4225/4224: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 4225/4224
| Monzo = -7 -1 2 0 -1 2
| Cents = 0.40981
| Name = leprechaun comma
| Name = leprechaun comma
| Color name =  
| Color name = 3oo1uyy1, thotholuyoyo unison
| FJS name =  
| Comma = yes
}}
}}
'''4225/4224''', the '''leprechaun comma''', is an unnoticeable [[13-limit]] [[superparticular]] interval. It is the difference between the following pairs:  
'''4225/4224''', the '''leprechaun comma''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[superparticular]] interval. This [[comma]] is the difference between the following pairs:  
* [[65/64]] and [[66/65]]
* [[65/64]] and [[66/65]]
* [[169/168]] and [[176/175]]
* [[169/168]] and [[176/175]]
* [[325/324]] and [[352/351]]
* [[325/324]] and [[352/351]]
* [[385/384]] and [[847/845]]
* [[2080/2079]] and [[4096/4095]]
* [[2080/2079]] and [[4096/4095]]
* [[3025/3024]] and [[10648/10647]]
* [[3025/3024]] and [[10648/10647]]
It factors into [[4375/4374]] and [[123201/123200]].


== Temperaments ==
== Temperaments ==
Tempering it out in the full 13-limit leads to the rank-5 '''leprechaun temperament'''. It is also tempered out in lower-rank temperaments including abigail, newt, decoid, donar, etc.
[[Tempering out]] this comma in the full 13-limit leads to the rank-5 '''leprechaun temperament'''. It is also tempered out in lower-rank temperaments including [[abigail]], [[newt]], [[decoid]], [[donar]], etc. Tempering it out also splits the [[33/32]] Alpharabian quarter-tone in two, each part corresponding to 65/64 and 66/65. You may find a list of good equal temperaments that support this temperament below.
 
[[Subgroup]]: 2.3.5.7.11.13
 
[[Mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 1 || 4 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 1 || 1 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 2 || 1 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~65/48
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0205{{c}}, ~3/2 = 701.9418{{c}}, ~5/4 = 386.2411{{c}}, ~7/4 = 968.7849{{c}}, ~65/48 = 524.6851{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9442{{c}}, ~5/4 = 386.2465{{c}}, ~7/4 = 968.7910{{c}}, ~65/48 = 524.6866{{c}}
 
{{Optimal ET sequence|legend=1| 41, 46, 53, 80, 87, 103, 121, 130, 183, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1308, 1448, 1578, 1889, 2756, 2843, 3337, 4421f, 4915f, 7758cff, 10514cdff }}
 
[[Badness]] (Sintel): 0.244
 
== Etymology ==
This comma was named by [[Margo Schulter]] in 2012 as a tribute to [[George Secor]]’s 2002 short story ''A Bit O' Blarney''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_105598.html Yahoo! Tuning List | ''EDO alternatives, Leprechaun comma (4225/4224)'']</ref>.


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


[[Category:13-limit]]
== References ==
[[Category:Unnoticeable comma]]
 
[[Category:Superparticular]]
[[Category:Leprechaun]]
[[Category:Leprechaun]]
[[Category:Commas named after mythology or literature]]

Latest revision as of 01:50, 19 March 2026

Interval information
Ratio 4225/4224
Factorization 2-7 × 3-1 × 52 × 11-1 × 132
Monzo [-7 -1 2 0 -1 2⟩
Size in cents 0.409808 ¢
Name leprechaun comma
Color name 3oo1uyy1, thotholuyoyo unison
FJS name [math]\displaystyle{ \text{P1}^{5,5,13,13}_{11} }[/math]
Special properties square superparticular,
reduced
Tenney norm (log2 nd) 24.0891
Weil norm (log2 max(n, d)) 24.0895
Wilson norm (sopfr(nd)) 64
Comma size unnoticeable
S-expression S65
Open this interval in xen-calc

4225/4224, the leprechaun comma, is an unnoticeable 13-limit superparticular interval. This comma is the difference between the following pairs:

It factors into 4375/4374 and 123201/123200.

Temperaments

Tempering out this comma in the full 13-limit leads to the rank-5 leprechaun temperament. It is also tempered out in lower-rank temperaments including abigail, newt, decoid, donar, etc. Tempering it out also splits the 33/32 Alpharabian quarter-tone in two, each part corresponding to 65/64 and 66/65. You may find a list of good equal temperaments that support this temperament below.

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨ 1 0 0 0 1 4 ],
⟨ 0 1 0 0 1 1 ],
⟨ 0 0 1 0 0 -1 ],
⟨ 0 0 0 1 0 0 ],
⟨ 0 0 0 0 2 1 ]]
mapping generators: ~2, ~3, ~5, ~7, ~65/48

Optimal tunings:

  • WE: ~2 = 1200.0205 ¢, ~3/2 = 701.9418 ¢, ~5/4 = 386.2411 ¢, ~7/4 = 968.7849 ¢, ~65/48 = 524.6851 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9442 ¢, ~5/4 = 386.2465 ¢, ~7/4 = 968.7910 ¢, ~65/48 = 524.6866 ¢

Optimal ET sequence: 41, 46, 53, 80, 87, 103, 121, 130, 183, 190, 217, 224, 270, 494, 684, 764, 935, 954, 1178, 1308, 1448, 1578, 1889, 2756, 2843, 3337, 4421f, 4915f, 7758cff, 10514cdff

Badness (Sintel): 0.244

Etymology

This comma was named by Margo Schulter in 2012 as a tribute to George Secor’s 2002 short story A Bit O' Blarney[1].

See also

References

  1. ↑ Yahoo! Tuning List | EDO alternatives, Leprechaun comma (4225/4224)
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