TAMNAMS: Difference between revisions

This section's running example will be 3L 4s.

Note that a unison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. 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.

To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.

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 are [[Distributional evenness|distributionally even]], every interval (except for the [[1/1|unison]] and [[2/1|octave]]) will be in no more than two sizes.

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The idea of [[Recursive structure of MOS scales|mos recursion]] may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and [[Modal UDP notation|UDP]] may be of help for a mos's modes.

*To find the mos whose order of steps represent the mos's brightest mode, follow the algorithm described here: [[Recursive structure of MOS scales|Recursive structure of MOS scales#Finding the MOS pattern from xL ys]].

*To find the generators for a mos, follow the algorithm described here: [[Recursive structure of MOS scales#Finding a generator]]. Be sure to follow the additional instructions to produce the generators as some quantity of mossteps. Alternatively, produce an interval matrix using the instructions here ([[Interval matrix#Using step sizes]]) for making an interval matrix out of a mos pattern. The generators are the intervals that appear as one size in all but one mode. The interval that appears in its large size in all but one mode is the perfect bright generator, and the interval that appears in its small size in all but one mode is the perfect dark generator.

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.

**''m-moskleisma'': |mosdiesis - s|

**''p-moskleisma'': |mosdiesis - (L-s)|

=== Naming neutral and interordinal intervals ===

For a discussion of semi-moschroma-altered versions of mos intervals, see [[Neutral and interordinal k-mossteps]].

Individual mos degrees, or ''k-mosdegrees'' (abbreviated ''k''md) 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. 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-mosdegrees'' may also be shortened to ''k-degrees'' 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.

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 [[TAMNAMS#Proposal:%20Naming%20mos%20modes|below]] for the convention we have used to name the mode).

This temperament is supported by {{Optimal ET sequence| 3, 10, 13, 16, 23 and 26 }} equal divisions, with non-patent val tunings including 6eg, 7e*, 19eg, 20e, 29g, 32egq, 33ce, 36c.

(Note that ''q'' in the above is a placeholder symbol meaning that the generator 21 is warted.)

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 ===

'''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.