Nexus comma

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Interval information
Ratio 1771561/1769472
Factorization 2-16 × 3-3 × 116
Monzo [-16 -3 0 0 6
Size in cents 2.0426516¢
Names diatonic-paradiatonic nexus comma,
nexus comma,
nexisma,
nexuma
Color name Tribilo comma
FJS name [math]\text{A}{-2}^{11,11,11,11,11,11}[/math]
Special properties reduced
Tenney height (log2 nd) 41.5115
Weil height (log2 max(n, d)) 41.5132
Wilson height (sopfr (nd)) 107
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~2.41653 bits
Comma size unnoticeable
open this interval in xen-calc

The diatonic-paradiatonic nexus comma, otherwise known as the nexus comma, 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 Axirabian diatonic semitones falls short of a 32/27 minor third, or equivalently stated, the amount by which a stack of three 121/96 intervals exceeds the octave. It is also the amount by which a stack of six 12/11 neutral seconds fall short of a 27/16 major sixth, the interval that separates rastma and the Alpharabian comma, and the result of stacking the schisma and the parimo.

Temperaments and Name

Tempering out this comma in the full 11-limit leads to the rank-4 nexus temperament, or, in the 2.3.11 subgroup, the rank-2 tribilo temperament. Either way, it 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 Aura considering the temperament that tempers out this comma to be some sort of "nexus temperament", which it turn gave rise to most of this comma's names. Furthermore, the names have since turned out to be justified in light of the comma's additional functions, such as splitting the Pythagorean comma into three instances of the rastma, splitting the perfect fourth into two semifourths, and in turn splitting the Pythagorean limma into two as well – all functions that contribute to both diatonic and paradiatonic significance, hence the longer descriptive name "diatonic-paradiatonic nexus 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 of various ranks that temper out the nexus comma, see the Nexus.

See also