for naming MOSes descended from the main ones listed here, as well as a few others. These extensions are currently being worked on mainly by Frostburn.
Non-mos scales
Intervals in arbitrary scales
Zero-indexed interval names are also used for arbitrary scales, so we can still call a k-step interval a k-step and the corresponding degree the k-degree. But instead of k-mosstep and k-mosdegree, we use k-scalestep and k-scaledegree for arbitrary scales.
Proposal: Naming 3-step-size scales' step ratios
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.
For step ratios where one ratio is unspecified:
- 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.
- 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.
3-step scale pattern names
Naming MV3 intervals
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.
Appendix
Derivation of the step ratio names
The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the mediant (aka Farey addition) to adjacent fractions.
- 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 collapsed, as the small scale steps collapse to zero cents and disappear.
- 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 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.
There are also tertiary names beyond the above:
- 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 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's edges
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 pseudocollapsed by adding a step ratio corresponding to semicollapsed. Thus:
Ultrasoft is between supersoft and semiequalized and pseudoequalized is between semiequalized and equalized.
Ultrahard is between superhard and semicollapsed, and pseudocollapsed is between semicollapsed and collapsed.
Then all that's left is to decide what the step ratios for semicollapsed 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 semicollapsed 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 semicollapsed 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 semicollapsed is ultrahard, the region may be split into two sub-ranges:
superhard (L/s=4/1) to extrahard (L/s=6/1) is hyperhard (4 < L/s < 6).
extrahard (L/s=6/1) to semicollapsed (L/s=10/1) is clustered (6 < L/s < 10).
With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.
Extended spectrum
Equalized: L/s = 1/1 (trivial/pathological)
- (Pseudoequalized range here.)
- Semiequalized: L/s = 6/5
- (Ultrasoft range here.)
- Supersoft: L/s = 4/3
(4/3 < L/s < 4/1 range here, called the nonextreme range, detailed by central spectrum.)
- Superhard: L/s = 4/1
- (Beginning of ultrahard range here.)
- (Hyperhard range here.)
- Extrahard: L/s = 6/1
- (Clustered range here.)
- (End of ultrahard range here.)
- Semicollapsed: L/s = 10/1
- (Pseudocollapsed range here.)
Collapsed: L/s = 1/0 = infinity (trivial/pathological)
Terminology and final notes
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.
Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of 12L 17s represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of 12edo and those of 29edo, which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.