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m →As harmonic segment subsets: Linking to the glossary entry "Harmonic series segment" and to the redirect "Harmonic mode".
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Did not change any of the definitions or meanings of any of the concepts, but made it more clear that harmodal is the rule and non harmodal is the exception. Also listed “near equal” as an alternative interpretation of the acronym as it felt wrong to not include the original meaning of the acronym as one of its possible readings
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A '''neji''' or '''NEJI''' (pronounced /nɛdʒi/ "nedgy"; for "near-equivalent [[just intonation]]") is a [[JI]] scale that is "nearly-equivalent" to an associated ''target scale'' (or ''target'' for short).  Nejis are designed with a focus on the harmonic resonance of the pitches in mind w.r.t their foreseen musical use. If the target scale is an [[edo]], a neji is technically also a type of [[well temperament]] for that edo.
A '''neji''' or '''NEJI''' (pronounced /nɛdʒi/ "nedgy"; for "near-equivalent [[just intonation]]" or "near-equal just intonation") is a [[JI]] scale that is "nearly-equivalent" to an associated ''target scale'' (or ''target'' for short).  Nejis are designed with a focus on the harmonic resonance of the pitches in mind w.r.t their foreseen musical use. If the target scale is an [[edo]], a neji is technically also a type of [[well temperament]] for that edo.


== History ==
== History ==
The concept behind nejis is probably first proposed by [[George Secor]] in 2002<ref>[http://www.anaphoria.com/secor-blarney.html GEORGE SECOR - A BIT O' BLARNEY]</ref>, where he called it a '''quasi-equal rational tuning'''.  
The concept behind nejis is probably first proposed by [[George Secor]] in 2002<ref>[http://www.anaphoria.com/secor-blarney.html GEORGE SECOR - A BIT O' BLARNEY]</ref>, where he called it a '''quasi-equal rational tuning'''.  


The idea has also been suggested by [[Paul Erlich]] and a feature to produce them has existed in [[Scala]] for generating them since some time in the 1990's.
The idea has also been suggested by [[Paul Erlich]] and a feature to generate them has existed in [[Scala]] since some time in the 1990's.


The term ''neji'' was coined by [[Zhea Erose]].
The term ''neji'' was coined by [[Zhea Erose]].


Due to the influence of Zhea Erose who originated the term, the term "neji" has often been used in connection with naming [[#Harmodal nejis|harmodal nejis]] specifically, with other types of nejis being less common.
== Approaches to neji construction ==


== Approaches to neji construction ==
=== In primodality ===
=== 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 and maximizing unique ones specific to that prime. 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.
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 and maximizing unique ones specific to that prime. 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.


=== As harmonic segment subsets ===
=== As harmonic segment subsets ===
Generalizing from the primodal use case of the term "neji", one might choose any relatively low [[Glossary#H|harmonic series segment]], ''not necessarily in a way associated with primodality'', 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 [[Glossary#H|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:
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.
: 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 ====
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.
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 mode|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 ====
==== Harmodal vs non-harmodal 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.
If the neji belongs to a ''relatively not large'' harmonic segment, and has a [[period]] equal to some positive integer harmonic (like the [[octave]], [[tritave]] or [[pentave]]), it is a "harmodal neji" (a contraction of "[[Harmonic mode|harmonic modal]] neji"). An octave is the most often used period.
 
When the term "neji" is used, it is automatically assumed to be referring to a harmodal neji unless stated otherwise, in the same way that when "equal division" is used, it is assumed to be referring to "equal pitch division" unless otherwise stated. (That’s why there is no P in EDO, but there is an F in AFDO). So, if a neji is harmodal, this is not usually explicitly stated. It is assumed.
 
If a neji does not belong to a ''relatively not large'', period-repeating harmonic segment, then it is a "non-harmodal neji". In this case, it is advisable to explicitly name non-harmodal nejis as such, seeing as they are the exception rather than the rule.


=== Detempering ===
=== Detempering ===
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 [[regular 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.)
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 [[regular temperament]] for a tempered scale, equal-division or otherwise.  
 
Importantly and nontrivially, detempering is stricter than merely requiring that the target scale has a ''scale logic'' (that is, a ''[[mapping]]''), as many nejis may approximate the target scale without following its associated mapping. Only if a neji obeys the mapping, is it a detempering.


=== Building edo nejis ===
=== Building edo nejis ===
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