1701/1700

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Interval information
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
FJS name [math]\text{P1}^{7}_{5,5,17}[/math]
Special properties superparticular,
reduced
Tenney height (log2 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:

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.

See also

References