Nexus comma: Difference between revisions

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The '''nexus comma''', otherwise known as the '''nexisma''', or – in the earliest materials where this comma is named outside of [[color notation]] – the '''nexuma''', is an [[11-limit]] (also 2.3.11 subgroup) [[unnoticeable comma]] with a ratio of '''1771561/1769472''' and a value of approximately 2 cents. In color notation, this same comma is referred to as the '''tribilo comma'''. It is the amount by which a stack of three [[128/121]] Alpharabian diatonic semitones falls short of a [[32/27]] minor third, or equivalently stated, the amount by which a stack of three 121/96's exceeds the octave. It is also the difference between the [[243/242|rastma]] and the [[Alpharabian comma]], or the sum of the [[32805/32768|schisma]] and the [[parimo]].

Tempering it out leads to the joining of the [[11-limit]] and the [[3-limit]], a fact which, in light of the importance of both p-limits, led to [[User:Aura|Aura]] ~~initially dubbing~~ the temperament that tempers out this comma "nexus temperament", which~~, in~~ turn~~, is the source of~~ most of this comma's names~~. However, the term "nexus", as a temperament name, has since been restricted to the Rank-4 temperament that tempers out this comma~~. While the importance of the 3-limit is generally accepted (see [[Pythagorean tuning]], [[circle of fifths]], [[FJS]], [[Helmholtz-Ellis notation]]), it can be derived mathematically that the 11-limit is an excellent basis for quartertones in terms of ratio simplicity, and the 11-limit can be shown to host a clear sequence of intervals in which every other member is the octave complement of what is effectively a stack of 128/121 diatonic semitones (see [[Alpharabian tuning]]). For a list of temperaments that temper out the nexus comma, see [[nexus family]].

Tempering it out leads to the joining of the [[11-limit]] and the [[3-limit]], a fact which, in light of the importance of both p-limits, led to [[User:Aura|Aura]] considering the temperament that tempers out this comma to be some sort "nexus temperament", which it turn gave rise to most of this comma's names. While the importance of the 3-limit is generally accepted (see [[Pythagorean tuning]], [[circle of fifths]], [[FJS]], [[Helmholtz-Ellis notation]]), it can be derived mathematically that the 11-limit is an excellent basis for quartertones in terms of ratio simplicity, and the 11-limit can be shown to host a clear sequence of intervals in which every other member is the octave complement of what is effectively a stack of 128/121 diatonic semitones (see [[Alpharabian tuning]]). For a list of temperaments that temper out the nexus comma, see [[nexus family]].

== Temperaments ==

In rank-2 environments, tempering it out in the full 11-limit leads to the [[nexus family #nexal family|nexal]] temperament, enabling [[nexal chords]], or, in the 2.3.11 subgroup, the [[nexus family #nexal family|tribilo]] temperament, enabling [[tribilo chords]].

[[Category:11-limit]]

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 The '''nexus comma''', otherwise known as the '''nexisma''', or – in the earliest materials where this comma is named outside of [[color notation]] – the '''nexuma''', is an [[11-limit]] (also 2.3.11 subgroup) [[unnoticeable comma]] with a ratio of '''1771561/1769472''' and a value of approximately 2 cents.  In color notation, this same comma is referred to as the '''tribilo comma'''.  It is the amount by which a stack of three [[128/121]] Alpharabian diatonic semitones falls short of a [[32/27]] minor third, or equivalently stated, the amount by which a stack of three 121/96's exceeds the octave.  It is also the difference between the [[243/242|rastma]] and the [[Alpharabian comma]], or the sum of the [[32805/32768|schisma]] and the [[parimo]].
-Tempering it out leads to the joining of the [[11-limit]] and the [[3-limit]], a fact which, in light of the importance of both p-limits, led to [[User:Aura|Aura]] initially dubbing the temperament that tempers out this comma "nexus temperament", which, in turn, is the source of most of this comma's names.  However, the term "nexus", as a temperament name, has since been restricted to the Rank-4 temperament that tempers out this comma.  While the importance of the 3-limit is generally accepted (see [[Pythagorean tuning]], [[circle of fifths]], [[FJS]], [[Helmholtz-Ellis notation]]), it can be derived mathematically that the 11-limit is an excellent basis for quartertones in terms of ratio simplicity, and the 11-limit can be shown to host a clear sequence of intervals in which every other member is the octave complement of what is effectively a stack of 128/121 diatonic semitones (see [[Alpharabian tuning]]).  For a list of temperaments that temper out the nexus comma, see [[nexus family]].
+Tempering it out leads to the joining of the [[11-limit]] and the [[3-limit]], a fact which, in light of the importance of both p-limits, led to [[User:Aura|Aura]] considering the temperament that tempers out this comma to be some sort "nexus temperament", which it turn gave rise to most of this comma's names.  While the importance of the 3-limit is generally accepted (see [[Pythagorean tuning]], [[circle of fifths]], [[FJS]], [[Helmholtz-Ellis notation]]), it can be derived mathematically that the 11-limit is an excellent basis for quartertones in terms of ratio simplicity, and the 11-limit can be shown to host a clear sequence of intervals in which every other member is the octave complement of what is effectively a stack of 128/121 diatonic semitones (see [[Alpharabian tuning]]).  For a list of temperaments that temper out the nexus comma, see [[nexus family]].
+== Temperaments ==
+In rank-2 environments, tempering it out in the full 11-limit leads to the [[nexus family #nexal family|nexal]] temperament, enabling [[nexal chords]], or, in the 2.3.11 subgroup, the [[nexus family #nexal family|tribilo]] temperament, enabling [[tribilo chords]].
 [[Category:11-limit]]