User:Ganaram inukshuk/Notes/TAMNAMS: Difference between revisions

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 == Sandboxed rewrites ==
+=== Finding and naming mos intervals ===
+To find what mos intervals are found in a mos xL ys, start with the pattern of large and small steps that represents the mos in its brightest mode. This section's running example will be 3L 4s, with the pattern (or string) LsLsLss as its brightest mode. A k-mosstep is reached by going up k mossteps up from the root, and can be represented as the first k steps of the pattern. Note that a mosunison, or 0-mosstep, is reached by going up 0 steps, so the pattern for that is no steps. Similarly, a mosoctave is reached by going up x+y steps up from the root, which encompasses the entire mos step pattern. This process finds the specific sizes for all the mos intervals. Repeat the process as described with the pattern that represents the mos in its darkest mode, which can be obtained by reversing the order of steps for the brightest mode. This produces all of the mos intervals in their specific sizes, where the process using the brightest mode produces the large, or major intervals, and the process using the darkest mode produces the small, or minor intervals.
+To make these sizes more clear, the mosintervals produced this way can be rewritten as a sum of large and small steps iL+js, where i and j are the number of L's and s's in the interval's subpattern. Note that the difference in size between an interval's large and small size is basically the replacement of one L with one s.
+{| class="wikitable"
+|+Specific interval sizes for 3L 4s
+! rowspan="2" |Interval
+! colspan="2" |Large size (LsLsLss)
+! colspan="2" |Small size (ssLsLsL)
+|-
+!Steps
+!Sum
+!Steps
+!Sum
+|-
+|0-mosstep (mosunison)
+|none
+|'''0'''
+|none
+|'''0'''
+|-
+|1-mosstep
+|L
+|'''L'''
+|s
+|'''s'''
+|-
+|2-mosstep
+|Ls
+|'''L+s'''
+|ss
+|'''2s'''
+|-
+|3-mosstep
+|LsL
+|'''2L+s'''
+|ssL
+|'''1L+2s'''
+|-
+|4-mosstep
+|LsLs
+|'''2L+2s'''
+|ssLs
+|'''1L+3s'''
+|-
+|5-mosstep
+|LsLsL
+|'''3L+2s'''
+|ssLsL
+|'''2L+3s'''
+|-
+|6-mosstep
+|LsLsLs
+|'''3L+3s'''
+|ssLsLs
+|'''2L+4s'''
+|-
+|7-mosstep (mosoctave)
+|LsLsLss
+|'''3L+4s'''
+|ssLsLsL
+|'''3L+4s'''
+|}
+The mosunison and mosoctave appear as only one size, as 0L+0s and xL+ys respectively, and are referred to as perfect. All other k-mossteps produced this way should be in one of two sizes; the smaller of the two sizes is referred to as a minor k-mosstep, and the larger of the two a major k-mosstep. However, the generating intervals of a mos use the labels augmented, perfect, and diminished instead. Every mos has a pair of generators known as the bright and dark generator, and can be found using this algorithm. (Add link to algorithm). The bright generator will have a large size that's referred to as perfect and a small size that's referred to as diminished. Similarly, the dark generator will have a large size that's referred to as augmented and a small size that's referred to as perfect. These are named such because, for the bright and dark generators, there will be only one mode that contains an augmented generator and only one, different mode that contains a diminished generator. In other words, across all modes, the generators will appear as one size in all but one mode each.
+Intervals that are more than x+y mossteps above the root share the same designation as the same mosstep that is octave-reduced. Given our example of 3L 4s, if there is a 10-mosstep, it is the same designation (in this case, either major or minor) as a 3-mosstep. Octave-reduction on a general k-mosstep can be done by finding the remainder of k divided by (x+y).
+Additional consideration is needed for multi-period mosses. If the mos is a multi-period mos, there will be at least one additional interval only seen as one size rather than two. These intervals occur every period and such intervals, specific to multi-period mosses, are referred to as perfect. This is to say that multiples of the period are perfect, just like multiples of the mosoctave are perfect. Generators of a multi-period mos that are raised or lowered by some amount of periods are also perfect, just like generators raised or lowered by multiples of a mosoctave are perfect. If, however, the mos is of the form nL ns, the generators use the labels of major and minor, rather than augmented, perfect, and diminished. The reason for this exception is to prevent ambiguity over every interval being referred to as perfect.
+In summary, bright generators are either a perfect k-mosstep or diminished k-mosstep, dark generators are either an augmented k-mosstep or a perfect k-mosstep, and all other intervals that are not the mosunison, mosoctave, or an integer multiple of the period (for multi-period mosses) are either major or minor k-mossteps. If the mos is nL ns, then the generators are either major or minor instead, and all other intervals are perfect. If context allows, "k-mosstep" may be shortened to "k-step", and to refer to generic intervals, the modifiers of major/minor or augmented/perfect/diminished are omitted.
+==== Naming mos degrees ====
+Individual mos degrees are based on the labels assigned to intervals using the process for naming mos intervals. Mos degrees 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 "k-mossteps" being shortened to "k-steps" if context allows, k-mosdegrees may also be shortened to "k-degrees". The modifiers of major/minor or augmented/perfect/diminished may also be omitted when clear from context.
+==== Naming intervals and degrees raised by a chroma ====
+Mos intervals are altered by raising or lowering it by a moschroma, the difference between a large step and a small step. This process is exactly like using sharps and flats in standard notation. Raising and lowering a major k-mosstep by a chroma produces an augmented and minor k-mosstep respectively, and raising and lowering a minor k-mosstep by a chroma produces a major and diminished k-mosstep respectively. The same applies with raising a perfect mos interval (such as mosunison, mosoctave, or generator) by a chroma to make it augmented, and lowering a perfect mos interval by a chroma to make it diminished. This process of raising and lowering by a chroma can be repeated multiple times; after an augmented and diminished mos interval is a doubly-augmented and doubly-diminished mos interval, and after that a triply-augmented and triply-diminished mos interval. These additional labels also apply to mosdegrees.
+A mosunison or mosoctave that is itself augmented or diminished may also be referred to a mosaugmented or mosdiminished unison or octave.
+{| class="wikitable"
+|+Table of alterations, with abbreviations
+! colspan="4" |Major and minor mos intervals and mos degrees
+|-
+!Number of chromas
+!Name of alteration
+!Mos interval
+!Mos degree
+|-
+|3 chromas
+|Triply-augmented (AAA, A³, or A^3)
+|Triply-augmented k-mosstep (AAA''k''ms)
+|Triply-augmented k-mosdegree (AAA''k''md)
+|-
+|2 chromas
+|Doubly-augmented (AA)
+|Doubly-augmented k-mosstep (AA''k''ms)
+|Doubly-augmented k-mosdegree (AA''k''md)
+|-
+|1 chroma
+|Augmented (A)
+|Augmented k-mosstep (A''k''ms)
+|Augmented k-mosdegree (A''k''md)
+|-
+| rowspan="2" |0 chromas
+|Major (M)
+|Major k-mosstep (M''k''ms)
+|Major k-mosdegree (M''k''md)
+|-
+|Minor (m)
+|Minor k-mosstep (m''k''ms)
+|Minor k-mosdegree step (m''k''md)
+|-
+| -1 chroma
+|Diminished (d)
+|Diminished k-mosstep (d''k''ms)
+|Diminished k-mosdegree (d''k''md)
+|-
+| -2 chromas
+|Doubly-diminished (dd)
+|Doubly-diminished k-mosstep (dd''k''ms)
+|Doubly-diminished k-mosdegree (dd''k''md)
+|-
+| -3 chromas
+|Triply-diminished (ddd, d³, or d^3)
+|Triply-diminished k-mosstep (m''k''ms)
+|Triply-diminished k-mosdegree (ddd''k''md)
+|-
+! colspan="4" |Perfect mos intervals and mos degrees
+|-
+!Number of chromas
+!Name of alteration
+!Mos interval
+!Mos degree
+|-
+|3 chromas
+|Triply-augmented (AAA, A³, or A^3)
+|Triply-augmented k-mosstep (AAA''k''ms)
+|Triply-augmented k-mosdegree (AAA''k''md)
+|-
+|2 chromas
+|Doubly-augmented (AA)
+|Doubly-augmented k-mosstep (AA''k''ms)
+|Doubly-augmented k-mosdegree (AA''k''md)
+|-
+|1 chroma
+|Augmented (A)
+|Augmented k-mosstep (A''k''ms)
+|Augmented k-mosdegree (A''k''md)
+|-
+|0 chromas
+|Perfect (P)
+|Perfect k-mosstep (P''k''ms)
+|Perfect k-mosdegree (P''k''md)
+|-
+| -1 chroma
+|Diminished (d)
+|Diminished k-mosstep (d''k''ms)
+|Diminished k-mosdegree (d''k''md)
+|-
+| -2 chromas
+|Doubly-diminished (dd)
+|Doubly-diminished k-mosstep (dd''k''ms)
+|Doubly-diminished k-mosdegree (dd''k''md)
+|-
+| -3 chromas
+|Triply-diminished (ddd, d³, or d^3)
+|Triply-diminished k-mosstep (m''k''ms)
+|Triply-diminished k-mosdegree (ddd''k''md)
+|}
+==== Naming intervals smaller than a chroma ====
+A type of interval that is sometimes used is the diesis, which is the absolute difference between a large step and two small steps. In our running example of 3L 4s, we have a major 1-mosstep, equivalent to a large mosstep. Lowering a major 1-mosstep by a chroma produces a a minor 1-mosstep, and lowering again produces a diminished 1-mosstep, which is equivalent to a mosdiesis. Depending on the step ratio, the diminished 1-mosdegree will either be higher, if soft-of-basic, or lower, if hard-of-basic, than the tonic by a mosdiesis. If the step ratio is basic, then the size of the mosdiesis is 0.
 === Reasoning for names ===