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

← Older edit Newer edit →

VisualWikitext

@@ Line 2: / Line 2: @@
 == 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''. The terms mosoctave and mosunison may be used interchangeably to denote an n-mosstep (where n is the number of tones in the mos) and 0-mosstep respectively.
+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'', producing a 0-mosstep or ''mosunison'', 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.
-The prefix of mos- in the terms mosstep, mosunison, and mosoctave may be replace for the mos's prefix, which can be found in the section for mos names below.
+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, such as strict-variety-3 scales, which have three specific interval sizes rather than two. Additionally, for non-octave scales, the term ''mosoctave'' is replaced with the term ''mosequave''.
-This produces the ''generic'' interval classes for a mos, intervals that lack the designation of major, minor, perfect, etc. The following section shows how to find the sizes of these interval classes, and thereby label them as major, minor, or perfect.
+=== Rationale for 0-indexed intervals ===
+Note that a mosunison is a 0-mosstep, rather than a mos-1st; likewise, the term 1-mosstep is used rather than a mos-2nd. 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, and the use of ordinal indexing is discouraged when referring to non-diatonic mos intervals.
-=== Rationale for 0-indexed names ===
+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.
-Note that a mosunison is a 0-mosstep, rather than a 1st. 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. Instead of having the mosunison be a mos-1st, it is a 0-mossteps. With our previous example of stacking two fifths to erroneously make a tenth, we now have two 4-mossteps that stack together to make an 8-mosstep.
 === Finding and naming specific mos intervals ===
-TAMNAMS uses the designations of '''major''', '''minor''', '''perfect''', '''augmented''', and '''diminished''' to refer to specific 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 sizes for all the mos intervals, specifically their large sizes. 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 - to find the sizes of all the mos intervals in their small sizes.
+Note that the phrase ''k-mosstep'' by itself does not specify whether an interval is major or minor. To refer to specific intervals, the familiar designations of '''major''', '''minor''', '''augment''', '''perfect''', and '''diminished''' are used. As mosses are [[Distributional evenness|distributionally even]], every interval will be in no more than two sizes, except for the mosoctave and mosunison, which only has one size.
-To make these sizes more clear, the mos intervals 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 pattern. Note that the difference in size between an interval's large and small size is basically the replacement of one L with one s.
+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. Since a k-mosstep is reached by going up k mossteps up from the root, to find every mos interval, we consider the first k steps of the mos pattern to find each interval's large size. To find the intervals' small size, we repeat the same process of finding mos intervals with the step pattern in the mos's darkest mode, which is the pattern of steps in the brightest mode reversed. 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 pattern. Note that the size difference between a large interval and small interval corresponds with replacing an L with an s.
 {| class="wikitable"
 |+Specific interval sizes for 3L 4s
@@ Line 78: / Line 74: @@
 |'''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, one of the defining properties of a mos; 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 there will be only one mode that contains an augmented dark generator rather than a perfect dark generator, and another, different mode that contains a diminished bright generator rather than a perfect bright generator. In other words, across all modes, the generators will appear as one size in all but one mode each, hence the generators' designation as perfect.
+The labels of major, minor, augmented, perfect, and diminished are assigned in the following manner:
+* The '''mosunison''' and '''mosoctave''' are '''perfect''' because they only have one size each.
+* The '''generators''' are referred to as '''perfect''' by default. However, the generators have two interval sizes, and all mosses actually have two generators: a bright and dark generator. The following subsection explains how to find the generators for a mos. For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. Specifically:
+** The large size of the bright generator is '''perfect''', and the small size is '''diminished'''.
+** The large size of the dark generator is '''augmented''', and the small size is '''perfect'''.
+* For all other intervals, the large size is '''major''' and the small size is '''minor'''.
+* 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 is perfect. Additionally:
+** Multiples of the period are perfect, just as multiples of the mosoctave are perfect.
+** Generators that are raised some multiple of the mosperiod are also perfect, just as generators raised by some multiple of the mosoctave are perfect.
+* 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.
+* For k-mossteps where k is greater than the number of pitches in the mos, those intervals have the same categories as an octave-reduced interval.
 {| class="wikitable"
 |+Names for mos intervals for 3L 4s
-!Generic interval
+!Interval
 !Specific interval
 !Abbreviation
@@ Line 149: / Line 157: @@
 |P7ms
 |3L+4s
-|}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. In this case, 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.
+==== Finding a mos's generators ====
+To find the generators for a mos, follow the algorithm described [[Recursive structure of MOS scales#Finding a generator|here]], and follow the additional instructions to produce the generators as some quantity of mossteps. Alternatively, produce an interval matrix using the instructions [[Interval matrix#Using step sizes|here]] 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.
-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 if applicable - all of which are perfect - 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 mos degrees ====
+==== Naming mos chords ====
-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.
+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).
-==== Naming alterations by a chroma ====
+=== Naming alterations by a chroma ===
 TAMNAMS also uses the designations of augmented and diminished to refer to alterations of a mos interval, much like with using sharps and flats in standard notation. However, mos intervals are altered by raising or lowering it by a moschroma, 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, and lowering a major mos interval makes it minor. A major or perfect mos interval can be raised by a chroma repeatedly to produce an augmented, doubly-augmented, and (uncommonly) a triply-augmented mos interval. Likewise, a minor or perfect mos interval can be lowered by a chroma repeatedly to produce a diminished, doubly-diminished, and (uncommonly) a triply-diminished mos interval. The names of alterations also apply to mos degrees.
-A mosunison or mosoctave that is itself augmented or diminished may also be referred to a mosaugmented or mosdiminished unison or octave.
+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.
 {| class="wikitable"
 |+Table of alterations, with abbreviations
-! colspan="4" |For intervals whose unaltered form is major or minor
 |-
 !Number of chromas
-!Name of alteration
+!Perfect intervals and degrees
-!Mos interval
+!Major/minor intervals and degrees
-!Mos degree
 |-
 |3 chromas
 |Triply-augmented (AAA, A³, or A^3)
-|Triply-augmented k-mosstep (AAA''k''ms)
+|Triply-augmented (AAA, A³, or A^3)
-|Triply-augmented k-mosdegree (AAA''k''md)
 |-
 |2 chromas
 |Doubly-augmented (AA)
-|Doubly-augmented k-mosstep (AA''k''ms)
+|Doubly-augmented (AA)
-|Doubly-augmented k-mosdegree (AA''k''md)
 |-
 |1 chroma
 |Augmented (A)
-|Augmented k-mosstep (A''k''ms)
+|Augmented (A)
-|Augmented k-mosdegree (A''k''md)
 |-
 | rowspan="2" |0 chromas (unaltered)
+| rowspan="2" |Perfect (P)
 |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 (m''k''md)
 |-
-| -1 chroma
+|  -1 chroma
+|Diminished (d)
 |Diminished (d)
-|Diminished k-mosstep (d''k''ms)
-|Diminished k-mosdegree (d''k''md)
 |-
-| -2 chromas
+|  -2 chromas
+|Doubly-diminished (dd)
 |Doubly-diminished (dd)
-|Doubly-diminished k-mosstep (dd''k''ms)
-|Doubly-diminished k-mosdegree (dd''k''md)
 |-
-| -3 chromas
+|  -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" |For intervals whose unaltered form is perfect
-|-
-!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 (unaltered)
-|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)
 |}
-=== Mos intervals smaller than a chroma ===
+==== Mos intervals smaller than a chroma ====
-* Mosdiesis (a generalization of the diesis as commonly found in meantone theories): |L - 2s|
+* Mosdiesis (a generalized diesis for use with mosses): |L - 2s|
-* Moskleisma (a generalization of the kleisma, also found in meantone theories): |L - 3s|
+* Moskleisma (a generalized kleisma for use with mosses): |L - 3s|
 == Other sandboxed rewrites ==