1701/1700
Ratio | 1701/1700 |
Factorization | 2^{-2} × 3^{5} × 5^{-2} × 7 × 17^{-1} |
Monzo | [-2 5 -2 1 0 0 -1⟩ |
Size in cents | 1.0180736¢ |
Names | palingenetic comma, palingenesis, palingenesma |
FJS name | [math]\text{P1}^{7}_{5,5,17}[/math] |
Special properties | superparticular, reduced |
Tenney height (log_{2} n⋅d) | 21.4635 |
Weil height (max(n, d)) | 1701 |
Benedetti height (n⋅d) | 2891700 |
Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~2.40193 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:
- 81/80 and 85/84
- 126/125 and 136/135
- 273/272 and 325/324
- 351/350 and 442/441
- 441/440 and 595/594
- 936/935 and 2080/2079
- 1089/1088 and 3025/3024
- 1225/1224 and 4375/4374
It factors into the following superparticular pairs:
Temperaments
When tempered out in a linearly independent fashion, the resulting temperaments are called palingenetic temperaments, and are characterized by the presence of essentially tempered chords called palingenetic chords in the 27-odd-limit.
Etymology
This comma's names ultimately come from the Ancient Greek word "palingenesía" (meaning "rebirth", "regeneration" or "renaissance"^{[1]}), 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.