Neji: Difference between revisions

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=== In primodality ===

In Zhea Erose's [[primodality]] theory, nejis can be used to explore a prime family (see [[primodality]]), while keeping the transposability, scale structures, rank-2 harmonic theory, notation, etc. associated with the target scale (usually an [[edo]]). (The neji's denominator need not be prime but primes may be preferred for sake of minimizing lower-complexity intervals. Zhea often uses semiprimes ''pq''.) Zhea Erose's theory also deals with modulations between different prime families, and combining different prime families into one scale.

~~=== Detempering ===~~

A more structural, ([[JI subgroup]]) lattice-based approach is [[detempering]]. Detempering entails that the neji has the property of being [[epimorphic]] (obeys the appropriate mapping logic) with respect to a temperament for a tempered scale, equal-division or otherwise. Importantly and nontrivially, this is stricter than merely requiring that the target scale has a ''scale logic'' (that is, a ''[[mapping]]''), as the neji may approximate the target scale without following its associated mapping! (And as aforementioned, the target scale is not ''required'' to have a mapping, although in many cases it does.)

=== As harmonic segment subsets ===

~~One~~ might choose a low harmonic series segment and select notes therefrom in order to build a neji with extra [[concordance]].

Generalizing from the primodal use case of the term "neji", one might choose any relatively low harmonic series segment, ''not necessarily in a way associated with primodality'', and select notes therefrom in order to build a neji with extra [[concordance]].

This distinction is more process-based than formalized (because any JI scale trivially occurs as some (possibly very high) harmonic series segment subset); sometimes the pool of intervals is chosen to be a specific "relatively not large" harmonic series segment before notes are selected from it while other times it is guessed at in some "relatively not large" range and then later revised based on what fits (or other considerations). The "relatively not large" focuses on the following key observation of Zhea's that others have agreed with as a valuable observation in the design of JI scales:

: If you focus on choosing intervals as representing different notes of your target ''without regard'' for the growing size of the implied denominator of the scale as a single chord, then the denominator will often grow ''rapidly'' to absurd numbers, ''especially'' if you are repeatedly stacking the same interval to reach some of the notes.

Technically, this is a simplification of the observation, as you can take any incomplete harmodal neji and add some intervals that are "awkward" w.r.t the denominator to complete it and thus increase the denominator massively, but the scale will overall likely still sound pretty coherent, ''especially'' if those intervals simplify w.r.t other intervals of interest to the composer, so it is rather the spirit of the observation that your notes of interest should cohere with each-other within reason and that where they don't should ideally be intentional harmonic tensions in the scale accessible for musical use. This logic also shows why the line between [[neji]]s (in the general sense) and '''harmodal nejis''' (defined below) is even more "fuzzy" than one might initially think.

===== Harmodal nejis =====

==== Harmodal nejis ====

If the target scale has a [[period]] equal to some positive integer harmonic (like the [[octave]], [[tritave]] or [[pentave]]), a fitting name for this type of neji is a "harmodal neji", a contraction of "harmonic modal neji". This period is typically an octave, as that's the most common ~~usecase~~. '''This term is new, however; it was proposed to resolve an ambiguity in the stand-alone term "neji"''', therefore please see the directly below alternative, which may be preferred for consistency/"backwards-compatibility".

If the target scale has a [[period]] equal to some positive integer harmonic (like the [[octave]], [[tritave]] or [[pentave]]), a fitting name for this type of neji is a "harmodal neji", a contraction of "harmonic modal neji", making primodal nejis a type of harmodal nejis. This period is typically an octave, as that's the most common use case. '''This term is new, however; it was proposed to resolve an ambiguity in the stand-alone term "neji"''', therefore please see the directly below alternative, which may be preferred for consistency/"backwards-compatibility".

==== Nonharmodal nejis ====

An alternative to using the term "harmodal nejis" is to ''assume'' that a "neji" is harmodal ''by default'', and instead specify that a neji is "''nonharmodal''" only when necessary, in order to preserve the colloquial usage of the term "neji". This is consistent with the sentiment that there does not need to be a new term "harmodal neji" while retaining clarity.

===~~== Nonharmodal nejis ==~~===

=== Detempering ===

~~An alternative~~ to ~~using~~ the ~~term "harmodal nejis" is to~~ ''~~assume~~'' that a ~~"neji" is harmodal~~ ''~~by default~~'', ~~and instead specify that a~~ neji is "''~~nonharmodal~~''~~" only when necessary~~, in ~~order to preserve the colloquial usage of the term "neji". This is consistent with the sentiment that there~~ does ~~not need to be a new term "harmodal neji" while retaining clarity~~.

A more ([[JI subgroup]]) lattice-based approach is [[detempering]]. Detempering entails that the neji has the property of being [[epimorphic]] (obeys the appropriate mapping logic) with respect to a temperament for a tempered scale, equal-division or otherwise. Importantly and nontrivially, this is stricter than merely requiring that the target scale has a ''scale logic'' (that is, a ''[[mapping]]''), as the neji may approximate the target scale without following its associated mapping! (And as aforementioned, the target scale is not ''required'' to have a mapping, although in many cases it does.)

=== Building edo nejis ===

@@ Line 19: / Line 19: @@
 === In primodality ===
 In Zhea Erose's [[primodality]] theory, nejis can be used to explore a prime family (see [[primodality]]), while keeping the transposability, scale structures, rank-2 harmonic theory, notation, etc. associated with the target scale (usually an [[edo]]). (The neji's denominator need not be prime but primes may be preferred for sake of minimizing lower-complexity intervals. Zhea often uses semiprimes ''pq''.) Zhea Erose's theory also deals with modulations between different prime families, and combining different prime families into one scale.
-=== Detempering ===
-A more structural, ([[JI subgroup]]) lattice-based approach is [[detempering]]. Detempering entails that the neji has the property of being [[epimorphic]] (obeys the appropriate mapping logic) with respect to a temperament for a tempered scale, equal-division or otherwise. Importantly and nontrivially, this is stricter than merely requiring that the target scale has a ''scale logic'' (that is, a ''[[mapping]]''), as the neji may approximate the target scale without following its associated mapping! (And as aforementioned, the target scale is not ''required'' to have a mapping, although in many cases it does.)
 === As harmonic segment subsets ===
-One might choose a low harmonic series segment and select notes therefrom in order to build a neji with extra [[concordance]].
+Generalizing from the primodal use case of the term "neji", one might choose any relatively low harmonic series segment, ''not necessarily in a way associated with primodality'', and select notes therefrom in order to build a neji with extra [[concordance]].
 This distinction is more process-based than formalized (because any JI scale trivially occurs as some (possibly very high) harmonic series segment subset); sometimes the pool of intervals is chosen to be a specific "relatively not large" harmonic series segment before notes are selected from it while other times it is guessed at in some "relatively not large" range and then later revised based on what fits (or other considerations). The "relatively not large" focuses on the following key observation of Zhea's that others have agreed with as a valuable observation in the design of JI scales:
 : If you focus on choosing intervals as representing different notes of your target ''without regard'' for the growing size of the implied denominator of the scale as a single chord, then the denominator will often grow ''rapidly'' to absurd numbers, ''especially'' if you are repeatedly stacking the same interval to reach some of the notes.
 Technically, this is a simplification of the observation, as you can take any incomplete harmodal neji and add some intervals that are "awkward" w.r.t the denominator to complete it and thus increase the denominator massively, but the scale will overall likely still sound pretty coherent, ''especially'' if those intervals simplify w.r.t other intervals of interest to the composer, so it is rather the spirit of the observation that your notes of interest should cohere with each-other within reason and that where they don't should ideally be intentional harmonic tensions in the scale accessible for musical use. This logic also shows why the line between [[neji]]s (in the general sense) and '''harmodal nejis''' (defined below) is even more "fuzzy" than one might initially think.
-===== Harmodal nejis =====
+==== Harmodal nejis ====
-If the target scale has a [[period]] equal to some positive integer harmonic (like the [[octave]], [[tritave]] or [[pentave]]), a fitting name for this type of neji is a "harmodal neji", a contraction of "harmonic modal neji". This period is typically an octave, as that's the most common usecase. '''This term is new, however; it was proposed to resolve an ambiguity in the stand-alone term "neji"''', therefore please see the directly below alternative, which may be preferred for consistency/"backwards-compatibility".
+If the target scale has a [[period]] equal to some positive integer harmonic (like the [[octave]], [[tritave]] or [[pentave]]), a fitting name for this type of neji is a "harmodal neji", a contraction of "harmonic modal neji", making primodal nejis a type of harmodal nejis. This period is typically an octave, as that's the most common use case. '''This term is new, however; it was proposed to resolve an ambiguity in the stand-alone term "neji"''', therefore please see the directly below alternative, which may be preferred for consistency/"backwards-compatibility".
+==== Nonharmodal nejis ====
+An alternative to using the term "harmodal nejis" is to ''assume'' that a "neji" is harmodal ''by default'', and instead specify that a neji is "''nonharmodal''" only when necessary, in order to preserve the colloquial usage of the term "neji". This is consistent with the sentiment that there does not need to be a new term "harmodal neji" while retaining clarity.
-===== Nonharmodal nejis =====
+=== Detempering ===
-An alternative to using the term "harmodal nejis" is to ''assume'' that a "neji" is harmodal ''by default'', and instead specify that a neji is "''nonharmodal''" only when necessary, in order to preserve the colloquial usage of the term "neji". This is consistent with the sentiment that there does not need to be a new term "harmodal neji" while retaining clarity.
+A more ([[JI subgroup]]) lattice-based approach is [[detempering]]. Detempering entails that the neji has the property of being [[epimorphic]] (obeys the appropriate mapping logic) with respect to a temperament for a tempered scale, equal-division or otherwise. Importantly and nontrivially, this is stricter than merely requiring that the target scale has a ''scale logic'' (that is, a ''[[mapping]]''), as the neji may approximate the target scale without following its associated mapping! (And as aforementioned, the target scale is not ''required'' to have a mapping, although in many cases it does.)
 === Building edo nejis ===