1701/1700: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = palingenetic comma, palingenesis, palingenesma
| Ratio = 1701/1700
| Color name = 17uzgg1, suzogugu unison
| Monzo = -2 5 -2 1 0 0 -1
| Comma = yes
| Cents = 1.01807
| Name = palingenesis comma, <br> palingenetic comma, <br> palingenesma
| Color name =  
| FJS name =
| Sound =  
}}
}}
'''1701/1700''', the '''palingenetic comma''', also known as the '''palingenesis''' or '''palingenesma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] [[comma]] with a size of roughly 1.02 [[cent]]s. It identifies the [[21/17|septendecimal submajor third (21/17)]] by a stack of two [[10/9]] intervals, therefore making it comparable with the [[325/324|marveltwin (325/324)]]. It is, in fact, the difference between the marveltwin and the [[442/441|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]].


'''1701/1700''', the '''palingenesis comma''', also known as the '''palingenetic comma''', or '''palingenesma''', is a [[17-limit]] [[unnoticeable comma]] with a size of roughly 1.02 cents. It identifies the [[21/17|septendecimal submajor third (21/17)]] by a stack of two [[10/9]] intervals, therefore making it comparable with the [[325/324|marveltwin (325/324)]]. It is, in fact, the difference between the [[273/272|tannisma]] and the marveltwin.  It also arises as the amount by which a stack consisting of [[27/16]] and [[28/25]] exceeds [[17/9]], and the the amount by which [[85/84]] falls short of [[81/80]]. When tempered out in a linearly independent fashion, the resulting temperaments are called "'''palingenetic temperaments'''", and are characterized by the presence of [[Dyadic chord|essentially tempered chord]]s called "[[palingenetic chords]]" in the [[27-odd-limit]].
In [[Sagittal notation]], it is the default comma represented by seven [[tina]]s.
 
== 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]]
 
<nowiki/>* 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 chord]]s called [[palingenetic chords]] in the [[21-odd-limit|21-]] and [[27-odd-limit]].
 
=== Palingenetian ===
[[Subgroup]]: 2.3.5.7.17
 
{{Mapping|legend=2| 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 tuning]]s:
* [[WE]]: ~2 = 1200.0180{{c}}, ~3/2 = 701.8238{{c}}, ~5/4 = 386.3748{{c}}, ~7/4 = 968.7188{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8252{{c}}, ~5/4 = 386.3913{{c}}, ~7/4 = 968.7278{{c}}
 
{{Optimal ET sequence|legend=1| 27g, 39dg, 41, 46, 53, 72, 99, 171, 472, 525, 571, 643, 742, 913, 1556, 1727, 2351, 2469, 2640, 2994, 3165, 3907, 4078 }}


This comma's names ultimately come from the Ancient Greek word "palingenesía" (meaning "rebirth", "regeneration" or "renaissance"<ref>[[Wiktionary: palingenesis #English]]</ref>), 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 [[24edo]], where many microtonalists get their first start, despite 24edo itself not tempering out this comma.
[[Badness]] (Sintel): 0.115


In [[Sagittal notation]], it is the default comma represented by seven [[tina]]s.  
=== Palingenetic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 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 tuning]]s:
* [[WE]]: ~2 = 1200.0180{{c}}, ~3/2 = 701.8238{{c}}, ~5/4 = 386.3748{{c}}, ~7/4 = 968.7188{{c}}, ~11/8 = 551.2639{{c}}, ~13/8 = 840.4736{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.8252{{c}}, ~5/4 = 386.3913{{c}}, ~7/4 = 968.7278{{c}}, ~11/8 = 551.2862{{c}}, ~13/8 = 840.4937{{c}}
 
{{Optimal ET sequence|legend=1| 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 }} *
 
<nowiki/>* [[optimal patent val]]: [[4649edo|4649]]
 
[[Badness]] (Sintel): 0.855
 
== Etymology ==
This comma was named by [[Aura]] in 2020. Its names ultimately come from the Ancient Greek word [[Wiktionary: palingenesis #English|''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 ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


== References ==
<references/>
[[Category:17-limit]]
[[Category:Unnoticeable commas]]
[[Category:Superparticular]]
[[Category:Palingenetic]]
[[Category:Palingenetic]]
[[Category:Commas named by translating something into another language]]

Latest revision as of 12:17, 28 March 2026

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
Open this interval in xen-calc

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:

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.

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

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