1225/1224: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Sectioning
→Temperaments: + relation to the breed temperament
 
(11 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{Infobox Interval
{{Infobox Interval
| Name = noellisma
| Name = noellisma
| Color name = 17uzzyy1, subizoyo unison
| Color name = 17uzzyy1, subizoyo 1sn,<br>Subizoyo comma
| Comma = yes
| 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 ==
== Commatic relations ==
In terms of commas, it is the difference between the following superparticular pairs:  
This comma is the difference between the following superparticular pairs:  
* [[273/272]] and [[351/350]]
* [[273/272]] and [[351/350]]
* [[325/324]] and [[442/441]]
* [[325/324]] and [[442/441]]
Line 25: Line 25:


== Temperaments ==
== Temperaments ==
Tempering out this comma in the 17-limit results in the '''noellismic temperament''', where [[18/17]] is split into two equal parts, each representing 35/34~36/35.  
[[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 ===
[[Subgroup]]: 2.3.5.7.17
 
{{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
 
[[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 ET sequence|legend=1| 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d }}
 
[[Badness]] (Sintel): 0.0985
 
=== Noellismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || -3 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || -2 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || 2 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || 2 || ],
|-
| ⟨ || 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 = 1200.0477{{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 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 }}
 
[[Badness]] (Sintel): 0.578


== Etymology ==
== Etymology ==
The name derives from ''Noel'', for the numerator or the denominator, when written in decimal system, is reminiscent of the date of Christmas.  
The noellisma was named by [[Flora Canou]] in 2022. The name derives from ''Noel'', for the numerator or the denominator, when written in decimal system, is reminiscent of the date of Christmas.  


== See also ==
== See also ==
* [[Unnoticeable comma]]
* [[List of superparticular intervals]]
* [[List of superparticular intervals]]


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

Latest revision as of 10:25, 25 March 2026

Interval information
Ratio 1225/1224
Factorization 2-3 × 3-2 × 52 × 72 × 17-1
Monzo [-3 -2 2 2 0 0 -1⟩
Size in cents 1.413829¢
Name noellisma
Color name 17uzzyy1, subizoyo 1sn,
Subizoyo comma
FJS name [math]\displaystyle{ \text{A1}^{5,5,7,7}_{17} }[/math]
Special properties square superparticular,
reduced
Tenney norm (log2 nd) 20.516
Weil norm (log2 max(n, d)) 20.5171
Wilson norm (sopfr(nd)) 53
Comma size unnoticeable
S-expressions S35,
S49⋅S50
Open this interval in xen-calc

1225/1224, the noellisma, is an unnoticeable 17-limit (also 2.3.5.7.17-subgroup) comma measuring about 1.41 cents. 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

This comma is the difference between the following superparticular pairs:

It factors into the following superparticular pairs:

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.

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 that adds prime 17, though it loses accuracy when compared to breed.

Noellic

Subgroup: 2.3.5.7.17

Subgroup-val mapping: [⟨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

Optimal tunings:

  • WE: ~2 = 1200.0477 ¢, ~3/2 = 701.9872 ¢, ~5/4 = 386.0466 ¢, ~7/4 = 968.4796 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9993 ¢, ~5/4 = 386.0746 ¢, ~7/4 = 968.4911 ¢

Optimal ET sequence: 22, 27g, 31, 41g, 46, 53, 68, 72, 99, 171, 581, 653, 752, 824, 995, 1576, 1747, 1918d

Badness (Sintel): 0.0985

Noellismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 0 -3 ],
⟨ 0 1 0 0 0 0 -2 ],
⟨ 0 0 1 0 0 0 2 ],
⟨ 0 0 0 1 0 0 2 ],
⟨ 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 = 1200.0477 ¢, ~3/2 = 701.9872 ¢, ~5/4 = 386.0466 ¢, ~7/4 = 968.4796 ¢, ~11/8 = 551.1747 ¢, ~13/8 = 840.3844 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9993 ¢, ~5/4 = 386.0746 ¢, ~7/4 = 968.4911 ¢, ~11/8 = 551.2309 ¢, ~13/8 = 840.4346 ¢

Optimal ET sequence: 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

The noellisma was named by Flora Canou in 2022. The name derives from Noel, for the numerator or the denominator, when written in decimal system, is reminiscent of the date of Christmas.

See also

Retrieved from "https://en.xen.wiki/index.php?title=1225/1224&oldid=226814"