1225/1224: Difference between revisions

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'''1225/1224''', the '''noellisma''', is a [[17-limit]] (also 2.3.5.7.17 subgroup) [[comma]] measuring about 1.41 [[cent]]s. It is the difference between [[35/34]] and [[36/35]], and between [[49/48]] and [[51/50]].

'''1225/1224''', the '''noellisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.5.7.17-[[subgroup]]) [[comma]] measuring about 1.41 [[cent]]s. It is the amount by which a stack of two [[7/6]] subminor thirds exceeds [[34/25]], and the amount by which a stack of two [[35/24]] subfifths exceeds [[17/8]], one octave above [[17/16]]. It is also the difference between [[35/34]] and [[36/35]], and between [[49/48]] and [[51/50]].

== Commatic relations ==

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== Temperaments ==

Tempering out this comma in the 17-limit results in the rank-6 '''noellismic ~~temperament~~''', or in the 2.3.5.7.17 subgroup, the rank-4 '''noellic ~~temperament~~'''. In either case [[18/17]] is split into two equal parts, each representing 35/34~36/35. You may find a list of good equal temperaments that support these temperaments below.

[[Tempering out]] this comma in the 17-limit results in the rank-6 '''noellismic''' temperament, or in the 2.3.5.7.17 subgroup, the rank-4 '''noellic''' temperament. In either case [[18/17]] is split into two equal parts, each representing 35/34~36/35. You may find a list of good equal temperaments that support these temperaments below.

Since 1225/1224 factors as (2401/2400)⋅(2500/2499), it would make sense to temper them both out, so noellic can be further tempered to a simple extension of [[breed (temperament)|breed]] that adds prime 17, though it loses accuracy when compared to breed.

=== Noellic ===

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{{Mapping|legend=2| 1 0 0 0 -3 | 0 1 0 0 -2 | 0 0 1 0 2 | 0 0 0 1 2 }}

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0477{{c}}, ~3/2 = 701.9872{{c}}, ~5/4 = 386.0466{{c}}, ~7/4 = 968.4796{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9993{{c}}, ~5/4 = 386.0746{{c}}, ~7/4 = 968.4911{{c}}

[[Optimal ~~tuning]] ([[CTE]]): ~2~~ = 1\1, ~~~3/2 = 702.0440~~, ~~~5/4 = 386.1228~~, ~~~7/4 = 968.5468~~

{{Optimal ET sequence|legend=1| 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d }}

~~{{Optimal ET sequence|legend=1| 19g, 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d }}~~

[[Badness]] (Sintel): 0.0985

=== Noellismic ===

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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]

|}

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

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0477{{c}}, ~3/2 = 701.9872{{c}}, ~5/4 = 386.0466{{c}}, ~7/4 = 968.4796{{c}}, ~11/8 = 551.1747{{c}}, ~13/8 = 840.3844{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9993{{c}}, ~5/4 = 386.0746{{c}}, ~7/4 = 968.4911{{c}}, ~11/8 = 551.2309{{c}}, ~13/8 = 840.4346{{c}}

[[Optimal ~~tuning]] ([[CTE]]): ~2~~ = 1\1, ~~~3/2 = 702.0440~~, ~~~5/4 = 386.1228~~, ~~~7/4 = 968.5468~~, ~~~11/8 = 551.3179~~, ~~~13/8 = 840.5277~~

{{Optimal ET sequence|legend=1| 22, 26, 27eg, 31, 41g, 45efg, 46, 68, 72, 103, 121, 140, 171, 190g, 212g, 217, 224, 270, 311, 414, 441, 460, 581, 995, 1265, 1648cd, 1846g, 1918d }}

~~{{Optimal ET sequence|legend=1| 19eg, 22, 26, 27eg, 31, 41g, 45efg, 46, 68, 72, 103, 121, 140, 171, 190g, 212g, 217, 224, 270, 311, 414, 441, 460, 581, 995, 1265, 1648cd, 1846g, 1918d }}~~

[[Badness]] (Sintel): 0.578

== Etymology ==

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== See also ==

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:Noellismic]]

[[Category:Commas referencing a famous use of a number]]

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
-'''1225/1224''', the '''noellisma''', is a [[17-limit]] (also 2.3.5.7.17 subgroup) [[comma]] measuring about 1.41 [[cent]]s. It is the difference between [[35/34]] and [[36/35]], and between [[49/48]] and [[51/50]].
+'''1225/1224''', the '''noellisma''', is an [[unnoticeable comma|unnoticeable]] [[17-limit]] (also 2.3.5.7.17-[[subgroup]]) [[comma]] measuring about 1.41 [[cent]]s. It is the amount by which a stack of two [[7/6]] subminor thirds exceeds [[34/25]], and the amount by which a stack of two [[35/24]] subfifths exceeds [[17/8]], one octave above [[17/16]]. It is also the difference between [[35/34]] and [[36/35]], and between [[49/48]] and [[51/50]].
 == Commatic relations ==
@@ Line 25: / Line 25: @@
 == Temperaments ==
-Tempering out this comma in the 17-limit results in the rank-6 '''noellismic temperament''', or in the 2.3.5.7.17 subgroup, the rank-4 '''noellic temperament'''. In either case [[18/17]] is split into two equal parts, each representing 35/34~36/35. You may find a list of good equal temperaments that support these temperaments below.
+[[Tempering out]] this comma in the 17-limit results in the rank-6 '''noellismic''' temperament, or in the 2.3.5.7.17 subgroup, the rank-4 '''noellic''' temperament. In either case [[18/17]] is split into two equal parts, each representing 35/34~36/35. You may find a list of good equal temperaments that support these temperaments below.
+Since 1225/1224 factors as (2401/2400)⋅(2500/2499), it would make sense to temper them both out, so noellic can be further tempered to a simple extension of [[breed (temperament)|breed]] that adds prime 17, though it loses accuracy when compared to breed.
 === Noellic ===
@@ Line 31: / Line 33: @@
 {{Mapping|legend=2| 1 0 0 0 -3 | 0 1 0 0 -2 | 0 0 1 0 2 | 0 0 0 1 2 }}
+: mapping generators: ~2, ~3, ~5, ~7
-: sval mapping generators: ~2, ~3, ~5, ~7
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0477{{c}}, ~3/2 = 701.9872{{c}}, ~5/4 = 386.0466{{c}}, ~7/4 = 968.4796{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9993{{c}}, ~5/4 = 386.0746{{c}}, ~7/4 = 968.4911{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.0440, ~5/4 = 386.1228, ~7/4 = 968.5468
+{{Optimal ET sequence|legend=1| 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d }}
-{{Optimal ET sequence|legend=1| 19g, 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d }}
+[[Badness]] (Sintel): 0.0985
 === Noellismic ===
@@ Line 56: / Line 61: @@
 | ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
 |}
+: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
-: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.0477{{c}}, ~3/2 = 701.9872{{c}}, ~5/4 = 386.0466{{c}}, ~7/4 = 968.4796{{c}}, ~11/8 = 551.1747{{c}}, ~13/8 = 840.3844{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9993{{c}}, ~5/4 = 386.0746{{c}}, ~7/4 = 968.4911{{c}}, ~11/8 = 551.2309{{c}}, ~13/8 = 840.4346{{c}}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.0440, ~5/4 = 386.1228, ~7/4 = 968.5468, ~11/8 = 551.3179, ~13/8 = 840.5277
+{{Optimal ET sequence|legend=1| 22, 26, 27eg, 31, 41g, 45efg, 46, 68, 72, 103, 121, 140, 171, 190g, 212g, 217, 224, 270, 311, 414, 441, 460, 581, 995, 1265, 1648cd, 1846g, 1918d }}
-{{Optimal ET sequence|legend=1| 19eg, 22, 26, 27eg, 31, 41g, 45efg, 46, 68, 72, 103, 121, 140, 171, 190g, 212g, 217, 224, 270, 311, 414, 441, 460, 581, 995, 1265, 1648cd, 1846g, 1918d }}
+[[Badness]] (Sintel): 0.578
 == Etymology ==
@@ Line 67: / Line 75: @@
 == See also ==
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Noellismic]]
+[[Category:Commas referencing a famous use of a number]]