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 | [math]\text{P1}^{7}_{5,5,17}[/math] |
| 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, [math]\sqrt{nd}[/math]) |
~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:
- 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
- 729/728 and 1275/1274
- 936/935 and 2080/2079
- 1001/1000 and 2431/2430
- 1089/1088 and 3025/3024
- 1225/1224 and 4375/4374 *
It factors into the following superparticular pairs:
- 2601/2600 and 4914/4913
- 2401/2400 and 5832/5831 *
- 2058/2057 and 9801/9800
- 1716/1715 and 194481/194480
* 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.