1701/1700

 Ratio 1701/1700 Factorization 2-2 × 35 × 5-2 × 7 × 17-1 Monzo [-2 5 -2 1 0 0 -1⟩ Size in cents 1.0180736¢ Names palingenetic comma,palingenesis,palingenesma Color name 17uzgg1, suzogugu unison FJS name $\text{P1}^{7}_{5,5,17}$ Special properties superparticular,reduced Tenney height (log2 nd) 21.4635 Weil height (log2 max(n, d)) 21.4643 Wilson height (sopfr (nd)) 53 Harmonic entropy(Shannon, $\sqrt{nd}$) ~1.20879 bits Comma size unnoticeable S-expression S18 / S20 open this interval in xen-calc

1701/1700, the palingenetic comma, also known as the palingenesis, or palingenesma, is a 17-limit unnoticeable comma with a size of roughly 1.02 cents. It identifies the septendecimal submajor third (21/17) by a stack of two 10/9 intervals, therefore making it comparable with the marveltwin (325/324). It is, in fact, the difference between the tannisma and the marveltwin. See #Commatic relations below. It also arises as the amount by which a stack consisting of 27/16 and 28/25 exceeds 17/9.

In Sagittal notation, it is the default comma represented by seven tinas.

Commatic relations

This comma is the difference between the following superparticular pairs:

It factors into the following superparticular pairs:

* both of these commas are also within the 2.3.5.7.17 subgroup.

Temperaments

When tempered out in the full 17-limit, the resulting temperament is called the palingenetic temperament, or in the 2.3.5.7.17 subgroup, the palingenetian temperament. Both are characterized by the presence of essentially tempered chords called palingenetic chords in the 21- and 27-odd-limit.

Etymology

This comma was named by Aura in 2020. Its names ultimately come from the Ancient Greek word palingenesía ("rebirth", "regeneration" or "renaissance"), a fitting name since people often hope for a new start after each year. The name is also appropriate in light of how certain essentially tempered chords generated by this comma are evocative of the kinds of chords heard in 12edo, which, oddly enough, actually tempers out this comma.