1701/1700: Difference between revisions

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| Comma = yes

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'''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 '''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 [[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 ~~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 [[tina]]s.

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* [[1716/1715]] and [[194481/194480]]

<nowiki>*~~</nowiki>~~ both of these commas are also within the 2.3.5.7.17 subgroup.

<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 [[~~Dyadic chord|~~essentially tempered chord]]s called [[palingenetic chords]] in the [[21-odd-limit|21-]] and [[27-odd-limit]].

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 }}

[[Badness]] (Sintel): 0.115

=== 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 ==

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== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:Palingenetic]]

[[Category:Commas named by translating something into another language]]

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
+'''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 '''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 [[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 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 [[tina]]s.
@@ Line 29: / Line 28: @@
 * [[1716/1715]] and [[194481/194480]]
-<nowiki>*</nowiki> both of these commas are also within the 2.3.5.7.17 subgroup.
+<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 [[Dyadic chord|essentially tempered chord]]s called [[palingenetic chords]] in the [[21-odd-limit|21-]] and [[27-odd-limit]].
+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 }}
+[[Badness]] (Sintel): 0.115
+=== 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 ==
@@ Line 38: / Line 81: @@
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Palingenetic]]
 [[Category:Commas named by translating something into another language]]