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

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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''.

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

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

=== Reasoning for 0-indexed 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.

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, 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 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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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). 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); for our running example of 3L 4s, this is LsLsLss. 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, and ~~perfect~~ are assigned in the following manner:

|}The modifiers of ''major'', ''minor'', ''augment'', ''perfect'', and ''diminished'' (abbreviated as M, m, A, P, and d respectively) are assigned in the following manner:

* The mosunison and mosoctave are '''perfect''' because they only have one size each.

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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. 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.

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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 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.
+This section's running example will be 3L 4s.
 === Reasoning for 0-indexed 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.
+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, 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 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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 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). 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); for our running example of 3L 4s, this is LsLsLss. 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, and perfect are assigned in the following manner:
+|}The modifiers of ''major'', ''minor'', ''augment'', ''perfect'', and ''diminished'' (abbreviated as M, m, A, P, and d respectively) are assigned in the following manner:
 * The mosunison and mosoctave are '''perfect''' because they only have one size each.
@@ Line 182: / Line 184: @@
 === 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. 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.
+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.