1701/1700

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	1.018074¢
Names	palingenetic comma, palingenesis, palingenesma
Color name	17uzgg1, suzogugu unison
FJS name	[math]\displaystyle{ \text{P1}^{7}_{5,5,17} }[/math]
Special properties	superparticular, reduced
Tenney norm (log2 nd)	21.4635
Weil norm (log2 max(n, d))	21.4643
Wilson norm (sopfr(nd))	53
Comma size	unnoticeable
S-expression	S18/S20
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1701/1700, the palingenetic comma, also known as the palingenesis or palingenesma, is an unnoticeable 17-limit 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 marveltwin and the tannisma. See #Commatic relations below. It also arises as the amount by which a stack consisting of 27/16 and 28/25 exceeds 17/9, and as the difference between 63/50 and 34/27.

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.

Palingenetian

Subgroup: 2.3.5.7.17

Subgroup-val mapping: [⟨1 0 0 0 -2], ⟨0 1 0 0 5], ⟨0 0 1 0 -2], ⟨0 0 0 1 1]]

mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

• WE: ~2 = 1200.0180 ¢, ~3/2 = 701.8238 ¢, ~5/4 = 386.3748 ¢, ~7/4 = 968.7188 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8252 ¢, ~5/4 = 386.3913 ¢, ~7/4 = 968.7278 ¢

Optimal ET sequence: 27g, 39dg, 41, 46, 53, 72, 99, 171, 472, 525, 571, 643, 742, 913, 1556, 1727, 2351, 2469, 2640, 2994, 3165, 3907, 4078

Badness (Sintel): 0.115

Palingenetic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨	1	0	0	0	0	0	-2	],
⟨	0	1	0	0	0	0	5	],
⟨	0	0	1	0	0	0	-2	],
⟨	0	0	0	1	0	0	1	],
⟨	0	0	0	0	1	0	0	],
⟨	0	0	0	0	0	1	0	]]

mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

• WE: ~2 = 1200.0180 ¢, ~3/2 = 701.8238 ¢, ~5/4 = 386.3748 ¢, ~7/4 = 968.7188 ¢, ~11/8 = 551.2639 ¢, ~13/8 = 840.4736 ¢
• CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.8252 ¢, ~5/4 = 386.3913 ¢, ~7/4 = 968.7278 ¢, ~11/8 = 551.2862 ¢, ~13/8 = 840.4937 ¢

Optimal ET sequence: 27eg, 39dfg, 41, 46, 58, 72, 111, 130, 145, 152fg, 159, 171, 183, 217, 224, 270, 354, 400, 441, 460, 571, 597, 624, 643, 684, 742, 814, 1084, 1385, 1609, 1826, 2423, 3211, 3435g, 4249b *

* optimal patent val: 4649

Badness (Sintel): 0.855

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.

See also

• List of superparticular intervals