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| == Sandboxed rewrite: Naming mos intervals and mos degrees == | | == Sandboxed rewrite: Naming mos intervals and mos degrees == |
| Mos intervals are denoted as a ''quantity'' of mossteps, large or small. An interval that is k mossteps wide is referred to as a ''k-mosstep interval'' or simply ''k-mosstep'' (abbreviated as ''k''ms). A mos's intervals are a 0-mosstep or ''mosunison'', followed by a 1-mosstep, and so on, until an n-mosstep or ''mosoctave'' is reached, where n is the number of pitches in the mos. The prefix of mos- in the terms mosstep, mosunison, and mosoctave may be replaced with the mos's prefix, specified in the section mos pattern names.
| | Already deployed on main TAMNAMS page: [[TAMNAMS#Naming mos intervals]] |
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| In contexts where it doesn't cause ambiguity, the term ''k-mosstep'' can be shortened to ''k-step'', which allows for generalizing terminology described here to non-mos scales. Additionally, for [[non-octave]] scales, the term ''mosoctave'' is replaced with the term ''mosequave''.
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| This section's running example will be 3L 4s.
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| === Reasoning for 0-indexed intervals ===
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| Note that a mosunison 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.
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| 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.
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| 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.
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| === Naming specific mos intervals ===
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| 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'', ''augment'', ''perfect'', and ''diminished'' are used. As mosses are [[Distributional evenness|distributionally even]], every interval (except for the mosunison and mosoctave) will be in no more than two sizes.
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| To find what mos interval sizes are found in a mos xL ys, start with the patterns of large and small steps that represents the mos in its brightest mode (the following subsection explains how to do this) and its darkest mode (which is the reverse pattern for the mos's brightest mode). For our running example of 3L 4s, this is LsLsLss and ssLsLsL. To find the large sizes of each k-mosstep, consider the first k mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each k-mosstep's small size. To make these sizes more clear, we can denote the mos intervals 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 step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.
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| {| class="wikitable"
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| |+Specific interval sizes for 3L 4s
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| ! rowspan="2" |Interval
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| ! colspan="2" |Large size (LsLsLss)
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| ! colspan="2" |Small size (ssLsLsL)
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| |-
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| !Step pattern
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| !Sum
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| !Step pattern
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| !Sum
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| |-
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| |0-mosstep (mosunison)
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| |none
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| |'''0'''
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| |none
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| |'''0'''
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| |-
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| |1-mosstep
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| |L
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| |'''L'''
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| |s
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| |'''s'''
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| |-
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| |2-mosstep
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| |Ls
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| |'''L+s'''
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| |ss
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| |'''2s'''
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| |-
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| |3-mosstep
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| |LsL
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| |'''2L+s'''
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| |ssL
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| |'''1L+2s'''
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| |-
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| |4-mosstep
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| |LsLs
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| |'''2L+2s'''
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| |ssLs
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| |'''1L+3s'''
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| |-
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| |5-mosstep
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| |LsLsL
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| |'''3L+2s'''
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| |ssLsL
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| |'''2L+3s'''
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| |-
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| |6-mosstep
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| |LsLsLs
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| |'''3L+3s'''
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| |ssLsLs
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| |'''2L+4s'''
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| |-
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| |7-mosstep (mosoctave)
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| |LsLsLss
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| |'''3L+4s'''
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| |ssLsLsL
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| |'''3L+4s'''
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| |}The modifiers of ''major'', ''minor'', ''augment'', ''perfect'', and ''diminished'' (abbreviated as M, m, A, P, and d respectively) are given as such:
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| * The mosunison and mosoctave are '''perfect''' because they only have one size each.
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| * 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. 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; specifically:
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| ** The large size of the bright generator is '''perfect''', and the small size is '''diminished'''.
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| ** The large size of the dark generator is '''augmented''', and the small size is '''perfect'''.
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| * For all other intervals, the large size is '''major''' and the small size is '''minor'''.
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| * 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 a mosoctave are perfect, as are generators raised by some multiple of the mosoctave.
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| For multi-period mosses, the additional rules apply:
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| * For multi-period mosses not of the form nL ns, there is an additional interval that occurs periodically that only appears as one size. This interval, the mos's period, is '''perfect'''. Additionally:
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| ** Multiples of the period are '''perfect''', as are multiples of a mosoctave.
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| ** Both the bright and dark generators appear in every period, not just every octave. Generators that are raised some multiple of the mos's period are also '''perfect''', as are generators raised by some multiple of the mosoctave.
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| * For multi-period mosses that are of the form nL ns, the generators are '''major''' and '''minor''', rather than augmented, perfect, and diminished. This is to prevent ambiguity over calling every interval perfect.
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| {| class="wikitable"
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| |+Names for mos intervals for 3L 4s
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| !Interval
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| !Specific mos interval
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| !Abbreviation
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| !Interval size
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| !Gens up
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| |-
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| |0-mosstep (mosunison)
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| |Perfect mosunison
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| |P0ms
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| |0
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| |0
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| |-
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| | rowspan="2" |1-mosstep
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| |Minor mosstep (or small mosstep)
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| |m1ms
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| |s
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| | -3
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| |-
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| |Major mosstep (or large mosstep)
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| |M1ms
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| |L
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| |4
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| |-
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| | rowspan="2" |2-mosstep
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| |Diminished 2-mosstep
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| |d2ms
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| |2s
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| | -6
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| |-
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| |Perfect 2-mosstep
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| |P2ms
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| |L+s
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| |1
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| |-
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| | rowspan="2" |3-mosstep
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| |Minor 3-mosstep
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| |m3ms
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| |1L+2s
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| | -2
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| |-
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| |Major 3-mosstep
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| |M3ms
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| |2L+s
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| |5
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| |-
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| | rowspan="2" |4-mosstep
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| |Minor 4-mosstep
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| |m4ms
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| |1L+3s
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| | -5
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| |-
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| |Major 4-mosstep
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| |M4ms
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| |2L+2s
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| |2
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| |-
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| | rowspan="2" |5-mosstep
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| |Perfect 5-mosstep
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| |P5ms
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| |2L+3s
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| | -1
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| |-
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| |Augmented 5-mosstep
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| |A5ms
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| |3L+2s
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| |6
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| |-
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| | rowspan="2" |6-mosstep
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| |Minor 6-mosstep
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| |m6ms
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| |2L+4s
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| | -4
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| |-
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| |Major 6-mosstep
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| |M6ms
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| |3L+3s
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| |3
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| |-
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| |7-mosstep (mosoctave)
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| |Perfect mosoctave
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| |P7ms
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| |3L+4s
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| |0
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| |}
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| ==== How to find a mos's brightest mode and its generators ====
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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.
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| * 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]].
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| * 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.
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| === Naming alterations by a chroma ===
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| 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 mosunison or mosoctave that is itself augmented or diminished may also be referred to a ''mosaugmented'' or ''mosdiminished'' unison or octave. Here, the meaning of unison and octave does not change depending on the mos pattern, but the meanings of augmented and diminished do.
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| 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.
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| {| class="wikitable"
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| |+Table of alterations, with abbreviations
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| |-
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| !Number of chromas
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| !Perfect intervals
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| !Major/minor intervals
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| |-
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| | +3 chromas
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| |Triply-augmented (AAA, A³, or A^3)
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| |Triply-augmented (AAA, A³, or A^3)
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| |-
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| | +2 chromas
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| |Doubly-augmented (AA)
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| |Doubly-augmented (AA)
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| |-
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| | +1 chroma
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| |Augmented (A)
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| |Augmented (A)
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| |-
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| | rowspan="2" |0 chromas (unaltered)
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| | rowspan="2" |Perfect (P)
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| |Major (M)
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| |-
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| |Minor (m)
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| |-
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| | -1 chroma
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| |Diminished (d)
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| |Diminished (d)
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| |-
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| | -2 chromas
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| |Doubly-diminished (dd)
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| |Doubly-diminished (dd)
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| |-
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| | -3 chromas
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| |Triply-diminished (ddd, d³, or d^3)
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| |Triply-diminished (ddd, d³, or d^3)
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| |}
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| Other intervals include the following:
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| * Mosdiesis (a generalized [[Diesis (scale theory)|diesis]]): |L - 2s|
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| * Moskleisma (a generalized [[kleisma]]; uncommon): |L - 3s|
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| === Naming mos degrees ===
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| 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. The modifiers of major/minor or augmented/perfect/diminished may also be omitted when clear from context.
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| ==== Naming mos chords ====
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| 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 [[TAMNAMS#Proposal:%20Naming%20mos%20modes|below]] for the convention we have used to name the mode).
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| == Other sandboxed rewrites == | | == Other sandboxed rewrites == |