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

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== 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''~~, 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.

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 start at 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.

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, scales with three specific interval sizes rather than two~~. Additionally, for [[non-octave]] scales, the term ''mosoctave'' is replaced with the term ''mosequave''.

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''.

This section's running example will be 3L 4s, with the pattern (or string) LsLsLss as its brightest mode.

=== Reasoning 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.

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. 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.

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.

=== ~~Specific~~ mos intervals ===

=== Naming specific mos intervals ===

The phrase ''k-mosstep'' by itself does not specify whether an interval is major or minor. 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 will be in no more than two sizes, except for the mosoctave and mosunison, which only has one size.

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 (the following subsection explains how to do this)~~. 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 using 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.

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 (the following subsection explains how to do this). 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 using 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

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|ssLsLsL

|'''3L+4s'''

|}The modifiers of major, minor, ~~augmented,~~ perfect~~, and diminished~~ are assigned in the following manner:

|}The modifiers of major, minor, and perfect 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 these). For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. ~~Specifically~~:

* The generating intervals, or generators, are referred to as '''perfect''' by default. Note that a mos actually has two generators - a bright and dark generator - and both generators have two sizes each. The following subsection explains how to find these. For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. When people talk about the generating intervals, they are usually referring to their perfect form; 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 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. Similarly, multiples of a mosoctave are perfect, as are generators raised by some multiple of the mosoctave.

* 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.

For multi-period mosses, the additional rules apply:

* 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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=== 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'' (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, 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.

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. 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, 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. It's typically uncommon to alter an interval more than three times, such as with a quadruply-augmented and quadruply-diminished interval; the notation for such an interval is to duplicate the letter "A" or "d" however many times, or to use a shorthand such as A^n and d^n. Repetition of "a" or "d" is usually sufficient in most cases.

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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=== Naming mos degrees ===

Individual mos degrees are based on the modifiers ~~assigned~~ to intervals using the process for naming mos intervals and alterations. ~~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.

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.

==== Naming mos chords ====

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 == 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'', 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.
+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 start at 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.
-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, scales with three specific interval sizes rather than two. Additionally, for [[non-octave]] scales, the term ''mosoctave'' is replaced with the term ''mosequave''.
+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''.
+This section's running example will be 3L 4s, with the pattern (or string) LsLsLss as its brightest mode.
 === Reasoning 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.
+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. 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.
 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.
-=== Specific mos intervals ===
+=== Naming specific mos intervals ===
 The phrase ''k-mosstep'' by itself does not specify whether an interval is major or minor. 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 will be in no more than two sizes, except for the mosoctave and mosunison, which only has one size.
-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 (the following subsection explains how to do this). 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 using 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.
+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 (the following subsection explains how to do this). 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 using 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
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 |ssLsLsL
 |'''3L+4s'''
-|}The modifiers of major, minor, augmented, perfect, and diminished are assigned in the following manner:
+|}The modifiers of major, minor, and perfect 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 these). For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. Specifically:
+* The generating intervals, or generators, are referred to as '''perfect''' by default. Note that a mos actually has two generators - a bright and dark generator - and both generators have two sizes each. The following subsection explains how to find these. For our running example of 3L 4s, the generators are a 2-mosstep and 5-mosstep. When people talk about the generating intervals, they are usually referring to their perfect form; 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 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. Similarly, multiples of a mosoctave are perfect, as are generators raised by some multiple of the mosoctave.
+* 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.
 For multi-period mosses, the additional rules apply:
 * 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:
@@ Line 180: / Line 182: @@
 === 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'' (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, 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.
+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. 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, 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. It's typically uncommon to alter an interval more than three times, such as with a quadruply-augmented and quadruply-diminished interval; the notation for such an interval is to duplicate the letter "A" or "d" however many times, or to use a shorthand such as A^n and d^n. Repetition of "a" or "d" is usually sufficient in most cases.
 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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 === Naming mos degrees ===
-Individual mos degrees are based on the modifiers assigned to intervals using the process for naming mos intervals and alterations. 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.
+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.
 ==== Naming mos chords ====